Molecular Thermodynamics

MOLECULAR THERMODYNAMICS John D. Simon University Science Bouks Sausalito, Califurnid lllrivrrsity Science Books Pr...

Author: Donald A. McQuarrie | John D. Simon

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MOLECULAR THERMODYNAMICS

John D. Simon

University Science Bouks Sausalito, Califurnid

lllrivrrsity Science Books

Preface

551) (;,11c I:~vcKodd S , ~ u \ ~ l ~CA t o . 94965

xi

Acknowledgments

x i ii

CHAPTER 1 / The Energy Levels ot Atorns and Production manager: Susunna Tadlock Manuscript ed~tor:Ann McCuire Designer: Robert Ishi Illustrator: Johtt Choi Compositor: Eigent)pe Printer & Binder: Edwards Brothers. Inc T h ~ sbook is prlnted on acid-free paper Copyright 01999 by Un~versityScience Books

+

Reproduction or translation o f any part o f 1111swork beyond th,~t~ ~ c r n i ~ thy ted Sect~on107 or 108 of the 1976 U n ~ t e dStates Copyr~ghtAct wlthout the pcrlnissron of thc copyr~ghrowner IS unlawfill. Kequcsts for p u r n i ~ s s l ~orn further inlornlation should be addresaed to the Perni~sstonsDepa~tmcnt, Univers~tyScience Books.

Library o f Congress Cataloging-in-Publ~cationData

M A T H C H A P T EAR / Numerical Llethods

McoJarr~e,Donald A. (Donald Allen) Molecular thermodynamics I Donald A. McQuarr~c,John D.Simon. p, cm. Includes bibliographical references and index. ISHN 1-891389-05-X I ' I ' l i ~ ~ r ~ ~ i ~ ~ t l y n aI .mSimon, i c s John D. (John Douglas), 195711

1111LI~IIIIIL(YI 2 1-2. Tlic Allowcd Energies o f d System arc, Ol)t,~~r~eel Iron1 tlie Sc lirotlinger L c ( t r , t l t ~ ~ t t 1 , 1-3. Atonis Have Translationdl Energy 111A t l d ~ t i o nto F l c ~I r o n ~ cEnergy 7 1-4. The Vil~ratlorlalMotion ot a D i a t o n i ~ cMoleculc C,rn t)cx hlc~tleledIly 1' I I.IIIIIIII~II Ov i I l , ~ l o r 9 1-5. 1 lit, I Ilcq!,y I (>vtsls(il ( ] L I J I I ~ L I I ~ ~ - M ~ ~ I I IcI ~~r IlIi~~ I( C O\( ~ ~~l ~~~ cl l ~ ~Art' t(lr I l l ~ ( t , I ;) \.\.1111 1' = 0. I. 2. . . . 1 .$ 1-0. I I I ~I, l . t r ~ ~ r o r iO n j 1 il1,itor Accourils for the Infrarc,cl Sptrc Irunl (11 '1 1)1,1toni11. 1111. 1-1 h111lts1 1-7. 1 lit, I ) I \ ~ o I , I I I ~ 1I ~rlcrfiy I~:Equ,it~onot State Call [ k s c r ~ l ) cI311fti the c;,~itaous ;ind I I C I L I IStlit(,i ~ 2-4. T11c' VJII dcr l'V,>.ils E c l ~ ~ a ~ant1 i o n tlit' Kcell~cliti\\.orig Erlu,it~on C)I~tsytilt, l .I\\ of C'orrcxsptrn(lingSt,itc>s 67 2-5. Sccoriti t'irial ( . o c ~ i i i c ~ e ~Cdn ~ t , Bc Ilsccl III D c t c ~ r l ~ i i rIlnr ~ l ( ~ r l i i r ~1l 1i 1~~ 1r1 I ' I I ~ I , I I I I . I ~ ~1 ~

I-(,. I ( I ~ I ~ I II I ~ II \ ~ , 1,111 ~ ,4.h IC I \ ~ uI*)~II,II ~ ( ~ 1111, I . ~ ~ c i *( \ it~ t ! l r ~ I ) ~ ~ tot iIo~ n IrC" ~Tt>rriii r i lhr! 2

I f ~ r l l l , l l ( l l#11!1,\ ~ ~ ~ l t ~ ~ l l l' ?l . b l I I i I I I I I

f'rol)lth1n>

I ~r

l

I

T

I of o

rr

i

r

H2

ll48

M A T H C H A P T EBR/ I'ruI7ahility and Stalislics

9;

Prrrtdr~ms lIll

C H A P T E R3 / Tho Roltzrndnn Factor and Partition Functivns

I 05

3-1. Tlw Bollzlnann Factor Is Cjne of tht- Mo+t Irriport,~ntQudntilies in the Phvslcal 5r iences 106

3-2. T t ~ Protwbililv r Tlwt a System In an tnst,mk>lr. Is i n thr. Stdtr! wilh Energy E , ( ! V . V ) IC I ~ ~ , ~ ~ ~ ~ >10 I I(,~ C'lY'I" I I I w ~' n r 106 3 4 . i.2.t-I'r~j 1 I 0 3.1. I l ~ e H t ~C.q).~r t irv ,it Ct>rl+r;lr)lVolulne Is ihe lbnlperat~lreLlerivativr nf thr Avel ace Ener~t 1 14 '1-5. i'r:r- ('.iri Fxprrqr lhr Prrswrr In Tellns of a Partition Fun(-tirlri I1 h 3.6. rI.r, l',)~ tillon I - ~ l l ~ r t nf ~ o,In5ystrrii of Ir~rlc*l>mtlt~~jl, Di~tinjiuisl)ahleMolecules I\ i t i t , Prrlrlucr ni hlolecular Partition IKunctiuns 1: 19 3-7. 1114. !',~rtitiori Funr tior) rlf a Syslenl of Indepe~tdent,lndlstinguishable Atorr~srlr .klr~IIic.arl5 that the tlirnencionlcc:, nurllbcrs on that :lxis :,rc cnrrgicv divided hy l0-ld J. We will u\e [ h ~ c notation to label colunins in tablcs and axc? in figures bccause of its unambiguous nature and algebraic ccr~menience.

-

-2. 8,1

'.-

-

Notc that F , e, P , . . . hecause of (he ncgativc sign in Eqr~ation1.1. The stilte of 7 ~ r oerlergy occul-v when 1 1 -+ w in Equation 1.1. In this state, the prototi and thc electroll arc so far apart that they do not attract cacli other at all. so wc take their intrraction energy to he zero. At cloacr distances (snlnller values 01' I ! ) , the proton and the clccrron atlract each othcr heciu~seof their opposite charses. i\ statc that has negative energy i s Inore stable rhan nnc thal has Leru energy. The a l l o ~ e dellcrgy states yiven by ['quation I . 1 are called srarir?nay stittcr. I'hc statc of lowesl energy (n = 1 ) is called the gmtmtl srutu. The othcr states itre called rxr,iitd .rrarcs; the state with n = 2 is called thcjirar excited stntr. that with n = 3 is ci11Ie~l the .rrr-o~~de.~rit~.dedt~tte, and so on. When an electron makes a transition from one sti~tic)rlnrystntc to another, i t emits or abhurhs electrntnagnetiu rildii~tior~. Wc piclure clectrurnagnctic radiatiurl as consisting 01- packets of cnergy called phoronr. wht~se cncrgy is equal to h v . where v is the ftequcncy of- the radiation and h is the I'lnnck constant ~h = 6.6261 x 10." J.s). Consider a transitinn frtlrn the state rt = 1 tn n = 2 (see Figure 1.1 ). Because e? > F , . cncrgy in the lor111of a photon must he sbsnrhcd. and we kave e, = F , -th v, .+: by cotlscrvation ot ene1,gy. or h 11, = e, - e l . For thc 2 4 I transition, conservation c ~ fenergy gives F~ = xI hvz +,. or Iru141 = s,- - E I ' Notice thal h v I + Z= k v I , , ; the freque~~cy depends unly upon the magnitude of the difference between the energies of the two states. The general result &,

+

..

WIICIV 1111. c~11:111111111 11~1111l1t.1. 1 1 . I \ I I + \ I I 11 I C 10 ~ till: integer values 1, 2 , 3, .. The units of rnergy 111 kl~cItlrctrl;~l~rl~~,~l S!\lr.ll~ (11 I! I I I ~~;~l\hr~vi;ited \ SI) fronl tlie French Systkrne 1nter11ation:llc11' l l ~ l i t c;IT\. ~ I I I ~ I ~ L -~lr\ipll:~tviI ~. f>y I. Clnc ,jt111lci s equal to the kinetic encl-gy of a mahs 01 lwr~kilcrp~;lrn\Ilnr\111gwith ;I qwcd 01-one nlcter per second. If you remember that kinet~ce n c y y I S ] rrr r*'. lllcrr y t r ~c;lrl %ccth;if I J = 1 kg.m2.s- The clcctrunic cnergy of a hydrorct~;1111111 p1vi.11 I1y ~ : ~ I ~ I : 1I II I iqI Iplc~ttcd I~ ill Figure 1.1.

where Afi j~ tlie (positive) d i f f c i c ~ ~ in c e rhe energy nf the two slates invnlvcd in tl:e tmnsitiori, is billcd the BoI!~-jrr~qrtt~nr:y motditior~. Figure 1.2 shows tllc oh.cerved emission spectrum of atotnic hydrogrn irl the vihihle i111tlnear ul~ravioletregion of the elcctrornagnetic spcctru~n.Note that the lincs

Chapter 1 / The Energy Levels of

I

Alnrns and Molecules

E X A M P L E 1-2 Expre\s the alloucd electrt~nicencrpiec or atomic hydrogen given hg Equation 1.1 in unrh oicrrl ' instead ofja~ulcs.

I

SO1 L'TI O N . The conversion fromi joule\ to c~n-Iis given hy Equatio11 1.5.

Thus. the encrgies are II

-F,,

/c~n-l

1-3, 4toms Have Tran~l.?iionalFncrgy rn Addiiirjn

IO

Flectrunic Energy

A wavc function has the physical interpretation that + ' ( x ) d x i s the probability that the par~iuleis located between x and x J.r. A wave function i s the most tnn~plete description c~fa particle that is pussihle, and is said to specify the state of the panicle. Quanlum mechanics (and the Schrijdingcr cquatiun) forces us to ahiindun our intuitive notion lhat we can locate a particle a.c wcll as we wish and replace that notinn wit11 a prubilhilisriu i~~tcrpreration inatesd. It turrls out that for the types of particles that w e work with in everyday life, the prcjhahilities are sn -c:~llcd hliluting results from rnagrletic elTects associated ed,ith the spin rlS thc clectron I ~ , I I. I I C I I ~ I I~rlcludedin Equation I.h.

'Fmm C.E. Mrlrrw, Alumic h p r g y h v d i , Ndll. Office. W ~ h h l n ~ t u lIK'. i. IWY I.

Bur. Sid

C'lrc.

No. 467 (LmS Gr>bcrnu~cr~l Plintlng

1-4. The Vibrational Motion of a Diatomic Molecule Can Re

Modeled by a Harmonic Oscillator In the previo~~s scctiun, wc discussed the trarlstntio~ni~l and c l e c l r o ~ ~cnergies ic of atrmls. The expression for the trarlslational encrgy ul' a diatomic molccule i s thc same as t h a ~

(.tlq)lrr L ! I he Lnergy Lcvcls oi Atoms and tviolcr ules

5 0 L U T I C I N : Recall that the u n i h r i k arc N . I ~ I -i~~ldrhat '. I N z I kg rn s = mass rime\ accelernliun). Thercforc, thc unlts of V itre given by

' Clorcc

1-5. The Energy Levels of a Quantum-Mechanical Harmonic Oscillator Are E , , = h v ( v {) with v = 0, 1, 2 , . . .

+

Whun thc ScItriHlirrgcr cquation for ;l one-dirncnsiond harrr~or~ic nscillalur is snlvcd. physicull y wcll-hchlrvcd w:~vcL'unctionscan he obtaincd only il- the energy is reslricted ~llr q u u ~ ~ t i l rvrilura d

Rvfo~r:wc disuuch thc qu;~r~(u~~i-~nc.ctiilriic'ill lrcuilllrnt irf u ltur~nr~niu rlbcillulirr, wc should discuss how grnxi an approx~~nirirm it i s fur a vibraling diritonl~crnc~lcuulc. The inlemuclear potential li)r a di;lt~,~nic n~r)lcculuis illus~rarcdby ~ h csolirl line in Figurc 1.5. Nuticc thal thu curvc rlxcs ~ c p l yto thc left (I( the ~ninirnurii,iriilicating the rlll7iculty r)T pushirig the two tluclc~closer tugether. Thc curve to the righi s ~ d uf e the rquilibriu~npoaitiun rihe* iritially hut eventually levels off. 'I'hc potential cncrgy at large separations is cssenlially the bond energy. Thc dashed line shows the potel~tial j k ( K - kc)' associated with Hooke's law. Although the harmonic-oscillator potential may appear to be a terrible approximation lo the experi~nentalCUI-ue, nore thal i~ is, i~ldced,a go(d approxinlation in the rcgion of the minimum. It turns out that this region is the physically important region lor many n~olcculcsat room temperalure. Although t h t harmonic oscillator unrealistically allows the displacement to vary froin - x to +,m, ihese large displacements produce poler~lialenergies that arc so large thal they do IIOT oftcn nccllr in priotiue. The harrnor~icoscillator will hr a guod approximation for vlhratiuns with snlall arnp1itude.c (Problem 1-24).

where

Each encrgy level of a harmonic oscillatc~ris nondegenrate. In uther words, the degeneracy, gL,,i s equal to one for all values of u. The allowed energies of a harmonic oscillator are plotted in Figure 1.6. Note that the encrgy levels are equally &paced,with a separation ku. Note also that the energy of the grclund statc, thc slate with I J = 0, i s

i h v and is not rero as the Iowesl classical energy is. Thc energy of this lowest energy Icvcl is called the :em-pint twcrgy of the harnmonic c>scilhkor,and the fact that it is not zero i s stric~lya quantum-mechanical resull.

F I G U R E 1.6

l'llc cllcrpy Icvcl+~ r ;lI~l~~;tr~~~~~n-~i~ccha~~rc.al Ililr~uri~~ic rnr ~llaklr. w F I C U K E 1.5 A colnpiwson of the harmonic-oscilla~t>r putentin1 (kx2/2; dashed line) with ~ h ccon~plelc ~ntcrnuclearpote~ltial(solid line) of a dia~umicmolecule. The harmonic oscilla~urpotential is a satislactory approxllnation at small displacernunt~.

E X A M P L E 1-5 (iivcn th:!~ttnc v ~ l u rul . ~ h ciorct.vt,n\~an~ rlitl?1s 575 N-m-I, calculate the iurda~nc~~t;~l vihri~liolli~l frcqucncy of H,.

Chapter 1 / The Energv Levels of Aroms and Molecules SOLUTION: According to Equatio~~ 1.19, v = ( k / p ) [ ' 2 / 2Thc ~ , reduced mass of H, is given by (Equation 1.13)

1 6 . Thc tiii~mon~c C)~rillnla~r Ar r rlrlnt, for

thc Infrared Sperirun~or ,I rli,rt#rm~c Mvlccule

accord with cxperirrlenl. and this linc i s called (hejundurnetltd vibra~ionrriii-eq~~rni-?'. Fur diatomic ~ n t ~ l e c u l these c ~ , lilies clccur at around l o 1' Hz 10 10'" [IT, which i s irl thc infrared regioti. Eqtralion 1.22 or 1 23 enables us lo deterlninc force corlslanls it-

the furtdilrnental vibrational frcqucncy is known. Fur cxa~nple,for H ' ~ c I ( ~ ) , 2886 cni and so, according 10 Equillion 1.23. the force constant o ~ H ' ' c I ( ~ is ) > ,,\;

'

is

arid s o

-481 kg-s ' = 3 X I N

1-6. The Harmonic Oscillator Accounts for the Infrared Spectrum of a Diatomic Molecule

111

'

Forcc cr)nst;lrlts for din~orrncr~lolcculc\:Irth 1y11ic:lfly~illbc orilcr 01 l r l ' N .In I . Tahlc 1 ..lliqts (he iurldanitntal vih~:ilitrri;~I I r t + t ~ ~ ~ rr o ~ tt; v~ it ~ r~\ constnrlts ~. ~ : ~ l (Scction 1-7). bond lengths, dcpcr~cr:icir\. ;illcI clcctru111c.cricrpich 01' illc g r ~ j u t i slales ~l ot. some diatomic molecules.

A diatomic molecule can make a transition from one vibntio~~al energy state to another by absorbing ur emiiting electromagnetic radiation whose observed frequency satisfies the Bvhr frcqucncy condition T A B L E 1.4

11tunls out 1hn1the hartnonic-oscillator model allows transitions only between adjacent cncrgy stitcs. \(I that we havc the candirion that Av = &I. Such a condition is called I:

I.UIE. kor absorption to occur, ALI=

srlectiotz

+ 1 and so

Thus. rlte observed lreyuency oT thr. radialion absorbed is

or, i n aauenu~nhcrs,

xlicre thc tilde indiuatca that thc units arc c m I . Furthermore, because successive cllcrgy statcF of a harmor~icoscillator are separated by the same energy, As is ihe same lor all allowed transitions, so this rntxlel predicts that the spectrum consists of just one line whoce frequency is given by Equation 1.22. This prediction is in good

The turldattlctital vibrational Ireql~cnc~c\ (I,). ~ah;rtiorl,tlwn%lulll% (fib, h~ltl Icl~gllih(K?j. degeneracies, and equilibrium ellergie\ I I>< I 17f tlw prarlr~ltl%~uic\ 111 wrii~cJ~iirt~rliic ~nolcculrc Thcse purunieterc were ohtailled ~II~III ;i Y:II ~ r . l y t11 WIIIILC* :hilt1 11t1 11111 rcI)rt\Cnt lhc 11!ost accurate d u e s hecau~ethcy wcrc r~hr:~lr~ctl u l ~ t l r~~l ~ rr t .~ 1 ' 1 rtrl;btrjr t t : ~ t r i ~ o~~~\ ~+t ~ , ill:~Ior approuimatiun. Molecule

:/CI~I-'

~/CIII

H,/~I~II Ikprr\cr:rcy

I)?/kJ IIIOI I .-.

1-7. The

1

EXAMPLE

nl\sr~cial~on Fntmrgy and

thc

(;round-Stale Elcttrorilc LrIcrRy ui d [ I I ~ ~ ~h10lt.r OI~ 1111, IIC

I

1-6

Llse the data in Tahlc 1.3to calculate lhr value ( I ( the force cul~stanlr)1 ' Y ~ " i ~ l g ) . SOL U - IICj N : The force crlnskint is given by

The rcduced rrlass of

is

' 9 ~ ' Y ~

(38.964 alnu)(3X.464 amu) (1.661 x 10 P = (2)(3R.964amu)

Lh

kg.ilrl~~i I )

w :' I",

'I#

The hnmionic-oscillii~or S L ' I ~ U ~ ~nllu I ~ I sirya ~ not osily 111;ti A t ) = 11, hut that the thc rtu)lucl~lcvihratc5. 'Thus, the dilx~lemomen1 of the ~rlrhcculc11lus1ch;lrIpc harmonic-oscillator rrlodcl prcllicts 1 1 1 ~ 1li('l(g) ;thsorhs In thc infrared but N,(g) does not, which is in good agreement with cnprrllilcnt. Tliere are, indeed, deviaticlns from the liaminnic-oscillatclr niodzl, bur thcy arc fairly small, and we can systematically irltroducc corrcctiorls and exten\ions to ;kccount for them (see Problelns 1-31 through 1-33),

1-7. The Dissociation Energy and the Ground-Statc Electronic Energy of a Diatomic Moleculc Are Rclated by D, = U, h u / 2

+

k'ipure 1.5 ahowed the pu~cntialenergy ul'a riiatornic molecule as a function of the internucle~rseparation, R. This potential cnergy that the two nuciei experience results from the distribution of the cleclrons around thc twu nuciei and is called the electronic ?nergy u l thc nioleculc. The complete curve can he calculated from the Schriidinger equation by holding the nuclei a fixed dis~anceK apart and then solving for the distribution nf electrnns and the correspondit~genergy. Cnlculainns l ~ k ethik must be llotlc on a conlputer, hut they have been cnmed nut for a number of ~noleculcs.The in~crnuulei~r separiit icln at the minimum of the curve in Figure 1.5 is the eyuilibriuln i1ltcl.nuclear scparaiioll. RL,, or thc bond length of the mulecule. Figure 1.7 s h n ~ , ttlz s pr)tent~al-energycurves of thc ground and first excrted elccironic statcs ol a tyllical d i ; ~ t o m ~molecule. c Note that the bond length ill the excited electron~cstate is not ncc5ssarily equal to the bond length in the gruund electnlnic stale. As for alorns, usually the energy of the first excited electro~iicstate of adiatomic lnolecule lies so much higher than the ground electronic state that we need nut consider it in thc themullynamic cillculrt~ionsthat we carry out in later chapters.

I),,

;

1, v, 2

F I G U R E 1.7 The grt)und-alateand llrct cxc~tcd-stateclcclrr)r~~c rnclgy ti! a ~ I ~ I I C ! I I I IIIOICCIIIC IC ~ I O I I C L I ilpalrl.;l the lnlernuclcar scparatlml. H, showing that he equil~briu~~~ grou~~d-htatc clcclronkc trlcrgy, /Ir. and the dishmiation cncrgy, I>(,,am rcbtcd by De = I),, + h v / 2 .

According to Figures 1.5 and 1.7, the elecironic energy of the grtlund elccrronic state ~ ( R to S zero as the internuclear separation increases. Energies must alwnys be reckoned relative 10 sonic (arbitrary) rcrO of encrgy. In this caw, we arc tithing tlic zero of the gnlund-state electronic energy of a dialoniic n~c~lecule to bc the sey,u;)tcd constituent atoms in their ground electronic states. Consequently, thr. gro~lnd-sli~te elcclronic cnergy of the molecule at its etjuilibriuni ~nterr~r~clear sep;iriition IS - I ) , , . n.c shown in Figure 1.7. For exa~nple,lhr valuc of DLfor H,(g) is 458 kJ.fl101 I , which m e w s that thc ground-state electronic encrgy of Hl(g) ak its cquilihriu~nintcmu~lenr separatic~n(74.1 prn) is 458 kJ.rnol-l below that of two vrpararcd hydn)gen atoms in their ground electrur~icstate (Is). or thal the elcctroc 158 k J . i n ~ l - ~ . This is not the energy required to dissocinte H,(g) into , . ....; .I:LIZ hycllr)gzri atoms, howcver. Recall that Ihz Iowcst v~hratio~lal cncrgp ktatc. hdk .I ~clro-l>olnter~e~,gy s,, = h v / 2 . Consequznlly, thc energy required clihsocia~zH, rg) liorn it3 ground slare into two ground-state hydl.c>gcn ;ktoIrl:, ic I ) , - I r l ] / ? . which wc. dc.\~gn;htuh y I),, 'I'l~us. wc WI-ilein at1 cquatlon

Givcn that v = 4401 C I I ~' fair Il,(gl (i'ron~L b l c 1-41,we see that D,,, the dissociatior~ energy of Hz(g) is 458 kJ-IIIOII - 2h kl.tl101 I = 432 kJ.rnol-I.

Chapkr 1 i The Encr~yI.ev~Is of Atoms and Molerules

I EXAMPLE 1-7 Givcn that the d~ssocintionenergy of F,(p) is 154 kl ,moi and that the vihraliorlol frequency i, 892 cln I. ralculatc rhe value oi the ground-atale clcctro~~ic energy.

'

S C l I l l T l Oh': We are givct~values or FI-o~mEquation 1.24. we have

freq~lcncyof vrG,[cycles per second, then ~ h cvelocities of the two lnasscs are u , = 2 r R II J ~ , , and v, = 2rr R,um,,. We define the anguhr v ~ l o c iw~, ,by o (radians per second) = 27r v,,,, so w c write I ) , and vz as u, = R, cd and a, = K,a). The kinetic cnergy of the r i ~ i rrot:ttor l i\

13,and 5 and wish tocalc~llntethe value of De.

L

whcrc I, ttir. jtlranltbrltr l j irtr.riro, i s p t v l - i ~ hy

1-8. The Energy Levels of a Rigid Rotator Arrg

F,,

-

whrrc

h'.l(.l t 1 ) / 2 1

sccticll~wc will cli\cucs :I sirllplr rilcnlcl M o l c c u l r ~rotate as wcll ;IS vihrii~c,and i n for :I rotatirlg diaton~iurrlo~cculc.'The ~lioclclconcists 01' two puini II;I>\CI nt I i111Cl 111, a1 6xcd distance5 K , and R, l'rolr~ thcir ccrltcr ul' iii;isc (ct. f:igurc I .H} neuausc the distance b e t w c c ~thc ~ two Iiussu\ IS liacd, thi5 mtxlcl is rrI'Urrc~Ito ils the rigid-n)lr~ror nmdrdl. Even though a diatolnic mr,lecule vibrates as i t nllatcs, the vibrational amplitude is sn~nllcoll~prurd&ill) thc bond length. s o considering the bond length fixed is a good ;~l>p~-nximutioli. A rigid rotntor rotates around its center of mass, which is given by the condition In, R, = nt,K, (see Figure 1.8). IF the molccule rotates about its center of mass at a

-

H I t K, ( ~ h ciixed

\cpalation of the two

niassesj and p is tllc rrdu~erl

t11(1.1.1

Whcr~we sulve the Schrudinger cquatiorl for a rigid rotator, the e a p w s s i n ~for ~ the allowed energics comes out to be

fil = -J(J

E,

21

+ 1)

J = O h I. 2. . . .

( 1 .2X)

Once again. we ()blain a set of discrete c n e g y levels. In addition to the allowed eiicrgies given by Equation 1.28, we also find thal each energy Icwl has a degeneracy g, given by

'l'hc seleu~ionrule for il rigid rtltatur says that trarisitinns are allowcii clnlp fro111 adjacent rtates or that A J = +I.I n addition tn the rzquiretrlcllt that A J = fI , the moleculc nus st also possess s pcrlnoncnt dipole mornrrlt to :thsorb eleclrnmagnctic radiation. Thus, we say that HCl(g) has a pure rotntional s l x c t s ~ ~hut ~ t Nl(g) ~, does nol. In the case of absorption of electronlagnetic radiation, the molccule g w s from n state with a quantum nrirnher .I to one wit11 .I I. 1-hc cnel-py d~llcrcncc,then, is

+

TI'

h'

I

4n'I

= --(.I+\)=

-(.I t 1 )

U s i r ~ gthe Bohr frequency ct~tltlit~rm At- = 1; I!, the I'req~~uncies at which thc ahsorption transitin~~s ouc111-;!re C

F I G U R E 1.8

'1-wrr tllncic., Jn, and r t l , shown rr)Ulir~gabout their center of Inass

I tr The. Fncrc) I t ~ v c l br,l

'l'l~coornrnon practice

ill

.j

K~jil~ K+)l;rlur l Art.

a , --A' 1 IJ : L!/II

ttticrowilve ~prctrriscupyis to write tqu;iliurt I .3 1 as

is cnlltd the romtiond ron.rtunt of thc rnolecllle. Alscl, the rrancitiort frcylrrrr~y13 o i ~ c n expressed in terms of wave numbers (urn-') rather thil11 hertz (Hz). If we usc ihc rclaticttl P = r/r, thcn Equalion 1.32 becorncs

whcre B is thc n~talionalconstant cxprzsscd in uni~col wavc rtu~~~lrvr\

mc)~lelprcdiuth that the Frcwn tither Equatio11 1.32 i)r I .3-1. wc KC th;~ttl~c~rigtrl-ro~i~tor ~nicrrjwavespeurrunl uc a diatulnic moleculr cortsists a scries of equally spaccd lines with a \epiuatinn o f 2 3 H~ or 2 8 crn I , a\ shuwn in Figure 1.9. Let's use Equation 1.32 to calculate thc values nf the ahsorotiun frequencies of H"cI(~) tising the equilihnurn bond length given in Table 1.4 ( 1 27.5 pni). Recause I = K:, we must tirst calculatr Lhe vnlric rjl' lhz rcduczd Inass ot H~'CI. "1

and

(1.008 arnu)(34.97 amu) -1.008 amu -t31.97 antu- ( l . h h l x 10 "kg.amu

I)

F I G U R E 1.9 The cnergy lcvels nttd ahcotption transi~ionhoI a rip~drotalor. l'he ubsorphon t~-aris~~ionh r)cclll belwecn adjacent Icvcls, so the ahsorpliun spe5trum shuwn helow the energy lcvels consisis uf a series of equally hpoced lines. The quantity H i\ / t / 8 r r 2 c - I (Fquatlrln I .35).

t;~t~r)n;~l constiillr arld hence the Iunmerit ofitlcrtia o f t & 111olec~1~. Furtherrntrrc, ~ L . C ; I I I C C -I -- 11h':. n.hrrc RLis the inrcmuclcnr distartcc or bond lrrtgth, we cxn dctel-r~tittc~ l t chotlrl Icltpth. I'hih ~>lricedurc i\ illurtl-ated irt Exarnple 1-8

C

As .I lakes on the valrlcs 0, 1. 2, . . ., rl takes on the virlues 6.343 x 10" HI, 12.686 x 10" Hz, 19.029 x 10" Hz. and so on. By I-ekrrirtg to F~gureI . 1 1 in Problem 1-1, wc see that these kecluc~~cies lie in the rr~icrowaveregion. Consequently, rutational

EXAMPLE 1 4 To :I g t ~ ) d~ I ~ ~ [ o x Itllc I I t~t ~~ i I~ Ir ~t ~I'~~wctruul w~ , ~~ ~ . ~ of . "N~~"CI(E c.ol!slsl\ ] r,l ;I \rries riTurlually qvaccd blur!, >cp;lratcdhy 0.4322 cm '. Cnluulste lht. bond Icrtglh ot " ~ a " ~ ' l ( ~ )

C h a p t ~ r1 I The Ln~rgyLerrlr c ~ Aromr f

arid M#>l~rules

SOL UTI 0 N : According to buation 1.35, the spacing of the lillcs ill the lnicrowave spcctrutn of "~n"Cl(g) is given by

and so

Solving hiir eq~~stir,i~ f(>rI. we have

The reduced masv r ) l " N ~ " C I is

1 Y The Y~hratrr,r~\ rlf Pnlyatr,mic

Mvlcr 1 1 1 ~ 5Arc

Represented by

Nrrrrri.~!Modes

three coordinates corresponds to translational motion o f the center of Inass ol' the molecule, and so we call thcse three coordinatcs tran.rlationul degrees nffrrcdrirn. ' h o conrdini~tesare rcquired to sprcil'y thc rjricntation of a linear molecule about its ccntcr uf mass and three coordinate\ to spcci t'y thc oricntatirm u l a n o ~ i l i ~ ~molecule car about its center of nlasq. Because rnutiotl alonr thew coordinatcs corresponds to rotatic~nal rno~ion,wc say lhai n linear ~noleculch;i\ two rlt,grt'e.s of~mtr~tionrrlJrePdnm and [hat a nonlinear rilnlcculc h;~sthree dcg~ccsof ~ol;i~iijrj;~l f'rccrlr>tn.Thc remaining coordinatrs (3rr - 5 [or a Lil~carmolecule and 3r1 - (I for ;I ~lulili~~ciw tr~ulcc~ilc) specify thc rclativr positions of the n n~rclci.Bcuairsr rrlotior~ long thrcc c-trc~rrlin:~tch correhpontl< to vibrational motiun, we say that a linriir n~olcculchas 3rr - 5 l,ihrrrtiot~ctlrlrxrrc..r rd' frpedonr and that a norjlincar mulccult: hii* 311 - - h vil~~:i~iijn:~l tlcprcck of trc.t.tlol~i. These rcsults are .s~i~nmnrizcd in Tirblr 1.5.

E X A M P L E 1-9

Deterrr~iricthe 11urnht.rul- various degreev (yi frcctl+,~n111 Il('1. ('0:.111( ). Nll,. and CH,. Using tl~clact that I = p R', wc obtni~~

5 0 1 IJTION:

Problclns 1 3h, I 38, and 1-39 ~ i i olljcr c cxarnple+ trl thc tlctcr~~~i~i:itiu~~ trl hrlr~d 1e11gth~ l'rtun tllicrtrwavc data.

A diatolnic rnoleculc ih not truly a rigid rotatub; bccat~srir s i r ~ ~ l t ; i ~ ~~ihr;itcs. uo~~ly however srn;tll the iunplitude. Cun.sequentlg, we might expect th:~t a1thr~ugh~ h cr ~ i cruaave speutnlm oL H diatomic lnr~lecuIeconsists nf a scrics of cqually spaccrl lint*. their crparation is nut r,x(~crlycunstant. There is a straightforward prncedurc tu correct fur thc fact t t ~ the t bond is riot exactly rigid (Problem 1 4 1 ) .

1-9. The V i b r a t i o n s of P o l y a t o m i c M o l e c u l e s Are Represented hy

L l ~ ~ d the c r hi~rrlro~~ic-o~cill:~t~~r ; L I I I ) I O ) ~ I I I \ : ~ [ I ( ~IIIL. I ~ . Y ~ ~ > I , : ~~IlIl ~ o Il iInIof~; I~;I~ 1)1iIyatumic n~nlcculecar1 be picturcd as ~ h ~ c ~ l o t i rll u t ~I I ~ , , ~nilvprr~dcn~ harmonic ohcillators, where II+,, is the number of vihr;1tio11;11dcgrccs 01. freedom. Fur example, a water moIecule has three degrees of vihr;i~iol~;i! t ~ c ~ d o(7n r r ~ (7 = 3 x 3 - 6 = 3), and the

N o r r n a l Mndes Poly;~turriicmoleuule.s have translatir~nal,vibrational, rotational, and electronic energics. Thc tral~slatinnalcl~crgyis the satne as that of an atom or i1 diatomic moleuule, and the electronic e,nergies of Iriany pulyalornic molecules are wcll tabulated. We will discuss thc vibrational cl~crgyof polyntnrnic molecules in this secticm i ~ n dtheir rotnti(~nal erieIgy iri the next section. Cunsidcr a lnolecule corltaitlitig r l liuclei. A complete specification of this inolec~ile rn space requires 317 coordil~ates,thl.ce Cartesian coorditintes lor each nucleus. We say that the n-atomic nlolecule has a tutal of 3n d q r e e s offreednm.Of these 3n coordinates, thrcc can hc uscd to specify the center uf mass of the molecule. Motion along these P

TABLE

2.5

'l'he nunibcr of viiriou, tic~rccbuf Crcctiuln of a polyaton~icmolecule contairi~rlgn :Il#>nl\. Degrws (if I'rrctlt~in

--

1,inear

Nunlinear

-

Ch;qitrr I i Ihe Energy Levcls uf Atun~r,lrhtl

Molecules

kihratiunal motioll of a water molecule can bc broken down into the thrcc cIi;~r;~cterlstiv vibrational motions ahuwn below.

T A B L E 1.6

l'hc flndntllcntal vihrxt~r>nal ~requenciesand the rotatiunal consr;ujts r h f home pol!~atnnllc nrolrcule~.7.k n~ilrthcrsin pmenlheser indicate the degeneracy uT the ilnrrrlal mrhdc. Thesc prnnttttrlr wcrc t~blilil~e~l frt)111 il variety of sources and do not reprt'scnt Ihc most xcuruto

valtlr>~TVUIIMI

Sylnmetric slrclch 3 5 8 6 crn

'

Asymmeiric s l r t t c l ~ 3725 cru

'

t l c ~ r d ~ rl t li i~r l l t r n ISVS rln I

C'O!

N?cl

CH,

These three charactcristic vibritional modeh of a w;iivr 11lulcc18luibrc cxul~~plva of normul modes. The arrows in the above drawillgc iudicatc thc rtirrciic~n111alcuch atrlln rnnvcs in the nurmal ~rlude.Bccausu thc norrr~;tltrltnI~'\i i ~ ta h ~ I I ~ C I ) C I I ~ ~tli~r~tlt)nic CII~ uscillntors, the vihrationi~lenergy of .I ~roly,ltr>~r~lr. ruulrr,ttlc ir p t v c ~by ~

II~\L*;I~

lu~car \phcrical I U ~ '

2 43Tb I3 i 3. blj,$t2) 222.1. 1270. 594).8(21

0.390 0.41 0 5.27

?Ht)N, I5 1 S(3). 1{K12(3). l 3(l(Y3)

CCI,

spher1c;il ~clp 458.7.215.5(?). 77X 4 11. 312.N( 11

NH,

symnlrlric (up

CH,CI

where v, is thc vihratic~nalI r c i l u r ~ ~01~ yrllc jth I I ~ ) I I I I 11lralc. ~~~ 'l'hc\u frcquvncies are indicated ior the normal ttltdcies clt a walcr rtlulcculc htiown ithovc. I:.acl~type of ~x)lyatr>mic moleculc has n+,,charactcristic riorlt~i~l ~utnlcs.f41r ('0:.for c x a ~ t l p lthere ~, ;Ire i0ur nul'rnal modes ( 3 n - 5 = 3 x 3 - - 5 - 4):

t ~ h l ~ i li~ntlcr i ~ d the rig~d-rotatorharmonic-osci1latr)r opproximaliu~l.

~ wtrc P ~

H20 So?

(1 1157

3336.945.2. 330?(21.

1619I2) syn~me~ric top 2968, 1355, 729.8. 3U44(2), 1487(2), 1015 ( 2 ) asym~letrictop 3725. 3586, 1545 asymmetric top

1 153, 1362, 521.3

h.?(l

'1.45

9-45

5 04

0.343

11 443

27 0

14.6

2.03

0.344

t)

71

0.243

-1-1 0, The Rotational Spectrum of a Polyatomic Molc.cule Depends Upon the Moments of Inertia of the Molecule The rocation;rl ~notlunof a I l n e ~ polqatornic r ~x~olec~rlc (such as C'O, ur HCN) ic gwcn hy the same expression as thal for a diatomic mulecule. Thus,

'l'he ~lnrlnalmode shown h v e at ihe Far right indicates vihl-.lt~or~;~l lno~ior)pcrpcndicul;~rto t t ~ cpage. Note that the hcnding mode i b doubly rlcgcrlcr;ltu; t11u twtrlling motiuns in thc plane of the page and prpmdivular 10 it art: thc halnc I ~ I O I I ~l ~I ~i occur t~ in ditkrent directions. A selev~ionrule for vibratitic~nalabsorption spectroscopy is th;it t h ~tllllcllc . Ii\onicnt ot-ihe molecule Inu.st vary during the motion ofthc normal mode. When rhls I > .+(I. thc normal modeissaid to hc itlfmrcdactive. Ott~erwise,it is infruredit~nt,lir'c,. Nurc 111;tt the dipole Inotnellr changes during the [notion of ail three nurmal modcs o f a wdtcl-~nolcclllc (showr~above), so all three normal modes of H,O arc infrared active. Thclcfrlrc, ;ill thrcc line\ are observed i r ~its infrdrrd spectrum. Nnte that there i s IIO ch;il~gcit1 tliliole ItluIrleIlt during the sym~netricstrctch ofCO,, huwever, so this mode is lnlrared ~r~;~r.tivc. Tht other modes arc infrara active, but the bending Inode 1s doubly dcgcncr~tu.s o il leads m only one infrared line. Tablc l .ti list.+ihe characteristic vibrational l'requel~cics of the nornlal nlodcs of some pulyatornic molcculcs. The numbers in parc~~thcces ii~dicatethe degeneracies of thc modes.

with g , = 2 J -1- 1 , whcre I is the moment of-incrila. In this case. hovievcr, I is giwn by

where m, is the mass of the j t h atom and ma\s of the moleculc.

EXAMPLE 1-10 Given that K,,, = 1M.h pm n r d #, thc

mnlnent OK 111enia.A. and

6.

t, -

,= 115 -1

(,ni

rtatir,nalct>nspantexpressed i r ulrl ~ I . Koiiltiotial lrallsitions occur in thc ~nicrowavtand far infiraced region uf the apzctrurn and yiclrl illformation ahnut thr molnent of inertia and hence the equilibrium hunrl Icrlgth\ ot nlolecules (Example 1-8). The vibration;~lmolion oKa di;11~111ic 111ol~c11lt+ i\ I~~(uIcIc(I xs i t I~;I~III~!I~c owillatorb whose energy is given by

1-1 1. The tncwy oi ,h hlolccule Irl the Rigid-Rut.llor Harmoni( -Oscillator A[l[rrt)ximat~on

To conclude h i s chapter, we prcsenl a sununary of thc energy Icvclx of aturns and lnuleculcs thal we w i l l use i n later chapters.

E clcr

= see l'ahlcs I.I and 1.3, for cxalnple

Diatomic molecules

where p is the reduced rnass of thc rnolcc~~lc, A is thc Iorcc ctlrihtunt, al~tt11 is the natural vibratio~~al frequency of the moleuulc. V~hr;~t IOII:LIIri~nhit~tlrih i ~ c i In ~ rthe intrared region uK the spectrum ;md car1 yicld int'cu-~il;~~ioti ; ~ l n > thc \ ~ t ftlrcc ct)l1sfitnfs of n~olcculzs(Example 1-h). As for atr,m.c, lherz is no analylic 'xplcccltrll 101tht. c l i - t ~ l r t r rlll-rpic* ~ u ~ ~ ol d ~ : ~ ~ t r r ~ i i c but thcy arc wcll tahulil~ctl f.'~)rIIII>\~ (..Iw>. wc IICC~I ( ~ r i l yItic rllcrgy and ~r~c>lecliles, the dcgzneri~cyclf- the proulld c l r ~ l ~ ' u l\l;llr* l~c The cncrgy of a ~ n ~ l y ; ~ t oIII~>IL~L.IIIL. r l l ~ ~ 111 lhtnI I ~ I OrtrI,tI(Ir II;LIIINIII~~. ~ h ~ . ~ l l ; ill)~ttlr ~I-LLXIIII:I~~OI! i\ ;IIUI g i w r ~tjy I ~ . ~ ~I 4I1. I'[~ tic ~ t rL i ~~ t ~; ~I l i ~ ~t ~i cor ~gI> ~ ytllc i ~ l\;11iic as lhat or a11 i j l o i l ~or I~~;I~~IIIIC III(~IC.CIIIC .l'hc ~t~l.ili(lli;hIclicrcy ~IL~~K'III!.~ ~ I [ U I~~ IL ' \Ii;i~w c ~ tlic f molecule. tior ;I 1111cii1. IHI~~;II{>IIIIC IIIOI~CIAII~, tllc r t ~ l ; ~ l i oI'IICIX~ r ~ ~ ~ l I\ \;t1111- ;I\ ~ t m tof a diaton~ic~ncll~.culc, w h c ~ cthc rllr)tllcllr r j l IIIC~II;I 1h p 1 1 ~ 1Ily 1 I:L~I~,~IILII\ I IH. For a nonlinear polyalorlllc ~liolrck~lu. lllc rol;ilio~l;~l cncrgy tlclwclcI\ uInrn w lirllichr lhc molcculc is a sphel-ical 1c1p (all Ihrcc rn0111c11ts01' l~lurlirtcqual), ;I xylnlllclrlc hrll Ilwcl r l f the three molnentx (11 irlcrtl;~cqu;~l). or LII; asytt~lt~cl~,iu tup (all ihrcc riiomcflts o f inerlia differ en^). In the harmonic-cl\cillatl,t. a l ~ l ~ ~ c j x i ~ n athe ~ i crihl-ational j~i+ r n o ~ i c ~of-a n poly;~lorn~c molecule decomposes into 3 n - 5 (Ilnear) ur 3n -- 6 (nonlinear) nor~nalniodcs, cach of which acts as all indepcncicnt 11:~rrllonicc~sciIlntorThe vihratir,nal energy is given

hy

where

I I ~ ,is ,

the number of vibralioni~ldegrees uf- Ireedom.

(Equal~or~ I .2X)

( E q u ~ ~ i o1.18) n (Table 1.4)

Chapter 1 IThe Energy Levels of Atom ,h

F v ,

Chv,(z!, + f)

each

vJ = 0, 1, 2, ...

(Equation 1.36)

1-1

1-7. 'I hv J M ~ W C 011111411 I of' II I:~rcr I\ I~IC:I\II~C~ 111 1111111 ut wath ( W J . whcrc m e watt is cquul 11) t~ihc~lnalcpcr wad ( 1 W = I J , h I ) . Illlw 111:lrlyphutoilh ~ I P crnitrc~ll)er ~ e ~ obty ~ad I (HI I~IW nlrrljgcn I ~ ~ u r r " ' l wilvclr~~gth l~c crnlltcd by a ~ ~ i t r o j i I:~\cr c r ~ i?337 rrrll.

1-8. Il+rILloill~onI .I 10 ualculatc {he vnluc uf the ioni7atiun enelgy r,f a hydrrlgen alolri in itq ~rllllllr!rlct'~ruriic'dutc

Problems

t -9. A llrir

1-1. Kadialinn i n the ultmviulet regiorl o f the clectro~nagr!tlicspcclrum is usually dcscribcd In ~ givcn d in nanonieters (IO-~ m). Calculi~tethe val~iesOC I,, i,. lcrms 01- wavelcngih. 1. a ~ is and F for ultra%ioIcl radiation with h = 200 nil), and compare your results with tl~rrrcin k i g ~ ~ I. r cI I.

1-10, A pruut~tl-htatrhydrogen atorrl absorhs a photon of light that has a wavelength ryf 97.2 nnl. 11 t t ~ r r givcx l oft' a photon that has a u.avelength elf 4x6 nrn. Wh:~t i s t t ~ cfinal ccopicphnicles is gorerncd by Newton'c equnriun of can be written in the 1i)rnl

drs

I I I ~ I ~ I ~ which HI.

wllzre x ( t ) is ihe pusition of the mahs In and f (s) is the Force acting on the partlcie. Fiwdtiun I i h a differential equation whose solu1ir)n ylres x ( l ) , the tralectrjry of the mass. I n ihe case of a harmrlnic oscillator, rn is the wduccd Inass and ,f( x ) = - k x (Iioclke's law). so that Ncwlrm'x eyLlatlon is

wl~cre11 IS thc reduced Illass of an electron arid proion. c is the prolutlic charge, E,, I, Ihe pcrmittir~tyof Cree hpauc, and h is the Planck constant. IJs~ngthe valuus given in khe inhide f in terrncofn2. Cnn!pafe your rehult with Equation I I . Irtrl~lcover. CHICI~IXIC Ihc v a l u c ~ 6"

1-24. It) ~ t ~ 111artrlc111, i h wc ~nvc\tljwic11111 h:lrnion~c riscill~torpoteniial as i l ~ eleading temi in n 'I'ayl~lrcxpanrluli ol Ihc achlal ~ntcrllu~lcar plentla!, V(H), about its cqu~llhriurnposiric~n. K . A c c u n l i ~ ~tog l'rrrhlrni ( ' 1). Ihc tirht icw kcrnls in thih cxpanqicln are

d'x

p-- , - -k. dt2 -

Show lhal ~ ( r = ) A cos2xvr suli+hes this equaticln i C v = ( 1 / 2 x ) ( k / ~ ) ' ~T' .h ~ result s is ralld only for a aacruscopic nscilliltrlr, called a clas.~icolhrrrmvniu usrillntor.

1-18. 'l'he kinetic energy of a clas~icalharmonic rlscillalur is

Usillg Equalio~ls1.11 and 1.12, shuw that

If R is always close to Kt,then R - He ia alwayr >111illl. ( ' ~ I I W ~ U C I I I I ~ . I ~ rCe r ~ ~on~ tht h right side of Etjualion 1 gel brnaller and smaller. The Krs~Icrrn in 1;yuatlon 1 i c a ctlllstanl and depends upun where wc chorw thc 7ero u i energy. It ~h crmvcnicnt 10 chocl\c thc /.cro ol~energybuch that V ( R w )equals zero wd to relaw V ( X ) I r ) lhih col>vt.nticm.Ihplain why the linear knn in the displacement vanishes in Equulio~~ 1. (Nutc rhdr - d V / r l N ir ttlu hircc

lrlturprel this rehull phyhlvally

acling betweerl the two nuclei.) Denote R - y by x , ( d ' ~ / r l ~ ~ )by , =k , and ~ ( r l 3 ! ' / t l ~ ' ) , = , ,hq y to wrirc Equation I as

1-19. Thc solution Ftlr ~ h clahzical c h:~rlilun~c ozci1latr)r ~cn ( I ) = A c o b 2rr PI (Equation 1.1 1). Shrlw that the ~iicplacclncl~~ ~rscill:~lclr httwrtn + A and - A with a frequency v cycle.s-'. !What is the pritd of the oscillat~tu~&. Ihi11I\. 11rm l u 1 1Ir ~ unc cyclc? 1-20. From Prr)hlcn~1--1'1, we \ct. tt~:it ~ h cIWI I I I ~r ~ :It ~ ~ , I ~ I I I rltnitla)t~ I I I I I C ~h r = I / v . The iirrrage r ) i the k l ~ ~ c tenercy is uvcr orlc cycle. 15 glvcn I,y

Stww that { K )= E / 2 where E 1s thc total encrgy Sllriw ~ 1 5 t thdt l (V) instantaneous potential energy is given by

-

t / 2 . wllcre the

Argue (hat if we restrict otlrselres to slnall dlsplncements, then s will be s m d l anti wc can n g l e c l thc terms beyond thc qundratic term 111 t..quatlon 2, showing that the gencrdl potential cllergy function V { R j can bsappruxitnated hy u harniuri~c-obcillatorpolcntial. Wc can co~lsiderC O ~ I - ~ L ~ ~ CorI Iextcrlsionc IS o f the h a r r ~ ~ c ~ ~ ~ i ~ - ~ l s cniudcl i l l a t cby > r thc higher-ur,der ,i,m!c. terms in Equntirm 2. These terlns art. caller! ur~hurtrloffic An analytic cxprzssion thal is a gcxd npproxi~llation to ~ I inlrrlnoleculnr I po~enlial energy c~irvt.15 B Mnr.vc put~r~tiul

Illterpret he result (Kj= { V } .

1-21. Calculate the value of the reduced mass of an electron in a hydrtrgcrr , I I ( ~ I I I 'I'ilkc thc macses of the eleclrrjn and protor1 to bc 9.113390 x 10-" kg and l.h72021 x I0 kg, wcpectlvely. What ic tllc percent difference k t w c e n t h ~ sresuit and thc rcrl I ~ I : I \ ~(11- iltl clcct~,on'!

''

C

1-22, ljuanlurn rricchanicq gireh that the electrorilc c n e g y of a hydrogcl~atom

15

and $ are pwameters that dclmul uptw the ~ntilecule.The pamlncter DLis the where ground-st& clec1rr)nic energy of the rnuleculc 1ne:lrured from thc minimum of I : { R ) , and fl is a rnedsule (>I khc curvature o f I 1 ( H ) at 11s minirn~lnl.Drrivc a reliltiul~k l w c e i ~[he force constanl and the parameter!, LlC and p . Giren tll:it /Ir = 7.3 1 x 10 lLJ~.rriolzculu-'. B = 0.0181 pn-'. ant1 R,, = 127.5 prn ior t ICI(g), calculute the forcc cun>Pallt of HCl(g). Plol thc hlvrhe potenlial for lICI(g), and plot thecorrcspt)nding harnlonic ncclllat~irpotcrlti.~l un the same graph (cf. Figure 1.5).

I:hapter 1 / The Encr~yL~cvelsuf Atclrns and Molecule$ 1.7 The vibrational spxtrum of ~ ~ ' C l ( g ) . TABLE

1-25. Uw the refult o f Problcm 1-24 and Equation 1.23 to show that

;obx/cnl-l H arrnonic oscillatur

Given thnl i, = 2886 ctll-l nrld DL= 440.2 k l - m o l ' for 1 1 ' ' ~ t ( p ) ,calcul:~tctllc vnluc of P . rompare your r e ~ u lwith t that ill Ploble1111-24 1-26. Cnrry out thc Taylor erpanqicln o i rhc Murhc p o ~ c n ~ iin a l Problen~1-24 through terms in (X - K:)'.1;rpresu y in Equnriull 2 ol Ilrt,l>lc~rl1-24 in tcrnms o f IIc and P .

1-27. I t lurrl!, r)ut that thc solution 11I'rhr Schrlnlinger equation for the Morse potential (ProbIcul 1-24) call he expressed ua

Transition

--

(I -. 1 (fundanlental) 0 + 2 (fin1 overtone) 0 -. 3 (second overtone) [) -. 4 (third c>vertonc) O + 5 (fourlh ovcnone)

C(,/cm

'

2885.9 5(10H 0

1 = 2X85.90~ 2885.9 5771.8

8.741 0

H657.7

10 413.1

1 1 543 b

13 396.5

14 429.5

whcrr

Given that fi = 2x86 rrn and 4 = 440.2 kJ. ~ n o l - 'for H ~ ' C I ( ~calculate ), the value\ o f I and 1.4.Plut rhc tibrat~unnlcncrgics of 11"Cl(~) for a Murse prllential.

with i~lcrcnsing71. The scJcction rule for ari anhermonic. oscillntr)r 1s that 11I: can havc any ir~tcgralvalue. although the intensities o f tllc A t l = *2. f3, . . . transitions arc rlluch lers ttinrl for the A I I= fI transitinns. Shuw that ifwe recrlgnizc t h a ~most diatort~icmolecules arc i n ihe ground vibrational slate nt room ternpralurc, tllc frequencies of the uhrerved 0 4 II trnnsitions will be given by

'.

1-28. In the infrared spectrum of H"Rr(g), there is an intense linc at 263U ccln U x 1hc hnr~nonic-osciilatr,rapprt>ximatiunt o calculale ihe valucs of thc forcc con\t;ittt t ~I11"13rlp) f ;jnd rhc p r i o t i (>rvihrafion ur H ' " R ~ ( ~ ) . 1-29. 1-he force conslant o f ' " ~ r "~ r ( g ~h ) 2 4 ) P4.m-I. Use thc harrllrlr~ic-oucillatorilpproximalion l o ca1cul;lle tllc vulucs rjf thc t i ~ n d i ~ ~ r l vihritiun:~l c ~ ~ ~ : ~ ! f~t.rlut.~~< = H 75 x 10 l u ~ . t i ~ oI . l F r 1556 cm-', and Re = 12U.7 pm for 0,. plot a Ma,rse pc~tcr,ti:ll IOI. 0, Plot the cor1.rspontling kurnonic-oscilla~(~r potcntirtl on the same graph.

1-35. Show that the momcnt of inertia for a rigid diatomic rotator can be written as I= @ K:. where Xr = R, -t R, (the fixed separation of the two masses). HI and R? arc the divlanccs o f t t ~ ctwo malise+fro111t l ~ ccenter o f rrioss. n ~ J I~ i\dthe wduccd mass. 1-36. I n [he tar-infrared cpxtrum o f H ' ' U ~ ( ~ ) ,there is n scries of Iinec sepnratcd by 16 72 crii - ! . Calculutc thc values or [he rllonlent uf inertia n t ~ dthe internurlenr separation in H'"B~(€).

where i is cnllcd the cmltrrm~onrt~~~v c-rrr~rtrrr~r 'l'l~e:~l~harli~rll~ic corrccrion i n Equalion 1 is r n ~ ~ smaller ch than the harmonic Ierm hec:~uscinal cnnxtallt of " ~ a ' ~ C l ( ic ~ )6503 Mllz.

i n the J

1-41. The rigirl-rotator 1ntdc1predlclc th:~tthe line!, ill thc rotaiionul spctrum of a diatomic rr~olt.culeshould he equally spaced The following rable lists sorne o f the uhcerved lines in thc rowtiunal hpectmm o f H ' ~ U I ( ~ )

C C ~ = , ~Z B ( J

Tranhi~ion

fi,,,h/~~~~

A , c m

B

+ 1)

1-43. Given that

= 8.4hS c ~ n ;~rttlI ) I1 I)IXtIJI\ C I I ~ I l;)r t i " ' l l r ( g ~ ,C~IIUUIA~C thc imquencyc)flhcJ=O-Jz1.J 1 * I -?.J-:2 . I - 3 . --.J=6-J=7 tra~~sitions ill thc rutat!unal hpcIrtIriI (11 1t l U l i r l p ~

= 111.243 crn

- .-

3+4

83.03

82.72

1-44. neter~ninethe nulnbcr ut *,lrliulr t l c ~ ~ r *I! c \ trccthr~~b trt N:, (':ti,,

21.07

4+5

103 40

104 I 0

124.311

6+ 7

145.0'3

7-8

1h5.51

C': ti,, arid

1-45. nulerniine llle tr>tal 11umhc1i ) t 116rr11r,1l ~ r ~ i 0 1~ 1 rkII~#,~~IIIII r 111 J1('N. ('I>,.SO,. Sl,,,, .III~~

20.211

5-h

(':ti,,

C,H,.

(CH,),CO.

124.011

20.73

1-46. Using the dala i n Tahle I.tj, calc111.11rtllc \uluc III t l r rrrrb ykurlr 1hrilliun:11C . I I C . I ~ ~

143.7h

~ j t,I

water molecule.

20.4K 16.5 44

1-47. Using the data in Tablc 1.h. ralculiitc ~ h cr.ill~c+>I IIIC trrur methane n~olevule.

186.12

1-48. Givcn that K,, = 112.8 ~ I Iand I Hh,, I I N .I

20.35 H

,-+

9

185.86

tllc 1llolIIeIlt r d inertia ant1

211.52 9-

10

206.80

206.38

11

226.50

227.48

I. c , ~ hb~lrtctllc rdluc, or'

i.

t ~ , ~~ l! T ~ T I I < 1j1rI) ~ 111~111~ ~ I C I Ir ~ I : i l ~ v>c~ ~ ; 1 r ~ t i o l l mcnsiorl, inleracling thr(111ghrl p t r t ~ , ~ lI)I.II ( A , , 1 1 = 1 iI I , I I;I\~II 111.111tn- I011(.-11 t111gII~R~II 111~ j l h parllcle ( . -K 0 t / , \VII.LI 1.1% I, itair ' is = -(aL'/?I r , 'I.ahrm 111:11 1, Newtcln'x erluatlnns fur 111, ,irltl rrr . .dl parllt Ir.. ( ' a l ~ l r ~ lwrr t ~ r Iimurwr, m, mil r r l , , ill unr d i -

20.12

10-

prllll

ot thc Solve fur

A,

al~rlA, 10 ~IM,IIII

Chapler 1 / The Energy Levels or Arorns and Molecules

Show that Newton's equations i n Ihece cvordinales are d ? ~m m d 2 x

M

dr-

"$

dr2

A

Dx

' +-aU M

d' x - m nt, d 2 x 'lt'

MATHCHAPTER

au

=--

-

dl2 -

DS

Now arid thew In(?ctluatint~sto hnd

lulerl~crhi.; rehull. NOW divide thc lirs! cqu:ltio~~ hy m ,and thc F ~ C O I hy I ~ 111! :lnd '.uh1rilct ohlain

I(1

d'x dl:

/.I--- =

JL7 ax

--

whcrc ,A is the reduced mass. Interpret (his result and discusc how Ihe original IWII-hxly has been reduced to two one-hdy prnblerns. ~mhlutl~

You learned in high school that n quadratic equation r r I : given by the so-called quadratic fornlula:

+ hk

t

(,

= 0 h;~stwo nlols,

Thu\, the 1wo value, o l x (called mots) that satisfy the equation xZ t 3.t

2

: 1)

ijrc

Al~hougI~ Ihcl-c al-c ggcerrll furmul;~cfor the ror>t\ 01 cubic ;111dqu;~rticr t j \ ~ ; l l i o ~th~.) ~\, vcry inuunve~iientto use, ant1 t u ~l~el'n~orc, f tIlc~cilrc nr) f'r>r~llul:l\lur ty t~:~lio~ic or' ~ h tifth c degree or higher. Unll)rtunatzly, in pri~c!ice we cncourltcr ~ u c rtlu;~tirlr~s t ~ freque~ltlyand must lcarn to dcal with thcm. Fortunately, wirh Ihc advent 01- 1l;lnd raluulatun and personal compulers. h e t\uitlerical ~ o l ~ ~ofl jI ) ~I ~ nI ~ I ~ ( ~ I I I ~ C< II II I I : I ~ ~ O ~ S and other t y ~ ot' s equi~~iur~s. such as x - cus T = 0, is routine. A l t h ~ ) u r llhc\c ~ ;III~ other equations can hc snlvcd by "brute force" trial and error, much more o r ~ ; l n l ~ e d procedures ran arrive at an answer lo almost any desired degree ui'accu~auy.1'crh;lps thc mosl widely known procedure is the Newton-Raphsnn ~ncthod,which is hcsi illustr~ted by a figure. Figure A.1 shows a I'unv~ionf ( x ) plol~edagainst .r. The s u l u ~ i c l10~ ~the equation ,f ( x ) = 0 is denotcd hy x * . The idca hchind thc Newton-Rnphcon method is to guess ;in initial value of .x (call it x,,) "sufficiently close" to x * , and rlr;~wlhr IarlgerH to the curve f ( x ) at x,, as shown in Figure A. 1 . Very or~eri,the extension the Iilrlgcrlt line through:he horizontal axis will fie closer to x* [hall docs x,,. iVc dcl~otctl11svaIue of .r by x, and rcpcat thc plnccss using x, to get a new value uf x,, which will lie even clr~serto x*. By repeating this process (called iteration) we can approach x* to essentiully any desired degree of accuracy. :ilC

3')

T A B L E A.1

Ihe results 01- the :II~~IIC~II~+)II o E the Ncwto~~-Kaphhr)rl rllcthod f ' ( . x ) = 4 . r 1 - ~ . 7 2 r ' fK.721 - 2 . 1 8 = 0 . tl . -.

x"

0

0.251HI

1

0 3442

?

11.355Y

1

015hl 0 35hl

4

f (1,;) -4.825 -4 855 -6.281 1.703

x 10

.

'

x ICI-'

Ihc >olutic>nuK ttlc etlualio~~

f'c-T,,) 5 . 1 10

3.139

x 10-I x 10

1~

'

4.033 4.031

F I G U R E A.1

A grilphical il1ustr;ition 01 lhc N ~ w ~ r r t l - K ; ~ l ~111rth~u1 h\cl~~

Wc can use Figure A.1 to derivc a convenient formula for the itzrativc valucs 01'x. The slope of f ( x ) at x,,, f '(x),), i.c given by

,

EKAMPLE A-l 111( ' ~ l i l ~ 2~, tWl' ~l~

1 1 '%tlI%c 1 1 1 L ~' U ~ ~C Cl ~ ~ l b l ~ l ~ ~ l l

Uhc thc N t ! w r ~ r i ~ . K a ~ r tilcthtd h ~ i ) ~ ~ 1~ llrltt flrc rcill rtwh t r I 1111, ctlit,rtltrrl tigureh.

Itr

Ilrc rrg~tllicanl

Solving this cqurlticln for x,,, gives f ( x ) = x'

which is the ~terativeformula for thc Newtun-Kaphct~nmethod. As an application of this brinula, consider the chemical equation

whosc relared equilibrium constant is 2.18 ; ~ ta certairl lempcrature. (Chemical equilibrium is discussed in Chaptcr 12, but we'rc simply using the algebraic equation below as an example at this p i n t . ) 11' 1 .OO atm i>l'NOCl(glis intruduccd into a rei~ctionvcsscl, then at equilibrium PK,,l = 1.00 - 2.r. Ph0 = 2x. and PC,, = r; thusc pressures satisfy the equilibrium-constar~texpression

+ 3x' + 3x - 1 =

By inspcction. a solutlo11lies trelwcco O and 1. 1Jsl11gx0 = 0 5 rcs11Itsi l l Ihc it>IIow111:! table:

The answer to tiire t.iglliticant figures is A = 0 25992. Nihk I~:II1 (Y,,) is s i g ~ l ~ f l ~ f i n t l y cmaIlcr at each s~cp,as ir should be as we approach thc valuc, of r t h a ~satistiec 1 ( r ) = 0,bur [hat j'(q1ducs not vary spprcciahly Tho same khar,ior can he >ern in W l c A. I.

which we write as As powerful as it is, the Newtun-Kaphson method drlcs not always work: when it il is trtwious the mcthod is working, ;I~IL~ whcn it doesll't W O I - ~ , i t irlay tje even m o l t obviul~z. spectacular failure is prrlvidrd by thc eqo:tliort / ' ( r) - r I." = 0. for xhict~.r,= 0. 11-we b c ~ i nwith .w0 = I , we will ohtairi r , = -2. .r2 = 1 4.1, -. - 8 , and su on. Figure A.2 shows why tile neth hod rs h i l i n g to convclgc. Thc lrjrssagc here is that you shoillrl always plot f(x) first to gel all idca c ~ f where the relevant moth

does work,

Because u i the stoichiometry of tht: reaction eyuation, the value of-x w e arc seeking nu st he between 0 and 0.5, so l e r ' ~chnose 0.250 as our initial guess (.you,). Table A.1 .shows thc results of using Equalion A.I. Nutice that we have converged tu threc cignilicant figurcs in just three steps.

then

Thc t u ~ ~ c t iF(.r) o ~ ~ ir so~netimescallcd thr ;~n~ulcri.;nive of f ( x ) . 11- therc i s nu v l c l i ~ r l ~ l : ~function ~-y F ( x ) whose derivutivc i\ j [ r ), w r m y II~;II111eiritegral of f (.r) c i ~ n r ~hr c ~cv;lluatcd t :inalyticdly. By elclr~e~~f;iry 1'tlrlc.tion, wc Illcan a lunctinn that rat1 hc cxprc*xcd

S I; fir~itccomhinntinr~n l p o I y r ~ o ~ ~ nI~-igrjr~c>r~~~-tr.ic, ;~l. cxpc~rlcntii~l. ;ltld h ~ ~ n r i ~ l i lu~lctions. ~r~ic. 11 Iurli* ~ > uthat l numerous integrals cannot hc cv;du;ltcd ~ r i : ~ l y t ~ c . iA ~ lp:miualnrly ly. Irrlpmunt cxiit~rpleof an integral that canllot hc cv;~lu:ltcrl i t 1 tcrrlls 111. s l t . r ~ ~ c ~ r ~ ~ ; ~ r y fulwIit~n\ir

FIGURE A plot

A.2

=

I,',

illuslraling that the Newton-Raphson method fails in this case.

lie and to see ~ h athe l lunc~iondoes nut havc any peculiar proprlies. You should do P~t,blc~n\,%-I In A-8 m bccolnc proficient with the New~on-Kaphsonmethod. 'l'here arc :ilso nulncricnl method\ lo evaluate integrals. You learned in ualculus that an integral is the area between a curve and the hori7.ontal axis (:uea under a curve) hetween the integration limits, so that the vnlue c~f

i s givell

by the shaded area in Figure A.3. Rccall a fundamental theorem uf calculus,

which says that if

I ~ u u ~ ~A..1 r r wrvrs n (r>dclincrhc(nonclementary) I ~ l r ~ c t()1{ i u1~) ~'I'tr~v;~luc nf 4t.t)for iuiy YUIIK. tnt I is ~ ~ V by ~ the I I area undcr thc curve ,/ ( , I ) - t. " hurl, rc z I) to 11 = x. Irl'\ ~otlhidcrl h nxli-c ~ general casc givcrl hy Erlu;ltit)~iA ? r ~ itlc r rhadcd :lre;i i l l i.'ipnrc A 1. We cun approximate this area in ;I ~ ~ u l l l h(I!r rwily\. f:lr\l Jlvldr rhc i n t c r v ~ l

,

( ~ J , / I ) iritar n cqurrtly spaced suhinterr;;ils u , - : I , , , r t : II . , , , ida ith wit11I,,, = (1 rlrld 11" h. Wc w i l l Ict h = u, - u, l i ~ ,Ir x 1 ) . I. . . . rr t I:ipurv A.4 ultrlws II ~rla#ftifirlblirm(1'one of thc suhiritcrvals. h;~y illc i r , , i r , , ~ r ~ h i l ~ c r r .011c i l l . wily 10 under thc curvc 1s to cotirlcct t l ~ cpc~irl~s !'(I, 1 a r ~ d((rc,, ) hy $: Ilpprruisialr ~ h arca c ~ t l l r i p hIIIK ~ 11% SJIOWR i n Figure A.4. The area uridcr the 5tmight line ; ~ p p r u x i r ~ ~ a t i ( ~ r ~ iu / t u l Itr tlrr itticrvill is the sum of the xrce of The reutanglc [A f ( u , ) ] and the area af

.-

.

,

,

Ihr ~ r t n r y k1 Ihl I( l r , + ,) - j ( u , ) l ) . Using this app~nximatior~ for all intervals, thr total r m a ur~clrrr l b r curvc fmrn u = (1 to u = b is given by the sum

.

N~)tt-1h;~1thv ~ . r ~ ' l l ~ ~ in ~ iEquation c ~ i f s A.4 go as 1, 2. 2, . . . 2. I.Equatiun A.4 is for n = 10 or so and on a personal con~puter ;Ig \prc:~(lclicc~lirr I;~rpervalues of n. The approximation to rl~eintegwl given hy uci~~ I~l'.r rule. whose fnrmula is

AI I' = 103 K, thc h u h s 1111~gr~l 1 0 cvilluulc ~lu~twrl~.ally 15

(e*

- 1)'

rlr

IJsing the tlapzzr>idnl approximalio~l(Equatiun A.5) and Simpcu~l's rule (Fquution Ah), we Iind the follouring values of I : II

It

i,, (~rape~oidal)

I," (S~mpsun'\rule)

in Nule that the coel'ficieilts go as 1, 4. 2, 4, 2, 4, . . . , 4, 2. 4, 1. We write Equaiion A.6 hccnuse S i ~ n p s u n ' srulc requires rhac there k a n even number 01-interirnls. Tahlc A . 2 show\ the values oT $( I ) in Equation A.5 for n = 10, 100. and 1000. Notc

T A B L E A.2 The application c ~ l - t t~,.~lx./uiti:~l h~ :qq~rc>x~riiation (Equatiurl A.4) and Simpson's rille (Equalicm A.b) to the evaluatiorl of @ ( I ) pvcn t>y 1Lqu:llion A.5 The exact value tr) eight decimal places I S 0.74682413.

-.

I7

11

10

0.1

109

iwo

-

I,, I

I I ~ I ~ / . ~ ~ I L ~I!,, ~ (~ ~) ~ 1 1 1 ~ 5 0l t1l ~l ~~ )9

. .

.

.--

..

I 1 7.1(r218{Kl

I I fJOH?4'1-1

0.01

Il.lJ(1HlH~Hl

11

o.w~

[I

7-1(>~2.107

7.111H?.l I J

o 7,1t1~2-I I \

+,r(', = l h . 5 J . n ~ u l ' , K-' , ~nagreeme111H ith the cxperirricntal value

Although the Newton-Raphson nicthod atid Simpson'r, rule can he irriplemcntrd casily on a spreadsheet, therc are a nunilxr of ensy-to-use nu~neriualsuftwarc plickagrs wch as MnthUud, Krxleidagruph, Mclthenlutiru. or Maple that can he i ~ w dto evaluate thc roots o f algchraic equations and integrals by eken inorc hophisticateti n u ~ n e r ~ c a l methods.

This inlegral can be evaluated analytically; it is given by tan-'(1). which is cqunl to x/4, so I = 1).7R539816to eight decimal places.

Problems A-1. Solve the eqtrati#>t)ur betweell U and 1 .

+ 2 r 4 -t41 = 5 to tour significal~tiigurcs ior the rlMlt that lie';

A-10. Evaluate In2 to six dccimal place!. by vvaluating

In2=/;'f

A-2. IJve lhe Newtun-Raphscln rnethd lo derive the iterative formula

What must n be to assure six-digit accuracy?

A-I 1. IJsc Simpsun's rule to evaiuale for Ihe value o f *h Thiv formula was discovered by a Babylonian mathematician rrlorc that) !lKMJ yc:lr\ :)go.Use this tor~nulato cvaluate f i to live significant figure,. A-3. Ckc thc Y\lcwrnl~-R;~ph?rw n~etllndto snlvc thecquatlon fipurc>.

t,-'

+ ( . ~ / 5=) 1 to Coursignificant

and cornpale your rcsult with the exact value, &/2. A-1 2. Use Simpson's rule to evaluate

A-4. ronridcr the chemical reaction described hy the equation

at 3tH) K. II 1-00 atln of CfI,,(g) and ll,O(g) are introduccd into a rcaction vesseI, the prccaurch at equilibrium nhry the equation

10

six duci~nalplaces. The exact value is rsd/l 5.

A-1 3. Use a nu~nericdsoftware package such as MafhCud, Kaleidagmph, or Muflrcrncrfi~ato evaluak the integral

for value:, u i u between0.200 and (1.300and show that S has o maxirnurn value at a

A-5. In Chapter 2, we w ~ l solve l 11icrul>lc cquntlon

= 0.27 1.

A-14. Use Si~npcon'srule to evaluate the integral (.we Fyuation 2.31)

( A % 'I h ~ t! 12.1 - l = lJ Usc the Nexvtrrn-Knpllson 111cthodto f i t ~ t fhc l o111yrt:ll rrwf ot thi+ctpiltion to five significant figui-ec

A-6. Solve the equnlinn .r'

-

71- + 1 = (1 (or ;1l1 rl~rcru T i i \ rr*rlr I,, faiur. dccirilal places

A-7. I n Example 2-3 we will solve the cl~hlcctluiitltirl

Ucc tllc htw~orl-Raphsonmelhrd lr) find the rtwl rnrhi< r011.1114111 111:11

I \ 111.~11 \'

0 I

A-8. In Sertir)n 2-3 we will sol%ethe cubic e q u a t i o ~ ~

USCthc N C W ~ O I I - K ~ ~~ncthod I I S O ~to I show [hat the three rrjuts t o hi\ rqu;ttlcrrl urc 11.11707>. 0.07897. and 0.2167. C

A-9. Use hc trape~oidalapproximation and Simpson's rule to cvaluatc

for T * = 2 . W and 3.W. The accepted values are -0.6276 and -U.1 152, respectively

CHAPTER

/

The Properties of Cases

Wc kgin clur study of thermodynamics with the propertics of ga5e.c. First, we will Jiwusr the ideal-gas equatinn and then some extensions of this equation. of which !llc vim Jcr Wuitls cquaiion is the most famous. Although the van der Waals equation ucuounlb in p;~r!for deviations Tmm ideal-gas hehimior, a nlrlrc cya~ernaticiind accurate uppn~uchIS to k15c il so-callcd virial expansion, which is itn expression tur thc pressure ut'u guh UR B p ~ l y n i ) ~ t ~Ini arhc l density. We will relate the cr,efficients in this polynt~niial trl ttr energy 01' intcracliun k t w c z n the molecules o f the gas. This relation will take ir+ ~ I I ~ u I , disc.us.\irm r)f how lnolecules interact with o n e another. Wc will sce that ilcvtalic~rlst'rorn idci~l-gnak h a v i o r teach us a great deal about molecular intcrxlions.

2-1. All

(;,IWS

Ht-I,,~vcIdcally If They Arc Sufficiently Dilute

It a pur Ir c u l Y l ~ , ~ r ~ rl~lutc ~ ! l y t h a ~its c~)n\titucritm o l t c t ~ l c sare s o f i r apart h l n e a c h c~thcrtnrrllc rlvrrupc 111~1w t ciln Ignore lhcir jnttr;~ctit)ns,it obeys the cquation of state

Johannes Diderik van der Waals was born in Lciden. the Netherlands, OII Nove111brr 23. 1837, and died in 1923. B e c a ~ ~ shee had not learnd Latin and Greek. he was at lirrt not able to continue wilh university studies and so worked a? a schr~11~eacherin a scco~iriaryschool. Aitcr pashape of new leg~slation.Iiowcver, van dcr Wials rjbtained ail cxcr~lptionlruln the unlbersity requiremen~s111 clacsical larlguageh n11d defended hls docturnl disscWatior1 ot I.eydr11 of the Ll~liirerhity111 1873. In his discertation. he proposed an t.xplsnatir)n 01- the crinlin~~ity p \ e o u s and liquid phases and thc ]~henr)~r~unon of the critlc:iI point. as well as a derivatiori of a 11twequatirln of s1;ite 111 gases, now callcd the ran der iVxals cquatirin A tcw years latcr. llc p r t ~ p b ~the d law of corr~spondil~g states, whlzh reduces thc propertics ot' all gabcs tu rrne common dcnomiwtor. Although his d~srertationwar writtcn in Ijutch, his wnrk quickly camc tu the aucrltion IIC Maxwell, who published a r w i e w o i it ill English in the British journal Ndiure in 1875 and so brolighk thc work to the ancntion of a luuch hrrwdcr alirliencc. In 1876, van der U'aals waq appointed the iirst Prufe~sorof Physics at the newly crcatecj Ilnlvcrsity u T Anwterdam. The University bccanic a c e ~ ~ tior e r both thetxetical and expcrimenral rcscarch un Buids, largely through van der Waals' inflilence. \'a11 der Waals was awardcd rhr Nohcl Prize lur physics in 1910 "for thc work r,n thc cquation uf state for gases and liquids."

II wc tllvldc. t n > t lhides ~ ol'lhis equation hy n , we obtain

w l ~ c r tv = V / n i s the molar volume. We will always indicate a nrolitr quantity hy dr;iwi~igi~ l i ~ i eabovc the symbol. Either of Equatiul~s2. I , h ~ n i l i a even r t o high school htl~ti~tits, IS called the idmi-gus eqac~tionof lar re. Eyuations 2. I are called i i ~ equi~lion i ul >t;ttc becausc they scrve as a relatiun hctween the pressure, volume, and tcniperature r l t ' the gas fur a given quantity of gab. A gas that obeys Equat~crns17.1 is callccl i i n idtit1 g i i c , o r the gas is ?aid to bchave ideally. The distinction between V and 7 illustra~esan important ul~iiraclerof the quat~tihes or the variables used to describe macroscopic systems. These qunntities are of twri types,

49

5U

(

t1,qitr.r ? i Thc Propcrries elf (idres

c21lIcd cxtcnsive quantities and intensive quantities. F~~rrtsivr q u o ~ ~ r i r i ror . ~exrri~sive , vririr:blrs, are directly proportional to the size of a system. V r > l u ~ ~mass. ~ c . and energy are exilmples of extensive quantities. Itrtetrsive q~rantitir.r, or. irlrr,r~sil~r ~.~~ri(~hle.v, do not depend upon thc si7c of the system. Pressure, temperature, and detikity ;Ire cxamplec of intc~lsivequar~tities.If we dikidt anextensivequantity hy thc riur~~hcr ot p;~rticlcsor ihe ~~urnhcrnf nloles iri a system, we obtain all intcnsive quantity. Forun:i~i~l~lc., V ( r l i l i ' ) is ;in cxte~iai~c quantily but (rlrn'.mol ' ) is an intcnsive quatitlty. Dislinguishinp htwtlen 11ic p r o l s r t i c ~of e x t c ~ ~ s i and r ~ e intenqive rluanlities is often irnportarlt i ti desc~-ihi~lg cf~emicalsysiems. The reason Equations 2.1 are encuuntered so frcqueti~lyin chc~iiisirycilurscs is thrtl all gascs nhcy Equatio~~s 2.1, as h ~ n gas thcy are sufficiently tlilu~c.Atiy ir~cliviiluiil charactcri~ticsof the gas, such as thc shape or size o l i ~molecu1c.s s or huw thc rti~>lrcuIcs interacr with each othcr, are lost in Equations 2.1. Jn a setlse, lhesc cqu;~tion\am a crlnirnon dct~ornitlilturfor all gaqcs. Experimenlatly, most gascs s;t~i\l'yI+li~:~tit,rir2 , I l o approxirnatcly 1%) at one altn and O'C. tquations 2.1 require us to discuss the system of units (SI) ;~drq~tcd t ~ yihc Iritcrnational I!nion of Pure and Applied Clicmistry (IUPAC). For cx:~~nlllv, i~ltl~rrt~ph thc SI 1111it01- \ , ~ I U I T I C is 111' (meters cubed), thc unit L (liter). which is dctirictl ilk cxr~ctly I rlrli' ( d e r i r n e ~ ecubed), r~ is an acceptable unit of volume in the Il!PAT' cycic~lb.'Ibr Sl unit of pressure is a pnscal (Pa). which is equal to onc newlor1 p r \~ll~:irrr i i ~ t ~ r (PJ= N.m = kg,m-' .s-'). ~ e c a lthat l a newton is the SI unit o f i'orcc. ~u wr w1.c ~ t i ; t ~ pressure i s a ~ K per P unit area. I'ressure can be mcasured experin~ent;~lly hy ol>ccrvi~tg how high a column of liquid (usually mcrcury) is supported by the gas. I!'III i c rl~r 1 1 1 : ~ ~ of the liquid and 8 is the gravitational acceleration constant, the pressurc is givrn b y

2-1.

/\I1

(;~ct.r I h . h ~ v cldcatlv IiThey h r c 511fflt ~t.~nlly Dilurr

Strictly speaking, new textbooks should use the IUPAC-suggested SI units, but thc units of pressure are particularly prohle~nalic.Although a pascal is the SI unit of pressure and will prubitbly sce incrc;~singu w , the alniosphere will t~ndouhtlycontinuc to he widely urcd. One ~ , i ~ ~ r o . t(at~il) p I ~ ~i~~defined r ~ ~ as 1.01325 x 10' Pa = 10 1.325 kPa. lone atmosphere uscd lo he dchned ;IS lhc prcssure that supports a 76.0-cni colunirl of mcrcury (see Example 2.1).1 Notu that nllc kPn i s ;~pproximately1% of an atmosphere. One iitrr~osphcreused tt) he thc c~:ind;~rd 01- preshure, ill the sense that t:~bul;lted prnpcrtiea nf substances werc preser~tctli1t c~ric:LIIII.Will1 the change to SI unils. Ihe standard is now one bar, which is equal to 10' [';I. c>r0.1 Mt'i~.Thc relation bctween ban and atmospheres is I atm = 1.01325 bar. Onc otlicr c ~ > ~ t ~ i s ~uscd o n l unil y uf Ilrcssure is a torr, which is the prcssure that supports ;I I .(M)-lu~ii coluirln I' mcrcury. Thus 1 iurr : :( 1 1760) attn. Because we arr experiencing ;I tr;11ihitio11period hclwccn the wirlespread use of atm and tnrr on the OIIC hand and the l'uturc u\r ut h;lr and k I 4 OII thc other halid, students of physical chemisiry must bc prolicien~it1 both \el:. or plcacurt' units. The re1ation.s betweell the various units of pressure are collectctl ill 'rahlc 2.1 Of the three quantities, volurne. prcssure, and tcmperature, ternpcr;~turei s the most difficult to cunceptualize. We will present a molecular i~~rcrpretaliorl of ternprature later, but here we will give an operational definition. The fundamental tempcraturc scale is based upon the ideal-gas law, Equationh 2.1. Specifically, wc detine T to he

'

where A is the babe area nf the colunln, p is the dcnsity of the fluid, and h is thc hcr~lli ut- the ctllumn. The gravitatjonal acceleration constant is equal to 9.8067 111 \ .'. Ilr 980.67 cn1.s '. Nntc that the area canccls nut in Equation 2.2.

because all gaqes behave ideally ill the limit u l P + 0. Tile unit of temperature is the kelvin, which is dcnoted K . Note that we do not use a degree synthnl whr.11tl~c temperature is expressed in kelviil. Rccuuse P and t'cannot takc on neeativc v:tIucr. lhe luwest llossible \,al~ieofthe temperature is L! K. 'rrmperatures ar l' it110 ihc (rppcr cast 7' for K . Note lilsrt t h a ~tbc degree symbol ( ' 1 i s associ;itcd with valucs 01. rhc tctnprralure in tllc I'clsius s c ~ l c . Equatic~n2.4 l e l h US ihat O K = 773. IS I.. t)r ttiitt (1 (' = 273.1 5 K . Htxitk~acof thc gcncral use r>l C i r i I;thor;ttur~c\. ii higuiiic;~r~carriuirtit of thcrrlltxlyu;~u~~c data are tabuli~tedfor suhstanccs at 0 C (273.15 K ) and 25'C' (2YH. 15 K):this latlur valuc 1s

whcrr wc hwc uacrl (tic h;rclttiut I h,m' = I N , n i = 1 J. Because of the change of rhc lrtundunl 01 prehhirrc I r t ~ t ruttnosphrrcb ~ ti) hirs, i t is also convenient to know the vuluc ctf H in u n ~ (11, h I..bur. 111t1l I .K I . Ilring ttlc k ~ c thi~t t I atm = 1.01325 har, we

Using the fact that I atm = 1 .O 1325 x 10' Pa and that 1 L = 10

' nl', we have

~ C lhal C

conurnonly called "rtnjrn tcmperaturc." 11' we lneacure at 273.15 K for any gas at a sul~ticienilylow prcswrt. that ils hehawor is ideal, then

PV

22.251 (I

_ I

I

0.2

0.4

0.6

(1.8

I

P / arm T/K FIGURE 2.1 1* Expcrir~lcntalmolar volume!, (urdid lirich) o f Ar(g) an-eplotted as a function of TIK at 0.040 atm, U.020 attn. and 0.010 alm. ,411 three pressures extrapolate to Ihe origin (da.rhc,dlines).

F I G U R E 2.2 A plot of experinlcnpl values of

PF

P for H,(g, tclrj\cc\). Nl4

and CO,(g) (circles) at T = 273.15 K. The dala fur PV = 22.414 L.atrn as P + O (ideal behavior).

(rlia~~lo~~dh),

all thrcc p;i..cr cutr;~lu)latrIIIa value of

T A B L E 2.2 'I'he values of the molar gas constant R in

variuur unit?.

H = 8.3145J rnol I.K.] = 0.083145 drn'.bar,mol-l . K - ' = 83.145 cni'.bar.r~lol-l.K-I = U.082058 L.atm.nlu1-' .K-I = 82.058 cni'.a~rn.rno~-l .K-'

2-2. The van der Waals Equation and the Rcdlich-Kwong Equation Are Examples of Two-Parameter Equations clf Stdtc The idral-gas equation is valid for all gasc5 at sufficiently low pressures. As the presstlrc o r 1 n gjvcn quantity nf gas is ir~creased,however, deviations iinm the ideal-gas e q u a ~ i r ~ n appear 'fl~cccdcviatinr~scan be displayed graphically by plotting P ~ / R as T a fullctio~l of pressli~t,as stiuwri iri Figure 2.3. The qurtntitj P V / R ?is- callcd the cclmn~)ressihility #u(.torand is denoted hy Z . Notc that % = I under all co~ldiiionsfur an ideal gas. For I-calgnccs. % = 1 at Inw pressures, but Jevia1ion.c frurn ideal behaviol.(Z# I ) arc sec11 as the pressure increases. The extent of the deviations from ideal behavior al a given pressure depends upon the tempcraturc and the nature of the gas: The closer the as is to [he point at which ir begins to liqueiy, lhe l a g e r the deviations from ideal behavior wrlf hc. Figure 2.4 shows Z plotted against P for methane at various temperatures. Note that Z dips below unity at lower temperatures but lies above unity at higher telnper2lurt.s. At lower temperature5 the molecules are moving less rapidly, and so

P l bar F I G U R E 2.4

The cnmpressihility factor o f methane versus pressure ;it various temperature\. I h i h 11gurt shuwf that the eltect of molecular attraction bccomes ~ E inlponant S al highcr lclllpl.r:llurr\

are more influenced hy their attractive lurccs. - Because of these attractive forces, thc molecules arc dri1~11 topcther, rhus i~lakingVmallcss than Vidri,,,which in turn causes Z to t x less than unity. A si~nilareffect can be seen in Figure 2.3: (he oirlcr ol. the curves sl~owsthar the ctfecl of molcculi~rattractions are in the order CH, > N, > He at 300 K. At higher letnperaturea, the niolccules arc moving rapidly enough thar their attraction is much s~nallerthan k,T (which wc will see ill Chapter 4 is a measure of their tbcrmal energy). The n~nleculesare influenced primarily hy their repulsive I-orces I. at higher temperatures. which lend to make I '-, 1',,lt4,1, and so Z Our picture of an ideal gas views the molecules ;I.; rnr~vi~lg it~dcpenderlllyt~l~c;lch other. not experiencing any intermolecular intrrnu~ion.;,P'igurcc 2.3 and 2.4 allow rhnt this picrure fails a1 high pressures, and thar the altlacrivc ;trld rcpi~l\ivcintermulecular interactions tnust be taken into account. Many equation\ c.xtentl the idcrtl-gas equi1lio11 to account for the intermolecular intcl-actions. 1'crh;lpc thc most well known is the vat? der Wi~alscquntiun,

Ideal gas

?OI1

U

3011

I'

M 1101) 800 I000 /11;1r

2.3 11plot of P ~ / R Tversus I' fur one mole of hcliurii, t\ilrt>pen,and methane at 30U K.n i s tifurc shows thnl the ideal-gas equation. fur w h ~ c hI ' V / H ~=' 1 , is "(71 ralid at high pressure. FIGURE

p

where designates moiar vt,lulllr. N r l ~ i cIII;SI ~ l:quutir,n 2.5 reduces to the idealgas equation when 7 i~ large. i~ Inuhr. Thc con.clants cr and h it1 Equatin~l2.5 are called vrm der 1N1ul.r c ~ o t ~ . r ; l t ~ l ~whtrcu .\-, vi~lucsdepend upon the particular gas (Table 2.3). We will see in Suction 2-7 thai the value u l a reflecls how stnlngly the mol~culcsof' u $35 altr;wl C:ICII other and the value of h reflects the size of the n~oleoules. Lct's use F.quation 2.5 10 calculate the pressure (in bars) cxened by I .(XI mol of CH,(g) thxt occtlpicc ;I 250-mL cuntai~lerat O"C. From Table 2.3, we find thal

;,\

.

57

2 - 2 I l l e van dcr r;\/,l,ll> Lquat~unand the Redlich-Kwrlg Lquat~on TABLE

2.3

At h ~ g hprc\surcs, thc lirst tcrm in Equation 2.6 dorninatcs because b l ~ l a l lilnd , ut low prebhurcs the sucond tcrm dtllninntes.

Fall der Waals constanlu for vanou\ substance!. Species

a/dmh.bar-~nol-'

a / d n ~ ~ ~ a t r n - m o l - ~b/dln7.mul

Heliur~~

0.U3.1598

0.034145

0.023733

Neon Arg011

0.21 666 1.3483

CI.21382 1.3307

0.017383 0.U3 1x30

Krypton Hydrogcrr Nirrngcn

2.2836

2.2537

0.24646 I ,3661

0.24324 1.3483

(3.038650 0.0266hS

1.3820 1.3734

1.3639

Oxygen

Carbon ~nonoxide

C'arbori dirjxide Arn~nut~is hlerhane Ethane Ethene Prrjpane Buhne 2-Methyl propane

Pentune Benzene

3.6551

1.4541

4-6112

3.6073 42181 2.2725 5.5088 4.5509

4.3919

9.2601

4.3114-1

2.3026

5.5Kl8

I' - b hectlrr~es

13 888

13.706

13.328 19.124 18.876

13.153 18.874

18.629

EXAMPLE 2-2 IJw ~ h van r tkr Wuith C~~U;III l r u ~ t l y~ h I$KIC ~ t #trcI T ~ ~ ~i lI~C~ u r aequiltl~ll'i le of SMC.

\

I By comparisnn, the ideal-gas equation predicts that P = 90.8 bar. The prediction o f the van der Wdals equation is in much better agreement with the exprirnental value of 78.6 bar than is the ideal-gas equiltion. The vim dcr Waals equation qualitatively gives the hehavior .chowri it1 Figures 2.3 ntld 2.4. We can rewritc Eqni~tion2.5 i n thc form

Iu{, ~ ~ t t l ~r.l,rt~rcly rr ~111qilr t+(IuUllllnh 01' state thiit are ~ n u c hrtlorc accurate and ~I-.~.I~II 1t1~1r1 ~ I I C \ it11 LII'I, W:I,I~'~ cquiltio~lare the Kcdliclr-k'wor~g~qrrution

t 1 t . 1 1 ~r 1tit11r

I'

K 'I'

r/ p

- -

u

v(vtp)+flcV-B)

58

2-2. Thc rCandcr LVFqllnlinn and the Rcdl~ch-KflungEq~f,ilir)ri T A B L E 2.4

S O L U T I O N : SubstituteT = 300K,P = 20Uatrn. A = 97.539drn6. a t m mol-I, K'," and 8 = 0.045153 drn7.rr~ol-lillto Equation 2.9. r r l (>blain

'I'hc Rcdlich-Kwong equation parameters for various substances. ~/dm"har.mth 2,K1!2

Spec ics

~ / d r n ~ , a t m o l - ' , ~ ~ : ~R/dln'.lntrl-'

--A.

I Irliurn Neon Argc~r~ Krqpturi tlydro~r~~ Nilrngc~~

0hygfl1 Carhnn monoxidc ritrhrln rlic>xidc An~rnoni;~

lfethrri~e Etl~anc Elhcne Prr>p;~~~c Hutnnc 2-Meihy l propane

Penta~~e Urnzcnc

0.079905 1.363 1 lh.7Rh 13.S7h 1.4333 15.551 17411 17.21)8 64.597 87.808 32.205 9K.Rl l 78512 183.02 240.1 h 272.73 419.97 453.32

0.078860 1.4439 16.566 33.137 1.4145 IS 348 17.183

16.983 63.752 86.660 31.784 97.539 77.486

180.63 286.37 269.17

414.48 447.39

0.0 I 0450

0.01 21WU U.02201>2 O.OZh780 0.0 18482 0.026738 0.022082 0.027394

0.029677 0.026232 0.029850 0.WS153 0.040339 0.U62723 0.08I)h8 0.0807 15 0.10057 0.082996

where we have supprc~sedtllc u-~ ~ ifor t s convenience Sulving thir crluation by thc Newtun-Raphson method gives I' = 1) 075U dlr~'.lnol I. cr,rnp:~rcd nit11 the rail d ~ , r W a a l ~I-erultof V = 0.04h dn~'.mol-l and the experimental recult of 0.07 1 d l n l -r11r)l ' (see Example 2-2). The prediction of the Redlicti-Kwtlng cquation i\ ne:~rlyquantitnlivc, ulllike the van der Waals equotlci~~. u'h~cllpredicts s value (IT th:!t 1s alwut ?Or, luo Iargc.

1:ipurc 2.3 colnpares experimental pressurc vcrsus density d a ~ for a ethane at 400 K with t h r predictions of the various equations uf state it~trnduccdin this chilplcr. Note that the Kcdlich-Kwong and Peng-Robinson equations are nearly yuantitntivc. whcreas [tic von d c r Waals equation fails colnpletely a1 pressures greatcr than 2(H) bar. One of t l ~ ci r ~ ~ l l ~ c \ \ ifeatures vc of the Redlich-Kwnng a l ~ dPcng-Robinson equation'; i \ Ihat ~ t ~ itrc c y 11ci1rlyquantitative in regions uhcrc the gas liquefies. For ex;~rnple.Figure 2.6 rlli,w\ r~ri.h\urc vcrhtts dcnsity datn f ( ~ethane r at 305.33 R. whcre it liquefies at around 4 0 h;~:. 'l'ltch t ~ r i ~ t t n t region al in the hgurc rcprcscnts liquid and vapor in cquilihriu~n will1 P I I C ~rrlhcr. Noic th;~t the I'el~g-Rnhinwn equatiorl is better in the l i q ~ ~ i d - v a p o r rcgirlri 1w1 that lhc KcdIich-Kwor~gequatior~is better at high pressuEs. The van dcr Wuuls cquotirtn ir not shown hcc;~usci t gives negative values of the pressure under thew cc~t~diliiuis.

where A. R , ct, and b. itre prtlamelers Ihi11 depend upon the gas. The vi~luesnf A and R i n Ihe Rcdlich-Kwong equation are listed in Table 2.4 for a variety of suhstances. 'I'11epi~rillnctclLY it1 the Peng-Robinson equation i s a sumrrvhat complicated function of lernpcraturc, F(I ~ t b:i l l not t a b u l a ~ evtllucs of a and 8. Equations 2.7 end 2.8. like the van der Waals equaliun ( E x a ~ r ~ p2-2), l e can be w r i ~ t e nas cubic equations in V .F o r exitmple, the KedIich-Kwong cqtin1iun hecomes (Prublern 2-26)

Problem 2-28 has you show that the Pcng-Robinson equation ol' statc is also a cubic equation in 7.

EXAMPLE

2-3

U s c thc Redlich-Kuo~~g equation to cillculute the molar volume of etl~arieat 300 K and 100 ntni.

t l ~ . ~ I U2.5 l

tixywrn~~c~~l;~l pic,w~revcrsus density data for ethane at 400 K (sol~dlinc) is compared r i t h Iht. cri the van der Waals equation (dot-dashed line), the Redlich-Kwong equation (long prctiuc~~c~~i\ tie\l~crIIinc). :~ritithe Pcng-Robinsan equation (short dashed line).

CICURE 2.6 Thc txperimen~alpreshure versus den\ity data (solid I~ne)for ethane at 305.33 K IS compared wilt1 he prediction!. of the Redlich-Kwong equation (long dashed line) and the Peng-Robinson equation (short dashed line). 'The liquid and vapor phascs are In equllibnurn in the hori~ontal

,

!I

region.

0 !

l o /l . . l b l ' b l

Although Figures 1.5 and 2.6 show compitri~unsclnly for ethane, the conclusions as to the relative accuracies of the equations art. genernl. In gcncral, the Rcdlich-Kwong cqilation is superior at high preasulrs, whcrcas the Peng-Kobinson equation is superior i n the liquid-vapor rcgion. In fact, these two equ;ltions of stale have been '-con~trucred" so that this is SO. There i r e mtlre sophisticated cquatiol~so f state (sorne containing Inore lhan 10 pnrnmeterh!) that call reproduce the experiniental dat;~tu a high degree o f accuracy over a large range o C pressure, den~ity,and tcniprature.

2-3. A Cubic Equation of State Can Describe Both the Gaseous

..

L ---J

li~pcrilntntnipressure-volume isc~thermsof curtkln d~un~dc around i t b critical tempcmture. MI V) ( ' I'O~III> (;. A, 0, and L are discussed $11 11ict c x ~

lincs in Figure ? 7 15 unllnl the rht#~Drmtr c,un.r. trrcuuw ally p o i ~ l within t this curvc comspc~ndsto InlrilJ unrl p n ct*rln~lnb in r q i ~ ~ l i h r i u with r n ciucl~other, At any pairit o n or o~ilcitlcthi\ ctbrvr, cmly t n e p h w 1% prcwnr I:trr ca:ltr~l~lc. ;it lwlir~t(i 111tJ1~ligtiw wc havc {rldy i t p,rr plhnr I[we lum kfurt ul ( i :illti crllliltlc\\ thr, G;I\ iltorlg llrc I.t.? C' 1hoth~1.111. l ~ t l l t l t wall l hrnt u1rpc.r w l ~ r l wc l ICIKIII ~ L Ch r ~ r ~ / ( ~ r ~ 11111t ; t l;!I 1njjri1 A . 'l'hr plccclil,c w i l l rclt1r1111t,rtnncmr IU wc r.ta~lrnrctlic Eikr 111 l u o l ; ~vr>li~rnc ~ 0. 1 1 ,111u1 (poilil A ) i t r Ilittutl rbl 111u1nt VIIIIIIIH.r l l ~ ~ p p r l x ~ l ~ 1107 h i ~ lI~ t111u1 y ( ~ n ) i t lI)). l Ahtr

'

rt.;~clllrlglrrllil 1). It& ~ncnburcr m r c u r r rlhiirllly w ~ t ;li ~ lunhcr dv~rc;i*c 111 vo~tltlle. t)cc.lu\c wc Iltrw llrlvr all Ilqutd UIHL I ~ vtrlt~rt~c F q d ;t I I ~ I~'1l.lrlg~'h ~ I ~ VCI-y littlc wllll I11I'CIIIII'

and Liquid States

;I-,ihr 1~111lk'rututc ~ ~ l i ~ r r qowirrrl : ~ u ' ~t h L,I ~ 11lc;tl tc~ul~ur;ttill.r.lhe horiIIIIC'I L ~ ~ M P~Hl i rdl i h a p p r ut the c v i l i ~ : ~ t~ I f l l l x r ~ t i At l r ~Ihlh , point, the nletlls-

Nirlr ih,it lrllllill

A reniark~blefeature of equntions o f state that can be written as cubic equalions i n V is that they describe both the gaseous arld the liquid rcgions o f a substance. To understand this fcaturc, wc start by discussing some exprimentally determined pluts of- P as i t fur~ctlunut V at constdnt T , which are cotnlr~onlycalled isotlrrrms (i.50 = cunstant). Figure 2.7 .chows cxl?crimentnI P versus V isotherms fbr carbon dinxidc. The isother111sshow11are iri the neighborhtxxl o f the critical tcnq>craturc, 'I, which 1s the ternperiturt: ahove which a ga5 cannot be liquefied, regardless of the pressulr. Thc critical prcssure. t', and the critical vulurne. are the corresponding prc.c.culc and thc molar volurpe ;I( the r,rilic,~rl pnir~r.Fur exrtmplc, for carlmn dioxide, 1: = 31M.1 1K {3O.YY C), q = 72.9 iltln, and = 0.094 L..lnol I. Notc t t ~ a t~ h r i.sotherlns in Figurc 2.7 flattrrl 0111 SI; 7' + TLtrum above and that tllcle arc h~rri70111ul regions when T is less than Tc. In the horizontal rcgions, gas and liquid coexirt 11) ecluilihrium with each other. The daahcd curve connecting the ends uf the h o r i ~ o l i ~ d l

8.

vL

lr~urrli ~ l t cIiqiritl illnl 11- vnprr tll\;l~~r;irs:tnd thur,c i.\ 110 ~lihtinctionbctwcun Iltluld ;tilt1 p.1~. ltw rut lrti,c tclirltbll dl~ill)lsilrh,1110 t JIG p ? r ;i11i1 Ilquld phases both l ~ ; t \ r i l r r w ~ r ~ t~cc t l u c ill 1 tlrthtly Wi. u 111 t l ~ r ~ ,~ i~ l i \cc~ \ 11ic;tl lx!irll i~rll)r)rc det;til in t 'I1.111lcr0 I +yurt .' H \ ~ ~ I H .\11111l;tt 'I I . ~ ( I ~ ~ C I I T \h~ \ thc' Y~III dcr Wad\ cqulirio~liltld the RedlichKuilli~ rqurhllrlrt Nd~tbrci1t:~i the twi) quiltions of state give fairly similar plots. The ~MIIIOLI\ hurpr ~ l ~ i : u i ~ttw c t l1' -r 1; rcsult from the approximate naturc of these cquat l t r l l c r r l ~ t l l t ct.apilrc1 2.') ?rhi)w~ a single van der Waals or Redlich-Kwurlg isotherm lor I - 1 : . '1-tic curvc. ( ; A l l ia the curve that would bc observed cxpenmer~tallyupon ~~crlillirca\lng the g i b . The hurizontal line D A is drawn so that the iilras of ihe Ioc)p hrlth ;ihcivt and below DA are equal. (This so-called .Moxwrll equrrl-urcrt

>-'I

A

(

ulj~rFtitm of S~dtcClili 111,511 rilw Roth lhr (

: l r l ~2nd ~ (Liquid ~ ~ ~51,1t('~ ~

is consistent with the fact that the van der Waals equation call bc wrilten as a cubic polynomial in the (tnolar) volunle (see Exalnple 2-2). The volume corrrspondi~~g to point D is the tnolar volurrje o f thc liquid, the volume corresponding to p i n t A is the molar vulurne of the vi~pori r cquilihriur~~ ~ with the liquid. and the third root, lying between A and D i s spurinur. At 142.69 K and 35.UO Gtln. argon exists ;IS two phases ill cquilihri~lrnwith each (111111' : i i r l c 1 thc dcnsitiw 01. thc liquid ;rnd v;IIwr ~ ~ : I S L I S ;II-c 22,491 rnol.L and 5.29 1 tllul.I . I , respectively. 1,ct's sec wh;it thc van dcr W,I:I~~cyuntion predicih ill this case. As we saw i n Exarrjple 7-2, wc can write ~ l j cvat1 dcr W,r;ils equation as (1 (I

0 2

V/

L. rnol

0.4

'

0.0

0.2 0.4 V / t.+~nv~-'

I

FIGURE 2.8 Prcrsure-vnlunle ihutherms uf carhun dioxide ar-oundits critical temperamre, au calculated from (a) the V~UI drr Waals cquation (Equaliol~2.5) and (b) thc Redlich-Kwong equation (Equalion 2.7).

U s i t ~ gthc values of ri and h irorrl Table F4~1ati1,n2.10 becorncs

7.3. T = 142.69 K. and P = 35.(XI atm.

whcn.. for ~(~nveniencc. we have supresscd the units of the coefliciellt~.Thr three nwtr ot tlli* cyuation arc (Problem 2-22) 0.07073 l . . r n t ~ l - ~0.07897 , L,rnr)l-I. and 11.2 l h7 I.-11101 I . The smallest nlot represents the molar volumc of liquid argon, and the I ; ~ r g r rr :I\ ;r c\rtbr~rtlu.ltlurl In 1 ' , I.r~ 2.19. \(I Hquatir~n2.19 is an example of the Inw nfcnrrespc~ndingstates, that ;rll p;iaes have the same - properties if they we compared at corresponding condition\ (salnc viiluibs r ~ l ' P,. V,, and T,)

,,,!

E X A M P L E 2-6 Usc t l ~ ccritica1-constan1data i n Table 2.5 10 evnluate A and 8 , the Redlich-Knong const,lllts Tor ethane SOLUTION:

(U.UX31.15 drrh'-har,~nol .K-')'(305.34 K)':' ----48.714 bar = 98.831 dtnh-hnr.111ol - K ' ' = ~ 97.539 dln6.a1Tn n l ~ l - ' . ~ ' ! ~

I

v,.

A = 0.42748 -

(0.083 145 dm7,bar-mol- -K-')(305.34 K) B = 0.OAh640--. 48.714 bar

E X A M P L E 2-7 Express the Kedlich-Kworlgequation i n tsrrns u i reduced qual~lilirq.

S(1I I J T I O N : Equalions 2.18 show that

Chapter 2 ! The Propert~rGr ~Gases f

Substituting these fquivalencies inlo Equation 2.7 givcs

r l i v i h the nurneratrjr and thc demn~inatorof the first term on the right bide by the second by to get

vL2

i/t and

14. The v.m rler bVaal;ialsEqudllnll and the Hedl~ch-KwongEquat~onObcy thr I .w of (:urreqiunrIing 5 t d t h

Equations 2.10 arid 2.2 I express Z as a universal function of 7, and T,, or o f any other two reduced quantities. such as P, and T,. Althc~ughthese equalions can he used to illustrdte the law ot corresponding states, they are based on approxiinate eq~~ntions oi' state. Nzvenhcless, the law US corresponding slates i s vlilid for a grcut variety of gi15e.c. Figure 2.10 shows rxpcrimzntal data for Z plotted against P, at u a r i o ~ ~vnlues s of I , for 10 gases. Note that the data fur all 10 gnses f i l l OII the came cli~vcs,thus i l l u s t m t ~ n ~ the law of corresponding states in a more g r i ~ c rway ~ l that1 either Equatiot~2.20 or 2.2 1. Much more extensive graphs are available, particularly in the engineering litcralurr, and are of great use in pmctiual applications.

-

Divide both sides by {,nl~duce the Fact that 1: V L / R 1 ; = 1/3 in the cccond tcrm tr, gel

Finally, multiply and divide the numeralor of the first lerm on the righl side by Tc to obtain

Thus, we hee that the Redlich-Kwong equdtion also obegs a law of corresponding

states.

Thc cr>mpressibility factur, Z , asswiatzd with the van der Waals equation also obeys the law of corresponding states. To dclnt~nslratzthis point, we start with Equation 2.6 and substitute the second of Equations 7.12 fur a and Equation 2.13b for b to get

pm , d,'

x

Nbtrogen

Melhane

(I,?

Now use Equation 2.15 for pLVL i n the secnntl tcrm and introduce reduced variables to

.r,= 1 .ou

n-Bulurle

lrnpcnlnnz

Ethylcnc

A

rr-Hcptarle Cerhun dioxide

Propane

m

Water

a Ethanc

i.

o s c

get

Similarly, the cunipressibiljty lilctor for the R~dlich-Kwongequation is {Prt~hlern 2-30)

F I G U R E 2.10

,

AII illustration of the law of corresponding stale$. The compressibility faclur, Z, is plcltled ugdinsr the reduced pressure, P R ,uf each of the 10 indicated gahes. Each curve rcpresenls a ~ I V C I reduced I tcmperatuw. Note that fur a given reduced renlperaturc, aH 10 gasec KaI1 on Ihc aamc curve because reduced quantitiec are uscd.

f,q

Chapter 2 E X A M P L E 2-8 l i s e bigure 2.10 to cstirrlalc lhc ~rlcil:~rvr~lunlcof' :lrnlilcrni;l

1 Thc Properties of Caw5

1 ,IT? 15 ( ' :tr\tl

4(A) har

5 0 L U T I 0 N: Ucing [he critici~l-con%~:~~~i d:ita 111 7bhlc 2 3, wc l i r t t l Ihat 'I; = 1.20 T h e rmllu and P, = 3.59. F i g ~ ~2r c10 chilwx thar 7 ?- 0 . M ) ur~tlcrtllcrc ~,crntlil~rmr volu~neis

Tl~cIilw o t ' c t > r r ~ * s l > c ~st;lttns ~ l d ~ II~IS ~ ~ g;I I~~c'c. ~)hy\iculiliturprvtatio~~. Ally tcnlpcrii1ui-c sc;~lc wc urc to dc\crjhc ;I F;IS i\ ~ i ~ c c c w r iarbitrary. ly Even the Kelvin .cuale, with its futlJarllc~~t;~l fcro 1e111pcra1ul-c, is arl~itraryin the sense that the size of n degrcc 011the KClvi~1 scale is arhilrary. 'I'hus, the numerical value we assign to the temperature is rnca~linglcsras far as the gas i~ uuncemed. A gas docs "ki~ow"its ctidual temperaturc, and theretnre i s '-itware" u f its temperaturc rrlurivr to its cnliual temperature or its rcduced relrlperature, T, = T / ?;. Similarly, pressure and vulunie scales arc ilnposed hg u ~ but , the reduced pressure and the reduced volume arc quantities t h a ~art. uf significance to a particular gas. Thus, any gas that has a certain reduced temperature, pressure, and volume w i l l behave i n the sanic mallner as another gas under the same conditions.

2-5. Secorld Virial Coefficients Can Be Used to Determine

Intermolecular Potentials Thc most fundatnel~taleqt~ationo f state, i n thc sense that i t has h e most xou~ldtheorctical foundation, i s the virinl equurion clj'srrrrr.The virial equ~tionof state expresses the compressibility factor as a pulynomial i n 1

/v:

The coefficients in this expression art: functirms of temperature only and arc called vrrrrxl corb/"fit.imrs.In part~vular,B,,(T) ir called the second virinl ro~@riclrtt,B,,(T) thc t h l ~ d and , so nn We w i l l see later that other propenleh such as energy and entropy can he exprcwcd as polyriorrli:~ls i n I/V.and generally these reIalions arc called viriul rbxpunhion.\. We can also cxpress the cornpress~b~Ilty factor as a polynomial in

P

2 5.

5t.t r b r ~ r l ' i r ~ a l Cric,ft~cienrs Can

Bc I!rcrl to nctr.rm~nt*Ir~lt.rrr~r,lrtu l a r t'rller~ri;ls

F ~ i ~ a t i o2.23 l i i s also caIlcd a virial expansion or a virial equation of state. The virial cc~.fficicnlsH,,, (7') ;ind R , , , ( T ) :rrc wlijtcd by (Prnblc~n2-36 )

Nutr ill llq(~;~!~nib 2.22 or 2.2 1 Ih:r~ % - + I SI; 7 becomes large or as P becomes LIII~~II, jirrt 11% 11 shrrultl. 'luhlc 2.h gives ;in idc;~of the magnitudes oi the terms i r l t$uatii~li 2,2? UA II IUIK.I~IL 01prthh~ln.I'or ;hrgrlii it! 25 C. Nrjticc that even at lo() b i ~ r ~ip thr lirmt lhrtc lrrnlr nre sunicicnl f i r c ~ ~ l c u l u t i %,

I A A l t 2.6 'l'l~c~ , t ~ t ~ I r i h i01'~ tI ~~aC!~I+I ~ r ~tcw Icrril\ III t l ~ ck~rialtxpdnhlnn c>f1,tiquatiorl 2.22, for argon a1 23 C'.

I + B'','T' . =_..+ I'

ZI1y\T'+ rcrnaining terms

-

v-

'I'hc sccnnd virial coefficient is the most important virial cocfticient heciiuse it reilccts the first deviation from idcality as the pressure oT the gab is itlcrcascd (or the volurijc is dccrt.;tscdj. As such, i t i s the most easily ~lieasurctlvirial crxffic~entand i s wcll lahulatrd I-or many giihus. Auvtwtling to f.qu:ltion 2.23, il ciin be deterlr~ined c x p c r i ~ ~ j c r ~ t iL'lon~ ~ l l y lhc slopc of ;I plut of % ;~K;LII~lccul;ir interiictions. Consider two inleracting rnulecules as ~ h t l 11 n 111 1.1g11rc2 1.1. 't'hc intcravtion of the two molecules dcpcnds upon the disti~nce twtwccr~tllcir c.cnlcr\, r , and upon their c~rienlations.Because the molccule~are ruaverage out, so for simplicity we assume that the laling, thcir ~ i r l c r ~ t ~ t l opartially ns inlcr;~cl~orl tlrpctids only u p n r . This approximation t l ~ r n sout to hc satisfactory f i ~ r

FICUUt

0.998

1

(1

I

0.U2

0.04

I

I

I

0.06

0 08

0.10

2.13

TWOliilcraullng l ~ m wriiolecules. Gencrully, thc interrnrdecula interaction bcrween h a rnolcculcx cLcycdx upm Ihc Jtrtuncx herwccn their cenlers (r) and upon their orientations (1)' , It:, 1 1 d $1

P / bar FIGURE

A

2.11

plot of Z Venus P at luw pressure? for NH,(g) at O'C, I W C , and 200°C. The alopes of

the lines :ire equal lo B , , ( T ) / R T according tn Equations 2.23 and 2.24. The respectice slopes give B1,(O C) = -0.345 dm'.mol-'. B I , (100'C) = -0 142 drnJ.mol I. and B1,(200 C ) = - O 075 dr~~'.mrll-'.

whcrc W, is (tic A c I I @ ~ cttnatml ~ und k , i s thc Hnlumann constant, which i s equal El divided by the Avtbgadro constant. Note that B I , ( T ) = O it' u ( 1 . ) -- 0; IIItlhcr wdn, rhcm arc ntl Jcviatii)ns l'rotn idcal hchavior if- there arc I ~ O illlr+r~ntrltculur ~r~lcrwlii~~ln. l!quatitrli 2.25 h t w a lhxl cmc u ( r ) l a Lrowll, i~1s ;I *il!l~,lc Inaitcl' 11) c;llculatc HI,. (7')aa rl l u ~ i c l l i ~ofn ~empcmture,or cr~rlvcncly.14) dctc.rmine u ( r ) I!' &2C. ( T ) IS known. [ti priric~plc,r r ( r ) can k c~~lculutcd I'rorri quantutn rncch;~nics, hut this is ;I dillic~bllcn~iipututior~al pnlhlclt~.11Gun be 5hijw11, h 20 3 3 7U 358

86.')

]II

37.0

N!

95.1

0:

I IX

C'U

ICW

CC),

189

Ct4

152

CIi, C?H4

149

CI H,

243

C~HR C(CH,),

242 232

199

31 7 63.V

57 .o

376 449 47U 378 452 395 5M 744

67.0

114.2 131.0 68.1 1 16.5

77.7

226.3 519.4

If we substitute the Lennard-Junes potet~tialinto Equation 2.25, ae obtain Equation 2.30 may Icluk cumplicated, but it ran he simplified. We first define o reduced

temperature T' by T * = k , T / € and l e I~- / l r =

whul,tH3,(7-'1

f I G V R E 2.14 A plot o t i ~ ( r )= /~ 4 [(:)I'

w

-

( f ) 6 ] versus

T/D lor the

Lennvd-Jones palentiri Note i b t the

d c p ~ hof rhe putcntial well i s c and that u(r) = 0 at r/n = 1.

-

I

try g u ~

H,,~I'~/(!~r~'~~/.~~.1~~1;1tioti2.J1showsthatthereducedsecond

~IIW>II only the reduced temperature, T '. The integral viriill ~ . ~ w l - f i ~/I,:, ~ i t ~1.'r ~1. lIIC~WIIII\ , in I Q u ; ~ t i t ~2r ~11 IIIII\I tw TV:IIII:IIC~ ~ i ~ ~ ~ r ~(Mathchapter ~ r i ~ n l l y Anl for each value of 7 ' * . I'.ut~-rl\~it+ t ; ~ t ~ l t .111 \ It:, I I' I vcr\u\ 7" are available. I,qu:ltlulr 2 . 1 I i\ ; ~ ~ l t r ~r tx~; ~c ~r i l p lof c the law of corresponding states. If we take cnpcl ~lllc.r~t;~l \:lluc\ 111 H : , I 1 ' ) - divide them by 2rsu3~ , / 3 and , then pIot thc data vrrcllc I' X ,I I/ r , t l ~ r u s i ~lor l ~ nll gases will fall on one curve. Figure 2.15 shows rucll :I l ~ l t r t ' 0 1 5 i a gi~scs-Conversely, a plot such as the one in Figure 2.15 (or better yrl. l~ulllc.r!c;~l 1:ihlcc) c:u~he used to evaluate B , , ( T ) for any gas.

2 6. London Oispers~onForr es Arc Otten the

Largest Cuntributkin lo thcr

y,cdl at pressures such that the contrihulion r,l

and the ideal-gas value cwfticient i s ncgligihle.

I

k' ,,b -I

I

E X A M P L E 2-10 Estilnnte H,,{T)for N,(g]at 0 C'.

--

-

5CILLJTLON: 'L'dhle 2.7givcs f / k , 95.1 K and 2 r r t r ' ~ , / . l hB 0 c.111' ~ r l c , l Nllg). 'Thus. 7' = 2.87. and Figure 2.15 givcs Or, (7")s - 0.2 Therefore,

lor

If wc had used numerical tablcs fol- H ; , ( T ' ) iristcad of Figure 2.15, we would hare uhkniried B;,.(Y'') = -0.16, or U?,. (1')= - 10 cln3.mol--'. I

I

The value of Ll,,.(T) has a slrriple intcrpretrttic~n.Consider Equalion 2.23 under uondiriorls whcrc we ran rgnoIe thc terms in P' and higher PV k,.(T) = I H2,,(7')P= 1 +KT' RT

+

By ~riultiplyingthn~ughby K T - / P and usirlg

TI,%,= R T / P , we can rewrite this

equation in the Form

-

H,, = V

KT.

I'

-

-

\I,,,

I

,F

(O.lln.7145 ~.r l . ' , h -: a.,mol --~ ')(300.0 K) . .. . I har

= 24.31 drn'-l~~ol ' - 24 04 tlll~'.nirll '

= -0.b7drll' mol

= - -hlO c n l ' , lnul

'

Although we have been discussing calculating EL,(7') in t e r n s of the LennirrdJones potential, in practice it's the other way m u l i d : Ler~nard-Jonespanttleters are usually detennined from experimental values o f Dz,(T). T h ~ dcturminatiun s i< usually made through trial and error using tables of B , * , ( T h )'l'he . valucs o l lhc Ennard-Jotlcs para~netersin Table 2.7 were determined frum cxperimcntitl second virial c o c t l ~ u ~ t . r ~ ~ d a u . Because the swond virial coefficient rcflccts the initial deviutluns from idcal behavior, which are caused by intermolecul~interactions, experimental F -V-7' k i l a turn out to bc a rich source of information concerning i ~ ~ t e m ~ o l e c uintzr~cliol~s. l:~r Once Lcnnard-Jones parameters have been determind, they can he uccd I ~ ualculi~tr I many other fluid properties such as viscosity, lherrnal col~ductivity,ttriits o!' vi~p~riiation, ;rrltl various crystal properties.

2-6. London Dispersion Forces Are Often the Largest Contribution t o the r P 6 Term in the Lennard-JonesPotential In the previous secrion, we used the Lennard-Joncs pr)tc1111;1l(l.:rli~;ttiui~2.29) to represent the inter~nc~leculnr potential between rnvlccl~les. I'hu r- '' tcnrl accuunts fclr the repulsion at short distances, and thc r-"ern1 nucclullt\ t t r ~thc u ~ r a c ~ i oilli l larger disw~~ccs. The actual fr,m of the rcpulsivc terrrl ic no1 well r.\~;ll>llshcd, but the Idcpende~rceof the allractive term is. 911 1hi5 \cc.tiorl. wc will dihc.i~\\three cc~ntributitlns 10 the r - h attraction and comparc thcir ~elativcI I I I I I O I I.IIIC,L,. Conbidcr twu dtpuliir tnoleculcs, nhc>+e~li(xllc.~rrrrrn~.r)l\ ~trc11 ilnd p , . 'The rtltcraction of thc\e dlpolcs del)cn~lsrllxln hclu lllcy :LIP orlcr~~cd with rehpcct tu cach otl~cr. The energy will v;try Iron1 rcpulsrvc, W ~ I C I Ir t ~ u pill-c cjricnted Itcad-to-head as sh(>wrt

,

Thus. we see that I 4 , ( T ) represent< the difference betwccn the actual value nf

the third virial

E X A M P L E 2-11 Thc molar volulrn: c>f 1sc7hulancat 300.0 K and onc bar i.; 24.31 drn'.rn~l-~. kstir~late the valuc of H ! , lrrr i\ohu~uncut 31Ml.O K. 5 0 1 . I J T I ( 1 N . 'I hc itlcitl-g;~\rllli~r v~~lurt~c at 3(K) O K and o ~ l bar c

F I G U R E 2.15 A plot of the reduced secolid virial corffioient 8 ; , . ( T X )= B , , { T ' ) / { ~ ~ u ~ N (rtllid , / ~linc) ) ~ T' = k,T/c. Expenmenpal data of six gahev (nrgull, nitrogen. again\[ the w d u c temperature oxygen, 'arbo~t dioxide, and sulfur hex,~Rur>ride) are alho piolted. 'lhis pto~ib another illurlralir)n of the law uf correspontling \tale$.

77

lern,

2 h.

I.r)rirlc)ri

T l i \ [ ~ t . r h ~ c > rFur3

HC'I

2 f>l

HI

3 44 1.47

*Hi

5.iK)

2 2
lcuule>.

where I, is the io~~izalion energy of alurn or rr)oleculc j . Nnle th;~tEquatlon 2.3h duch not involve a permanent dipole moment and that the interaction energy is propurtlonal tu thc pruduc~of the polari7ability volumes. Thus, the irnportuncc ~ I ~ I I , , , , ~ ( I -increases ) with lhc sires oi ~ h uloms c or jnolecules, and, in fict, is often the duininar~tcoritrihution l o the Fb ter~nin t y u a ~ i 2.29. o~~

E X A M P L E 2-15

Colc~llutcthe value of the cmflicient of r " for u,,,,,cr) Ibr two 11rItg3 rr~c>luculc.b.

Chapter 2 I The Properties of

82

Cases

This quantity is about six times greater than -r6u,,(r) and 20 times greater than - r ' ~ , ~ ~ , , ~Similar ~ ~ ( rcalculations ). show that the disperison term is significantly larger than either the dipole-dipole term or the dipole-induced dipole term except for very polar molecules such as NH,, H,O, and HCN. The total contribution to the r-6 term in the Lennard-Jones potential is given by the sum of Equations 2.33, 2.35, and 2.36, giving C C U(r) = I Z- A rI2 r6

2-7. The van der Waals Constants Can Be Written in Terms of Molecular Parameters

This potential represents hard spheres of diameter 0. Equation 2.38 depicts the repulsive region as varying infinitely steeply rather than as r-". As simplistic as this potential may seem, it does account for the finite size of molecules, which turns out to be the dominating feature in determining the structure of liquids and solids. Its obvious deficiency is the lack of any attractive term. At high temperatures, however, meaning high with respect to & / k , in the Lennard-Jones potential, the molecules are traveling with enough energy that the attractive potential is significantly "washed out," so the hard-sphere potential is useful under these conditions. The second virial coefficient is easy to evaluate for the hard sphere potential. Substituting Equation 2.38 into Equation 2.25 gives

with (Problem 2-53)

for identical atoms or molecules.

2-7. The van der Waals Constants Can Be Written i n Terms of Molecular Parameters Although the Lennard-Jones potential is fairly realistic, it is also difficult to use. For example, the second virial coefficient (Example 2-10) must be evaluated numerically and one must resort to numerical tables to calculate the properties of gases. Consequently, intermolecular potentials that can be evaluated analytically are often used to estimate the properties of gases. The simplest of these potentials is the so-called hard-sphere potential (Figure 2.17a), whose mathematical form is

which is equal to four times the volume of N , spheres. (Remember that a is the diameter of the spheren.) Thus, the hard-sphere second virial coefficient is independent of temperature. Note that the high-temperature limit of the second virial coefficients shown in Figuren 2.12 and 2.15 is fairly constant. The curves actually go through a slight maximum because molecules are not really "hard." Another nimple potential u d fairly often is the square-well potential (Figure 2.17b):

The parameter E is the depth of the well and (A - I)a is its width. This potential provides an attractive region, as crude a. it is. The second virial coefficient can be evaluated analytically for the square-well potential

.

F I G U R E 2.17 (a) A schematic illustration of a hard-sphere potential and (b) a square-well potential. The parameter u is the diameter of the molecules, E is the depth of the attractive well, and (A - 1)u is the width of the well.

Note that Equation 2.41 reduces to Equation 2.39 when A = 1 or E = 0, there being no attractive well in either case. Figure 2.18 shows Equation 2.41 compared with

83

2-7. The van der Waals Constants Can Be Written in Terms of Molecular Parameters

Comparing this result with Equation 2.22, we see that

for the van der Waals equation. We will now derive a similar result from Equation 2.25 and interpret a and b in terms of molecular parameters. The intermolecular potential that we will use is a hybrid of the hard-sphere potential and the Lennard-Jones potential

We substitute this potential into Equation 2.25 to obtain F I G U R E 2.18

A comparison of the square-well second virial coefticient for nitrogen. The square-well parameters for nitrogen are a = 327.7 pm. ~ l k = , 95.2 K, and A = 1.58. The solid circles

represent experimental data.

B2,(T) = -2n NA

I'

(-

l)r2dr - 2n NA

In the second integral, we assume that r6/k,Trb (MathChapter C)

experimental data for nitrogen. The agreement is amazingly good, but the square-well potential does have three adjustable parameters. We will finish this chapter with a discussion of the second virial coefficients for the three cubic equations of state introduced in Section 2-2. First, we write the van der Waals equation in the form

I"

semltntlrgy ot a Syrlern

We will learn in Chapter 4 hat fur a rnona!omic ideal gas, EXAMPLE 3-2 We will learn in the next chapter that fur the rigid rotator-harmtmic oscillator model of an ideal diatomic gas, the partition function is given by

where where

For a monatomic ideal gas in its electronic ground state, the energy of the system is only in the translational degrees of freedom. Before we substitute Equation 3.22 into Equ~tion3.20, we write In Q for convenience as a sum of terms that involve B and terrns that a r t independent of 8: InQ=NInq-lnN! 3A' 2

In this expression, I is the moment of inertia and v is the fundamental vibrational frequency of the diatomic molecule. Note that q ( V , B ) for a diatomic molc~uleis the same as the expression for q ( V , B ) fora monatomic gas (Equation 3.23, a tnnstational term), except that it is multiplied by a rotational term, 8 r r ' l / h 2 @ , and a vibrational term, e-ph"/2/(1- e-Oh'). The wd5m for this difference will become apparent in Use this partition function to calculate the average energy uf one mole of Section a. a diatomic ideal gas. S 0 L U T 10N : Once again, for convenicnce we write In Q as the hum of terms that

involve 8 and terms that are independent of 8:

3N

--In B + terms involving only N and 2

InQ = Nlnq - l n N !

V

Now we can see more easily that

+ terms nut involving fi Now 3NdInp

N d- l- n- b- N Nhv

dB

atid that (Equation 3.20)

For n moles, N = nNA and kaNA= R , SO (E)= ; ~ R T

This observation leads us to a fundamental postulate of physical chemisiq that the ensemble average of any quantity, as calculatd using the probability distribution of Equation 3.17.is the same as the experimentally observed value of that quantity. If we let the experimentally observed energy of a system be denoted by U, then we have

for one mole of a monatomic ideal gas. (We indicate a molar quantity by w overbar,)

2

3 Nhv U = ( E ) = -Nk,T+Nk,T+-+2 2

dIn(1-e-"") dB

~hve..~~"

For one mole, N = N, and N,k, = R , so

Equation 3.24 has a nice physical interpretation. The first term represents the average translational energy, the second term represent5 the average rutatiunal energy, the third term represents the zero-point vibrational energy, and the founh tenn represents the average vibratihal energy. The fourth term i s negligible ill low temperatures for most gdseb but increases with increasing temperalure as the excited vibrational states become populated.

3-4. The Heat Capacity at Constant Volume Derivative of the Average Energy

Is the Temperature

E X A M P L E 3-3 111 1905, Einstein proposed a simple lnndel for an alomic crystal that can be used to calculate the molar lwat capacity. He piclured an atomic crystal N atoms situated at lattice sites, with each atum vibraling as a three-di~nensiunalharrnorlic oscillator. Recause all the lattice sites are identical, hc further assumed that each atom vibrated with the same frequency. The partition functinn aswiated with this model is (hob-

The constant-volume heat capacity, C , , of a system is defined as

lem 3-20)

C, is then a measure of how the energy of the syslem changes with temperature at constant amount and volume. Consequently, C , can be expressed in terms of Q(N,V , T ) through Equation 3.21. We have seen that = 3 R T / 2 for one mole of a monatomic ideal gas, so The heat capacity

-

C, = ; R

monatomic ideal gas

SOLUTION : The average energy i s given by (Equation 3.20)

For a diatomic ideal gas, we obtain from Equation 3.24

5

2

where v, which i s characteristic uf the particular crystal, is the frequency with which the atoms vibrate about their lattice positions and U , is the sublirnatiun energy at O K, or the energy needed to separate all the atoms frvm one another at 0 K.Calculate the molar heat capacity of an atomic crystal from this partilion function.

diatomic

(ideal ) gas

(3.27)

Figure 3.3 shows h e theoretical (Equation 3.27) versus the experimental molar heat capacity of O,(g) as a function of temperature. The agreement between the two i s seen to be excellent.

NOW that U consists uf three terms: U,. the sublimation energy at 0 R,3 N h 1 ~ / 2 , the zcro-point energy nT N thrcc-dimensional hannonic oscillators; and a term that represents the increase in vibrational energy as the temperature increases. The heat capacity at constant volume is given by

where we have used the fact that

r i c u u E 3.3 l'hc r x p r i l n c ~ l ~and ; ~ l theoretical (Equation 3.27) molar heat capacity of O,(g) from 300 K to I(HWI K 'rhc ~hcorcti~al cunvc(wllid curve) is calculated using hulk = 2240 K.

N = N, and NAk,= R for one mole.

Equation 5-29 contains one adjustable parameter, the vibrational frequency v . Figure 3.4 shows the molar heat capacity of diamond as a function of temperature calculated with u = 2.75 x 1013 s-'. The agreement with e x ~ r i m e n tis wen 10 be fairly good considering the simplicity uf the model.

3-5. We Can Express the Pressure in Terms of a Partllion F u n c l ~ r > ~ ~

Using the facl that the average pressure is given by

we can write

This expression can k written in a more compact form. Let's stan with

F I G U R E 3.4

The vhserved and thoretical (Einste~nmodel) rnulnr heat capacity of diamond as a function of tempemure. The solid curve is calculated using Equation 3.29, and the circles represent expt.ri~ilentnldata.

and differentiatei t with respect to V keeping N and fi fixed:

Comparing this result with the second equality of lquation 3.31 shows lhat It is interesting tu look at the high-temperature limit of Equation 3.29. At high temperatures, h v / k , T is small, so we can use the fact that d I x for small x (Mathchapter C). Thus, Equation 3.29 becomes

+

Thix result predicts that the molar heat capacities of atomic crystals should level off at a value of 3 R = 24.9 J-K-'.mol-' at high temperatures. This prediction i s known as the law of Dulong and Petit, which played an important role in the determination of ittolnic masses in the I XOOs. This prcdictic~nis in good agreement with the data shown in Figure 3.4.

3-5. We Can Express the Pressure in Terms of a Partition Function We will show in Sec~ion54-that the pressure famacroscopic system is given by

Just as we equated the ensemble averdge of the energy with the observed energy, we equate the enselnble average pressure with the observed pressure, P = ( P ) . Thus, we see that we can calculate the observed pressure if we know Q ( N , V. 8 j. We can use rhis result to derive the ideal-gas equation of state. Firqt, considel :t monatomic ideal gas. Recall from Equation 3.22 that Q ( N , V , @ ) for a monatomic ideal gas i s given by

where

Let's use this result to calculate the przssurc o f a monatomic idcnl gas. 'To evaluate Equation 3.32, we write out In Q first for convenience:

118

Chapler 3 1 The Boltzmann racror and Partition Functions

Because N and

are fixed in Equation 3.32, we write In Q as

36. The Partition Function of a System of Independent, Distinguishable Molecules Is the Product of Molecular Partition Functions

In Q = N In V t terms in N w d /? only Therefore,

and substituting this rcsult into Equation 3.32 gives us

as you mrght have expected.

Nolice that the ideal-gas cquation results from the fact that In Q = N In Vf terms in N and f l , which comes from the fact that q ( V , T) is directly proportional to V in Equalion 3.22. Exampfc 3-2 shows that q ( V , T ) is directly proportional to V for a diatomlc ideal gas also, and x t ~P V = N k , 7' for a diatomic ideal gas. This is the case for a pc)lyatomic ideal gas as well, so the ideal-gas equation of state results for any ideal gas, rn~lllatc~rnic, diatomic, or polyatomic.

EXAMPLE 3 4 Calculate the equation of state mociated with the p d t i o n function

where

(I

The general results we have derived up to now are valid for arbitrary systems. To apply thcse equations, we need to have the set of eigenvalucs ( E j ( N ,V ) ] for the N body Schrdinger cquation. In general, this is an i~npossibletask. For many important physical systems, huwever, writing the tutal energy of the system as a sum of individual energies is a good approximation. This procedure leads ro a great sirnpliticalion of the partition function and allows us to apply the results with relative ease. First, let's consider a system that consists of independent, distinquishablc par~icles. Although atoms and molecules ;ire certainly not distinguishable in general, they can hc treated ax such in a number of cases. An excellent example is that of a perfcct cryxtal. In a perfect crystal, each atom is conlined to one and only one lattice site, which we could, at Icast in principle, identify by a set of three coordinates. Because each particle, then, is confined to a lattice site and the lattice sites are distinguishable, the particles themselves are distinguishable. We can treat the vibration of each particle about its lattice site as independent to a fairly good apprt~ximation,just as we did for normal modes of polyatomic molecules. We will denote the individual particle energies by ( E ; ] , where the superscrip1 denotes the particle (they are distinguishable), and the subscript denotes the energy state of the particle. In !his case, the total energy of the system E , ( N , V) can be written as

a~lrth are collstents. Can you identify the resulting cquation or slate?

SOLUTION: We usc Equation 3.12 to calculate the equation of shte. First, we evaluate In Q,which gives In Q

=N

ln(V

- N b ) + PUN' -+ terms irl hf and 9 , v

We now differentiate with respect

tc)

and the system partition function bcuomes

only

V , keeping N and constant, to get

Because the particles are distinguishable and independent. we can sum over i , j, k. . . . independently, in which case Q ( N , V , T) can be written as a product o f individual summations (Problem 3-21): and so

Bringing the last tern tu the left sidc and rnulliplying by V

- Nb

gives us

where each of the q (V, T) is given by which is the van der Waals equation.

120

Chapter 3 / The Boltrmann Factor and Part~tinnFunctions

In many cases, the (e,) is a set of molecular energies; thus y ( V , T ) is called a rnolecuhr purrition function. Fquation 3.33 is an important result. It shows that if we can write the total energy as a sum of individual, independent terms, and if the atoms or molecules are dis~rrtguishable,then the system parlition function Q ( N ,V , T ) reduces to a product of molecular partition funclions q ( V , T). Because q ( V , T) requires a knowledge of the allorved energies of only individual atoms or n~olecules,its evahatiun is often feasible, as we will sce For a number of ca.ws in Chaprer 4. H' the energy stater of all the atotns or mo1ecu)e~ are the same (as for a monatomic crystaf), tlien Equatir~n3.33 becomes

Q W , v , T ) = [q(V,T)]"

distinguishable atoms or rnoleculcs

)

0.35)

where

3-7, Thc Part~t~on Flmr 11onof a System of Independent. Ind~,l~nguishabIrAtvrrls ur Molcr ulc5

(note the lack (IT distinguishing supcrscripls, as in Equation 3.33) and the system partition function is

Because the particles are indistinguishablc, we cannot sum over i,j, k , . .. separately as we did in Equation 3.33. To see why, we must consider a fundamental property of all particles. You iemed in general chemistry that the Pauli Excluslun Principle says that no two electrons in an atom can have the same set of four quantum numbers. Anc~therway of saying this is that no two electrons in an atom can be in the same quantum state The Pauli Exclusion Princ~pieis actually more general than the above staiement, and applies to all particles o f spin 112, 312, 512, and so on. Such particles, vallcdJerennrons, have the restriction h a t no two fermions In a system can occupy the same quantum statc. Examples of fermions arc: electrons (spin 1/2j, protons (spin 1/2), and neutrons (spin 112). Particles that have spin of 0,1, 2,. ., called busnns, do not have any restri~rion regarding the occupancy uf individual quantum states. Examples of hosons are alpha particles (spin 0) and photons (spin 1). It turns out that fermions and bosuns constitutu all the known particles in nature. We must recognize the occupancy requirementh 01 fermions and b s o n s when we aKempt to c q out the summation in Equation 3.37. Let's go back now to the summation in Equation 3.37 for the case of fermions. Because no two identical fernions can occupy the same single-particle energy slate, terms in which two or more indices are the same cahnol be included in the summation. Therefore, the indices i , j, k , . . . are not independent of une another, \o a direct evaluation o f Q ( N , V, T ) by means: of Equation 3.37 p s e s problems for fenniuns.

.

The Einstein rnidel of atomic crystals (Example 3-3) considers rhe atoms to be fixed at lattice sites, so Equation 3.35 should be applicable to that model. Notice that the partition function of thal model (Equation 3.28) can be written in the form of Equation 3.35 if we let u, = U J N he the sublimation energy per atom at O K, in which

case we have

EXAMPLE

3-7. The Partition Function of a System of Independent, Indistinguishable Atoms or Molecules Can Usually Be Written as

3-5

Consider a system of two noninterxting identical fermions, cach of which has states with energies E , , c,, E,, and E,. Enumerate the allowed total energies in the summation in Equation 3.37.

[q(v,T I I ~ I N ! SOI.UTION: Forthissystem

Equaliun 3.35 is ail attractive resuit, but a t o m and lnolecules are, in genera!, not distinguishable; thus the utility of Equation 3.35 i s severely limited. The reduction of a system partition function Q ( N , V, T) to m o l e c u l ~partition functions q ( V, T ) bccornes somewhat more complicated when the inherent indisringuishahility of atnlns and mulccules cannot he ignored. For indistinguishable parlicloa. !he toral entrgy is

-

Of the 16 terms that would wcur in an unrebrricted evaluation of Q, only six are allowed for two identical fermions; these art! ihc tenns with energies

121

Chdpter 3 1 The Boltzmann ractor and Panition t'unctions

l'he six terms in which ihe el are wri tten in reverse order are the same as those above (because the particles are indislinguishable), and the four terms in which the R, are the same are not allowcd (bccau= ttle particles are fermiol~s).

Bost>nsd o nut have he restric~ionthat no two of the samc type can occupy the samc single-particle state, but h e summation in Equation 3.37 is still complicated. To see why, consider a term in Equation 3.37 in which all the indices are the same except for one; for examplc, a tcrm like

123

3-7, The Pnrriticm Funrtiun ul a Syst~mof Ind~-pentleut,lndirringuishableAtoms or hlolecu!m

same state. Although most of the quantum-mechanical systems we have studied have an infinite number of energy states, at any given temperature many of these will not be readily accessible because the energies of these states are much larger than k,T, which is roughly the average energy of a ~nolecule.If, howcver, the number of quantum states with energies less than roughly k,,T is much larger than the numher of particles, then essentially all the tertns in Equation 3.37 will c o n t ~ i nE'S with different indices, and so we can evaluate Q ( N , V ,T) to a good approxilnation hy summing over i,j , k , . . . independcntly in Equation 3.37 and then dividing by N! to get Q ( N 3V . T ) =

independent, indistinguishable

N!

(3.38)

(in reality, these indices might be enormous numbers). Because the panicles are indiswhere tinguishable, the position of the term E , is not important, and we could just as easily t ~ a v e e , , + ~ , + ~ ~ , + ~~, ,~, f~ + ~ , , , o r ~ , , , + ~ , , , + ~ , + ~ , , t ~ ~ ~ + ~ , ~ a n d s o o n . B e cause t h e ~ eterms all represent the same state, such a state should be included only ollce in Equiition 3.37, but an unrestricted summation over all the indices (summing oker I , j , k . . . . independently) in Equation 3.37 would produce R, terms of thiq type The number of translational states alone i s usually sufficient to guarantee that the (the F , can he located in any of the N positions). number of energy states available to any atom or molecule is greater than the n u ~ n k r consider the other extreme in which all the N particles are in different of particles in the system. Therefore, this procedure yields an excellent approximation moleculx states; that is, for example, a system energy of E, E, F, E, -t. . . s,. in many cases. The criterion that the number of available states exceeds the nurnher of Because the particles are indistinguishable. all N! arrangements obtained by permuting particles so that Equation 3.38 can he used is thesc N tcrrns are identical and shauld wcuronly once in Equation 3.37. Yet such terms lvill appear N !limes in an unresrricted summation. Consequently, a direct evaluation of &( N , V , T ) by menns of Equntiun 3.37 poses prublems for busons as well as ferrnions.

c)w

I

E X A M P L E 3-6 Rudo Example 3-5 for bosons

+ + +

+

instcad of fermiolls.

S U L U T I O N : In this case there are 10 allowed terms: the six that are allowed in Example 3-5 and the four in which the F, are the same (bosons do not have the restriction that no two can occupy the same state).

Notc that in every case, the tcrrns in Equatic~n3.37 that causc difficulty are those in which two or more indices are the same. 1f it were not for such terms, we could carry out the summation in Equatinn 3.37 in an unrestricted manner (obtaining [ q ( V , T)]" as in Section 3-6) atid then divide by N! (to obtain [ q ( V ,T ) ] ' / N ! ) to account for the over-counting. For example, if we could ignore terms like a, E , , E, E,, etc. in the evaluation of Q ( 2 , V , T), there would be a total of 12 terms, the six enumerated in Example 3-5 and the six rn which the energies are written in reverse order. By dividing hy 2!, we would obta~nthe correct number alluwed terms. Certainly, if the number of quantum states available to any particle is much greater than the number of panicles, it would be unlikely for any two particles to be in the

+

+

Notice that this criterion is favored hy large particle mass, high telnperature, and low density. Although our discussion at this poinl is limited t o ideal gases (independent, illdistinguishable particles), we show the valucs of ( N /~ ) ( k ~ / X r n k , ~ in ) ~'raahle " 3.1 even for some liquids at their boiling points, just to show that inequality 3.40 is easily satistied in most cases. Note that the exceptional systems include liquid helium and liquid hydrogen (because of their small masses and low temperatures) and electrons in metals (because of their very small mass). These systems arc the prototype examples of quantum systems that must be treated by special methods (which we will not discuss). When Equation 3.38 is vdid, that is, when the numkrofavailable m~lecularatates i s much greater than the number of particles, we say that the particles obey B o l ~ i t n u t ~ n statistics. As Inequality 3.40 indicates, Boltzmann statistics becorncs increasingly valid with increasing temperature. k t ' s test Inequality 7.40 for N,(g) at 2U"C and one bar, Under these conditions,

T A B l E 3.1 The quantity ( N /V ) ( h 2 / 8 r n k , ~ ) 3 tat 2 a pressure of one bar for a number of simple systems.

3-8. A Molecular Partition Function Can Be Decomposed into Partition Functions for Each Degree of Freedom In this section, we will explore the similarity hetwcen a system partitiun function. Equation 3.14, and a molecular partition function, Equation 3.39. We will stan by substituting Equation 3.38 into Equation 3.21 :

System

Liquid heliu~n Gaseous helium Ga%ous helium Gaseous hellurn Liquid hydrogen Gaseous hydrogen Gaseous hydrogen

Liquid neon Gaseous neon

But Equation 3.38 is valid only for independent particles, so

Liquid krypton Elecwns in metals (Na)

( E )= N {E)

-

(3.42)

where ( E ) is the average energy of any one molecule. If we compare Equations 3.41 and 3.42, we see that

and h2 -= 8mk,T

(6.626 x lo-" 4,s)' kgl(1.381 x 10-23 J.K-')(293.2 K)

(8)(4.653 x

We can conclude from this equation that the probability that a molecule is in its jth molecular energy state, x J ,is given by

which is much less than unity.

Let's test Inequality 3.40 for liquid nitrogen at i t s baiting point, - 195.8"C.Expermen tally, the density of N,(l) is 0.808 g . m ~ - at ' its boiling point. Therefore,

N

(28.02molgN?N, ) (6.022mol10")

V = (0.808 g . r n ~ - ' )

I

l

x

Note how similar this equation is to Equation 3.13. Equation 3.44 can be reduced even further if we assume that the energy of n mthecule can be written as

(10' .L)

I m"

and

Because the various energy terms are distinguishable here, we can apply the reasoning behind Equation 3.33 and write

where, for exampk

This, Equation 3.38 is valid, even for liquid nitrogen at its boiling point.

Chapter 3 / Thp Boltzrnann Factor and Partit~unFunLTions

Note that the partition function for a diatomic molecule we used in Example 3-2 was expressed as

1-8. A Mvlecular Partition Function Can Be Decornped into Partition Functruns

and

where EXAMPLE

3-7

Use the partition function for a diatomic molecule giver1 in Example 3-2 to calculate (E"'~).

SOL U T I 0 N : Acctlrding to Example 3-2. we can write

and and so

If we substitute Equations 3.45 and 3.46 into Equalion 3.44, we obtain in agreement

with Equation 3.24.

I

where rr,,,, is the probability that ti molecule is in the ith translational stale, the j t h rutationai stale, the kth vibrational state, and the Ith electronic st*. Now if we sum Equation 3.48 over i (all translational states), j (all rntatiunal states), and 1 (all electronic states), we obtain

I

To this point, we have written partition fi~nctionsas sumrna~ionsover energy strlfrs. Each state is represented by a wave function with an associated energy. Thus, wc write

We will call sets of states that have the sarnc energy, levels. We can write q ( V . T )as a summation over levels by including the degeneracy, g,. of the level:

wliere, as the notation suggests, Chis the probability that a molecule is in its k t h vibrational state. Furthermore, the average vibrational energy of a molecule is given by

In the notation of Equation 3.53, the ternis represcnting a degencratc levcl are rrpculed 3.54, lhey are written once and rnulliplid by #,. For example, we learned in Section 1-8 (Equations 1.28 and 1.29) thul the energy und degeneracy for a linear rigid rotator are gj times, whereaq in Equation

Again, note thc siail:vily-with I$uatit,n 3.2 I .

Of course, we also have the relations and

128

Chaprer 3 !The tloltzmann Factur

rartition Functions

Thus, we can write the rotational parlition function by summing over levels:

Including degeneracies explicitly as in Equatiun 3.54 i s usually more convenient, s o we will use Equation 3.54 rather than Equation 3.53 in later chapters.

Problems

and that

3-7. Generalize thc results of Example 5 1 to the case of a spin-l nucleus. &!ermine the low-ternperdure and high-temperature limits of (E). 3-8. If Nw i s the n u m k r of protons aligned with a magnetic field B2 and No is the numher of protons opposed to llic field, show that

Problems 3-1. How wouid you describe an ensemble whose systems are one-liter containers of water at 25'C:I

+

5-2. Show that Equation 3.8 is eyuiwlcnt to f (x y ) = f ( x ) f (y). In this pmblem, we will prove that f (x) a eU*. First, take the logarithm of the above equation to obtain

= 26.7522 x lo7 rad.T-' .s-' for a proton, cakulatc N,,/Nw its 3 function or temperature for a field strength of 5.0 T. At what kmperature i s No = Nu? Interpret this

Given that y

result physically.

3-9, In Section 3- 3, we derived an expression tor ( E ) for a nlonato~nicdeal gas by a p p l y i ~ ~ g Equatiticln 3.20 to Q(N,V, T) given by Equation 3.22. Apply Equatiun 3.21 to Differentiate both sides with respect ro x (keeping y fixed) to get

to derive the same result. Note that this expression for @ ( N , V. T ) is simply Equation 3.22 with fl replaced by I / k , T .

Now differentiate with respect to y (keeping x fixed) and show that

3-10. A gas absorbed or1 a surface can sometimes b modelled a.i a two-dimensiunal ideal gas. We will learn in Chapter 4 that the partition function uf a two-dime~lsionalideal gas is

For this relation to be true fur all x and y , each side must equal a ctmstant, say a. Show that f (x) a e"'

and

f(y)cr.P'

where A is thc area of the surface. Derive an cxprebsion fur ( f )and comparc your rcsull with the thm-dimensiuml result.

3-3. Show that a,/a, = ~ " ' 4 -" 1 ' implies that a = C@8$ 3-4. Prove to yourself that

C,

=

~ ' " 8

2,:,t.-'4.

3-11. Although we will not do so in this book, it is possible toderlve the parl~tionfunction fur a monatomic vun der Waals gas.

3-5. Show that the partition function in Example 3-1 can be written as

w h m a and b are the van der Waals uvn\tunts. Derive an expression for the energy of a

Ilw lhc fact that d C O S ~X

= sinh x to show that

monatomic VAIIJer Waals gas.

/ ~ X

3-6. IJ,wtrthtr the cxprr*rlr*nfor ( E ) ill Example 3-1 or Lhe one in Pruhlenl 3-5 to show that

3-12. An apprcmirnu~cpurrition function for a gas of hard spheres can be ubtilined from the paflition function 111s monatomic gas by replacing V in Equation 3.22 (and the lotlowing equation) by - b, where b is related to the volume of the N hard spheres. Uerivc errprc~siunsftlr the energy and the pressure of thif system.

3-13. l l e the partition function in Problem 3-10 to calculale the ticat cap:lcity of a ~ w o d~mensionuld e a l gas.

rhapter 3 1 Thc Boltrrnann Fartor and Partition Functiorls

130

3-14. Use the partition function for a monatomic van der Waals gas given in Problem 3-1 1 to calculate the heat capacity of a monalomic van der Waals gas. Compare your result with lhalol a nlonatornic ideal gas. 3-15. Using the partition function given in Example 3-2, show that the pressure of an ideal diak~rnicgas obcys P V = N k , T , juct a 1, the series diverges; and if r = 1, the test i s inconclusive. Let's apply this test to the geometric series (Equation C.3).In this case, ~ i ,,,= , xn+' and urn= r",so

w

Thus, we see lhat the scries converges if 1x1 < 1 and diverges if 1x1 z 1. It actuaIly diverges at x = I , but the ralio test dws not tell us that. We would have tu use a more sophisticated convergence test to determine the behavior at x = 1 .

and let x = 0 to find that c, = ( d f / d ~ ) ~ Differentiate =,. again,

and let x = O t o gut c, = (d2f/d~')~=,,/2. DiHi.rcntiatc oncc rrlrlrc,

MathChaprerC / S E R I E S A N D L I M I T S

and let x

= 0 to get c, = (d3f / d ~ ~ ) ~ _ ,The / 3 !general . result is

MathChaprerC / S E R I E S A N D L I M I T S

We can use the series presented here to derive a number of results used thruughc~ut the book. For example, the limit

sinx lim -

x r 0

x

su wc can write

occurs several times. Because this limit gives 010, we could use I'Hbpital's rule, which

tells us that Equation C.9 is called the Maclaurin series of f (x). If we apply Equation C.9 to f ( x ) = e", we find that

,

d sinx dx

sinx

lim -= lim -= lirncosx = I

I+n

x

.-.o

1-0

We could derive the same result by dividing Equation C.10 by x and then letting x + 0. (These two methods are really equivalent. See Problem C-14.) We will do one final example involving series and limits. According to a theory by Debye, the temperature d e ~ n d e n c euf the molar heat capacity of a crystal is given by

Some other important Maclaurin scries, which can be obtained from a straightforward application of Equation C.9 (Problem C-7) w e

In this equation, T is the kelvin tcmpcrature. R is the rnvlar gas constant, and (-1, is a parameter characteristic of the particular crystal. The parameter 0, has units of temperature and is called the Debye temperature of the crystal. We want to determine both the low-temperature and the high-tempcraturc limits of F, ( T ) .In the low-temperature limit, the upper limit o f the integral h o m e s very large. For large values of x, we can neglect 1 compared with ex in the denominator of the integmnd, showing that the integrand goes as x4e-" for large x. But x4e-" + 0 as x 4 co,so the upper limit of the integral can safely be set to m, giving Series C. 10 and C. 1 1 canvcrge for all values ufx, but asindjcated, Series C. 12 converges only for -1 < x 5 1 and Series C.13 converges only for x2 < 1. Note that if n is a positive inleger in Series C. 13, the series truncates. For example, if n = 2 or 3, we have

Whatever the value of the integral here, it is just a constant, so we see that -

and

C , ( T ) + constant

x T~

a

7' 10

This famous result for the low-temperature heat capacity of a crystal is callcd the T' law. The low-temperature heat capacity goes to zero as T'. We will use the T ' law

k q ~ ~ a t i aC.13 n for a positive integer is called the binomial expansion. If n is not a positive integer, the sergs continues indefinilely, md Equation C.13 is callcd the binr~mialseries. Any handbook of mathematical tables will have the Maclaurin series for many functions. Problem C-13 discusses a Taylor series, which is an extension of

a Mnclaurin series.

in Chapter 7. Now let's h k at the high-tempcmture limit. For high temperatures, he upper limit of the integral in Equation C.14 becomes very small. Consequently, during the integration fmm 0 to OD/T, x is always small. Therefore, we can use Equation C.2 for ex, giving

.- .- - - -- -

MathChapterC I S E R I E S A N D L I M I T S

-

lirn C , ( T ) = 9 R

T-tW

-

+ o(x2)idx

wTX ~ [ I

Problems

C-l 0. Evaluate the limit of

as x + 0.

C-1 1. Evaluate the integral

This result is called the Law of Dulong and Petit; the molar heat capacity of a crystal becomes 3 R = 24.9 ~ . ~ - ' . m o l - for ' monatomic crystals at high temperatures. By "high temperatures", we actually mean that T >> (-), which for many substances is less than 1030 K.

1=

1'

it-'

cos'xdx

for small values of a by expanding I in powers of a through quadratic rems. C-12. Prove that the series for sin x converges for all values of x .

C-13. A Maclaurin series is an expansion about the point x = 0. A series of the form

Problems C-1. Calculate the percentage difference between ex and l

+x

for x = 0.0050. 0.0100,

0.0150, . . . ,0.1000.

C-2. Calculate the percentage difierence belween In(] 0.0150, . . . ,0.1000. C-3. Write out the expansion of ( I

+x)ln

f x)

is an expansion about the paint x, and is called a Taylor series.Firsf show that c, = f (x,,). Now differentiate both sides of the above expansion with respect to x and then let x = x,, to show that c, = ( d f / d ~ )Now ~ ~ show ~ . that

and x for x = 0.0050, 0.0100.

through the quadratic term

C-4. Evaluate the series

and so

C-14. h e r on, we will need to sum the series

C-5. Show that

C-6. Evaluate the series

and

-

S2

=

C u2xU "4

C-7. Use Equation C.9 to derive Equations C.10 and C. I 1.

To sum the first one, start with (Equation C.3)

C-8. Show that Equations C.2, C. 10, and C. I 1 are consistent with the relation e'" = cosx f i sinx.

C-9. In Example 3-3, we derived a simple formula fur the inolar heat capacity of a solid based on a model by Einstein:

Differenbate with respect to x and then multiply by x tcr obtain

Using the same approach. show that where H i n Ik mrtlar gas cclnstant and (3, = h v / k , is a constant, called the Einstein Iclnpcrnturr, #hut i h uhurnc~crirticofthe solid. Show that this equation gives the Dulong ;~ntlI'ctil l i m ~ IC', t - - 3HI ut high Icltlpcrillurcs.

Partition Functions and Ideal Gases

In this chapter, we will apply the general results of the preceding chapter to calculate the partition functions and heat capacities of ideal gases. We have shown in Section 3-7 that if the number of available quantum states is much greater than the number of particles, we can write the partition function of the entire system in terms of the individual atomic or molecular partition functions:

This equation is particularly applicable to ideal gases because the rnr~leculesare independent and the densities of gases that behave ideally are low enough that the inequality given by Equation 3.40 is satisfied. We will discuss a monatomic ideal gas first end then diatomic and polyatomic ideal gases.

William Francis Giauque was born un May 12, 1895, in Niagara Falls, Ontario, Canada, to American parents and dicd in 19x2. After working for two years in the laboratory at Hooker Eleclro-Chemical Company in Ni~garaFalls, he entered the University of California at Berkeley with the intent of becoming a chemical engineer. He decided to study chemistry, however, and remained at Berkeley to receive his Ph.1). in chemistry with a minor in physics in 1922. tiis dishertation was on the khavior of materialsal very low temperatures. Upon receiving his Ph.D., Giiluque accepted a faculty position in the College of Chemistry at Berkeley and remained ~ticrufnr the rest uf his life. He made exhaustive and meticulous themochemical studies that cxpk)red the Third Law o f Thermodynamics. Tn particular, his very low terngerature studies of thc cntrlrpier uf substances validated the Third Law. Giauque developed the technique of u~liuhatic&magnetization to achieve low temperatures, He achieved a tempramre uf O.?S K , und othcr rescarch groups subsequently reachod ternperarures as low as 0 . 0 1 4 K using li~:i~lrl~ar*\ twhniquc. Togvthet with his graduate student Hemck Johnston, he spectroscopically i~lcr~ulictl thc iwo t$~iticn{~ unknown oxygen isotopes 17 and 18 in 1929. He was awarded the Nutwl I'rirr for clicrni\rry in 1q49 " h r his cuntributions in the fieldof chemical thermodynamics, prtictllilrly cuilccr1iit1g thc hchuvitrr of suhstnnces at extremely low temperatures.''

4-1. The Translational Partition Function of an Atom in a Monatomic Ideal Gas i s ( 2 j ~ n a k ~ ~ / h ~ ) " ~ 1 . ' The energy of an atom in an ideal monatomic gas can be written as the sum of its translational energy and its electronic energy

so the atomic partition function can be written as

We will evaluate the translational partition functirjn first

Chapter 4 !Partition Funclions and Ideal G s m

The translational energy states in a cubic conlaincr are given by (Equation 1 4 5 ) En", I

.

h2

=

(n:+n:+n;) 8ma2

n I , n y l n z = l , 2, . . .

(4.2)

We substitute Equation 4.2 into qtranh (Equation 3.47) to get

Because e"+h'' = e"ebe",we can write rhe triple surnnlation as a product of three single summations: F I G U R E 4.1 An illi~strdtionof thc approximation of a sulnrnalion C z ,f- by a11 inlegrat. The summation i \ equal to the areas af the mtangles and the integral is equal to the area undcr the curve obtained by letting n be a continuuus variable.

Now, each of these tllrce sirlgle sumniatio~~s is a!ike, hecause each one is simply

so we have that

Thus, we can write Equation 4.3 as

This summation cannot be expressed in terms of any simple analytic function. This situation does not present any difficulty, however, for the following reason. Graphically, a summation such as f R is equal to the sum of the m a s under rectangles of unit width centered at I, 2, 3, .. . and of height f,, f,, f,. . . . as shown in Figure 4.1. IF the heights of successive rectangles differ by a very small amount, the area of the rectangles is essentially equal to the area under the continuous curve obtained by letting the summation index n be a continuous variabte (Figure 4.1). Problem 4-2 helps you prove that the successive terms in the surnmatlctn in Equation 4.4 do indeed d~ffervery lrttle from each other under most uunditjuns. Thus, it i s an excellent approximat~onro replace the summation in EyuaIiun 4.4 by an integration:

where we have written V for a'. Note that q,,,,, is a function of V and T. We can calculate the average translational energy of an ideal-gas atom from this partition function by using Equation 3.51:

z,"=,

In T

in agreement with what

+ terms independent or 1'

we found in Section 3-3.

4-2. Most Atoms Are in the Ground Electronic State at Room Temperature Note that the integral starts at n = 0, whereas the summarion in Equition 4.4 starts at n = 1. For the small valueb: of @h2/8ma2we are considering here, the difference is z a,the above integral becomes negligible (Problem 4-41). If we denote ~ h 2 / 8 m aby (see Mathchapter B)

In this section, we will invesligatc the electronic contrihutiona 10 q ( V , T),It is more convenient to writs the elwtronic partition function as i~ sum over levels rather than a sum elver states (Section 3-8)- so we write

Chapter 4 / Parfitinn r u n r ~ i o r ~JIK! s Ideal LAWS

146

where gci is the degeneracy, and get the energy of the ith eIectronic level. We first fix the arbitrary zero of energy such that = 0; that is, we wit1 measure a11 electronic energies relative to the ground electronic state. The electronic contribution to q can then be written as

T A B L E 4.1 Some atomic encrgy ..

Atom

levefs."

.

Electron configumtiun

Degeneracy g- = 2J -C 1

~ner~~/crn-'

where E ~ i, s the energy of the jth elcctronic level relative to the ground state. Note that is n function of T but not of V. As wc have seen in Chapter I , these p's are typically uf the urder o f tens of thousands ol'wuve numhers.Using the fact that 1.986 x 1 W2' J = 1 cm-' ,the Boltzmann constant in wave numbers is k, = O.6950 cni '.K I. Thus, we see that typically

which i s equal to I0 even for T = 1000 K. Therefore, c-",? in Equation 4.9 typically is :trnund 10-qtilr most atoms at ordinary tetnperatures, so only the fist term in the suill~nutionli)r qclmi s significantly different from zero. There ate surne cases, however, such a q the halogen atoms, for which the first excited state lies only a few hundred wave numbers ahr~vethe ground state, so that several terms in qc,rrare necessary. Even ill these cases, the sum in Equation 4.9 converges very rapidly. As we learned in Chapter 1, the clcctronic energies of atoms and ions aredctcmined hy uromic spectrascupy and are well tabulated. The standard reference, "Moore's tables:' lists the energy levels and energies of many atoms and ions. Table 4.1 lists the lirst few levels for H, He, Li, and F. We can make some general observations tiu~rn t:~bleslike Table 4.1. The first exci~edstates of the noble gas atoms are of order of 10' cm-' or highcr than the ground states; ihe first excited statcs at the atkali metal atoms are of order u l 1 0 4 cni or higher than the ground states: and the first excited stiucs of thc Ii;~logc~i alolns arc only ol'ordcr of 10' cm- higher than the ground statcs. 'I'hus. at ordinary tempertltures, thc electronic partition function of noble gas atoms is essentially unity and that of alkali metal atoms is two, while those for halogen atoms ct~nsistof two terms. Using the data in Tablc 4.1, we can now calculate the fraction of helium atoms in lhc first excited state. This fraction is givcn by

'

"Fmm C.E. Moore, "Atomic Energy Levels" NatI. Bur Std. Cir. 1 467. U.S.Government Rinting OfFrce. Wa~hingtonD.C., 1949

be less than a few hundred cm-' or sn before any population of the excited level is significant.

EXAMPLE 4-1 Using the data in Table 4.1, calculate the fraction of fluorine atoms in the Rrsl exciled state at 300 K, IOOO K,and 2MH)K.

S O L U T I O F : Using the second line of Equation 410 with gel = 4, g,, = 2, and g*, = 6, we have

Ar 3(H) K. p f r : = 770, $0f, 10 '". Even at 3000 K, ji f ; This is typical of ~lic~iohlcc:kwh. 'l'l~cenergy separation between the ground and cxcited levets must

I

Chapter 4 / Partition Funclinns and ldcal Ga-

with s*, = 404.0 cm-' and E

~ = ,

102406.50 em-'. We also have

and

4-3. The Energy uf 3 Diatumic Mulecule Can Bc Approximaled a,

,I

Sum ul Srparate Terms

The first term represents the nvewg kinetic cnergy, and the second lerm represents Ihc average electronic energy (in excess of the ground-state energy). Thc contribution of the electronic degrees of Creedam to the average energy is small at ordinary temperatures. If we ignore the very small cmtribution from the electronic degrees of frmdam, the molar hear capacity at constant votume is given by

Clearly, we can neglect the third term in the denominator off,. The valuc of f2 fur the various temperatures is

The pressure is

Thus, the population of the firs1 excited state i\significant at these temperature.. and so the first two terms of the summation in Equaticln 4.9 nmst be evaluated in determining llc,*L(T). For most atoms and molecules, the first two terms of the electronic partition

Curiction are sufficient, or

At temperatures at which the second term is not negligible with respect to the fin1 term, we must check the possible contribution of higher terms as well. This completes our discussion of the partition function of monatomic ideal gases. In summary, we have

where

qClFf ( T ) = gel

+

+ ...

~ , ~ e ~ ~ @ ' ~ 2

V

+ terms not invulving V )

--Nk,T v

(4.15)

which is the ideaIgas equalion of state. Note that Equation 4.15 resuits because q ( V , T ) i s of the form f ( T ) V ,and only the translational energy of the atoms contributes to thc

pressure. This is expected intuitively, because the pressurc 1s due to bombardment ol' the walls of the container by the atoms and molecules of the gas. In the next few sections, we will treat a diatomic ideal gas, In add~lionto translational md electronic degrees of freedom, diatomic molecules also possess vibrat~onal and rotational degrees of freedom. The general procedure would be tu set up the Schriidinger equation for two nuclei and n electrons and to solve this equation for the set of eignvalues of the diatom~cmolecule. Fortunately, a series of vcry gclcK1 approximations can be used to reduce this complicated two-nuclei, n-elcctron problcnl to a set of simpler problems. The simplest of these appr13ximations i s thc rigid rtaalclrharmonic oscillator approximation, which we described in Chapter I. Mre will he1 up this approximation in the next section and then discuss the vibratimal and rotdtiunal partition functions within this approxitrlation in Sections 4-4 and 3-5.

4-3. The Energy of a Diatomic Molecule Can Be Approximated as a Sum of Separate Terms

Wc can now calculate some of the propeltics o f a monatomic ideal gas. The average 1'111'1,gy is

When treating diatomic or polya~otniumolecules, wc uac thc rigid rt~tatnr-har'moniu oscillator approximation (Chapter 1). In his case, we can write the lute1 energy of tt~c molecule as a sum of its wnslational, rotational. vibrational, and electronic energies:

Chapter 4 1 Partilion Functions and Ideal Gases

4-3. The En~rgyof a Diatomic

Mol~~ule Can Be Approximated a5 a Sum of Separate Terms

As for a monatomic ideal gas, the inequality given by Equation 3.40 is easily satisfied at normal temperatures, and so we can write

Furthermore, Equatiun 4.16 allows us to write

so the partition function of a molecular ideal gas is given by

The translational partition function of a diatomic moIecule is similar to the result we found in Section 4-1 for an atom:

Note that Equatiun 4.20 is essentially the same as Equation 4.6. The electronic partition function will be similar to Equation 4.9. We will discuss the vibrational and rotational contributions to the partition function in the next two sections. Although Equation 4.19 i s not exact, it is often a good approximation. particularly for small molecules. Before we consider q,,, and q\,b,we must choose a zero of energy for the rotational, vibrational, and electronic states. The natural choice for the zero of rotational energy is the J = 0 state, where the rotational energy is zero. In the vibrational case, however, we have two sensible choices. One is to take the zero of vibrational energy to be that of the ground state, and the other is take the zero to be the bottom of the internuclear potential well. In the first case, the energy of the ground vibrational state is zero. and in he second case it is 11~12.We will choose the zero of vibrational energy to be the bottom of the internuclear potential well of the lowest electronic state, so the energy of the ground vibrational state will be h v / 2 . La\t, we tdie the zero o f the electrrmic energy ro be the separated atoms at rest in their grt~undeleclronic states (see Figure 4.2). Recall that the depth of the ground electronic stare po~entialwell is denoted by Dc (4is apositive number: see Section 171,and so the energy of the ground electronic state is eel = -Dc, and the electronic partition function is

where D, and c12 are show11 in Figure 4.2. We also introduced in Section 1-7 a quuntity I),, that is equal to 4 - ikv. As Figure 4.2 shows, Do is the energy difference hctwce~itlic lowcst vibrational state and the disswiated molecule. The quantity Do can hc uwa*uwd spctroscopically, and values of Do and Dc for several diatomic molezules utr yrvcn in l'ahlc 4.2.

F I G U R E 4.2 The ground and firs1 excited electronic states as a function of the inlernuclear separation, illustrating the quantities 4 and Doof the ground state and E?,. The qUntlties 1)* and D,, are related by D* = D, h v / 2 as shown in the figure.

+

T A B L E 4.2

Moiecular cnnstnllls for ~everaldiatomic n~olecules.These parameters were obtained from a variety of sources and do not represent the most accuralc values k a u s e they werc obtained undcr the rigid rotator-harmonic uscillator approximation.

Degencraq of the ground

Molecule

Qv,,IK

HCI HBr Hr

4227 3787

Ntl,

229 133

%

3266

OWtlK

~,/kJ.rnol-'

q / k J ~ r n 0 1 - ~ electronic state

15.02

427.8

12.02 9.25 0.221 0.081

3hZ.h

445.2 377.7

294.7

30H.h

I

71.1

72 1

I

53.5

54 I

1

I

I

4--4. Mosl Muleculrs Are in the Ground Vibratiundl State dl R w m Tcmprdll~rc

4-4.

Most Molecules Are in the Ground Vibrational State at Room Temperature

We can calculate the average vibrational energy fmm qb,,(T)

In this section, we will evaluate the vibra~ionalpart of the partition function of a diatomic rnc~leculeunder the harmonic-oscillator approximation. If we measure the vibrational energy levels relative to the hotturn of the internuclear potential well, the energies arc given by (Equation 1-22)

with v = ( ~ / / A ) ' / ~ / ~where T C , k is the force constant of the molecule and reduced mass. The vibrational partition function q,,, becomes

JX

is itb

Table 4.2 gives (-Iv, for sevcml diatomic molccules. The vibrational conlribution to the molar heat capacity i s

Figure 4.3 shows the vibrational contribution of an ideal diatomic gas to the molar heit1 capacity as a function of temperature. The high temperature lin~itof ?v,vi, is A, and Cv,v,, is one-half of this value at T / O b i b = 0.34.

This summation can be evaluated easily by recognizing it to be a geometric series (Mathchapter C):

with x = FPh" < 1. Thus w e can write

so q,,,(T 1 becomes

F I G U R E 4.3 The vibrational contribution to the molar heat capacity of an ideal diattmic gaq as a funcliol~ol reduced temperarure, T/@v,b.

I Note that this is the vibrational term encountcred in Example 3-2, which presented the p i i t i o n function for the rigid rotator-harmunic oscillator model of an ideal diatomic gas. If we introduce a quantity, (-Iv,, = h v / k , , called the vibmtionul temperature, qv,,(T)can be written as

E X A M P L E 4-2 Calculate the vibrational contribution 10 the molar hcat cap:ioi!y The experimental value is 3.43 I.K-' .rnul-'.

of

N:(g!) at llWW K .

S O I . U T I O N : We use Equarlon 4.26 w~th (-I,,, = 3374 (lahlc 4 ?)

8",/T = 3.374 and

This is one of the rare cases in which y can he summed directly without having to approximate it by an integral, as we did for the translational case in Section 4-1 and will do shortly for the rotational case in Section 4 5 .

The agreement with the e a p e r i ~ l r ~ i l vr~luc ul 1% yultc yrrxl.

1

Ihur,

Chapter 4 1 Partition Functions and Ideal C;ases

An interesting quantity to calculate is the fraction of molecules in various vibtational states. The fraction of molecules in the vth vibrational state is

If we substitute Quation 4.23 into this equation, we obtain

The follnwing exarnplc illusrrates the use uf this equation.

EXAMPLE

I

4-3

Use Equation 4.28 to calculale the fraction of N2(g)mulecules in the v = 0 and v vibra~i~mal states at 300 K.

=1

F I G U R E 4.4 The population of the vibrational levels of Br, (g) at 300 K

S O L U T I O N : We lirst calculate exp(-(+",/T) for 300 K:

Table 4.3 gives the fraction of molecules in excited vibrational states for several diatomic molecules. Therebe, T A B L E 4.3

The fraction of molecules in excited vibrational states at 300 K and 1000 K.

and

eV,K

Gas Notice that essentially all the nitrogen m~leculesare in thc ground vibrational state at 3(X) K.

I],

6332 4227

HCl

N2

3374 3103

x:,

xr+

1.01 x

2.00 x lo-' 1.46 x lo-= 3.43 x lo-' 4.4'1 x I [ ) - I 4.47 x lo-' 7.35 x lo-'

8115

3.22 x 10-' 6.82 x lo-'

I,

308

3.58 x lo-'

4-5, Most Molecules Are in Excited Rotational StaJes at Ordinary Temperatures The energy levels of a rigid rotator are given by (Equation 1-28) F J

or simply

f,,, 1T = 1000 K)

7.59 I()-' 1.30 x

'3

CO

Figure 4.4 shows the pupulation of vibratiunal levels of Br,(g) at 300 K. Notice that most molecules are in the ground vibrational state and that the population of the higher vibrational states decreases exponentially. Bromine has a smaller force constant imd a larger m a s (and hence a smaller value of (-Iv,,) than most diatomic molecules, however (rf. Table 4.2). so the population of excited vibrational states of Br,(g) at a givcn temperature is greater than most other molecules. We can use Equation 4.28 to calculate the fraction of molecules in all excited vibrational states. This quantity is given by fU but because jV= 1, we can write

(T= 300 K)

=~

+ I)

~ I J ( J

21

J=O,

I,

2, . . ,

(4.3ou)

where I i s the moment of inertia of the rotator. Each energy level has a degeneracy of g, = 2J

+1

(4.30b)

Chap~er4 1 Panition Functiuos and Ideal Cases

156

Using Equations 4.30)~and 4.30b, we can writc the rotational partition function of a rigid rotator as

where we sum over levels rather than states by including h e degeneracy expliciLy. For convenience, we introduce a quantity that has units of temperature and is called the

157

&5. Most Molecules Are in Excited Kotat~onal Stales at Ordinary Tcmperalures

For simplicity, we will use only the high-temperature limit, because 8, molecules at room temperature. (See Table 4.2.) The average rotational energy is

0,Figure 5.8b represents an endother~nicrcaction; the enthalpy of the reactants is less than the enthalpy of the prcducls, so q , = ArH > 0, and cnergy as heat must be supplied to drive the reaction up the enthalpy "hill." Let's consider several examples of chemical reactions carried out at one bar. For the combustion of one mole of methane to form one mole of CO,(g) and two molcs of

I

Reactants

7

P ~ D ~ Y C ~ S

-

F I G U R E 5.7 The molar enthalpy of ben)tne [relative to H(O)jfrom O K to 500 K.

I

(:I)

(b)

F I G U R E 5.8

An cnthalpy diagram for (a) an exothermic reaction and (b) and endothermic reaction.

Chapter 5 / The First Law ul Thermodynarn~cs

H,O(I), the value of AtH i s -890.36 k.l at 298 K. The negative value of ArH tells us that the reaction gives off energy as heat and i s therefore exothermic. An example of an endothermic reaction i s the water-gas reaction:

5 - 10. Enthalpy Changes for Chemical Equations Arc Additive

As an exart~pleof the application of Hess's Law, consider the use of

and For this reaction, ArH = +I31 kJ at 298 K,so energy as heat must be supplied to drive the reaction from left to right. An important and useful property of ArH for chemical equations is additivity. This

property of A,H follows directly from he fact that the enthalpy is a state function. Jf we add two chemical equations to obtain a third chemical equation, Lhe value of brH for the resulting equation is equal to the sum o f the ArH for the two equations being added together. The additivity of A,H i s best illustrated by example. con side^ the following two chemical equations.

to calculate (he value of ArH for the equation

Wl,(I)

+ Cl,(g)

+

pcl,(s)

In this case, we add Equation 2 to the reverse of Equation 1 to obtain Equation 4:

Thas, from Hess's law, we obtain

If we add these two chemical equations as if they were algebraic equations, we get We now multiply Equation 4 through by 112 to obtain Equation 3:

The additive property of ArH tells us that ArH for Equation 3 is simply and so

In effect, we can imagine Equations 1 and 2 as representing a two-step process with the same initial and final states as Equation 3. The total enthalpy change for the two equations together must, therefore, be the same as if the reaction proceeded in a single step. The additivity propetty of A,H values i s known as Hess's Law. Thus, if h e values of ArH( 1) and A, H ( 2 ) are known, we need not independently determine the experimental value of ArH(3) because its value is equal to the sum A r H ( l ) ArH{2). Now let's consider the following combinaiion of chemical equations.

+

-

1 EXAMPLE

-

5-9 The molar enthalpies of combustion of isnbutanz and n-bulanc are - 287 1 kJ,mol and -2878 kl.rnol-I, respectively at 298K and one atm. Calculate ArH fur the conversion uf one mole of n-butane to one mole uf isobutrlne.

'

S 0 L U TI 0 N: The equaiiuns fur the two cumhustion reactions are

7

H - C , H , ~ (+ ~ ) Ol(gl -+ -1CO,(g) f 5 H,OfI)

and

Bwause Equation 2 is sirnpiy the reverse of Equation I , we conclude from Hess's Law that A t H (reve~se)=

-Ar H (forward)

(5.48)

(1)

I

Chapter 5 1 The First Law of Thermodynamics

If we reverse the second equation and add the result to the first equation, then we

obtain the desired equation

5-1 1. Weals of Reactions Can Be Calculated imm Tabulated Heats of Formation

vaporizatinn [e.g., H20(1) -t H,O(g)J. Table 5.1 lists many of the subscripts you will encounter. The stundard molar enthnlpy of formarion, A,H0, is a particularly useful quantity. This intensive quantity is the standard reaction enthalpy for the formation of one molc of a molecule from its constitue~~t elements. The degree superscript tells us that all reactants and products are in their standard states. The d u e of A,H0 of H,0(1) i s -285.8 W.mol-' at 298.15 K. This quantity irnpIies that the balanced reaction is written as

The heat or this reacliun cannot be measured directly because competing reactions occur.

5-1 1. Heats of Reactions Can Be Calculated from Tabulated Heats of Formation ' I ' l ~ ccr~thalpychange of a chcrnical reaction, A r H ,depends upon the number of moles of the reactants. Recently, the physical chemistry division of the International Union of IZrre and Applied Chemistry (IUPAC) has proposed a systematic procedure for tab~ilatingreaction enthalpies. The s t n d u r d reaction enthalpy of a chemical reaction is denoted by A, H" and refers to the enthalpy change associated with one mole of a

specified reagent when all reactants and products are in their standard states, which for a gas is the equivalent hypothetical ideal gas at a pressure of one bar at the temperature of ititerest. For example, consider the combustion of carbon to form carbon dioxide CO,(g). (The standard state of a solid is the pure crystalline substance at one bar pressure at the temperature of interest.) The balanced reaction can be written in many ways, including

because A , H refers to the heat of formation of one mole of H20(1). (The standard state for a liquid is the normal state of the Equid at one bar at the teniperature of interest.) A valuc of A,H0 for H,O(I) equal to -285.8 kJ.mol-' tells us that one mole of H,0(1) lies 285.8 kJ "downhill" on the enthatpy scale relative to its constituent elements (Figure 5.9h) when the reactants and products are in their standard states. Most compounds cannot be formed directly from their elements. For example, an attempt to make the hydrocarbon acetylene (C,H,) by the direct reaction of carbon with hydrogen

yields not just C,H2 but a complex mixture of various hydrocarbons such as C,H, and C2%, among others. Nevertheless, we can determine the value of A,H" for acetylene by using Hess's Law, together with the available A=H" data on combustion reactions. All three species in Equation 5.5 1 burn in oxygen, and at 298 K we have

and

The quantity ArH ' implies E q u a t i o ~5.49 ~ because only one mole of the (specified) reactant C(s) i s cornbusted. Thc d u e of AIHofor this reaction at 298 K is ATH" = -303.5 k~.rnol-'.The corresponding reaction enthalpy for Equativn 5.50 is

If we multipty Equation 1 by 2, reverse Equation 3, and add the results to Equation 2, we obtain

with

We see that drlI is an e x t e n h e quanlily, whereas ATHoi s an intensive quantity. The ;~rlunntagcof thc tcrrninology is that it removes lhe amhiguity of how the balanced reaulion corresponding to an enthalpy change is written. Certain subscripts are used in place of r to indicate specific types of processes. Frw cxainple, the subscript "c" is used For a combustion reaction and "vap" is used for

5-1 1. T A B L E 5.1 Common subscripts for the enthalpy changes of processes.

Subscript

vap sub fus

trs mix

SOLUTION: The chemical equations for lhe rhree conlbuslion reactions are as

follows:

Rei~tion

Vaporization, evaporation Sublimation Melting, fusion Transition hetween phases in general Mixing of fluids

ads

Adsurp~iun

C

Combustion Formation

f

Heats of Reaction$ Can Be Calculated from Tabulated Heals nf Formation

If we reverse Equation 3, multiply Equtltion 2 by 2, and add the rcsults to Equation 1. we obtain the equation for the formation of CH,(g) frum ils elements.

\

along with

Because Equatiun (4) represents the formation uf one mole of CH,(g) directly irrm its elements, we have A,H0[CH,(g)l = -74.81 kl-mul-' at 298 K.

I

I

F I G U R E 5.9

Standard enthalpy changes involved in the formation uf CO,(g), H,O(I), and C,H,(g) from their elements, based upon the convention that A,H0 = 0 for a pure element in its stable furm at oric bar and at the temperature of interest.

Note that h e s t o i c h i o m e ~ ccoefficients have no uniuj in the IUPAC convention. Because Equation 4 represents the formation of one mole of C,H,(g) from its elements, A,HG[C2H,(g)] = -t226.7 kJ.mol-' at 298 K (Figure 5 . 9 ~ )Thus, . we sce that we can obtain values of A,H' even if the compound cannot be fumed directly from its elettlents.

As suggested by Figure 5.9, we can set up a table of A f H o values for con]puunds by setting the values of A,H0 for the elements equal to zero. Thai is, for each pure element in i t s stable form at one bar at the temperature of interest, w e set A,H0 equal to 7em. Thus, standard enthalpies of formation of compounds are given relative to the dements in their nonnal physical states at one har. Table 5.2 lisls values of A f H Cat 25°C for a number of substances. If you louk at 'Fdble 5.2, you will see that A,Hn[C(diamond)] = + I ,897 kJ-mol-', A, Hc[Br,(g)] = $30.907 k l .mol-I, and A, HU[I,(g)l = +62.438 kJ.rnr11-' . The values of A, H" fur these forms uf the elements are not equal to zero because Cldiamond), Br,(g), and I,(g) are not the normal physical states of these elemeits at 25°C and one bar. The normal physical states of these elements at 25°C and one bar are C(graphite), &,(I). and I,(s).

E X A M P L E 5-11 Use Table 5.2 tu calculate the molar cnthalpy of vaporizatiun A+apH' of brorninc al

25°C. E X A M P L E 5-10 Given that the standard enthalpies of combustion of C(S), H2(g), and CCli,(g) are -393.5 k.l.rnol-I, -285.H kl-mol-', and -890.4 W.rnol-'], respectively, at 298 K, calculate the stmdard enthalpy of fnmaticln uf methane, CH,(g).

I

S 0 L U T I O N : The equation that represents the vaporizatiun of one nlolc of hrtrnilne is

Chapter 5 1 The First Law of Thermodynamics

5.2 Standard tnolar enthalpies of formation, A,H", for various subslances at 25°C and one bar.

5-1 1. Heats uf Reactions Can Be Calculated from Tabulated Heats of Formation

TABLE

Substance

Formula

Therefore,

A. H"/!d.~nol-'

Note that this result is not the value of AVqH'' at its normal boiling point of S8.8'C. The value of Av:pH" fi 558.T is 29.96 kJ.mol-I. ( h r c'cvill learn how to calculate the temperalure varration of A H in the next section.)

Carbon dioxide Carbun monoxide: Cyclohexanc Ethane Ethanol

Ethenc Cilucose Hexme

Hydrazinc

coz(g) co(g)

-393.504 - 1 10.5

c(,H~z(l) C, H, (g) C,HsOH(I) C p , (g) C&H,206(s) c6H14(lj N2H4(1)

We can use Hess's law to understand how enthaplies of formation are used to calculate enthalpy changes. Consider the general chemical equation

where a , b. y , and z are the number of moles of the respective species. We can calculate A r H in two steps, as shown in the following diagram:

II

N*H,g)

mu1 uf A

Products

A , H"

z n ~ o lo f Z

am)

Hydrogen bromide Hydrrlgen cl~loride Hydrogen fluoride Hydrogen lodide Hyrlrugen peroxidc lodine

H20,(I) 1, (g)

Methane

CH, (g)

Mclhanul

CH,OH(I)

-74.81 -239. 1

CH,OH(g)

-201.5

licl(g)

\

HFk)

-aAfH3[A] - 187.8

f Elcments in their

f 62.438

Nitrogen dioxide Di~iirrogcn~etr~oxide

No2(g} N,O,(g)

Octnnc

c8H

(1)

-250.1

PenPane

csH I I(]) C, HE(g) C I ~ H ~ (s) PII

-173.5 -103.8

First, we decompose compounds A and B into their constituent elements (step I ) : and then we combine the elements to form the ci~rnpoundsY and Z (step 2). In thc first step, we have

N20, (1)

P~opanc Sucrrlse

Sulfur dioxide Sulfur trioxide , Tclrachloro~netha~~t.

so,(g) so,(g) UCI,(I)

-22.2~ -296.8 -395.7 -135.44

We have omitted the degree superscripr on the ArH because this value is nnt necessarily referenced to one mole of a particular reagent. The minus signs occur here because the reaction involved is rhe reverse of the formation of the compounds from their elements; we are forming the elements from the compounds. In the second step, we have

ArH(2) = y A,H5[Y]

-+ zA, H , [Z]

The sum of A,If ( I ) and AtH ( 2 ) gives ATH for the general eql~ation:

2 16

Chaptcr 5 / The F~rsiLaw of Thermodynam~cs

Note that the right side of Equation 5.52 is the total enthalpy of thc products minus the total cnthalpy uf the reactants (see Equation 5.47). When using Equation 5.52. you need to specify whether each substance is a gas, liquid, or solid because the value of A,Ho depends upun the physical state of the substance. Using Equation 5.52, wc determine A r H fur the reaction

5-12. The Temperature Dependence OF A, H Is Given in Terms of the Heat Capacities of the Reactants and Products Up to now, we havc calculated reaction enthalpies at 25°C. We will see in this seulion that we can calculate A,H at other temperatures if we have sufficient hcat-capacity data. Consider the general reaction

We can express A, H at a temperature T, in the form

Using the data in Table 5.2, we ubtain where, from Equation 5.45,

Note that A, H0[O,(g)l = 0 because the Af HI' value for any element in its stable state at 298 K and one bar is zero. To determine A r H for

we niuitiply ArH = - 1299.58 kJ.mul-' by

I

etc. Similarly, A r H ( T , )is given by

2 mnl lo obtain ArH = -2599.16 W.

EXAMPLE 5-12 IJw thc A$' dava in Tahle 5.2 to calculate the value ol-A,H' for the comhrlstion clC liquid ethanol, C,H,OH(I), at 25°C:

ctc. Ifwe substitute Equation 5.54into Equation 5.53 and Equation 5.56 init1 Erlu~ation 5.55, and then subtract the resultant A , H ( T l ) from ArH(T,),we obtain

SOLUTION: Reremng to Table 5.2, we find that A , H 3 [ C 0 , ( g ) ]= -393.509 kJ,mol-I; A, H'[H,O(1)] = - 2 ~ 5 . ~ 3 k l ~ m o l - ' ; A,HU[O,(g)J = 0 and A, Hn[C,H,OH(1)J = -277.69 kJ-mu]-' . Application of Equation 5.52 yields

where, as the notation suggests, AC, ( T ) = y C,,, ( T )

+ zC,,,(T)

-uC,,(T)-

bC,,,(T)

(5.5gj

Thus, if we know AtH at TI,say 2SnC,we can clilculnte ATH a1 any other temperature using Equation 5.57. In writing Equation 5.57 we have assumed there are no phabe transitions betwegn TI and T,. Equation 5.57 has a simple physical interpretation given hy Figurc 5.10. 'Lb cnlculate the value of A,H at some temperalure T, given the value of A r H at T I ,we can follow the path 1-2-3 in Figure 5.10. This pathway involves taking the reactants

5-1 2. The Temperature Dependence

219

1

E X A M P L E 5-13

'

The standard molar enthaply of formation, A,H0,of NH,(g) is -46.1 I kl-mol- at 25°C. Using the heat capacity data given below, calculale the standard molar heat of formation of -(g) at 1000 K.

a

+ 7.'

F I G U R E 5.10 An illusvation of Equation 5.57. Along path 1 we take the reactants from 7; to TI. Along path 2 we let the reaction occur at TI.Then along path 3, we bring thc prducts from T,back to T,.Because AH is a state function, we have that AH&) = AH, AH, AH,.

+

+

from teinperaturc T? to T I ,letting the reaction occur at T,, and then taking the products f ~ w nTI back to T,. The mathematical expressions for AH for each step are

L:

iT

< 1500 K

S O L U T I O N : Weuse thteyuation

The relevant chemical equation for the formation of one mole of NH,(g) from its elemerits is

and so

AH, = ArH(Tl)

A H3 =

where 298 K

AC;(T)/J.K-' .rnol-' = ( I ) C;(NH,)

C p(products)dT

= -31.21

+ (30.88 x

-

l r 3 K-')T

(i) Ci(N,) - (:) - 15.895 x

Ci(H,)

lo-* K-')T'

The integral of A C , ( T ) i s

and so AH(I',) = AH, 4-AH,

+ AH,

As a simple application of Equation 5.57, consider

Let's calculate A,wingfum:

Do you see a pattern in going trom one row to the next? The triangular arrangement here is called Pzxal's triangle. E-6. I n how many ways can a committee of three be chosen from nine people?

E-7. Calculate the relative error for N = 50 using the formula for Stirling's approximation given in Example E-3, and compare your result with that given in Table E.1 using Equation E.7. Take In N! tn be 148.47776 (CRC Handbook of Chemistry a d Physics). E-8. Prove that x lnx + 0 as .T + 0. E-9. Prove that the maximum value of W ( N . N , )= N ! / ( N ( H ~ n rTreat : N, as a continuous variable.)

E-10. Pruve that the maximum valuc of W I N , , N,, R, =R, =N/r. 2 -

-

N,)!N, ! is given hy N, = N / 2 .

. . . ,Nr)in Eqmtiun E.5 i s given by N, =

.

E-1 1 Prnve that

E-12. The quanlily n 1 as we have defined i t is dcfined only for positive integer values of n. Consider now the function of x defined hy

and dv = s-'dt) tu get

Now use Equation 2 to show that I'(x) = (x - I)! if x is a positive integer. Although Equation 2 provides us with a general function that is equal to ( 1 1 - I ) ! whcn x takes on integer values, il is defined just as well for non-integer values. For example, show that r(3/2),which in a sense is is equal to 7r1'*/2. Equation I can also be used tu explain why O ! = 1. Let x = 1 in Equation I tu show that r ( l ) , which we can write a< O ! , is cqual to 1. Thc function T(x) defined by Equation 1 is called the gamrnabnction and was introduced by Euler to generalize the idea of a factorial to general values of n . The gamma function arises in many problems in chemistry and physics.

(i)!,

Entropy and the Second Law of Thermodynamics

In this chapter, we will introduce and develop the concept of entropy. We will see thal energy considerations alone are not sufficient to predict in which direction a pruceas or a chemical reaction can occur sponvaneously. We will demonstrate that isolated systc~ns that are not in equilibrium will evt~lvein a direction that increases their disorder, and then we will introduce a thermodynamic state function called entropy thal givcs a quantitative meaure of the disorder of a system. Onc statement uf the Sccand Law of Thermodynamics, which governs the direclion in which systems evolve to their equilibrium states, is that the entropy of an isolated systern always increases as a result uf any spuntaneous (irreversible) prmess. In the second half of (his chapter, we will give a quantitative m o l e c u l ~definition of entropy in terms of partition function. Rudolf Clausius was hurn in Koslin. Prussia (now Koszalin, Poland), on January 2, 1822, and died in Ig88. Although Clausius was initially attracted to history, he eventually received his Ph.D, in mathematical physics from the llniversity of Halle in 1847. Hc held a position for several years at the Un~versityof Zurich but returned to Germany and in 187 1 wtlcd at the University of Bonn,wtlerc he remained for the wst of his life. Clausius is credited with creating the carly fuundations of thermodynamics. In 1850, he published his first great paper on the theory of heat, in which he rejected the then-cumi~tcaloric theory and argued that the energy of a system is a thermodynamic state fundion. In 1865, he published his second lmdnlark papzr, in which he inlnlduced another new thermtdynunic stale function. which he called etltropy, and expressed the Second Law uf Thcr~nodynamicsin terns of thc entropy. Clausius alxo >tudicrl the kinetic thet>l-yr,f gases :~ndmudc i~npjrtantcr>~itributions to it. He wits ct\auvinistic :uld htrongly dcfended Gcr~nanachievcr~lcotsagainst ~ h n he t cunsidercd the iniri~~gclncnts of others. Muft of Clausius' work was done before I870 because of two events in his life. In 1870, he was wuunded while serving in an ambulance corps in the Eranco-PrussinnWar and suffered life-long pain from his injury. Morc tragically, his wife died in childbirth, and he asaunled the respons~bility01- raising six young children.

-

6-1. The Change of Energy Alone Is Not Sufficient to Determine the Direction of a Spontaneous Process For years, scientists wondered why some rcaclions or pmcesseh proceed spontar~eously and others do not. We all know that under the right conditions iron rusts, and that objects d o not spontaneously unrust. We all know that hydrugen and oxygen react explosively to form water but that an input of energy by means of cleutrotysis i s requircd to decompose water into hydrogen and oxygen. At one tilnc scientists believed tho1 :I criteriun for a reaction o r B process tu procccd s p o n ~ a n e o h l ywas Ihitl it shoultl bc exothermic, or evolve energy. This belicf was tnotiv;lted by the [act that the prorlucls of an exothermic reaction lie at a lower energy or enthall~qthan the reactants. After all, balls do roll downhill and opposile chagea do attract each other. Mechanical sy stcms evolve in such a way as to minimize their energy. Now consider the situation in Figure 6.1, however, where uiie bulb contains a gas at some low pressure at which it may be considered to behave idcally, and the othcr hulb is evacuated. When the two bulbs are connected by opening thc strjpcock between them, the gas will expand into thc evacuated hulb until the pressures in

6 2 . Nonequ~libr~urn lsulated Systems Evolvc in a Direction That Increases Their Disorder

-

each of these processes obeys the First Law of Thermodynamics, but using this law, we cannot tell why one direction occurs spontaneously and its reverse does not. Although mechanical systems tend to achieve their state of lowest energy, clearly some other factor is involved that we have not yet discussed.

F I G U R E 6.1

mo bulbs connected by a stopcock. Initially, one bulb contains a colored gas such as bromine and the other one is evacuatd. When the two bulbs are connected by opening the s t o p k , the bromine occupies both bulbs at a uniform pressure as seen by the uniform culor.

the two bulbs are equal, at which time the system will k in equilibrium. Yet a careful determination of the thermal processes of this experiment shows that both AU and AH are essentially zero. Furthermore, the unaided reverse process has never been observed. Gases do not spontaneously occupy only part of a container, leaving the other part a$a vacuum.

Anothcr cxa~npleof a spontaneuus prwess that is not e x o t h e m ~ cis depicted in Figure 6.2, where two pure gases are separated by a stopcock. When the stopcock is opened, the two gases will mix, and both will eventually become evenly distributed between the two bulbs. in which case the system will be in equilibrium. Yet once again, the value of AU or A H for this process is essentiafly zero. Furthermore, the reverse process has never been obsewed. Mixtures of gases do not spontaneously unmix. There are many vpuntrtneous endothermic processes. A simple example of a spontaneous endolhermic reaction is the melting of ice at a temperature a h v e O'C. This spontaneous prucess has a value of AhsHgequal to +6.0 kl.mol-' when the tempcrature is around WC. An espxially interesting endothermic chemical reaction is the reation o f solid barium hydroxide, Ba(OH),(s), with solid ammonium n i h t e , NH,NO,(s):

The energy ahsorbed by mixing stoichiometric amounts of these two reagents in a test tubc can cool the system tu below -20°C. These and numerous other examples indicate that spontaneous processes have a direction that cannot he explained by the First Law of Thermdynarnics. Of course,

F I G U R E 6.2 Two bulbs connected by a stopcock. Initially, each bulb is occupiad by a pure gas, say bromine and nitrogen. When the two bulbs are connected by opening the stopcock, the two gases mix u~liforti~ly, so each bulb contains the same uniform mixture.

6-2. Nonequilibrium Isolated Systems Evolve in a Direction That Increases Their Disorder If we examine the a h v e pmesses from a microscopic or molecular point of view, we see that each one involveh an increase in diwrder or randomness of the qystcm. For example, in Figure 6.1, the gas molecules in the final state are able to move over a volume that is twice as large as in the inilia1 state. In a sense, lwating any gas molecule in the final state is twice as difficult as it1 the initial state. Recall that we found that the number of acccssible translational states increases with the volume of the container, Problem 4-42,A similar argumenl applies tu the mixing ut two gases. Nut only is each gas spread over a larger volume, but they are alw mixed together. Clearly the final (mixed) state is more disordered than the initial (separated) state. The melting of ice at a temperature greater than O"C also involves an increase in disorder. Our molecular picture uf a so11J being an ordered lattice army of its constituent particles and a liquid being a more random arrangement dirzctly implies that the melting of ice involbes an increase in disorder. These examples SUggeFt that not only do systems evolve spontaneously in a direction that lowers their energy but that they also swk to increase their disorder. There i s a competition between the tendency to minimize energy and tc~maximize disorder. If disorder is not a factur, as i s the case fur a simple mechanicai system, thcn energy is the key faclor and the direction of any sponlaneous prwess is that whlch minimizes the energy. If energy is not a factor, however, as 1s the case when mlxing two gases, then disorder is the key factor and the direction of any spontaneous process is that which maximizes the disorder. In genera!. some compromise between decreasing energy and increasing diburder must be met. What we need is to devise some particular property that puts this idea of divnrdcr on a useful, quantitative basis. Like energy, we want this property to be a state functiun k a u s e then it will he a property of thc state of the system, and not of its previous history. Thus, we will rule out heat, although the transfer or energy as heat to a s j stem certainly does increase its dihr~rder.To try to get an idea of what an appropriate function might k,let's consider, for simplicity, the heat transfer a~sociatedwith a reversible, small change in the temperature and volume of an ideal gas. From the First Law (Equation 5.9). we have

Chapter b / Enbopy and h e Second l a w of Thermodynamics

242

6-3. Unlike qXv,Entropy Is a State Function

1

and

whcre Tz is given by (cf. Equation 5.21)

243

E X A M P L E 6-1 Calculate qmvand A S for a reversible expansion of an ideal gas at consrant pressure PI from T,,Vlto T,,V, (path D in Figure 6.3)followed by a reversible coaling of the gas at constant volume V, from P , , T,to P,. TI(path E). SOLUTION: ForpathD(cf.Exarnple5-4),

The poini here is that qm differs for the two paths, A and B t C,indicating that qw+i s riot a state fu~~ction. Now let's evatuate

and so

For path E,S?umv = 0, and so 6qmv,,= dU, = C,( T ) d T

for thcse two paths. For path A from P,,TI, V , to P,, V,,TI, we have, using Eqnation 6.6,

and

For the complete p e s s (paths D + El,

- qmv,,j+ q w v . ~= Pi(V2 - I:)

q r e r . n r ~-

For the reversible adiabatic expansion from PI,V,, T, to P,,V,, T, (path B) followed by a reversible heating at constant volume from P,, V,, T, to P,, V,, TI Cpath C), we have, using Equation 6.7.

and

To calculate AS for path D, we use Fquation 6.1 1 to write

To evaluate the second integral here, wc must knnw how T varies with V Tui thif procew. But this is give11by PI1' = nHT,so

But using Equation 6.8, AS, turns out to be

.

For path E, Aw,, = 0, and using Equation 6.12 for 6qmv gives

and so

The value of A S for the cornplete process (paths D + E) is

Thus, we see, that the AS, (Equation 6.9)is equal to AS,,, the value of AS is independent of the pnth.

(Equation 6.10) and that

A SI , , E = AS,

+ AS,

v

=n ~ l n 2 VI

Chapter 6 / Entropy and the Second Law of Thermodynamics

Notice that this is the very same result we obtained for paths A and B + C, once agaln suggesting lhaf S is a slate function.

I

6-2 We shall prove in Example 8 4 that siinilar to that found fnr an ideal gas, U is a function of only the temperature for a gas that obeys the equation ut' state EXAMPLE

P

But T,,thc temperature at the end of the reversible adiabatic expansion, can he found from

I

Using the fact that d a = ? , ( T ) ~ Tand 6qm, = 0 gives -

-

RT

-

C , ( T ) d T = - P d V = - V7 - db V

Divide through by T and inkgrate from thc ~nitial\tatc to the final state lo get

RT = w v-b

where b is a constant that reflects the size of the mulecules. Calculate y , both the paths A and B + C in Figure 6.3 for one mole of such a gas.

6 4 , The Second Law of Thermodynamics

and A S fur

,

Substituting this result inlo the above cxpreshion for AS,, gives

3OLt~TION: Path A reprewnls an sot hernial expansion, so dlJ, = O because U depends only upon the ternperaillure. Therefore, SyE,, = -Aturn,,

RT = PdV = r - - d V V-b

Therefore, we see that even though qmy,, $ qrCY B t (.,ncverthelefs, -

and

-

AS, = AS,?,

The enwopy change i s given by

For path B, a reversible adiabatic expansion, qw,,fl = 0,so AS, = 0

We will show several times in the following sections that the entropy is related to the disorder of a system, but for now, notice that if we add energy as heat to a system, then its entropy increases because its thermal disorder increases. Furthermore, nt~tice that because dS = Sqzv/T,energy delivered as heat at a lower tetnperature contributes mare to an entropy (disorder) increase than at a hjgher temperature. The lower the temperature, the lower the disorder, so the enegy added as heat has proponinnally mure "order" to convert to "disorder."

For path G, 6111=~,, = U, and SqFov,r = dUr = C , ( T ) d T

and

The molar entropy change is given by

and so

-

AS,,,

=

AS,

+ AS,

=-

& The I . Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process We all know that energy as heat will flow spontaneousl~ fi-rjm a region of high tenlperature to a region of low temperature. Lel's investigate the role entropy plays in this pmcess. Consider the two-compartment system shown in Figure 6.4, where parts A and El are large one-componen! systems. Both systems are a1 equilibriunl, but they arc not at equilibrium with each other. Let the temperatures of these t w t ~systems he 7, and 1;. The two systems are separated from each other by a rigid, hear-conducting wall so that energy as h>at can R(IW from one system to thc other, but the two-comparttncnt system itself is isolated. When we call a system i.volured, we mean tha~the sysiern is separated from its surro~~ndings by rigid walls that d o not allow malter or energy to pass through them. We may picture the walls as rigid, totally nun-heat cc~nducting,

H. The Second Law of Thermodynamics F I G U R E 6.4 A two-compartment system in which A and B are large, one-cumponent systems.

Each system is at equilibrium, but they are not a1 equilibrium with each other. The two systems are separated from each other by a rigid, heat-conducting walI. The total

two-compartment system itself is isolated.

and impervious to tnntter. Conscqucntly, the system c w d o no work nor can work be done on thc system, nor can it exhange energy as heat with the surroundings. The two-compartlnent system is describcd by the equations

U,

+ U , = constant

we wish to separate the effect due to a change in energy from the effect due to a change in entropy. Because the energy remains constant, the driving force for my spontaneous process in an isolated system must be due to an increase in entropy, which we can expmss mathematically by d S > 0. Because the system is isolated, this increase in entropy must he created within the system itself. Unlike energy, elitropy is not necessarily conserved; it increases whenever a spontaneous process takcs place. in fact, the cntrt~pyo f an isolated system will ct~ntinueto increase until nu mare spontaneous processes occur, in which case the system will be in equilibrium (Figure 6.5). Thus, wc conclude that the entropy of an isolated system is a maximum when the system is in equilibrium. Consequeiitly, d S = 0 at equiribrium. Furthcrmore, not only is dS = O in an isolated systcm at equilibrium, but dS = 0 for any reversible process in nn isula~rd systcm because, by definition, a reversible process is one in which the system remains essen~iallyin equilibrium during the entire process. To summarire our conclusions thus far, then, we write

Vg = constant

VA = constant

247

dS > 0

(spontaneous prwess in an isolated system)

riS = 0

(reversible process in an iqolated system)

(6.17)

s = s* + s, Because V, and V , are fixed. we have for each separate syskni

Because we have considered an isolated system, no energy as heat can fluw in or out of the system. For other types of syslems, however, cnergy ax heat can flow in or out, and it is convenient to view dS in any spontanerjus infinitesimal procers as consisting of two parts. One part of dS is the entropy created by the irreversible process itself, and the other part is the entropy due to the cnergy as heat exchanged between the system and its surroundings. These two contributions account for the entire change in entropy. We will denote the part of dS that i s created by the irreversible process by dS,, because it is produced by the systcm. This quantity is always positive. We will denote the part of d S that is due to the exchange of energy as heat with the sum~undings by dSrxL,because it is due to exchange. This quantity is given by Sq/ T, and it can bc positive, negative, or zero. Note that 69 need not be BqrtV. The quantity Sq will he Sqmv

dU, =JSqTN+J~~irtY = ThdSn (dV,, = 0 )

dUB=&qw-t6wrev=T,dSs (dV,=O) The entropy change of the two-compartment systein is given by

But dU, = -dU, hccause the two-compartment system is ist~lated.so we have

(i &)

d~ =duH

-

(ti. 16) Srner

...

Experimentally. we know that if T, > T,, then dU, < O (energy as heat flows I'mm sysiern B to system A), in which case dS r 0. Similarly, d S > 0 if T, < ?A because d U , > 0 in this case (energy as heat flows from system B ti] system A). We may interprel this result by saying ihat the spontaneous flow of energy as heat from a budy at a higher temperature Za a body at a lower temperature is governed by the condition dS > 0. If T, 5 T,. then the two-ct)mprtrtment system is in equilibrium and

(JS = 0. We can generalize this result by investigating the rule entropy plays in governing the directic~nof any spontaneous process. To be able to focus on the entropy alone, we will cansider an infinitesimal spontaneous change in an isolated system. We chwse an ist11att.d systern because the energy remains constant in an isolated system, and

......................

Spontaneous

.

---

Equilibriunl dS=O

S

F I G U R E 6.5

A schematic plot of entropy versus time for an isolated system. The entropy increases ( d S > 0) until no more spontaneous processes occur, in which cafe the system is in equilihriuln. and d S = 0.

2.18

Chaptcr h / Fntrupy and the Second Law uf Th~rmdynarnics

if the exchange is reversible and 6qInif the exchange i s imversiblc. Thus, we write for any process

Reversible

For a reversible process, Sq = Sqnv,dSpRd= 0, SO

F I G U R E 6.6

A cyclic process in which the system is first is;cllatcd and undergoes an imevcnible process from state 1 to state 2. Then the system is allowed to interact with iis surrt~undingsand i\ brought back to state I by some reversible path. Bcaufe entropy is a state functiun, A S = 0 for a cyclic process.

in agreement with Equation 6.3. For an irreversible or spontaneous process, dSp,, > 0, dSeXEl, = Sqirr/ T. and so

Equations h. 19 and

6.20 can be written

;ts

une equation,

where the equality sign holds for a reversible process and the inequality sign holds for an irreversible p m e s s . Equation 6.22 is one of a number of ways of expressing the Second Law of Thermodynamics and i s called the hequaliw of Clausius. A formal statement of the Secr~ndLaw o f Thermodynamics is as follows: 'Ihcru is a thermodynamic state function of a systcm called the entmpy, S,such [hat lor any change in the thermodynamic siaie of Lhe system,

where the equaliw sign applies if the change is carried out reversibly and thc inequality sign appltcs if the change is carried out irreversibly at any s l a p .

tPr. van US^ Equ~llio~i 6.22 tr) prove quile generally thut the entmpy 01' iun isolated sybrem always increases during a spontaneous (irreversible) process or that A S > 0. Consider a cyclic process (Figure 6.6) in which a system is tirst isulated and undergws an irreversible process from state 1 to slate 2. Now lei the system interact with its surroundings and return it to-statc I by any reversible path. Because S is a state function, A S = 0 for this cyclic prwexs, so accnrding to Equation 6.22,

The inequality applies because the cyclic process is irreversible from I to 2. The first integral here equals zero bxause the system is isolated, i.e., SqIm= 0. The second integral is by definition equal to S, - Sz,LO we have 0 > S, - S?. Because the final state is state 2 and the initial state is state 1 ,

Thus we see that the entropy increases when the isolated system goes from state 1 to state 2 by a general irreversible process. Because the universe itself may be considered tu he an isolated system and al\ naturally occurring processes are irreversible, one statement of the Second Law of Thermodynamics says that the entmpy of thc universe is constantly incre;lsinp. In fact. Clausius summarized the first two laws of thermodynamics by The encrgy of the Universe is constant;

the entrnpy is tending to a maximum.

6-5. The Most Famous Equation of Statistical Thermodynamics Is S = k , In W In this section, we will discuss the molecular inlerpretatiun of entropy more quantitatively than we have up to now. We have shown that entropy is a state function that is related to the disorder of a system. Disorder can be expressed in inurnher oI'ways, hul the way that has turned out to k the most useful is the following. Cmsider an el~semble of A isolated systems, each with energy E , volume V , and number of panicles N. Realize that whatever the valueof E, it must bc an eigenvalue of the Schrodinger equation for the system. As we discussed in Chapter 3. the energy is a [unction ol'N and V , so we can writc E 7 E ( N , V ) (see, fur example, Equations 3.2 and 3.3). Although a l l the systems have the same energy, they may k in diffcrenl rluantum states because of degeneracy. Let the degenewcy associated with the energy 6' be !i2(E), so that we can label the L?(E) degenerate quantum states by j = 1. 2, . . . , Q ( E ) . (The degeneracies

Chapter 6 1 Entropy and the kcond Law of Thermndynnn~ics of systcms that consist of N particles turn out to be enormous; they are numbers of the be the order of P" for cncrgies ]lot too close to the ground-state energy.) Now, let ~f. nu~nbcrof systertls in the ensemble that are in the state j. Because the A systems of (he ensernhte are distinguishable, the number of ways of having a,systems in state 1, 0: systelt~s in state 2, ctc. is given by (Mathchaplet E)

with

If-all A systcms are in one particular statc (a totally 'ordered arrangemetit), say statc 1, then a , = A, a, = o, = . . . = 0 and W = I , which is the smallest value W can have. 111 the other extreme, whcn all the uj are equal (a disordered ardngement), W takes on its largest value (Prt)ble~nE-10). Therefore, W can be taken to be a quantitative Ilwnsurl: ul'the disorder uf a system. We will not set the entropy proportional to W , however, but to in W according to

wIiere k , is the Bolt7mann constant. Note that S = 0 for a completely ordered system (0,= 1, ri, = a, = - . - Oj and achieves a maximum value for a completely disordered cystem ta, = la, = u , = . . .). Equativn 6.24 was formulated by Bnltzinann and is the most f;imr>us equation of stiltistical therniodyr~amics.In fact. this equation is the only inscription on a monument to Boltzmann in the central cemelary in Wenna. It gives us a qualititatirc relation belwecn thc thermodynamic quantity, entropy, and the statistical quantity, I+'. We set S equal to In W rather than W for the following reason. We want S to be such that the total entrupy o f a system that is made up of two parts (say A and B) is

given by

In other wurds, we want S to be an extensive state function. Now if W, is the value of W for system A and W, is the value of W for system B, W,, for the composite system is given by

' h e entrupy of the compbsite system is

b 5 . The Mn~lramnus Equation of Statistical Thcrmdynamicr ts S = k, In W

An alternate form of Equation 6.24 expresses S in terms of the degeneracy St. We can determine this exprcrsion in the following way. Given no other infonnation, there is no reahon to choose one of the 52 degeneratc quantum states over any other: each one should occur in an ensemble with equal prr~bability(this concept is actually one of the postulates of statistical thermodynamics). Consequently, we cxpcct that the enxemhle of iqolatcd sfktcms should cuntain equal nurribers of Fystems it1 each quantum state. Becau~eS is a maximum for an isolated sy'iterrl at eqriilibriu~n,W must illso hc n maximum. The value of W is niax~lniredwhcn all the a, art. equltl (Problerti E-10). Let the total number of systems in the ensemble be A = n Q and let each q = ~ t so , that the set of Q degenerate quantum states is replicated n timcs in the ensemble. (We will never need the value of n ) Uwng Stirling's appruximation (Mathchapter E) In Equation 6.23, we get n

Senrrmhlc = k HIn W = k,[AIn A

rr, 111a,]

1-1

L1

= k,,[nQ In(n0) - x ( n Inn)] = k,[1151 In(n0) - Q ( n I n n ) ]

Thc cntropy of a typical system in the enselnhle is given by SejlsenlMc = AS nQS5yrm,and st"

TYI,I.III

=

where we have dropped the suhxcript, system. Equation 6.25 is an alternate fonn nl Equation 6.24 and relates entropy to disorder. As a concrete example, consider a system of N (distinguishable) spins (or dipoles) that can bt: oriented i r ~one of two pn~sihle directions with equal probability. Then, each spin has a degeneracy of 2 associatcd with it, and the degeneracy of the N spins is 2". The entropy of this system is Nk, In 2. We will use this result when w e discuss the entropy of carbon mont>xide at 0 K in Section 7-8. As another example of the use of h u a t i o n 6.25, Problem 6 2 3 has you show that

for an ideal gas of N particles, where c ( N ) is a function of N and f ( E ) is a function of the energy. Now let's determine A S for an isothermal expansion of one mole of an ideal gas from a volume V, to V,.

Chapter 6 I Entropy and the Second Law of Thermodynamics

h4.

We Must Always Devise a

Reversible Process ro Calculate Entropy Changes

E X A M P L E &4 I n Exutlple 6-2 we stated that U is n function or only the tcmpcrature for a gas that obeys thc equation o f state

Nule t h a ~A S > O kcause V2 > V,. Thus, we see that the enlropy increases in the cxpnnsion of an ideal gas into a vacuum. Because Equation 6.19 tells us to calculatc AS by expanding the gas reversibly and isr,themally from V , to V,, Equation 6.28 holds for the reversible isothermal expansicln. Because S is a state function, hnwever, the value o f A S obtained hum Equnhon 6.28 is the satne a< thc value o f A S for ihe irreversible isothermal expntlsion from V, to V,. How, then, do a ~vzr.cihleand an irreversible isothermal expansion differ? The answer lic:, in the value of A S for the surroundings. (Remember that the condition AS > 0 applies 10 an isolated system. If the system is not isolated, thcn the condition AS 2 0 appliec to 1112 sun1 of the entropy changes in the system and its sur~nundings,in other words. the entire universe.) Let's look at the entropy change of the surroundings, ASww,for h ~ a hreversible and an irreversible isothermal expansion. During the reversible expansion, AU = 0 (the process is isothermal and thc gas is ideal) and the gas absorbs a quantity uf energy ax heat, qrev= -ulrw = n R T In V,/ V , , from its surroundings. The entropy of the surroundings, thcrcforc. decreases according to

whcrc b is a constant that reflects the size of the molecules. Calculale AT when one molc of such a gas at T and V , is allowed to cxpsnd into a vacuum tr, a Inla3 vulurnc of K.

SOLUTION: Westilrtwilh

Becausc U is a function of only the teniperature, and hence is independent of thc volume. dl' = 0 for the expansion. Thcrcfore.

and

The total cntropy change is given by

as it should be hecausc thc cntirc process is carried out reversibly. In thc irreversible expansicln, AU = 0 (the pnlcess is isothermal and the gas is ideal). The value of Prx,is also zero, sn loLrn= O and therefore, gin = 0.No energy as hent is delivered to the system by the surroundings and so

Once again, thc cntropy increases when a gas expands into a vacuum.

Let's look at the mixing of two ideal gases, as depicted in Figure 6.2. Because the two gases are ideal, each acts independently of the other. Thus, we can consider each gas separately to expand from VInltialto Vhnsl.Rlr nitrogen, we have (using Equation 6.28)

'Thus, the total cntropy change is given hy As N*

=n

N~

Rln

VN,

+ llg,

and s o A S =- O as we enpecl for an irreversible process. Did we use qhn= n to calculate 4S,umin this process?We actually did because no work was done by the process. In the general case of an isoihermal process in which 110 work is dune (Stu = Of. the process is one of pure heat transfer attd dU = 69 = d q , where dq is all exact diffcrcntial because U is a state function. Therefore, q is path indcpcndcnt and so w e can use qImto calculate the entrupy is this particular case.

and for brwmine,

=

R In

-11 ' 1

v ~ 2

+

"N:

VN2 ifBl2

255

256

Chapter 6 1 Entropy and the Second Law uf Thermodynam~cs

'I'hc total entropy change is

M .We Must Always Deviw a Reversible Proces to Calculate Entropy Change

There is essentially no work done, su hq,," = dU = C , d T . Therefure,

= -n

Rln ";

hl

v~r:

Rln

-n

v%z+VB,t

nr2

VN2fVBr2

Because V is proportional to n for an ideal gas, we can write the abave equation as

if we take C, to k constant from T, t c ~T,, then 1

(6.3 1)

AS = C , In 2 =1

If we divide both sidts by the total number of moles, nLuld, = 11,;

+ n,,! and introduce

mole fractions and

ynr2=

Now, for the initially hotter piece, TI = T, and T? = (T, + Tc)/ 2 , and so Th + Tc AS,, = C, In -

-

2Th

n,aal

Similarly,

then Equation 6.29 becomes

More generally, A,,,,xSfur the isothermal mixing of N ideal gases is given by

in agreement with Equation 6.26. Equation 6.30 says that amIx3 > 0 because the arguments of the logarithms are less than unity. Thus, Equation 6.30 shows that there is an incrcase in entropy whencvcr ideal gases mix isuthennally. Last. let's consider A S when two equal sized pieces of the same metal at different T, and Tc,are brought into thermal cuntacr and then isolated from their teml~erati~res, s ~ ~ r ~ - t ) u ~ ~Clearly, d i n p s . the two pieces of ~nztalwill come trl the same ti11;11telnprature, T , which can be calculated by

The total change in entropy is given by

We will now pnlve that ( T , + T ~ ) '> 4T,lC,and that A S z 0. Start with

(T,- T ~ ) '= T; - 2ThT + T,'

>0

Add 4ThTc to both sides and obtain

heat lost by hotter piece = heat gained by colder piece

Therefore, the value of the argument of the logarithm in Equation 6.32 is greater than one, so we see that A S > 0 in this irreversible process.

Solving for T gives T = - Th

+ + %ar,

= r r N 2 R In V

Therefore.

y~

S = - 5R + K n [ ( T ) 21t mk, T 2

-

2'3

=]

+ nEr2A In V + terms not involviog V Br2

wherc we havc wrirtcn N k R = i t R . The tirial state 1s given by

Fgc,

2'

= '2.N2 =n y2

E X A M P L E 6-6 Ust. Equation 6.45 try calculate the molar entropy of argon at 298.2 K and one bar. and

+ '2.kr2

+

Rln(VN1 VR5)

+ nBr2Rln(VN2+ VHr?)+ terms not involving V

Therefore

compare your result wilh thc experimental value o f 154.8 J ~ K - '.rnoi-l. S O L U T I O N : At298.2Knndonebar.

Because V is proportional ton for an ideal gas, wc have (6.022 x 10" mol-])(l hnr) (l).08314L.bar K .mu]-')(298.2K) = 2.429 x 1 02?L-' = 2.431 x 102' rn-?

-

-'

If we divide this result through by nN2+ trRrz, then we obtain Equation 6.26. t

I

and

2.rrnkDT

'!'

Z~(0.03995kg.rnol-' j(1.3806 x

(7 = [- ) (6.022 x 10" mo1-')(6.626 =

J , ~ - ' ) ( 2 9 ~K) .2 x lo-" l , e ) '

I

3i2

6-9. The Molecular Formula S = kg tn W 1s Analogous to the

(3.909 x l d ' m-2)'12

Therefore

Thermodynamic Formula dS = Sq,,/T In this last section, we will show that Equation 6.24, or its equivalent, Equation 6.40, is consistent with our thermodynamic definition of the entropy. As a bonus, we will finally pruve that p = I / k,T. If we differentiate Equation 6.40 with respect to p,. we get

But Thiv va1uc of 3 agree.; exactly wit11 the experimentally determined value

C dp, = 0 because C pi

= 1, so

Chapter 6 / Entropy dr,d the Second Lavv ol Thermodynamics

2 64

Now substitute Equation 6.41 intc~the In pi rerm in Equation 6.46 to obtain

e place in B accompanied by an exchange of energy a% heat GqEv(arbltrarg). 8 ~ a 1 s the composite system is isolated, the First law rtquires that

The lcrln involving In (jdrops u u ~becausc

Nuw use Equation 6.4 to prove that

Therefore, we can say that the definition given by Equalion 6.4 holds for any system.

and so

But we showed in Section 5 4 that X I E , ( N , V)dp>(N,V , B ) i s he energy as heat that a system gains or loses in a reversible process, so Equation 6.47 becomes

6-6. Calculate qm and AS for a reversible cooling of (memole of an ideal gas at a constant volume V, from P , , Vl, T,to P,, V , , T, followed by a revmihle expansion at constanl pressure P2 from P,, V,. T, to P?, Vt, TI (the final state fur all the processes shown in Figure 6.3). Cotnpare your resull for A S with dlose for paths A, I3 C, and D E in Figure 6.3.

+

+

6-7. Derive Equiltion 6.8 without referring to Chapter 5. 6-8. Calculate the value of AS,if one mole of an ideal gas is expanded reversibly and isothermally from 10.0 dm' Ip Zg,U dm3.Explain the sign o f AS.

Equdtion 6.48 shows, furthermore. that Bk, is an integrating factor of 6qm, or pk, = 1 / T , or p = l / k , T . Thus, we have finally proved that /3 = I / k , T . In t h e next chapter, we will discuss t h e experimental determinatinn of the entropies of substances.

Problems

6-10. Calculate the values of qmvand A S along tbc path U + E in Figuw 6.3 for one mole of a gas whose equation uf state ih given in Example 6-2. Cornpre your result with that obtained iri Example 6-2. 6-1 1. Show that ASn+, i s equal to AS, and AS,+, for the equation urctale gikc11In Example 6 - -

2.

6-1. Show that

6-12. Calculale the values of q,, Bnd AS along the path descr~bcd111Problem &6 For one 111ole of a gas whose equation of state is given in Example 6-2. Compare yuur result with that obtained in Example 62.

it Y is a state function.

6-2. Let z = ;(I, y ) and dz = xydx + y 2 d y . Although d z is not an exact differential (why not'!), what combinatiun of dz and x andlor y is an exact differential? 6-3. Use the criterion developed in MathChapter D lo prove that 6qm, in Equation 6.1 is not an cxact differential (see also Problem D-I 1). 6-4. Use the criterion developed ill MathChapter 1) to prove that SqJT cxirct differential.

6-9. Calculate the value of A S if one ~ n o l eof a11 ideal gas is expanded mvcrsibly and isothermall y from 1.00 bar to 0.100 bar. Explain the sign of AS.

in Equation 6.1 is an

6-5. In this problen~,wc will prove that Equalion 6.5 is valid for an arbitrary system 'lij drl this. cunsider an isolated systemmade up of two equilibrium subsystems, A and B, which are in thermal contact with each other; in othcr words, they can exchange energy as heat k l w e e n themselves. Let suh~ystemA bc an ideal gas and let subsystem R be arhilrnry. Suppose now that an infinilehirnal rcvcrsible process occllrs in A accampanicd by ;in exchange of c ~ l c ~ , gi ~y hh u t Gq,,,(idcal). Simultancrlusly, another infirii1cslmal rcvessiblc process takes

6-13. Shuw that

for a constant-pressure process if C, i~independenlof temperature. Calculate the change in entropy of 2.00 moles of H,0(1) ( U p = 75.2 J-K-'.mol-') if it is heated from 10°C to90C.

6-14. Show that

if one mole v f an ideal gas is taken from T, , V, to T,.V,,assumtng tha~?, is independen1 of temperature. Calculate the value of AS' ',f one malie of N,(g) is expanded from 20.0dm" at 273 K to 3Uodm3 at 400 K . Take = 29.4 ~ . ~ - ' . ~ n o l - ' .

c,.

Entropy and the Third Law of Thermodynamics

In the previous chapter, we introduced the concept uf entropy. We showed that entmpy is created or genemted whenever a spontaneous or irreversible process rxcurs in an isolated system. We also showed that the entropy of an isrbl:~tvrlsystem thut is no1 in equilibrium will increase until the system rzaclles equilihriuni, iiom which time the entropy will remain constant. We expressed t h i s condition mathema~icallyby writing d S 2 0 for a process that occurs at constant U and V. Although we calculated the change in entropy for a few proceses, we did not attemp1 'to calculate absc~lutevalues of the entropy of substances. (See Example 6-6 and Problems 6 4 1 through M 3 , --..

however.) In this chapter, we will introduce the Third Law of Thermodynamics, so that we can calculate absolute values of the entropy of substances.

7-1. Entropy Increases with Increasing Temperature Walther Nernst waq horn in Briessen, Prussia (now Wabrzetnu. Poland), on lunc 25, 1864, and died in 1941. Although he aspired to be a poet, his chemistry teacher kindled his in~erest in science. Between t883 and 1887, Nernst studied physics with von Helmholtz, Uoltzmann, and Kohlrausch. HE received his doctorate in physics at the University of Wurzhurg ln 1857. Nernst was Ostwald's assistan1 at he University of Leibzig from 1887 to 1891, after which he went to the University of GGGttingen. where he esrablishad he Kaiser Wilhellil Institute for Physical Chemistry and Electrochemistq in 1894. Upcm moving to the Universi~yof Berlin in 1905. Nernst h g a n his studies of the behilvirx of hubhvnnces ai very low tenlpratures. He proposed one of the early versions of the Third l a w uf Thermudynamics, which says that the physical activities of substances tend to vanish as the temperature approaches &solute rem. The Third Law made it possible to calculate them~odynamicquantities such as equiiibrium collstants from thermal dah. He was awarded the N o h 1 Prize for chenlistry in 1920 -'in rzrrugltition of his a,r>rkin themmhemistry." He was an early automoblile ellthusiavt and herved during World Wilr I a!. a driver. Nernst lost both of his s o ~ in ~ sWorld War 1. His anti-Nu.i stan* io the 1930s led to increasing isolation, so hc retircd to Ilis country home, where he died io 1941.

We stat with the Fist Law of Thennodynamics for a reversible process:

Using the fact that SqW = TdS and 6wlr> = - Pd V , we obtain a combination of the First and Sscond Laws of Therm4ynamics:

We can derive a number of relationships k t w e e n themdynamic quantities using the laws of thendynamics and the fact that atate functions are exact ditlerentials. Example 7-1 derivks the following twu important relationships

I!

6

2 74

Chapter 7 1 Entropy and the Third taw o f Thermodynamics

and

7-2. The Third Law nt Thermodynamics S a p That thc Entropy of a Perfect Crystal Is Zero at 0 K

and substitute Equation 7.1 for dU to obtain

Proceeding in a simitar manner as in E ~ a m p l 7-1 e (Problem 7-1). we obtain I

I

E X A M P L E 7-1 Express U a$ a function of V and T and then use this result and Equalion 7.1 111 derive Equations 7.2 and 7.3. SOLUTION: I i we treat U as a thaptcr H)

function o f V and T , its total derivative is (Math-

and

From Equation 7.7. we get We suhhtilute Equi~liun7.4 into Equation 7.1 and solve for dS to obtain

Thus, if we know C, as a function of T, we can calculate AS. Most processes we will consider occur at constant pressure, so we will usually use Equaticm 7.9tocalculate AS. If we let TI = 0 K i n Equalion 7.9, then we have

Using t l ~ cdefinition t h a ~( a U / D T ) , = Cy,we obtain

If we comparc this equation for d S with the Iota1 derivative of S = S(T. V),

S ( T ) = S(0 K) t

(constant

P)

Equation 7.10 tells us that we can calculale the entrnpy o f a substance if we know S ( 0 K) and C , ( T ) from T = 0 K to the tempenture of interest. (Notice once again that we use a prime on the variahle of integration to distinguish it from an integration limit.)

we see that

Equation 7.2 tells us Row S varies with temperature at constant volume. If we integrate with respect to T (keeping V constant), we obtain

(constant V )

Thus, i f we knt~wC, (T) as a [unction of 2', we can calculate AS. Note that because C , i s always positivc. the~ntropyincreases with increasing temperature. Equatiun 7.5 is restricted to constant volume. To derive a aiinilar equation for uonstanl prcssllre, we start with dH = d ( l l + P V ) = dU f PdV

+VdP

7-2. The Third Law of Thermodynamics Says That the Entropy of a Perfect Crystal Is Zero at 0 K Let's discuss S ( 0 K) first. Around the turn of the century, the' German chemist Walther Nernst, after studying numerous chemical reactions, postulated that A , S -+ 0 as T + 0. Nernst did not make any statement concerning the entropy of any particular substance at 0 K, only that a l l pure crystalline substances have the same entropy at 0 K. We have added thc "purc crystalIinc" condition here to avoid some apparent exceptions to Nernst's postulate that we will resolve later. In 191 1. Planck, who incidentally did agreat deal of research in hennodynamics (including his dwtoral thesis), extended Nernst's postulate by pustulating that the entropy of a pure substance approaches zeru at 0 K. Planck's postulate is consistent with Nernst's but takes it further. There are

2 76

Chapter 7 1 Entropy and the Third Law uf Thermcdynamics

several equivalent statements of what is now called the Thirdhw of Thermodynumicx, but the one we will use is

7-3. A,,S = A,,H/T,,, at a Phase Transition We made a tacit iisxurnptir,n when we wrote Equation 7.14; we assumed that there is no phase transition between O and T. Suppose there is such a transition at T,,, between 0 and T. We call calculate the entropy change upon the phase transition, A,,,S, by using the equatinn as, = % (7.15)

Every substance has a finite positive entropy, but at zero kelvin the entropy may become zero, and does so in the case of a perfectly crystalline substance. Thc 'Third Law of Thermodynamics is unlike the hrst two laws in that it introduces no new state function. The first jaw gives us the energy and the second law gives us the entropy; the third law provides a numerical scale for entnlpy. Althrlugh the Third Lw was formulated hefore the full development ul' the quanrunt theory, it is much more plausible and intuitive if we think of it in terms of ~nolccularquantum states or levels. One of our molecular formulas for the entropy is (Equation 6.24)

where W i s the number of ways the total energy of a system may be distributed over its various energy states. At 0 K, we expect that the system will be in its lowest energy stare. Therefore, W = 3 and S = 0. Anorher way to see this result is to start with Equation 6.40 frlr 3:

TI,,

A phase transition is a good example of a reversible process. A phase transition can be revered by changing the temperature ever so slightly. In the melting of ice, for example, at one atm, the system will be all ice if T is .just slightly less than 273.15 K and all liquid if T is just slightly greatcr than 273.15 K. Furthermore, a phase transition takes place at a fixed temperature, so Equation 7.15 hecomes (recall that A H = q , for a phase transition)

'

I

EXAMPLE 7-2 Calculak the rncdar entropy change upon melting and upon vaporizatiot~ at one atm fur H,O. Use ah5H = 6.01 k.l .rnol-'at 273.15 K and A ~ =~40.7 ~ kH ~-moi-' at 373.15

K.

S 0 L UT I0N : Using Equation 7.1 h, we have

where p, is the probabihry of finding the system in the jth quantum state w ~ t henergy E,. At O K, there is no thermal energy, so wc expect the system to be in rhe ground statc; thu?, p,, = 1 and all the other pj's equal Lero. Therefore, S in Equation 7.12 equals tsro. Even if thc ground state has a deger~eracyof n , say, then each of the n quantum states with cncrgy E, would have a probability of I/n, and S in Equatlon 7.12 would be

S(O K) = -k,

" 1

C ,=]

-

1

and

Note that A,,T is mnuch Imger than A,">S. This make< sense r~lolecularlybecause the difference in disorder between a gas and a liquid phase is much greater than the difference in disorder between aliquid and a solid phase.

In- = k , Inn n

Even if the degeneracy of the ground state were as large as the Avr~ydroconstant, 3 would be equal to only 7.56 x 10-12~ . ~ - ' - r n o',l which is well below a measurable value of 3. Because the Third Law oJThermodynamius asserts that S(0 K) = 0, we can write Equation 7.10 as

To calclllate .I'(T), we integrate C,(P')/'Tup to the first phase transition temperaterm for the phase transition, and then intcgrate C,.(7')/T from ture, add a Atr,H / the first phase transition temperature to the second, and so on. For example, if the substance has no solid-solid phase transit~on.we would have, h r 1' greater thm the boiling point,

2 78

Chapter 7 1 Entropy and the Third Law of Thermwlynamics

where T," , I ) ,tt constant temperature and volume because the system.is not isolated; a systcrn must he in thermal contact with ilthermal reservoir to be at constant temperature. If the criteriun d S 2 O does riot apply, then what I S the criterion for a spntaneuus prnrc* ,llh,ir we can usc fur a syq1t.m at constant temperature and volutne? Let's start with tht: expression of the F m t L a w oT Thermodynamics, Equalion 5.9,

.,

Because Su) = Pex,dV and dV = 0 (constant volume), then 6 w = 0. If we suhsti~ute Equation 6.3, dS 2 Sq/ T , and 6 w = 0 into Equation 8.1, we obtain

&I.

The equality holds for a reversible process and the inequality for an irreversible prmess. Note that if the system is isolated, then dU = O and we have d S > O as in Chapter 6. We can write Equation 8.2 as

If T and V are held constant, we can write this expression as

d(U - 7's) 5 0

(constant T and V )

Equation 8.3 prompts us to define a new thermodynamic state function by

so Equatian 8.3 hecomes

dA

0

(8.5)

(constant T and V )

The quantity A is calted the Helmholtz energy. In a system held at cotlstant T and V ,

the fIelmhr,ltz energy will decrease until all the possible spontaneous prtxesses have occurred, at which time h e system will be in equilibrium and A will he a minimum. At equilibrium, d A = 0 (gee Figure 8.1). Note that Equation 8.5 is the analog of the criterion that d S 3 O for a spontaneous prmess td vccur in an isolated system (cf. Figures 6.5and 8.1). For an isothermal change from one state to another, Equation 8.4 gives

Directinn nt R S l ~ ~ t l t n r l r ~ uf'rmt~~j5 !. in a Systcm at C(mstant i:olun~e 2nd Ic~riipr,~turc

where the equality holds for a reversible change and the inequality holds lor an irwversihle. spontaneous change. A process for which AA > 0 cannot t&e place xpontaneously in a system at constant T and V. Consequenrly, something. such as work, must be done on the systcm to effect the change. Notice that if A U < 0 and A S > 0 in Equation 8.6. then both the energy change and the cntn~pychange contribute to A A being negative. But if they differ in sign, some sorl of compro~nisemust be reached and the value uf AA is a quantitative measure uf whcthcr a process is spontaneous or nut, The Helmholtz energy represents this compromise between the tendency of a system to decrease ils energy and to increase ils enlropy. Because A S is multiplied by T. we see that the sign of rlU is rnore importan1 at low tempcratures but the sign of A S is rntll-e important at high temperatures. We call apply the criterion that AA < O for an irre\,crsible (spontaneous) pnxess in a system at constant 7' and V lo the mixing uf two ideal gases, which we discussed in Scction 1 7 4 . For that process, A U = 0 and ~3 =,!R In y, -- j2H In y2. Therefore, for the rnixirlg of Lwo ided gases at constant T and Y. AX = RT (y, In ,! J? In?;), which is n negative quantity because p, and q are Iess than one. Thus, we see once again that the isolhermal mixing of two ideal gases is n spontarleoua

+

pnxes';. In addit~ontu scrving as our criterion for spontaneity in a systcm at collstant temperature and volume, the Helmholtz energy has an important physical interpretation. Let's start with Equation 8.6

for a spontdncous (imevcrsible) process, so that AA < 0. In this process, the initial and final states are well-defined equilibrium states. and there is no fundamental reason we have to follow all irreversible path to get from onc state to the other. In fact, we can gain some considerable insight into the process if we look at any reversible path connecting these two states. For a reversible path we call replace A S by qrm/ T, giving

Using Equation 8.5, we see that

But according to the first law, A U

Spontaneous

AA = I

- qm> is

U , ~ ~

equal ta wmv,so we have

(isuthem~al,revcr~iblc)

(8.9)

Eqn~ltbrium dA = 0 - --

t

f F I G U R E 8.1 The Hclmhdtz energy. A , of a system will decrease during any spontaneous processes t h t c~ccurat const;lnl T and V and will achieve its minimum value at equilibriu~n.

If AA i0, thc process will occur spontaneously and turw rcpresents the work that can be done by the system if this change is canied out reversibly. This quantity is the maximum work that could be obtained. If any irreversible prvcess such as friction occurs, then thc quantity of work that can be obtained will be Iess than u ! ~If ~ AA. > 0, the process will not occur spontaneously and 11)- represents the work that must be done on the system tu produce the change in a reversible manner. If there is any irreversibility in the prmess, the quantity of work requircd will be even greater than I I J ~.~ ,

3 04

8-2. Direction uf d Sponbnmus Process for a Syrtem d t Constaml Pressure brlci Tcrnp~taturc

8-2. The Gibbs Energy Determines the Direction of a Spontaneous Process for a System at Constant Pressure and Temperature Most reactions occur at constant pressure rather than at constant volume because they are open to the atmosphere. Let's see what the criterion of spontaneity is for a syswnl at constant temperature and pressure. Once again, we start with Equation 8.1, but now we substitute d S > Sq/ T and 6 w = - PdV to obtain dU 5 l'dS

- PdV

or dU

-

TdS - PdV

0

Because both T and P are constant, we can write this expressinn as d(Il

-

TS + PP') 5 0

The equality holds for a reversible process, whereas the inequality holds for an irreversible (spontaneous) process. If AH < 0 and A S > 0 in Equation 8.15, both terms in Equation 8.15 contribute to A G being negative. But if A H and AS have the same sign, then AG = A H - TAS represents the ct>mprornisebetween the tcndency ul' a system to decrease its enthalpy and tu increase its entropy in a constat~tT wd P process. Because of the factor of T multiplying A S in Eyuatiun 8.15, the A H term can dominate at low temperatures, whereas the T A S term can dominate at high fcmperatures. Of course if AH > 0 and A S -;- 0,then AG > 0 at ail relnperatures and the proccss is never spontaneous. An example of a reaction favored by its valiic oT hut dishvorcd by its valr~e ArS is

(constant T and P)

The value of A,!{ for this reaction at 298.15 K and one bar is -176.2 kJ, whereas the corresponding value of ArS is -0.285 ~ J , K - ' ,giving n,G = ArH - TArS = -91.21 kJ at 298.15 K. Therefore, this reaction prmecds spontaneously at 298.15 K and one bar. A process for which h e sign of AG changes with a small change in temperature is the vaporization of a liquid at its normal boiling point. We represent this process by

We now define a new lhermodynamjc state functir~nby

so Equation 8.5 h o m e s dG 5 0

The analog of Equation 8.7 is

(constant T and P )

(8.12)

Note that Equation 8.11 is the analog of Equation 8.4. The quantity C; is called the Gibbs energy. In a system at constant T and P, the Gibbs energy will decrcace as the result of any spontaneous pmesses until the system reaches equilibrium, where dG = 0.A plot of G versus time for a system at constant T and P would be similar to the plot of A bersns lime tor a system at constant T and V (Figure 8.1). Thus,we see that the Gibbs energy, G, is the analog of the Helmoltx energy. A , for a process that takes place at constant temperature and prchsure Equation 8.1 1 can also he written as

L

The expression for the molar Gibbs energy of vaporiation, Ay3pG, for this process is

The molar enthalpy of vaporization of water at one atm near 100"C, AbdpH, is equal to 40.65 kl-mol-' and buap3 = 108.9 J.K. '.rnol-'. Thus, we can write A+4,,G as

+

where H = U P V is the enthdpy. Note thal the enrhalpy plays the same role in a constant T and P process that the energy U plays in a constant T and V process (uf. Equation 8.4). Note also that G can be written as = 40.65 kJ.mol

thus relating the. Gibbs energy and the Helmholtz energy in the same manner that H and U are related.

' - 40.65 kl.mol-' = 0

The fact that avavG = 0 means that liquid and vapor water are in equilibrium with each other at one atm and 373.15 K. The molar Gibbs energy ut'liquid water is equal to the molar Gibbs energy of water vapor at 373.15 K and one atm. The transfer of one ~riolc

!{-.I, Directinn uf a Spurilanmus Prncfis lur a Syrlptn nr Constant Prc5s~rrcontl r~rnp~rarurc

of liquid watcr to water iapjr under these conditions is a reversible process, and so A,,,pC; = 0. Now let's consider n tcmprature less than the normal hoiling point. say 363.15 K. At thls temperature, ALhpG= + I . 10 kJ.nivl-I. The positive cign means that the formation 01' one molc of water vapor at one ahn from one mole of liquid water at one atrn and 363.15 K is not a spontaneous proccss. Ifthe temperature is above the normal boiling point. however, say 383.15 K, then ArapG= - 1.08 kJ.mol-I. The negative sign means that the fwmation of one mole of w;iter vapor from one mole of liquid watcr at one atm and 383.15 K is a spontaneous process.

If bG < 0, the process will occur spontitneuuslp, and wnom, i s the work exclurive of P - V work that can be dune by the system it' the change is carried out reversibly. This i s the maxinlu~nwork that can be obtained tmm Ihe process. If any ~rreversibility occurs in the process, the quantity of work obtained will be less than the maxiniurtl. If A C > 0, the precess will not uccur spontaneously and u:"~,,,is the minimum work, exclusive of P-V work, that must he done on the system to make the process uccur. For example, it i s known experimentally chat A G for the formation of one mole of H,0(1) at 298.15 K and one bar from H,(g)and O,(g) at 298.15 K and ane bar is -237.1 kJ.rnol-I. Thus, a maximum of 237.1 kJrnol-' of useful work (that is, work exclusive of P - V work) can be obtained from the spontaneous reaction

E X A M P L E 8-1

Thc n~olarenthalpyof fusion o f ~ c c a273.15 t K anrlnne arm is A,~H= 6.01 kl .rnol". and then~olnrcntropyuf fusionunder lhe fanlcconditiuns ic Alu5S= 22.0 J - K - I ,rnol-l. Show that A,,,G= 0 at 273.15 K rind one atm, that A,$G < 0 when the tclnperature i s greatcr than 273. I5 K, ant1 thal A ~ >~ 0 when ~ E lhe temperature is lesu than 273.15 K. S O L U T I 0 N : Assulning that A , , . R and A,u,3 do not 273. I5 K,we can write

Crmverseiy, it would require at least 237.1 kJ.mol" of energy t o drive the (nnl3spr)ntaneous) rcaction

vary appreciiibiy around

I

E X A M P L E a2 The value 01- AE for the decornpnsition of one mole u i H,O(I) to H,(g) and O,(g) at mlc her and 298.15 K is +237.1 kJ.rnol-l. Calculate the minimum voltage required tu decnnlposc one mole of H,C)(I) to If,(g) and O,(g) at one hilr and 298.15 K by

If r = 273.15 K,(hen A,,~Z = 0, indicaringthat iceandliquid watcr are inequilibrium with cach other ilr 273.15 K and onc atm. If T -z 273.15 K, then A,*E > 0, indicating that ice will not sp)ntimeously melt under these conditions. If T > 273.15 K, then A,,,f H,n(l) can be determined from tllc chemical equalion of the reacliun

to get

1dS- SdT

6 1 0 , ~for ~

+ P d V -t V d P

dU to gel

+

The oxidation state of hydrogcn goes from 4-1 to 0 and that of oxygen gces from -2 to 0.Thus two electrons are Iransferred per YO(1) molecule, or two times the Avogadro constan1 of electrons per mule. The total c h a r ~ eof twu moles of electrons is

dG = - 6 d T t V d P - k S ~ o ~P~d V

+

Because ihe reversible pressure-volume (P- V) work is - Pd V , the quantity SIU,, Pd V is the reversible work other than F- V work (such as electrical work). Therefore, we can write dE; as

T h e minimum voltage, &, required to decompose onc Inole is giwn by -

AG E= 1.929 x Id c

whcre ~ U J ~ , represents ~ , , the total work exclusive of P-V work. For areversible process taking place at constant T and P, dG = B T U ~ ~ , ~ or ,, A(; = tunrm,,

(reversible, constant T and

P)

(8.16)

lo3 J-mo[-' - = 1.23volts 1.929 x 10' C

- 237. I x

-

where we have used rhc fact that one jrwle is o coulomb times a volt ( I l = ICV).

1

1

8-3. Maxwell Relations Provide Several Useful Thermodynamic Formulas

A number of the tbermodynan~icfunctions we have definedcannot be measured directly. Consequently, we need to be able to express these quantities in tcrnms nf others that can be experimentally determined. To do so, we start with the definitions o f A and G, Equations 8.4 and H. II . Differentiate Equation 8.4 to obtain

For a reversible process, dU = T d S

- Pd V , so

By comparing Equation 8. I7 with the formal total derivative of A = A ( V , T ) ,

we see that

( )

= -P

and

($),

F I G U R E 8.2 The molar entropy uf ethane at 4I)0 K plotted against density ( p = I /TI.The value of 4 l O K is 246.45 ~.mol-'.~-'.

s ' at~

Equation 8.20 allows us to determine the entmpy oC a substance as a function 01volume or density (recall that p = 1/ V ) from P-V-Tdata. I f we let V, in Equation 8.20 be very large, where the gas is sure to behave ideally, then Equation 8.20 bccorries

= -S

Now if we use the fact that the croas derivalives of A are equal (MathChapter D), Figure 8.2 plots the molar entropy of ethane at 400 K vcrsus density. (Problem 8-3 involves calculating the molar cnuopy as a function of density using the van der Wadls equation and the Redlich-Kwong equation.) We can alsu use Equatiun 8.10 10 deri~ean equatlc~nw e dc~ivedearlier 111 Scction 6-3 by another methnd. For un ideal gaq, (d P / a T ) , = t i R/ V , so

we find that

Equation 8.19, which is obtained by equating the second cross partial derivatives of A , is called a M u ~ c l relution. l There are many useful Maxweli relations involving various thermodynamic quantitics. Equation 8.19 is particularly uscful because i t allows us to deter~ninchow the entropy c ~ f asubstance changes with volume if we know its eqwition of stiite. Integrating Fquation 8.19 at constant T, w e have

''

):(;

dV

E X A M P L E 8-3 Calculatc AS Tor an isothern~alexpar~sionfrurn V , to V2 fur equiltion of state

(constant T )

Y

we have integrated ( a S / 3 V ) , ; in other words, T is held constant in Ihe derivative, so T must he We have applied the conditirln of constant T t c ~Equation 8.20 because

held constant when we inlegrate.

I

SOL U T I O I\;:.Wc use Equation 8.20 to obtain

I o gac

that

obey:.

t11u

Chapter 8 / I Ielmholu anrl Gihhq Energies

Note that we derived thisequation in Example 6 2 , but we had to be told that dU = 0 in an isothermal prtwess for a gas obeying the aFmve equatiotl o f ftatc. Wu did not need [hi:, l~~for~llatio~~ to derive our result hcrc.

We have previously stated that the energy of an ideal gas dcpcnrls only upon tempcraturc. This stntement is not gcncrally tl-uc for real gaxes. Suppose we want to know how thc cncrgy uf a g~ changes with volume at constant temwralure. Unfortunately, this quantity cannot bc measured directly. We can use Equatiun 8.19, however, to derive n practical cquation for (i3U/d V),; in other words, we can derive an equation thal tells us how thc cnergy OF a substance varics with its volume at ccunstmt temperature in tern~sof readily measurable quantities. Wc differentiate Equation 8.4 with respect to V at constant teinpcrature to obtain

,

Subqtituting Equalion 8.18a for {a A / a V ) and Equation 8.19 for ( a S / a V ) , gives

0

200

400

P f har F I G U R E 8.3 The molar energy U glutted against pressure for ethane 14.55 kJ. ~nnl-I.

I

at 403

K. The value of

EXAMPLE 8 4 In Example k 2 , we ftated we would prove later that the energy of n gas that obeys thc equalion t>C stale

is independent olthe vulume. Use Equatiun 8.22 to prove this

Equatic~n8.22 gives (ilU / 3 V ) , in terms of P-V-T data. Equations like Equation 8.22 that relate thermodynamic functions to functions of P, V, and T are sometimes called

SOI.U'I ION: Fur

P ( V - tr)

= R7-,

tl~ertnodyna~nic equations of state. We cat1 integrate Equation 8.22 with respect to V to determine U relative to the ideal gas value, and SO

where V'"S a large volumc, whcre the gas is sure to behave ideally. This equalion along with the P-V-T data gives us U as a function of pressure. Figure 8.3 shows U plnttcd againsl pressure for ethane at 400 K. Problem &4 involves calcultlting as a fu~lctionof volume fur the van der Waals equation and the Redlich-Kwong equation. We can also use F4uation 8.22 to show that the energy of an ideal gas is independent of thc volu~ncat constant iemperature. For an ideal gas, (3 P / a T ) , = n R / V , so

which proves that the energy uf an ideal gas depends only upon teinperature.

1

vdi s equ:~lro

E X A M P L E 8-5

Evaluate

[a~/av), for one molc of n Rcdlic11-Kwong gas

SOLUT tON: Recall that the Redlich-Kwong equatio~~ (Equation 2.7) ic

I

Chdpter 8 1 Hel~nhnltzand Cibbs Energie

&I The . Enthalpy of an Ideal Car Is Indupnden~of Pressurc

and

We derived the cquatiun The experimental value of C, is 24.43 J.K-] .rnul-'. Note that C, compared with

in Problem 3-27. Using Equation 8.22 for ( 3 U / 3 V ,,)

we obtain

Foranidealgas(aP/aT), = n X / V a n d ( a V / a T ) , = n R / P , a n d s o C , -C, = n R , in agreement with Equation 3.43. An alternative equation frx U p - C y that is more convenient than Equation 8.23 [or solids and liquids is (Problem 8-1 1 )

Each of ~e paniai derivatives hcre can be expressed in terms of familiar tabulated physical quantities. The isothermal compressibility of a substance is defined as

and the coefficient of thermal expansion is defined as

- c, is sll>all

z,(vr C,,)and i s alsu much sn~allerfor solids than for

gases,

a% you

might expcct.

I

I

8-4. The Enthalpy of an ldeal Gas Is Independent of Pressure Equation 8. i8r4 can be used directly tu give the voiurne dependence (IC the Hclmholtt energy. By integrating at constant temperature, we have

For the case of an ideal gas, we have

Notice that this result i s -T times l?qualiun 8.21 for AS. This result must be so hecausc AU = IIfor an ideal gas at constant T, so AA = - T AS. If we differentiate Fxpation 8.1 1, F = U - . T S F V , and subsritutc c l I J = 7-dS - P d V , we get

+

Using these definitions, Equation 8.24 becomes dG = -SdT

+VdP

(8.30)

By comparing Equation 8.30 with

EXAMPLE 8-6 The cwfficient of thermal expansion. a,of copper at 298 K is 5.00 x K-I , and its isothermal compressibility, K , is 7.85 x I()-' atm-I. Given that the density of copper is 8.92 g . ~ r n -at~298 K, calculate the value of C, - C, h r copper.

w e see that

I

SO I. U T I 0 N : Fur cupper, the molar volume, F, is given by

Note that Equation 8 . 3 1 ~says that G rlecreascs with increasing tempcralure (kS 5 0) and that Equation 8.31h says that G increases with increasing pressure (because V > 0).

cause

Chapter 8 1 Hclrnhultz and Gibbs Enelgies

1f we now take cross derivatives of G as we did for A in the previous section, we find that

31 5

W. The Enrhalpy of an Ideal Gas 15 Independen1of Pressure

EXAMPLE 8-7 Use the virial cxpansion in the pressure

to derive n viriaI cxpansion for A S fur a reversible isothermal change i n p s < u r c .

This Maxwell relation gives us an equation we can use to cdculate the pressure dcpcndence uf S. We it~tegmteEquation 8.32 with T constant to get

A s = - ( )

dl'

(constant T )

SOL U T I 0 N: Solve the abote equation for V :

and write

f

Eq~~atinn 8.37 can he w e d to nhtain the molar entropy as a functio~lof pressure by integrating (il V / i f T ) , data frnm some low pressure, where the gas i s sure to behave ideally, tu some arbitrary pressure. Figure 8.4 shows the molar entropy of ethane at 400 K obtained in this way plotted against pressure.

Substirute this result into Equation 8.83 : ~ n dintegrule h m P, to ?'f

tu obtain

We can alho use Equations 8.31 tu shuw that the enthalpy uf an ideal pas is independent of the pressure, just as its energy is independent of the volume. First. we differentiate Equation 8.13 with respect to P at constant T to obtain

Now use Equation 8.31b for (aG/a P ) , and Equation 8.32 for ( a S / B P ) , to obtain

0

200

400

600

800

P l bar F I G U R E 8.4

The molar ccntropy of ethane at 400 K plotted against pressure. The value of 246.45 ~.rnol.-' .K1.

S' at 400 K is

Note that Equation 8.34 is the analog of Fquation 8.22. Equation 8.34 is also called n therrnodyna~nicequation of state. It allows us to calculate the pressure dependence of H from P - V - T data (Such data for ethane at 400 K are shown in Figure 8.5). Fnr an ideal gas, ( a V / B T ) , = n R / P ,so ( i ) H / a P ) , = 0.

E X A M P L E 8-8 Evaluate ( a H / aP), for a gas whose equation of state is

For an ideal gas, (3 V / a T ), = n R / P , so muation 8.33 gives us

S O L U T I O N : Wehave

This results is not really a new one for us because if we let KT/ V , , we obtain Equation 8.21.

tr

P, = n RI'/V,

and P, =

Chapter 8 / Helmholtz and G~bbsEncrg16

so Equation 8.34 gives us

Notc that

%5. The Var~ousThermodynamic Fun~tionsHavc Ndlural Independt.n~Varidt>l@

takes un a simple form, ill the sense that the coefficients ol' dS and d V are simple thermodynamic functions. Consequenlly, we say that the nalural variables of U are S and V ,and we have

This concept of natural variables i s particularly clear if we consider V atid T instc;irl oF S and V to be the independent variables of U , in which case we g t (cf. Equation 8.22)

( ( a H / a ~= ) , 0 when B ( T ) = 0.

I

1 Certainly U can be considered to be a function of V and T, but its total derivative is not as simple as if it were considered to be a function of 5' and V (cf. Equation 8.36). Equation 8.35 also givcs us that a criterion for a spontaneous process is that dU < O for a system at constant S and V. We can write Equation 8.35 in ternla c ~ dS l rather than rill to get

which suggests that the natural variables of S are U and V . Furthermore, the critcriun for a spontaneous process is that dS 2 0 at constant U and V (Equation 8.2 for an isolated system). Equation 8.39 givcs us P I bar F I G U R E 8.5

The molar enthalpy of ethane a1 400 K plotted againsl pressure. Thc value of 17.867 kl.rnnl-'.

3'' at 400 K is The total derivative nf the enthalpy is given by (Equation 7.6)

8-5. The Various Thermodynamic Functions Have Natural Independent Variables We may seem to he deriving n Lot of equations in this chapter, but they can be organized neatly by recognizing that the energy, enthalpy, entropy, Helmholtz energy, and Gibbs cnergy depend upon t~aturalsets of variables. For exampie, Equation 7.1 summari~es the First and Second Laws of Thcrmodynan~icsby

which suggests that the natural variables of H are Sand P. The criterion of spc>nraneily involving H is that d H < 0 at constant S and P. The total derivative of the Helmholtz energy is d A = -SdT

-

PdV

,

from which we obtain

dU

= TdS

-

PdV

(8.35)

Notc that when S and V are considered to he the independent variables o f U . then the total derivative of U ,

(;),=-S

and

(g),

=

-.

18.42)

Equation 8.42, plus the spontaneity criterion that d A < 0 at constant T and V, suggebt that T and V are the natural variables of A. The hlaxwell relations obtained fmm Equation 8.43 are useful because the variables held constant art. more cxperime~~tally

Chapter 8 / Helmhul~z~ n Gihbs d Encrgies

con~rollabiethan are S a n d V ,as in Equations 8.37, or U and V, as in Equations 8.40. The Maxwell relation h m buations 8.43 is

which allows us to c~lciilaletbc vrlli~rnedependence of S in tenns of P-V-T daia (see Figure 8.2). Last, let's consider the Gibbs energy, whose toval derivative is dG = -SdT t V d P

A-6. Thc Standard Statc fur a GASat Any Tcrnperalure Is the Hyputhetical Ideal Gas a t Orle H J ~

If we add d ( P V ) and subtract d ( T S ) from Equation 8.48, or subtract d ( T S ) from Quation 8.49, or add d ( P V) tn Equation 8.50, we pet

The other equations of this section r011ow by comparing the total derivative of each function in trl ms of its natural variables to the above equations for d U , dFJ. d A , and d G . Tahlc 8.1 su1nmari7xx wrne of thc principal equations w e have derived in this and

previrlus ciiaptcrs.

(8.45)

Equation 8.45, plus the spontaneity criterion dG < 0 for a system a1 constant T and P. tell us that the natural variables of G are T and P. Equation 8.45 gives us

T A B L E 8.1 Thc four principal thcrmdynamic encrgies, their ditfcrcntinl expwsrions, and the corresponding Maxwell relations.

Thermodynamic enerev

Differential

Corrcsponding

expression

Maxwell relations

'l'lic Maxwell ~.eli~tion we obtain fruin Equations 8.46 is

d A = -SdT

which we can use to calculate the pressure dependence of S in terms of P-V-T data (F~gure8.4). This section i s mean1 lo provide both a sutnlnary of many of the equations we have derived so far and a way to bring some order to them. You do not need to memorize these equations because they can all be obuincd from Equation 8.35: d U = 7-(IS -

PJV

(8.48)

which is nothing m o r r than the First and Second Laws of Thermodynamics expressed as one equation. If we add d ( P V ) to both sides of this equation, we obtain

If wc subtract J ( T S ) from both sides of Equation 8.48, we have I

d ( U - TS)= TdS - PdV

- T d S - SdT

- PdV

(g)T(Elv =

8-6. The Standard State for a Gas at Any Temperature Is the Hypothetical Ideal Gas at One Bar One of the most important applications of Equation 8.33 involves the correction for nonideality that we make to obtain the standud molar entropies of gases. The standard molar entropies of gases tabulated in the literature are expressed in tcrms nf a hypothetical ideal gas at one bar and at thc same tetnperature. This correction i s usually s m d l and is obtained in the following two-step procedure (Figure 8.6). We first take our real gas from its pressure, nf one bar to some very low pressure P ' ~where , it is sure to behave ideally. We use Equation 8.33 to do this, giving

-

s[pid)- -$(l bar) = -

n-7. The GibkHelmholtz Equation 13escrlbt.s the Temperature Dependence of thc Lihbs Encrgy

Real gas = ideal gas

we have

r e r y low pressure ( P ' ~ J

f

\

E q n . 2 2 . 5 2 using

Eqn. 2 2 . 5 3 u s i n g ideal gas equation of s t a t e

cqunlir)n of state

Substituting this result into Equation 8.54gives

S.(at 1 bar) = ?(a I bar) +

,

Real gas bar

Hypothetical

_7 ideal gas Correct~un l o r non-ideality

I bar

F I G U R E 8.6 Tile scllcrne to bring the expen'menralentropies of gases to the sttandard s i a k I-a(hypathetical) ideal gas at the same temperature.

f!dT %

x (1 bar)

+

,,

.

(8.56)

where we have neglected P ' with ~ respec1 to one bar. The second term on the right sidc of Equation 8.56 represents the correctiun that we add to 5 to get S ' . We can use Equation 8.56 to calculate the nonideality comctio~ltu the entrupy of N,(g) at 298.15 K that we used in Table 7.1. The experimental value of ri B , , / d T for N,(g) at 298.15 K and one bar is 0.192 cm3.mot-' -K'I. Therefore, the correction for nonideality is given by corrcctinn for nonjdeality = (0.192 cm3.mol-'. K-I)( I bar)

The superscript "id" on P emphasizes that this value is for conditions for which the ga\ behaves ideally. The quantity ( a F / i ) ~can ) , be determined from the equation of state uf the actual gas. We now calculate the change in entropy as we increase the pressure back to one har, hul rrrking rhr g a . ~to hc idr,rzl. We me Equation 8.52 for thia prtxess, hul will) ( i l V / i ) ~ )= , K I P , giving

= 0.192cm3.bar.mol~' .K-'

which is what was used in Table 7.1 :The correc~ionin this case i s rather mall, but that is not always so. If second virial coefiicent data are nut available, then an approximate equation of state van be used (Problems 8-20 through 8-22). The superscript o of S'(1 bar) emphasizes that this is the standard molar entropy of the gas. We add Equations 8.52 and 8.53 to get

S m ( a 1t hu)- S(at I bar) =

(8.54)

In Erluation 8.54, 3 is the molar entropy we calculate from heat-capacity d a ~ aand heats of transitions (Section 7-3). and S" is the molar entropy of the corresponding hypothetical ideal gas at one bar. Equation 8.54 tells us thal we can calculatc the necessary c o m c ~ i u nto obtain the standard entropy if we kntw the equation of state. Because the pressures involved are around one bar. we can use the virial expansicln using only he second virial cwfficient. 2.22, IJsing Eqi~;~tic~n I

8-7. The Gihbs-Helmholtz Equation Describes the Tcrnperature Dependence of the Cibbs Energy Both of Equations 8.31 are useful because they tell us how [he Gibhs energy varies with pressure and with temperature. Let's look at Equation 8.31b first. We can use Equation H.3 1 b to calculate the pressure dependence of the Gihb~energy:

aG =

L:

VdP

For one mole of an ideal gas, we have

(constant 1.)

(8.57)

Chapter 8 1 Helmholtz ~ n Cd i b b ~Energies

We could have obtained the xmrle result hy using

G ( T , P) = G'(T)

+ HI' In( P / 1 bar)

The Git)t>s-tlelmholtz Equatiun Describes the Temper;lrure Depcnder~ceOIthe G~bbsEncrgv

Now differentiate partially with respect to T keeping P fixed:

For an ixolhennal change in an idcal gas, AH= 0 and AS is given by Equation 8.2 1. It is custo~naryto ICI PI = 1 bar (exactly) in Fquation 8.58 and to write it in the fonn -

8-7

(8.59)

wlicrc F (T) is called the standard molar Ciibbs energy. The standard molar Gibbs cllergq in this case is the Gibbs energy of one mole of the idea! gas at a pressure uf nrle bar. Note that G ' ( 1 ) depends upon only the temperature. Equation 8.59 gives the Gibbs energy c~fa n ideal gax relative to the standard Gibbs energy. Accr~dingto tiquation 8.59, E(T, P) - G o ( T )tncreaws logarithmically with P, which we have see11 is cntirely an entropic effect for an ideal gas (because H is independent of P for an ideal gas). We will see In Chapter 12 that Equation 8.59 plays a central role in

These last two terms cancel because o f the rdation (aS/d l'), = U , ( T ) / T (Equation 7.7),so we have

Equation 8.60 is called the Gibbs-Helmholiz eqruttion. This equation can be directly applied to any process, in which case it hecomes

cl~en~icnl equilrbr~ainbolving gas-phase reactions.

E X A M P L E 8-9 Solids and l~quidsare fdirly incompressible, so V in Equation 8.57 may be taken to be constan1 lo a good apprvritllntion in this case. Derive an expression fur G(T,P) analugous to Equation 8.59 for a solid or a liquid. S O L LIT IO N : We inlegrate

Equalion 8.57 at constant T to get

We let P, = I bar and E(I',= I bar. T ) = G ' ( T ) to get

This equalion is simply another form of the Gibbs-Helmholrz equaLion. We will use Equatioris 8.60 and 8.61 a number of times in the following chapters. Fur example. Equation 8.61 is used in Chapter 12 t o derive an equation for the temperature dependcnce o f an equilibrium constant. We can dctcrrnine the Gibbs energy as a function uf temperature directly from equations wc dcrived in Chapters 5 and 7. In Chapter 5, we learned how to calculate the e~tthalpyof a substance as a function of temperature in terms of its heat capacity and its varlous heats of transition. For example, if there exists only one solid phase, so that there are no solid-solid phase transitions between T = 0 K and its melting point, then (Equation 5.46)

where P must be expressed ill b m . In this ca@,e,Z ( T , P ) increases lineilrly wlth P,but k c a u l e the volumc of a condcnscd phaseis much <mailerthan that of a giu, the slope of C ( T .P) versuq P ,or ( ~ G /PB ) ] = V , is very small Consequently, at ordinary pressures G ( T ,P ) if almost Independent of pressure and is apprr)xirnatcly equal to I C ( T ) for a condensed phaw.

Equation 8.3 la determines the temperature dependence uf the Gibbs energy. We can derive a useful equation for the temperature dependence of G by starting with Fqualion 8.31r1 (Problcm K24), but an easier way is to start with G = H - TS and divide by Z'to obtain

for a temperature above the boiling point, where Figure 5.7 shows HIT) - H(0) versus T for benzene. We calculate H(T)relative to H ( 0 ) because it is not possible to caIculate an absolute enthalpy; H ( 0 ) i s essentially our zero of energy.

Chapter 8 1 Helmhultz and LiWs Energies

I n Chapter 7, we learned to calculate absolute entropies according tu Equation 7.17).

8-8. Fugacity Is a Measure of the Nonideal~tyof a tias

325

also shows that there is a discontinuity in the slope at each phaw transition. (Ben/.ene melts at 278.7 K and boils at 353.2 K ai me atm.) We can understand why there is a discontinuity ill the slopc of G ( T ) versus T at each phase transition by looki~lgat Equation 8 . 3 1 ~

Because entropy i s an intrinsically pnsitive quantity, the slope of C (T)versus T i s negative. Furthermore, because S(gas) > S(1iquid) > S(solid),the slopes within each single phase regionincrease in going from solid to liquid togas, s o the slope,{i)G/ilT) is discontinuous in passing from one phaae to another. Thc values of H " ( T ) - H2(O), S 3 ( T ) , and G'(T 1 - H " ( 0 ) arc tabulated for a variety of substances. We will use these values to calculate equilibrium constanls it1 Chapter 12.

,,

Figurc 7.2 shuws T ( T ) versus T for benxne. We van use Equations 8.62 and 8.63 to because calculate E ( T )-

n(0)

Figure 8.7 shows E ( T )- H(O) versus T for hcnzene. There are several features of Figure 8.7 to appreciate. First note that G ( T )- H ( 0 ) decreases with increasing T. Furthermore, E ( T )- H(0) i s a continuous function of temperalure, even at a phase transition. To see that this is so, consider the equation (Equation 7.16)

&8. Fugacity Is a Measure of the Nonideality of a Gas

I n the previous section, we showed that the molar Cibbs encrgy ofan ideal gab is givctl by Because A,,G = AmH - TbAvSS,wc see that L I , ~ $=G0, indicating that the two phases are in equilibrium with each other. Two phases in equilibrium with each other have the same value of G, so G ( 1 ) is continuous at a phase trtnsitiun. Figure 8.7

-

G ( T , P) = G 3 ( T )4- R T l n

Y , P

(8.W

The pressurc P" is equal to one bar and C"(T) is called the standard molar Gihhs energy. Recall that this equation is derived by starting with

and then integrating, using the ideal gas expression, R T I P , fur 7. Let's now generiilizc Equation 8.64 to the case of a real gas. We could start with the virial expansion,

TI K I

F I G U R E 8.7

A plot uf C ( T )- H(0) versus T for benzene. Note that G ( T )- H{o) is cuntinuous but its derivative (the s l o of ~ the curve) is discontinuous at the phase transitions.

and substitute this intu Equation 8.65 to obtain a virial expansion for the molar Ejihhs energy,

Chnlser A I Hclrnl~olr~ 2nd Gibbs Enrrgies

where we are intepnting from some low pressure, say Pid,where the gas is sure to behave ideally, to somc arbitrary pressure P. The result of the integration is

Now according to Equation 8.64, G(T,PId) = GV(T)+ R T 1n p L d / P 3where , GJ(T) is the tnolar Gibbs energy of an ideal gas at a pressure of P" = I bar. Therefure, Equation 8.66 can be writtcn as

Equa~irln8.67 is the gencralizatiot~o f Equalion 8.64 to any real gas. Although r gas, depending upon the vali~esof B,,(T), Equation 8.h7 is exact, it diirers f r ~ each B,,!T), and so011.It turns out to he much morc convenient, particularly for calculations involving chemical equilibria, as we will see in Chapter 12, t c ~maintain the form of Equation 8.64 by defining a new thermodynamic function, f (P, T). calledfugnciv, by the eqi~ation

8-8. rugnrily Is a Mcasure of the Nn~jide~litv uf a L a 5

suggested by Equation 8.69, because vtherwise f ( P , T ) would not reduce to P ahen B z p ( T )= B,,{T) = 0. This choice of standard state not only allows all gasef to be brought to a single common state, but also leads to a procedure to calculate f (P, T ) at any pressure and temwraturc. To do so, consider the schcme i n Figure 8.8, which depicts the dilferencc in molar Gibbs etiergy between a real gas at P and T and an ideal gas a1 P and T. We can calculate this difference by starting with he real gas at P and T and thcn calculating the change in Gibbs energy when he pressure is reduced to essentially zero (step 2), where the gas is certain to behave ideally. Then w e calculate the change in Gibbs etiergy as we compress the gas back to pressure P, but taking the gtas to hehave ideally (step 3). The sum of steps 2 and 3, then, will bc the difference in Gibbs cncrgy of an ideal gas at P and T and the real gas at P and T (step 1). In an equation, we have

Substituting Equations 8.64 and 8.68 into Equation 8.711, we hwe

But the standard state of the real gas has been chosen yuch that The nonideality is huried in f (P, T). Because all gases behave ideally as P -+ 0, f~ig;~t-ity must have the property that

= P" = I bar, so

We now use Equation 8.65 to calculate the change in the Gihhs energy along steps 2 md 3:

sn that Equatinn 8.68 reduces to Equation 8.64. Equations X.67 atid 8.68 are equivalent if

11 might sccm at this point that we are just going in circles, but by incorporating the nonirleality of a gas thruugh i t s fugacity, we can preserve the thermudynamic equations we have derived for ideal gases and write those corresponding to a real gas by simply rcplauing PIP' by f/SC. All we need at this stage is a suaightfurward way to determine the fugacity of a gas at any pressure and temperature. Before looking into this, however, we must discuss the choice of the standard state in Equation 8.68. Being a type of energy, the Gibbs energy must always be taken relative to some chosen

s t ~ i ~ d a rstate. d Nc~lethat the standard p o l a r Gibbs energy G - ( T )is taken to be the same quantity in Equations 8.64 and 8.hS. The standard state in Equation 8.64 is the ideal gas at one bar. so this must be the standard state in Equation 8.68 as well. Thus, the standard state of the real gas in Equation 8.68 is taken to be the corresponding ideal gas at une bar: ill other words, the standard state of the real gas is one bar after it has been adjusted t c ~idcal lxhavior. In an equation, we have that f " = Po. Note that this choice is also

Real gas

A El

Ideal gas (T.P)

(T+P)

Real gas (T.P+O)

-

Ideal gas (T,P-+O)

F I G U R E 11.8

An ilIustraiion of the scheme used to relate the fugacitj of a gas to its standard state, which is a (hypothetical) ideal gas at P = I bar and the temperature T of interest.

Chapter 8 1 tielmholtz and Libbs Energies

328 The sutn of

AZ,and AC;, gives anolher expression for AE,

Equaling this expression for AZ, to

AG,in Equation 8.71, we have

0

200

400

600 P /bar

SO0

1000

F I G U R E 8.9 A plot of ( 2 - I ) / P versus P for CO(g) at 2M) K. The area under this curve from P = 0 to P gives In y at the pressure P.

Given either P-V-T data or the equation of state of the real gas, Equation 8.72 allows 11s t ~ ci~l~uliltt: l the ratio of the fugacity t c ~the pressure of a gas at ilny pressure and temperature. Note that if the gas behaves idcally under the conditions of interest (in od~crwurds, if 7 = qd in Equation 8.72). then In f./ P = 0, or 1 =; P. Therefore, the extent of the deviation of j/P Cmm unity is a direct indication of the extent uf the deviation of the gas from ideal behavior. The ratio f / P is called the ,fugaciv

uutficirtlt, y ,

For an ideal gas, y = 1 . By introducing the compressibility factor, Z = P I / / H T , Equation 8.72 can be written as

F I G U R E 8.10

A plot of y = j / P against P for CO(g) at 200 K. Thew values ot f / P were obtained frurn a numerical integration of (Z - 1)/P shown in Figure 8.9.

Even though thc luwer limit here is P = 0, the integrand i s finite (Problem 8-27). Furthcrmoir, (Z - I ) / P = 0 for an ideal gas (Problem 8-27), and hence In y = 0 and f = P. Figt~re8.9 shows ( Z - 1 )/Pplotted ag;unsl P for CO(g) at 200 K. According lo Equation 8.74, the area under this curve from 0 to P ih equal to In y at the pressure P. Figure 8.10 shows the rzsuldnp values oF y = f / P plottcd againat the pressure for UO(g) at 200 K. We can also calculate th; fi~gacilyif w e know the cquatiun ul' stale of the gas.

1

E X A M P L E 8-10

Derive an expression fur the fugacity of a gas that obcys tbe equation o l

where b is a wnstant.

ht;lte

Chapter A I Helmhultz and Gihbs Energier

SO 1U TI O N : We solve for 7 and substitute into Equation 8.72 to get

Problems 8-33 through 8-38 derive expressions for In y Tor the van der Waalf equation and the Rcdlich-Kwong equation.

We can write Equation 8.74 in a form that shows that the f i ~ g a c i t ycoefficient is a function ( I € the reduced pressure and the reduced temperature. If we change the integration variable to P, = PIP,, where PL is the ciitical pressure of the gas, then Equation 8.74 takes the form

Now recall from Chapter 2 that, to a good approximation for must gases, the compressibility factvr Z i s a universal function of P, and T, (see Figure 2.10). 'rilerefore, the righ~side of Equation 8.75, and s o In y itself, is also a universal function of P, and T,. Figure 8.11 shows the experimental values of y for many gases as a family of curves ol'cunstnnt T, plotted against P,.

E X A M P L E &I1 Use F i ~ u r e8.1 1 and Table 2.5 to estimate the fugacity of nitrogen a1 ti23

K and FIGURE

S O L U T I O N : We find from Table 2.5 that Tc = 126.2 K and q = 33.6 atm for N,(g). Therefore 7, = 4.94 at 623 K and P, = 29.8 at 1 0 0 am. Reading from the curve!. ir! Figurc 8.1 1. we find that y w 1.7. At 1030 ahn and 623 K, the fugncity of nitrogen is 1700 atm.

8.11

The fugacity coefticientsof gases plotted against the Mduoed pressure. P/PL,Tor variuus values of the reduced remprature, T / Tc. I

and 82.4 J.K-'.tnol-', r e s p e ~ ~ v e l yCompare . your results with those you ohtained in Problem 8-1. Are any of your physciaI interpretatio~~s differem'!

Problems 8-1. The molar enihalpy of vapurization of benfxne at its normal builing point (80.09"C) iu 30.72 kl.rnul '. ,4ssu&ing that AmpH and A - ~ Sstay constant at their values at 80.09C, calculate the value of AvapG a1 7S.O'C. 80.09"C,and 85.UC. Interpret these I-esultcphysically. -

8-2. R d o Problem 8-1 without assuming that Avap H and dvwSdo tlot vary with temperature. Take the molar heal capacities of liquid and gaseous benwne to be 136.3 J-K-' -mul-'

8-3. Substitute ( a P / a T ) , from the van der WaaIs equation into Equation 8.19 and integrate from TIdto V to obtain

---

.Y(T> v)- F ( T=) R 1" V - b

-

-

V

-h

Chapter 8 I Hclrnholtr and C i l h Energies -ad

NOW

let V' = R T / P l d rpld= P = one bar, and V

>> b tu ubtain

Given that u = 327.7 pm, s l k , = 95.2 K, and h = 1.58 for N,(g), calculate the value of AUfur a pressure increase from 1.00 bar to 10.0bar at 300 K.

8-8. Deternine C, Gwen that

5''

= 246.35 J.rnol-'

.K ' for ethane at 400 K, show rhat

-

P(V- h ) = R T .

'

8-10. Use Equatiun 8.22 tu \how that

Show that (aC,/a

fur the Redlich-Kwong equation for ethane at 400 K.Calculatc 3 as a fi~nctivnof p = 1 / F and compare yc)ur results with the experin~cntalresults shown in Figure 8.2.

V), = 0 for an ideal gas and a van der Waals gas, and that

for a Redlich-Kwong gas.

8-1I. In this prohlem you will derive the equation (Equation 8.24)

8-4. Use the van der Waals equation to derive -

for a gas that obeys the equation uf s t i l k

8-9. The cwficient of thermal expansion af water at 25°C is 2.572 x K I, ,and its isulherma1 compressibility is 4.525 x 10 bar-'. Calculate the value of C', - C, fur one n ~ v l e or water at 25'C. The density of water at 25'C i s 0.997I15 g . n l ~ - ] .

Calculate 5 as a functiun of p = l/V fur ethane at 400 K and compare your results with ihe experimental results shown in Figure 8.2. Show that

-

C,

-id

U ( T . 1')- U ( T ) = -2

V

Use this result along with the van dcr Waals equaticln to calculate the value rlf U as a function of 7 for ethane at 4M)K, given that = 14.55 kJ.~nol"'.To rlo this, s p ~ c i f y7 (from 0.0700 ~ - m o l - to ] 7.00 L.rno1-I, see Figure 8.2), calculate both U(V)and and p l o ~U(V)versus ~ ( 7Compare ). your result with the exprri~ncntald i ~ t in ; ~Figure 8.3. Use the Redlich-Kwvng cqun~ionto derive

vid

~(v),

Repcat the above calculation for ethane a1 4(M)K.

8-5. Show that ( B U / d V),. = 0 for a gas that obeys an quation of state of the form Pf ( V )=

To start, consider V to be a function of T and f and wrile out d V . Now divide thrclugh by dT at constant volume (dY = 0) and then substitute the expression for ( a P / d l ' ) , that you obtain into Equation 8.23 trl get the above expression. 8-12. The quantlty ( W / d V ) , has units o f pressure atid is called the rr~ten~ul pre.rsure, wtirttl i s a measure of the intermolecular forces within thc M y of a substance. It i s q u a i to zcro For an ideal gas, is nonzero but relatively small for dense gases. and i s relalively large tor liquids, particularly those whose molecular interactions are strong. Use the frhlowing tlutn tocalculate the internal pressure ofethane as a functiun uf preswt: at ?NO K. Compare yhlur values with I t ~ evalucs you obtdn frum the van der Waals equation and the Redl~ch-Kwor~g equation.

RT. Give two examples of such equatiuns (11state that appear in the text.

8-6. Shnw that

8-7. IJse the result of the previvus problem tu show Lhnt 8-13. Show that

-

Use Equation 2.41 for the square-well potential tu show that

Chapter 8 1 Helmholtz and tiibbs Energies

USCEquation 2.41 for the square-well potential to obtain

8-19. What a l t the natural variables of the entropy?

8-20. Experimentally determined cnwopies me commwly adjufted for nonideality by using iln equation of state called the (niudiiicd) Berthclot equation: Given t h a ~o = 327.7 pm, e / k , = 95.2 K, and h = 1.58 for N,(.g), calculate the value of ( a H / aP), at 300 K.Evaluate AH = H(P= 10.0 bar) - X ( P= I .O bar). Compare your rcsult with X.724 J.rnu1-'. the value nf Z ( T )- H(o) for nitrogen a1 30() K.

8-14. Show that the enthnlpy is a function of only the temperature for a gas that obeys the cqu;ttiun orstotc P(V -- h T ) = R T , whcre b is aconstant. 8-15. Use your results Tor the van der Waals equation and the Redlich-Kwong equation in R o b l e ~ n8 4 to calculate H(T,V ) as a function of volumc for ethane at 400 K. In each casc, use the equation Ti = + PV. Compare your results with the experimental data shvwn in Figure 8.5.

Show that t h i ~cquatior~leads to the corrcftinn -

S ( a t one bar) = S(at one bar)

27 RT"

+ -+(I

32 %T

bar)

This result n c c d ~only the critical data for the substance. Use this equalion dung with the critical data in Table 2.5 to calculale the nonideality correction for N,(g) at 29H.15 K . Cntr~pareyour result with the value used in Table 7.1.

8-21. Use thc resuIt of Problem 8-20 along with the critical data in Table 2.5 to determine the

0-16. Use Equatio118.34 tu l,~r~r,r III l w n

: L.iquid ;

-:,-

Liquid

Phares rn Equilil)r~urb~ Arr

t

36 1

11l1al

The partla1 derivatives In Equatiutl 9.2 are central quantities in the trcatmenl oi cquil~hria.They art: called ckem~rnlputmt~ul.\and are denotcd hy 1 1 ~dnd IL':

-

G

Sol~d

Sulid

G

In tcrms of chen~icalpotentials, the^^, Equation '1.2 reads i

I

-

) d g

(corlstant

r and F')

(9-4)

If the two phases are in equilibrium with cach other, the11dU = 0. and because dtry# 0. we find that p L p ' . Thus, we find that if two phases of a singlc suhsvance are i11

----Solid

Sulid

-

FIGURE-9.12

h plot of G o ' ) aguins~P a1 a nunlhcr of ternperstures for a "normal" substance llkc benzene. III ( n l the ttmpcmturc is icss than the triple-point temperature; in (b) thc tenlperature is equal to the tr~l>lr-poii~t 1enlperatui-c;in (CJ the ternFrature is a little l e u than thc critical tclnpzruture; and ill (d) the

=(

te~nperalureis grcotcr than the cr~llcaltc!~~pc~-;$tu~-t..

equilihrii~rnwith each othcr, then the chemical pc>tcntialso T t h a ~substance in the lwo phases are equal. If the two phaaes are not in cquil~hriurrlwith cach utlirr. sporiktneous tran4c.1of matter from one phwe the other will uccur in the direction such that riC; < O. If pg > F 1 ,the term in parentheses in Equation 9.4 is positive, so rlnF must he negative in order that dC i0. In c~therwords, matter will transf r rrorn the vapor phase lo (he liquid phase, or from the phase with highcr chemical polential to thc phncc wilh lower chemical potential. If, on the other hand, p+ p l , ,hen drrQvill bc positive. nlz:~r~i~lg that matter will transfer from the liquid phase to the vapor phase. Once agarn, (he transfer occurs from thc phase with higher chemical potcntial to the phase with lower chemical polzr~tial.Notice that chemical putetitinl is analogt,us lo eleclric potcntial Jusl as electric current flows from a higher electric potcntial 11)a 1owe1 elcctric prltcnt~~l. matter "tlowh" from a highcr chemical potential to a lower chenlical potcntial (hcc Problerri 9-1 9). Although we have defined chcmical po~entiiilquite gel~crallyin Eqi1atio119.3, i t rakcs on a simple, F~niiliarform for a pure substance. Bccause G , I ~ k eU , H . and S, is an cxtenqive therniody~iamicfunctic~n,C; is proponional to the si1.e of a system, or I; cc n.We can express this proportionality by !he equa~ionG = tip( T . I ) ) . Note that this equatiorl is consistent with the def nition of p ( T , P) hecause

d t ~lnoles are transfemed from the liquid ptlnsc: to the vapor phase, whilc T aiid P are kept constant. The inliriitesimnl changc in Ciibbs energy for lhis process is

Pi-

Rut On1= -dn' fur t l ~ ct~ansfcrol'drr niolea tlum ~ h liquid z phase tu thc rapor phase, so Equatic~n'I.1 becomes

Therefore, for a single, pure substance, f i is the same quarltity as thc molar C;il,hs energy and pcf, P } is an intensive quantity like temperature aitd pressure. We can use the fact that the chemical potentials ot a single substancc in two p h a w in cqullihriunl are equal to dcrive an expression for the variation clf eqililibriurn preasurc with temperature fur ;my two phascs of a giver! pure substancc. I.et Lhr two phases be a and f l , s o th>~t p" (1'. P) = "I

(T,P)

(equilibrium hetweeri ptlases)

(9.61

0 I The

N o w take

I ~ I ?total

Chcm~cdlI'uterili;il? r)i a Pule S L I ~ S ~ I 111 I I LIWOV Phases 111 Equif~hr~urn Arr

Fqi~al

Wc can take rhe reciprocal n f this result to obtain

derivatiXmeof both sides nf Ecjuiltion 9.6

d7' dP

= 0.0292 K atm

'

Thus, wc see thal l t ~ ert~eltingpoint o l ' k n ~ e r l iticrcascs e hy 0.0292 Kper atmosphere of prevsure i~r~)urlcl otie atm. Tf A,",Z and were independetlt of prcssurc. we c t ~ ~ l d me this reqult lo predict that the rnelting point of hcnzene at 1000 atln is 29.2 K higher [hall it is at one atrn, r ~ 307.9 r K. The experimental value ir 306.4 K, so our asvumprion of constant h,,,pand A,>V is fairly satiqfilctory. Neve:ertbelesr. Equation 9. I 0 i \ strictly valid only ovcr a lirnilell mtlge hccausc A,",H and Atu,i7do vary with I' and P . Figure 9.2 rhows the cxperimerital melting point uf benzene vcnus prescure lip to IOO M atm. You can see from thc figure Ihat the slopc i q not quile constanl.

atu> 7'. so that wc can neglevl V ' uonlpared with in thc ~ l ~ n o m i n a t o r Erl~lation9. I I . Punhcrmtirc. if thc vapor pl-e.csureis nut too high (oncc agaln, if w e are no1 loo close to the a-itic;~l point), wecan asaulnc the vapor i s ideal and replace 'li"hy R T / P. s o thi~tEclu;~tit>rl 'j. I I

vY

hccon~es 1 dP

P

'l'he molar volume u r the vapor

~h

Ttw corresponding value Iro~nthe ideal gas equntlon is

uhich is slightly larger t h n ~the~ aclual value. I

In I' - ri--

dl'

-

AbdI>H -

i?r2

This equation, which was Rrst derived by Clauxiils in I XSO, is Lrluwr~as the C1r;u~itriClap~yrnneqrdation. Remember that wc havc neglected the rr~olarvolume of the liquid ctlmparcd with the molar volurr~eof thc gas and that we assurr~t'clrt~evapor can hc treated as an rdcal gas. Never~t~eless. Equatio~i9.12 has the advanrage of t,ei11g ~norr: corlvenient to usc than Equalion 9.10. As ~niphtbe expcctctl. however, Eq~ralio~! 9.10 is Inure nccuritc rhan Eqi~i~tion 9.12. The real a ~ d v i ~ ~ ~uft aEquation ge 9.12 is that it cikn be reiidily in~cgratcd.If wc assume A ~ does ~ nu1 ~ vary H with tcmpcrature over the integratiori limits of T , Equat~on9.12 becomes

Equation 9.13 cikrj be used to c;ilcularc rhc vapor pressure at some tetnpcraturc givctl rhc molar cnthalpy ofraporization and the vapor p ~ a s u r t :11 : snrrie other temperature. Fur ertnlnple, the 11ormal boiling point of benzene is 353.2 K and A%,,, H = 30.8 k l -lrlol I . Assuming A14,,,H dues not vary with temperalur't', let's onlculaic t l ~ cr,;ipor prcqxure ol

C - h ~ f ) l')~ r! Fhasc Equi1ihri.l

366

hcnzcne ai 371.2 K. We suhslitute PI = 760 tom, TI = 353.2 K.and T, = 373.2 K into Equation '1.13 to obtain

P

In= 760

(

30800 1,mnl-'

'

H.314 I - K .lrlol.

19.8 K

'

or P = 1330 turr: The experilrlental value is 1360 tom.

E X A M P L E 9-5 -Ihe vapor przshtrru watet.;tt 363 2 K i q 529 inrr Use Equation 9 13 10 dererminr the average value of A,,,p11 o f waler hctwecn 363.3 K and 373.2 K.

1 0 0 0 K 1 7' F I G U R E 9.13

A plot o f the logarilhm u T the v a p r preshurr r d liquid h e n m n e again\! the rrc~prr)calkelvin temperatrll-euvel a temperature mnge of 3 13 K to 353 K .

5 0 1 I I T I O N : W e 11ltethe tact t h ~ tthe nornlal b;oilirlg point of waler is 373.2 K (I' = 760 ton.) nr~du-rite --

where k is an ~r~tcgration constarlt. Equat10119.15 cxplrsscs flit- iarrntlon 01 kapor pressure obrr a larger temperature range than tquation 9.14. 'I'hus. a plul of Iri I' against I / T will nut hc cxactl!, I~ncar,ill aglcelncnt u ~ t hthc exper~mentnldata fur most liquids and solids ovcr an cxtcndcd tcmpcraturc range. For cxamplc. thc r:lpur pressure of solid ammonia in torr 1s found to ohcj thc cquatlon Ttlc valuc of A,,,pH[rir wntcr at its norrnal boiling p o ~ n i~ t M.65 kJ-mul-I. I

1

If wc intcgrale Equation Y. I2 indefinite1 rather than k t w e e n definite limits. u'e obtain (assuming A + x r His constant)

from 1 4 6 K t o IYSK

I

I

l:rl~r;~tion 9.14 says that a plot 01- the log~rithmof thc vapor presst~rcagninht - the rcciproual of thc kclvin telnperature should be a stmight line wirlr a slope 01- - A v z p H/ I ( . Figure 9.13 ahow:, s u c l ~n plot for herlzef~cover h e temperature range 3 13 K to 353 K. The slope r,l the line givcs A+dpH= 32.3 kl-tnol I. This value represcnts an average vnluc of Aydp%over thc given lernperature interval. Thc value of ~ % at~the ~ normal f l boiling point (353 K ) ia 30.8 k J , n ~ o lI . We car1 recogni~ethat AvopHvaries with temperature by writing AvaVHin the form

-

A , , ~ = = A t- RT

+

+ ...

\r>hert:A. B. C. . . . are constants. If this equation is substituted into Equatiun 9.12, lhen inlegratio11gwes

E X A M P L E 9-6 Use the CI:trlsius-C'lapcynm etpiatinn ; ~ n dEquation 9.16 to thnlpy nf uuhliln;~tionrir :llnrntmi;~from 146 K to 19.5 K , SOL CI T I O N : According to Equation 9.12

dete~nlinethe molar erl-

9-5. Chemical Poient~al Can UP Lualuaterl I-ro~r~ .r P,lrt~lirrn I unclll)n

'I'he Clausius-E:lapcyron

equatlon

can bc used to show that the slopt. of the solid-

501. U T l O N : T h e denvalives u i h l h expl-essiuns ,lt

the triple poillt are

ga5 coexistence cun7enlusr he grcater than

the slope of the liquid-gas coexistetlce curvc ucar the triple point. wherc thcse two curves mccr. According tu Equation 9.12, the slope ol' the solid-gas cun7eis given hy and

rl P' A,,,,iT = p'dl' R I.2

d PI dl'

-=

and the slope of the liquid-gas vurvc is given b j

so the ratio uT the a l u p

I!,

(2.303P,)

K (1330T )3.52 ton.. K ' =

4 3 1/3-52 = 1 22

I At the lnple point, P' and P I , the vapor p r e s s u u of the sulid and liquid. reqx.ctivcly, are equal, bn the ratlo of the slopes from Equal~rms9.17 and 9.18 is

at lhe triple point. Because et~thaIpyis a statc function, Ihe enthalpy change in going directly from the solid phase to the gas phasc is the same as tirst going from the solid phasc to the liquid phase and Ihcn going from the liquid phase to the gas phase. In an equation. we have

9-5. Chemical Potential Can Be Evaluated From a Parlition Function In this section, we will derivc a convcnicnt fortnula for the chemical potential in tern15 of a partition function. Recall that the currespunding formulas for the energy i111d entropy me {see Equations 3.21 and 6.43)

and whcrc the three A ~ must S all be evaluated at the lame tc~nperature.I f wc substitule Equation 9.20 into Equatior~0.19. wc see that

Using the fact that the fIelmholtz energy A i s cqual to I J

- TS. Equi~tinns9.21

and

9.22 give Thua, we see that the slope of the solid-gas curve is greater than t h a ~or the liquid-gas culvc at the triple point.

I

EXAMPLE 9-7 'l'lh: vapor pressulcs of cchid and liquid an\tnorhia near the triplc prlird art: given by

k t ' s nuw include

N in our discussiun of natural variahlcs, and writc

I

The last tertn i11 Equation 9.24 i s expre~sedin 1ernl.s ol' N , the number c ~ f ' ~ n o l c c ~111~ l c x the system. It i s mirlre conventiunal t c ~express this quantity in tcrms of n. thc numher of moles in the system. We can do thi.; hy noting that Calculntc the ratio o f tbc clrlpes of the sulld-gas curve and the liquid-gas ctlrve at the rriplc pninl.

hecausc 11 and N dilTrr by a constant factor of the Avogadro conscant. Therefore, we may uJrite Equation 11.24 in the for111

We'll now show thai (;I A / a n ) , , ,is just nnothcr way of wiling lhe chemical potential, 11. If we add d ( I 1V ) both sides of Equation 9.25 and use the equation G = A l1V , we get

+

rlC; = d A

+ d ( P I : ) = - S d l + VrlP + (

)

Recall now thal y ( V , T) cx V for an ideal gas, and so we can write Equation 9.28 as

wherr y (L'. 7')/ V is a functic~nor terriperature only. tyuatiun 9.2Y also gives us an equation Tor G because G = np. We can make Equation 9 29 look exact1 llke Equatiun X.59 iC we substitute L,T/P I'or \'/IV:

dri 7 C

But it-wc compare this resull to thc total deli%-ativeof G = I ; ( J , P . I ! ) ,

If we cotrlpare this cquntion wilh

ive see [hat

I'tlus, H C call u\e either I; or A to dttcrrnine fi as lung as !be keep thc natural variahler

of each olle tixed when we takc the partial derivative with respcct to tt. Uk can now suhslitutc Equatiori 9.23 inlo Equation 9.26 to ohtait~

We have gonr irom the secclrid lerm to the third tcrln by mul~ipyingk , and n by the Avogadro coriscant. Equat~ol~ 9.27 lakes on a lirirly qirnple form fur dn ideal gas. If we whqtitutc the ~rle~~l-ga., cxprcsrion

we see that

Once again. rccall that q / V is a function of T unly lor an ideal gas. To calculate '(T). we must remeniher that P is expressed relalive t~:,the standard state prcssurz P ', which is equal to one bar or 10' Pa. We elnphasize this conwrition by writing Eqr~ation9.3 1 as

If we compare Equntic~ng . 3 3 with kquation 9.30, u-e see that

The arguttlent of thc logarithm in Equatioll 9.34 is unitless, ax it must be. Equntiun 9.34 gives us a tnolccular formula lo calculate p C { T ) ,or G " ( T ) .For example, for Arrg) at 298.15 K: whcre we have used Stirling's approximation tor In N ! . If we subs~itutethis rcsult into

Erlr~atiotl9.27, w e c~btaitl

k 1' - RT H4-.P

-

NAP

'

(8.314 J 1no1 K ')(298.15 K ) -- -. (6.022 x I d 3 mol ')(1.00 x 10' 13a)

The panilion function .yn(v.T ) lor a diatomic molecule is

and Notice that this expression is the same ua Equation 4.39 except for the factor of e h u / L ~ T e " ~ '= k oelJoiR7 T in Equation 4.39, which account5 fur the ground-state cnelgy being taken to be - [ I , , . The gnlund-state energy associated wilh ril'(v.T ) ~ 1 ~ by ~ 1 Equation 9.37 is ~e1.0.1 . ~ 1 'use ~ Equation 9.36 along with Equation 9.37 tu calcublc p - Eo for HI{& at 298.15 K in the harnlonic oscillator-rigid rotator app~~onimallo!l. with HRI1= 9.25 K and 0>,,2 3266 K (Table 4.2).Therefore.

V This resull is in excellent agrecnicnt with thc cxpcrirt~cntalvalue of -3'3.97 kJ,mol-I. Being essenlially an cnergy. thc valuc of thc chcmica1 potential rtlust be based upon some choice of' ;I zcro of cllcrgy Thc cht~nicalpr)tentiul wc haveiust calcula~ed is I ~ i ~ >upon r d the gnluntl state of [he ; I ~ O I huit~g ~I x r o . For di;iio~l~iu rr~t)lecules,we haue choberi grourid-srale energy (vihratin~~al and clectro~~ic) to he -11,. as illus~rated in Figure 4.2. In tabulaling valucs of $(TI. ~t 15 customary to take the groutid-state CIlergy I)[ thr ~nr>leculc rather than the separated atorns as iri Fig~~rt: 4.2 to he the zuro of energy. To .see how this definition uf the zcro of energy changes the h)rm af thc partition fuoction, write

1"'

(1n)(0.12791;g.rnol ' ) ( 1.3X06 x 10-'' .l.K')(298.15 K) (6,022 x 10'' rnol ')(h.h2hx 1 (I-'" - s ) -

RT --

NAP'

- (8.314~.mol~'.~-')(298.15K1 (6.022 x 10'hmui-')(lO' Pa)

and

I f wc factor out .e

'l,''rl1,

we have

where wc havc writtet~q " ( ~7'). lo emphasize thal he grt~und-statecnegy of the molecule is takcn to be zero. Subs~ituling(his result into Equation 9.34 gives

Thc lileraturc value, which include\ nnhiirmonic and nunrlg~d rot:ktur et'Kects, 1.; -52.94 kJ.mol I. We will use balues of p ( T ) - E l , when we d~scusschcmici~l etluilibr~ain Chnptc~12.

Problems 9-1. Sketch the phakc diagra~nfor oxygen uh1n.g the follotving data: lriplc pr>lllt.54 3 K ;111il 1.14 torr: CI-iticnl puinl. 154.6 K and 37 828 torr: nnrrr~alrnelritlg point, -2 1 H 1'C: ;ud nu]-ma1boiling point. - 182.9 C, Drle~oxygen nlctt u~idcrurl applied pl'ess111-ear n;itt.ldoes?

('

whew E,, = N,?, and P: = 1 bar = 10' pa.

9-2. Sketch the ptalc diagm~nfor I, g~vcllrhe Ir~llu\vingdata: triple polllt. I I 3 :LIILI 0 I 2 atm; critjcal pink. 5 1?C and 1 16 atrn: nt)r~nalrneltj~~g point. I 14 C; nvrr~lnlbuilrng point. 184°C; and density uf liquid > clen~ityol'solid.

1

9-6. The s l o p of tllc melting curve uf melhanc is given by

Freezing liquid

dP (1 T

= (0.08440 bar, K

' ")T'"~

from [he triplc point to arbitrary tetnpcratures. Using the filct that thc telnperature and pressure a1 lhe triple point nrr 90.68 K and 0 1 I74 har, cnlculattr the rn'lling prewuic of tr~cthar~c nt 300 K.

9-7. The vapor prcssure uf methanol along thc entire lirluid-vapor exprerhc.rl vcry acc~lratelyby the crrlpirical cquntiorl

CklCS1FtCtlCP

cuwc c:lrl hr

F I G U R E 9.14

A dut~sily-temperaturepl~ascdiagram c~fbenzene where I. = T I T L ,and Tt = 512.60 point of mcthnnul i u 337 67 K.

9-3. 1;igllre 9.1-1 shuws n dc~~sity-lemperalure phase diagram for benrene. Using the follow~ng dala for the triple point and the critical point, interpret this phase diagram. Why i s the triple point irldicnted hg a line in this type r l C phase diagram?

Triple poit~t Critical poi111 Nurmal frccziijg pcyinl Norm;il boiling point

I'IK

Phm

278.681) 56 1.75 278.h8 353,240

0.04785 48.7575 1 .O 1325 1.01325

/)/rnul.L" Vapur Liquid '

0.002074 3.90

1 1.4766 3.90

0.035687

10.4075

9-4. The vnpor pressures of solid and liquid ch101-incwc giver1 by

K. Use lhis formula to show that Ihe norrr~nlI)oilin~

a I~quidis the temperature at which the v a p r prescure 1% 9-8. Thc standard huiling point exactly one bar, l!se the cmpil-ical formn1:r given in thc previous pl-ohle~nto ~ l i ~ ~ c p u i ~ of ~ t ben7ene.

9-10. Plot thc follow~ngdata for the drn\it~cscrf liquld and g:iseuuc rltiane in cqutlihrium with each uthcr as n function of tempruturc, nt~ddetermine the critical telnpxiturc of eth;~lle.

uhcre P is tllc abnolule temperaturc. C:alculate the ternperaturc and pressure at thc triple point uf c h l o r l ~ ~ e 9-5. The pressurc along the melting curve frnm the lriple-point tcmprature to an arbitrarq tempclaturc can be tit empirically by ~ h Sirnun c equation, which i s

and u are constant< w h ~ f eralues depend upon the fuhstance. Given that Ptp= O.(W7X5 bar. TP= 278.hH K , n = 4237, and a = 2.3 for hznzene, plot P against 1' and compilrc your. rcsuIl wilh that givcn in Figure 9.2. wherr

(i

9-11. Use the (lam 111 the preceding prc~blcirito plot ( ( I ' + p"/2 apiiinsl T -, T . wit11 1: = 305.4 K. I he res~iltirtgstraight line is rill enlpirical law called lhe luw rifrrcrllinen~.dium~ters. IT this curve is plotted on the same ligure a the criliciil pclinl. 11,

4

11.

-- constant

Nn\t show that h)i- I h i ~OlO f L I . rcspeclively. Estirnn~c[he hniling point nf ant1 baprlr are 0.9584 g,rr~I.V.al?r baric clcr~\l~re.:of liquid o l ~ dgaseous ethyl acetatc art. 0 826 g-inL- arid 1) 00319 g . ~ ~ t ~re\prctively. -l. a1 its rlorrrlnl hoillng point (77.1 1 ('1. The rate of change uf v;lpor prt'shure wit11 trII1pcrature i < 9.0 tc~rrK at thc norrnal hoilinp pnlnt. Eslimate thc rnolnr erilh:111>y uf rapori/airt>n of ethyl acetate at its normal boiling po1111.

9-28. The raprlr presnurc of benzaldehyde is 100 turr at 154 C and its normal hrliling point is 17Y('. Estimate its molar enthalpy 01- rapr!rrzatiorl. 't'l~ccxprl-ininnliil v;ilkle i c 42.50 kJ mol-I 9-29. Use rhe followirlg data lo ektirtlntc thc nortl~alboiling point nnrl the 111olilrtl\tll:~I~)y 111 vaporiz:~tiu~i of lead.

9-30. The vapor

preahurt

of solid iodi~lcis given hy

9-22, The v;lpc71 prcqsurt. uC rne~-c~iry from 4W'C lo t 300'C can k expreu\cd by

Jhc dencily of thc vaprlr at its normal b n ~ l i r ~pninl g is 3.82 E . ~ - and l [hat of tllc liquid is 12.7 g -nlL- I . Estirii;ite thc ~ r ~ u lrnthalpy ar 01- vaporizatiot\ 01- mercury at it< rlnrmal huilinp 1wi111.

-

9-23. The p r v w r e s nt the solid-liquid cwxister~ce boundary of prupune are givcn by [he empiric;~le q u a r i u ~ ~

Use this etluation tu calculate the nortl~nlsuhlin~ntiontemperature and the rrlolar et~tl~illpy of sublitnation of i:(sl nt 25 C. The experlrnrr~talvaluc of Ah,,, drr~\itg.l;T, ) will giw loops for subcritical tcnlpcratuws (T, c- I). ?'he vorrcct behur,ior its the. prcqsurc is increased is riven by the pattr a h d k in Figure 9.15. The honri-rtmt;it regioli bdt. rjo~given hy the rat1 dcr tVs11c equation. represents thc cotldencalion of thc gas to a IiqilirI n! a fixed preswI,e. VV"V' can tlraw Ihe hwrizontni lirlc lcnIled a !if line) at tllc corrrcl posilian by rccogniring [hat the chenlical potcntials oithz liquid and the vapor musl hr equal at the poir~tsh arid f. Using this rcquircmcnt. Maxnrll showed that the hori~ontal IIIIC rcprcsrr~firigcondellsotiot~shuultl he drawn such that the a r m of the loops above and below the line must be equa1. Tu prove MII.YHYII'Sf q w ~ I - ~ r CotISrrucrIofl e(~ mltv.integrate (lip,,';3P), : :V by parts alol~pIlle puth hcdel and use the fact [that ,I' ([he value of at puirit f ) = ~ ~ V t ~h ael u e l l i a( j ~p o i ~ h~ )t to obtain

Bccnust. ( 2 f /a V )7 = Oat thecrilicnl poillt, h, rli\trges thcrc. A question that has perier:lred a great deal of cxprriment:il and rhet)rer~calresearch 1% Itre question of the rnanner in w h ~ d ) K~ divc~fics nh 7' a p p r < ~ a c l ~ Tc.~D t w s i r divcrgc a< ln(7' - 7;) or ~ r t i a p sas ( I ' - -' wtlcrc y is arlme ct.ttrtut t,.rpo!ic.rlt? An tally thrrlry o f thc bc11uvirlr of thrrn~odyrjanlic furlcliur~\ such as K ) tery near the crtt~calpoinr was propwed hy van dcr Waals. w h o prediclrd that u , divuges as (l' - T')-'. To cer how van dcrW;ial> arl-ived at this pretlicrion. wc cr)r~\idrrthe (doublc) Taylor enpa~~siotl u i the pressure P(V. 1 . 1 :lhout 7;, and 1:

Why are thrre no terlrls it)

Now show that

and Ihal

e,

w l ~ c ~ c i $ Itie prersr~i-eccurrcspnr~din,vtn Ihe tie line. ll~terprctthis rehull

(V -

y)or (I' y)'.?W ~ i t cth~sTaylor series a$ -

how let

1.' = vt to obtain

'Thus, ihe van der W x a l ~thcory prcdictc that the cnlical cxponeur h e n shown expcri~nentilllythat

:iccuru~ee*pe~-ir~icnt,il n ~ r ; ~ > u r c t ~ l c . n t as s u1f ~+ , '/;,suggest t h : ~uT d i ~ ~ r g i~ e little b niom

clrongly than (P'- Ti,) I. I n particular, it is found thi~tK, 4 (I' - 7')- whcre y = 1.24. Thus. the thewry of van dcr Wnals. nlthough qualipativrly correct, i~ not quu~titatively corrcct. 9-48. We can use the ideas of thc prcvious problem to predict how thc diffcrerlcc in the dc~~sitles (pl and p G )of the crex~stingliquid and vapor states (urthubarir det~sinps)behave its T + Tc. Substitute

111

Itus v;lsc is 112. It

11~15

where 0 = 0.324. Thus, as in the previuu\ prohlcm, alrhouph qunlrtatively corrcct. the V:III der Waals theory is not quantitatively correcl.

9-49. T h e folluwing data give the temperature, ~ h t vapor . preuurc, ntld he density uf lhr coexisting vapor p h s c OF butane. Use lhc van der Wwls cquatiwn and the Kedlich-Kwim:! equation to calculntc the vapor prcssure and compare your rcsult with the cxperimcl~lal values given below.

Into the hlaxwell equal-urea c.un$truction (Prr>blern 9 4 6 ) lo gel

v',

For P < PL.Equation 1 g i ~ e sloops and s o has three rclotc, TL,a ~ i d7'fur I' = I;,. We can obtain u hrct approximation to these rrmts by assuming t h a l v , x 7')in Equatitln 2 and ucrlting

:(v'+

9-50. The Fullowillg data give the tempmture, the rapor pressure, and the density of the coexisting vapor phase of benzene. Use thc van der Wil5 cquation and the Kedlich-Kwong equation to calculate the vapor pressure and compare your rc.\1111 with the cxperirnc~~t:~l s arid 2.18 with TL= 5h1.75 K and Y = 48.7575 bar valucs given M o w . Use E q u a t i o ~ ~2.17 trl calculate the vall der Waals para~neterhartd the Kedlich-Kwong pmrrjuters.

To l h ~ approximalion, c the lhree roots tu Llualion 1 are ohpained from

Show tllnt the three rtmts are

VI = ,Ly-

v:

=

v,

h o w show that

and that this c q u a t i o ~is~ uqulvalcnt to

(d ) i z C

l

,

-

T)]!~

9-51. In Problems 8-57 to -3, wc considered only cases with conslanl n . Use the approach of Prohleln 8-57 to repartition n between two in~tiallyidentical comparlnle~lthto show that ( a ' ~ , ' i t ~ ' ) ,I > 0 and tl~nt( d p / a nj , . , > 0.

CHAPTFR

10

Solutions I: Liquid-Liquid Solutions

In this anri thc next chapter, w e will apply our rherrnudyliatt~icprirlciplcs to sr~lulinr~.;. This chaptcl- fbcu';cs on sulutic~nsthal consist oC Iutj volatile liquitls, such as nlcnholwater solutions. LVc will first discu<sparlial rrlr~larquatjtitirs, which probide thc Inort cotlvenic~ltsrt c ~ thcrmorlynamrc f kariahles to describr .soluticlns. l h i s diccussion will lead to rllc Gibhs-ntrlicm cquallt)l>,which gives 11sa relaliori betwrer~the chaoge i r ~ the prowrties nC utle con~ponet~t of n soluticln in rcrms ul- the change in the propertics of (he olher corrjpurient. '1 he si ttlplest 111odel of a solution is an icIcal solution, in which both co~nponenlsobey Kaoult'h law o$er the entire cnmpnuplin f i x st~lutionsbehave alrrlost ideally. triost solutions are not ideal. Just as nonidcal pi~xc.; can be described in tcrms of fu'ugacity. nonideal snluiinns can he described in tesfns of n q u a d ty called activity. Activity rnust k calculated with rcspect to ;I qpecific stanrhrd stale. and in Section 1 ( k X we ilitroduce two com~nonly-uscdstandart1 states: a wlvent. or Raoult"; Iilw standard state, and a solute, or Henry's law statidard state.

10-1. Partial Molar Quantities Are Important Thcrniodynamic Propcrt ips of Solut inns lvrl Hildebrantl 11 ill Citrtldcu, h'l. on Nr)vcrnhei- I h. I H X I , : I I I ~dicd in 1083, t lc 1x11 hiz, Ph.l) in fi<siot~u[ly octivr unlil his death, puhtishi~~g h ~ last s paper in 1981. Hildehrand ~i~;ictr siy~ihcnritcontnhritiul~sto thc 6cIdh or liquids and nunelectrolyte snluLirln\. He rctnincd ;i Iorip interest ill dcvlationc Iroin ide;~lqolutionc 1Kaoult's law1 and the tl~coryur regular sulutior~cHis hooky. Y hr Smluhilif~r!J'~T~~:,llzplrrtrv)Ivt~s and Ilr~gubrS o l u r i o ~ published ~~. wiih Rrlhert Scott. ncrc ?taritlartl rel'rrer~ccsit1 tl~cficld. Hildcb~lndwar 3 Fanicd. uxcellent teacher of p~'t~uraI C~CIIIIC i ~IU I[ -eyr k c l c ~Hi5 perieral chemisrry text. I'rit~r>iplear!f Ch~lt~isrrx, influcucetl ~ l t l ~ c . r>c.l~oolz, trr pl:~ceglc,Itcr c l ~ ~ p h ar~l ri ~kp~iric~plcs;III(I lcsr on thc ~rlcrllorualionuf cpccific 111:1rrrial ill tl~cttxhing :ri1e1:11 chctlli~try.Hildct~ronrl\%,a\a great luvcr Itit. ourduors nrtd cq'c'c~allyei~juyctlqkiil~gar~ilcamping. Ile managed thc U.S. Olympic Ski 'lcnlrl irr 1436, was I'residc~~t nT the Sierra Clrib fion~1437 to 1940, and wrote ;I book on catnpir~gwith ltis daugh1t.r 1 ruli+r.C-btt~p ( , ' ( I / C , I - ~ I I Kor. HoL~, ro r ~ i ~ ~r l ~< - , ~ i h l ihiktv-,~, / ~ roitIprrs, I ) I O I ~ I I I ~ I ~ I TC( (, PI I~~T( ,I ~ ~ S I , T , / l f l l l l l ~ l - ! ti;lnponc.r~~ syhtclnq. LVc will now diccuss lht. tIic.r~nr,dynamicxo1'rr1i1ltic.o111p011eril systerrls, althtli~yh. for si~npl~uily, w e will discusx only syslerns oC lwu crllnpoIirrlts. Most of tllc coilsystcms. L.ct's cepls and r ~ s u l l sw e will develop are npplicnhle to multicompo~~cr~r c~)risidera solution cor~sistit~g of H~ 1110les of cornponcnt I and 11: molcs of cumponcnt 2. The (iihhs energy nl' thiv solution is a functiorl of 7-and P and the two ttlule numhcrs n , and 1 1 , . We ernphasi~ethis dependencr of G ori ~hesevariables by ~ . r i t i n g G = G ( T . P,n , , n ? ) .Thc total derivative uf- G is givrn by

10- I . P , j r l ~ .?rllll;lr ~l Q L I ~ I ~ I I I Arc I ~ + lnlporta~itT h ~ ~ r n i ( > d v r ~I(' ~r (r ~t ~p~r ~ r l ~0 1~ S \ ~~Iul~otl~

IC the cutt~positic>nof t l ~ csolution 1.; I~xed,s o Lhat rill, is the sarne us Bquiitir)n 8.30. ant1 wc have

=

rl11, = 0, the11 L q ~ ~ i ~ t i10. u nI

At coilstant I' o t ~ fP . ruc hakc

Now, imagine that

w e increase the size of thc system unifi)rmly by Inearls of :i s c : ~ ! ~ parameter i, such that On, = I I , d l and dn? = r~,u'h. Notc that as we ~ ~ 1 i - fro111 y 0 11) I the nurnher of moles of componcnta I and 2 \ a n t s from O to 11, and 0 to 1 1 ,, I-especli\el). Bccausc G dependsex~ensivelyo n r t , andr;?, wc musl h;~vethul OG = Grlh. Tlicrc.icjrc. the total Gibbs cncrgy varies h ~ Om to some final value G as 1 I < va~-ied.Ir~truducit~y rlL into Eqilation 10.5 gives

.

.4s irl tlic prcvious chapte~;the parlial derivatives of C; wilh respect to lllolc numbers

are ci~lfedcl~emicalpotenti;ils, or partial rnolilr Gibba energies. The standard notatiun lor chrrrliv~lpotential is p , so wc can write Equatiol~10.1 a< n , . and tr2 are final valuca (antl so do no1 depzr~dupon h ) antl / I , and p , arc iintcnaive variahles (arid so do not dcpcnrl upon lhe size pamfnetcr >.). wc can wi-ltc. the abovc cquation as

Bccause G, where

We will see that thc chcmical potenlial ot- each componenl I r l llje solution plays a central role in dctcnnining the thern~odynaniicpmperlies of the solution. Other extensive thcrmt~dyna~nic variables havc associaled panial molar valucs, al1t11,ugtionly the partial molar Gibbs energy i.s given a special symhol and rlaIne. For cxsmplr. (aS/a,i, j, ",-, is called the partial lnolar entropy and is ticnoted by S,, and

7.

i i t , is callcd the partial molar vnlun~canrl is deno~zdby Gcncrally. f. if Y Y ( T . 1', , I , , n , ) is some ttx~e~lhive thcr~norlyni~rr~i property, thcn ilx ;icl in I-PI-opanol/watcrsoli~lionsat 20 U. Wt:cirn uhc Ihls I I ~ L I Ilo - r .e\ljrll;itc the final volu~ue'of solution when I(X) rt~Lof I-plnl>;rnol is rr~ixedr\ith 100 tnl. ol wakr at 20-C. 'The dcnsititc I' 1-propano1 artd watcr at 20. C' are 0.803 g.n-11. and O.LlY8 g.rriL-]. respcctir.cl> . .,ing thrse densiticq. we ste (ha1 100 mL ciicli of I propano1 and water corresponds to a niole fraction of I -prr,piulol of 0.194. Rct'crri~~g

10-2. The Gil,I)s-Duhpni Equation Rclatcs tlw Change in the Chemical Putcnt~al

301

Furthermore, by using the fact that cross second partial derivatives are equal, we gel

and

If

we

subsiitute these two rckults lnlo

10.1 The partial molar volumes uT I -propanu1 and water in a 1 -propanol/water solutivn a1 20LC plr~l~rrl against the mule fraction of I-propanol in the <elution. FIGURE

-

= 72 rnL,mol- and we see that this corresponds to roughly l/MdIrr = 18 ~ i i ~ . n i u l -Thus. '. the final volume of the solution is

to f-irurr 10.1,

-

which is an exlension of Equation 8.30 I O rnulticoniponent xysterns.

E X A M P L E 10-1

Derive an eyuatir)t\ for thc temperature dependence of p,(7-.P ) in analogy with t l ~ c Gibbs -Hellnholtz equa~inn(Equalion 8.W).

S OF L' 1 I (IN: The Eiihbs-Hcl~uhullz eq~ralinr!i s (Fquation 8.60)

rompared wilh il total vcllurne of 200 m1,before mixing. Problems 10-8 through 10- 12 irwc>lr.cthe deterrrlination of partial moIar voluines from solution data. Now differenhate nith ~ ~ s p eto c t11, ; ~ n dinterchanpc t t ~ corder o f differentiation oli Ihc

10-2. The Cibhs-Uuhrm Equation Relates the Change in the Chemical Potential of One Component of d Solutiun to the Change in thc Chernical Potential of the Other

left side to get

Most of our thermodynamic fi~rmulasfor single-component systems (pure substance>) have a~~alogous formu1;ts in terms of partial molar quantities. For example, if we start with G = H - TS attd diverentiate with reqpect to n, keeping T, P. and n,+, fixed, ~e obtain

where

g, is the partial molar el~thalpyo f component j . 1

We will now derivc one of the tnost useful cquatinns inv~dvingpanial molar quantities. First we differentiate Equation 10.6

an,

and subtract Equation 10.5 to get

rh;lpter 10

1

Solut~onsI: L~quid-L~qu~d Solutions

10

(

AI Equ~l~hriurn, thc

If w e riividc h t h sides by n , -tn,, we have t ,I

,

+/

=0

(constiint T attd P )

(10.11)

and n, are mole fractions. Either of tcjuatlons 10.10 or 10.1 1 1s called the Gibhb-T)~dhml ~qumttutt.The Gibbh-Duhem equation tclls us that ~f we know thc of one component as a fi~nctionnf colnposition, wc c;in dctcrminc chclnlci~l~lotent~al the c~tlicr.For example. suppose we were to know that whcrc

A,

over rhc wholc rangc of x2 (0 to 1 ). A superscript * is thc IUPAC notation for a property of a pure substance, so in (his equation, p; = p , ( x , = I ) is the chemical potential OC pure component 2. We call diffcrentiate 9, with respect to x? anti suhstirute into Equtltlon 10.1 1 to gct

But

= - d x , (because .il t I;, = I), so

a h e ~ eO 5 x , 5 I because 0 :- x1 5 1. Now ~nlegrateboth sides frotn n, = 1 (pure cutnponent I ) to arbitrary x , ti) gel

where 11; = p ,( x , = 1). We will see later in this chaptcr that this result says that if one compment uf a binary solution obcys Raoult's law over the complete cuncentration rangc, the other component dues also.

I

I

E X A M P L E 10-2 Derive a Gihh\-Tluhrm rype of equation

for rhc volu~ricof a hinary holr~liun.

S (71 (1.1 I C)N : We start with Equation 10.7. which is Ihc analog of Equation 10.6 -

V ( T , P , n , . isi)

= n l Vl

+ n, V?

and differentiate (at uollst.?l$ T ntid P) to oblalri dV =

r,,r~-ci +Yrln, +

rr2tfK

+Fdnl

Si1blr;ic.lIhe analog o f Equation 111.5

+

d 1' = y d t ~ , F d n 2

(cvnstilnt

T and P )

vJap

Chemrcal Pott.r~t~al # i f L A ~ Component I

n,dy

f n,dV, = 0

H.15

thc 5.1rr1c. l',>lurIn Larh Phasc

(conslant T and P j

This equation cays that if &c k ~ ~ othe w change i n lhe partial ~ ~ l o l wlurne ar uf one component of n hinary syhtcrn over a r a ~ ~ g e co~npo~~licjll, wc can determine Ihe change in the partial molar volulnc of the other cr,mponerit orcr thc same rangc.

10-3. At Equilibrium, the Chemical Potential of Each Component Has the Same Value in Each Phase in Which the Component Appears Consider a binary solution of two liquids that is In equilibriutn with its vapur phase, which contains both components. Examples are a solution of I-propanol and water or a solution of hcnzenc and loluene, each in cquilihriurll with its vapor. We wish to generalize our treatlncnt in the previous chaprcr, in which we treated a pure liquid in equilibrium with its vapor phase, and develop thc criterion for equilibrium in a 1~in;iry solutioti. The Gibhs energy of the solution and its vapor is

Let nyn, n;" and nYP,niq be the mole nutnbers of each component 111 each phase. Fur generality, Ict J dent~leeither componcnt 1 or 2, su n , denotes thc number id' ~nolesof cornpoilent J. Now suppose that dn, moles of component j are transferred ftonl thc solution 10 he vapor at constant T and P, so that dn? = +(in, and drt;'" = -dn, 'The accompanying change m the Gibbs energy 1s

If the transfer from the solution to the vapor uccurs spontancously. then d l ; c. 0. Furthermore, d n y > 0. so must be less than In order that dG < 0. Thrrefore, molecules of component J move spontaneously from the phase of higher chern~cal potential (solution) to that of lower chemical putentral (vapor). S i n ~ i l ly, a ~1Fp;"" 2 I L ~ ' " . then moleuules of component j move spo~itaneou~ly [rot11 the vapor phaw to the solut~onphdse (d11:"~ < 0). At equilibrium, ahcre d G = 0. we have that

Equation 10.12 holds for each componcnt. Although we have discussed a solution in equilibrium with its vapor phase, our choice of phases was arbitrary, so Equation 10.12 is valid for the etluilibriurn between any two phases in w h ~ c hcomponent .j occurc.

393

Cl~.~pter 10 / Snlutions I. Lialr~id-L~qu~d 5nlutinn~

'The irnporlaru rcqult here is that Equation 10.12 says that the chemical potential uf each colnponetlt in the liquid solution phase can be ~neasuredby the chemical potential of thal uoinponcnt in the vapor phase. If the pressure of thc vapor phase is low et~oupli that w e can consider it to bc ideal, then Equation 10.12 becomes

~vhcrethe sta~ldardstate is taken to be lion 10.13 hecornes

q"= 1 bar. For piire

FIGURE

&t~crcthe superscript * represents pure tlirluid) component j. Thus, for example, @ : ( I ) is the chemical potential and q"is the vapor pressure of pure j. If we subtract ~ ~ u a t i oI U. n I4 from Equation 10.13. \ve obtain

Equalion 10 15 1s a ccl~tralcquatlorl i n thr study of binary sc~lutiotls.Nnte thal 1~:1" + 11: rlb q -+ q*. Strictly spe,tki~lg,wc should lice I'ibgacitic'i (Section 8-8) irlstead of pressures i n Equation 10. IS. but usually the magnitudes of vapor pressures are such that prrssures ;Ire quile adequate. For example, thevapor pressure of water at 293.1 5 K i~ 17.4 torr. or 0.0232 bar.

10-4. The Comporlcnts of an Ideal Solution Obey Raoult's Law Cor All Conccntrations

10.2

h n~oleculardepiction of an ideal sulution. The two t y p s of molecules are distrihuled through nu^ the sulutiun in a random

component j , Equa-

manner.

j rnc~lcculcs.Rccausc the j 1nolecules o t ~the surfisr are (he molecules that can c\cnpr into the vapor phitsc, the palqial pressure is just .r, q*. According tu RanuIt's law (Equation 10.16) and Equalion 10. IS, the chcmical uC utmponent j in the solution is give11 hy poten~ii~l

7

Equation 10.17 also serves to d e h e an Ideal solutir,n ~t ~t 15 valid for all values of A, (0 5 .Y, 5 1) Fu~qhermore,we sho*ed in Seci~on1 0 . 2 that if one cu~nponentobeys Equation 10.17 from .r, = 0 to x, = I. then $0does the other The total vapor precsure over an deal qolution is givcn hq

Therefore, a plot of

Sot,, against x2 or.^,) will be s straight line as shown in Figurr 10.3.

A I'ew solutions have the property that the partial vapor prcssurt: of each cornpatient is given by the sirrlple equation

E q u ~ ~ ~ i10. o nI (1 is called Kuorilt '.I Iuw, ant1 a solution thal obeys Rar,ull's law over the elitire colr~l~ositic>n rarlgc is said lu be an ideal anluriolr. The ~nolecular~>icturt.behind an ideal hinary solution is that the twc~lypes of ~nulccr~les are ranrlomly distributed thrtlughout the solution. Such a disu-ibulion will occul- it ( I ) the ~llnlcculesm roughly the same size and shape. and (2) the inlermoleculitr fi~rcesin the pnre liquids I and 2 and in a ~nixturt.of I and 2 are all similar. iVc cxpect ideal-solu~ion6eh;jvior only when (he molecules of the two conlponents are sitnilar. Fnr example, benzene and toluenc, o-xylenc and p-xylenc. hexane and Iirptar~c,and hrorrrocthane and iodocrhane [urn] esserltinlly ideal soludo~ls.Figure 10.2 depicts an ideal solution, in which thc two types r ~ molecules f are randomly dihtributed. '1-l~e~imlefriic~ion1,reflects the Craction uf [he solution surface that is occupied by

F I G U R E 10.3

A plot of PtVIaI againsl x ~ Lor a~ soiutivn , uf ~ benzene ~ and tnluctie at 40.C. T h i s plol shows that a benzeneltnluc~~r solution is essentially ideal.

I

E X A M P L E 10-3 I -1)n)pilllol;~nd2-prol>arir)lCorn1 ehsenlially all idleal colutioil al all cuncc~~tr,llicln:, nt 25 C. Leiling thc ~LIbs~liptk I und 2 denote 1-pmpanul and 2-propanol, re\ptlcliveiy, ant1 gikc11that P i = 20.9 torr nod Pi =: 35.1 turr at 25 C. ci~lculalethc total vapur pcssure and tt~ccnrrlpusilirln oi the rapor phave at - 3 , = 0.75.

I

t:,,.,,,( A ? 1 0 . 7 5 ) = A , P ; + x 2 P ; = (0.25)(20.9tom)

+ (0.75)(45.2 tom)

= 39.1 torr

dentltc thc lilvle fraullor, ot cach co~!rponenlin lhc vapor phase. Then, by Llalton's law ut partial preswres. I ut

);

I

'-

' 1

-

-

, =.-=P,

-

p

~

39.1 lorr

P7" -=-~

--

., .

(0.75)(45 1 torr)

1,

- 2

+

, I ' = (0.?5>[20.9torr) . - 0.13

p,c>l,,~ pa~>~.l~

~ p ~ L~> ~ , 8d~

~

= 0.87

39.1 tom

Note (hiit y, y, I . Alho rmtu thal the vapur vr)latllc component.

ik

FIGURE

10.4

A pressure-conipr,sition diagram for a I -pmpwol/2-propanol solulir)n, uhich for-nn un essenlially iclcill solution a1 25'C. This figure can he calculated using the approach in Example 10-3. Ttre upper curve (called thc liquid c ~ ~ r v represcrlts e) ~ e r \ u cr , , thc lnolc fractio~iof 2-pl-c~panrd111thc liquid phace, and thc lower curve (callcd ~huvnprv LurYc) rcpIcserlts I:oldl vcrsus sl, the mclle lractiun of 2-prclpanol 111 ~huvapul p h a w Tlic rrrcl poinls marked fiy x repwscnt the values o f xl and !1 from Exarr~plc1I) 3.

of liquid and vapor phase in the fullowing way. The mule fractiot~sin the liquid vapor phases arc

rlchcr than the solution in 111e more

Proh1c.n) 10-1 5 has y o u expand Example 10-3 by calculating I:L>lol as- a t unction of A ? ([he molc flaclion of 2-prupannl in the liquid phase) and as a funclinn of y2 (the mule friiction nf 2-propanol in the vapor phirse), nrtd then plotting $t,lJi against x. and v2. The rest1111ng plol, which IS tart i l t ~ h tpuirl~ ' q,..I , iri Figure 10.4 :111dInwcr [Ire ~ ~ r t ~ + s u r c . At t t ~ cpoinl PC,,r(,, t t ~ cprrrsure cwcceds rhz rdpor p ~ s s u r cof the solu~ir>n,so the region above the liquid curve cunsiats ol'orle (liquid) phase. As (he pressi~reis lowered. we reach (he point A, where liquid starts to vaporize. Alnng [he line AD, [he system d in equilibrium with cach other. At the point B, all the liquid consista c~fliquid n ~ vapor ha5 ~ a p c ~ r i ~ atid e d . the regio? below the vapor curve consists c~fone (vapur) phase. Lct's conhider the poinl C in tl~cliquid-vapor region. Puint C: lies or1 a line cnnnectir~gthe composition of liquid (x, = 0.75) and vapor (y2 = 0.87) phases that we calculnted in Example 10-3. Such a litie is called a rir line. The overall vompusition uf the two-pl~asc(liquid-vapor) system is ,r',. We can determine the relative amounts

A', -

=

ni - 11;

n',

+

H!

n'

and

11

y 2 = -n;dP --

yr

+ Hyl'

i~~iil

>:,;,

H:

-

where nYnPand n1 are the total numhcr uf moles in thc vapor arid liquid phases, rcspcctively. The overill1 mole fraction at .x4, is given by thc total nurrtber of mtdes (11 component 2 divided by the tola1 nil~riberof moles

Using a material halance of the number of rnolcs of conlponent 2. wc have

Thls equatlon represents what is called the Irvpr nil? because I I ' " * ( V ~ - x u ) = ,,I( t a balance or each value ut "n" lime^ the distiance frorit each curve to the point C in Figure 10.4. Note that rrl = 0 when I,~-lnuI:~s

and

Use thesc fnrtnula~tocnlcu1atexI and values given I n Figure 10.5.

at 9.1.OC, and cumpare your rr\l~ltswith ihe

Tlie pcl~ntcorrcspottding 10t = 40 C and x, = 0.59 is labcled by poit~trr In F~gurc10.5. \lie can al\o calcr~latethe cnrrespundi~~g ccmtposition of the vapor phase as a function of

telnpratulc l'tte ~ n o l efrxtion of componerlt I in the vapor phase is given by Dalton's I~IW ),

r

=-.L-- X1 p;

760 torr

7M torr

or

P,' = 647 turr. Similarly. P; = 1150 lorr Tl~erefnre. XI

=

P; - 760 tom - 1 '

-

-

- 760 torr -. I1.77 l150lorr-6371orr

1 150 tom

I

10-5. Most

5ulutlons ,\re

Not 1dc.d

where G ; and G;are tlic Gibbs cnergies ot the pure components. LJsing Equalion 10.17 for an ideal solution gives

and

A

G ' =~H I P ; " '

ill agreement w ~ r h[he valurc st~ownin Flgulz 10.5.

'l'hc tcnlperiirure-composition diagram can k used t iIluslrate the process of fractional distillation, in which a vapor is condensed and lheli re-evaporated Inany time< (Figure 10.6). I f we wcrc 10 start wirh a I-propanul/Z-propa~lolsolutiun that has n mole Craction of 0.59 in 1-propanol (point a in Figure 10.5). the mule fractir~nof l pmpar~olin the vapor will be 0.45 (poitlt b). If this vapor iscondensed (point c) and thcn re-cvaporatcd, then the mole fraction rlf 1-PI-opanulin thc vapor phase wrll be abuut 0.30 (point d). As this pnlcess i s continued, the vapor becomes increasingly richer in ?-p~upanol,eventually resuIting In pure 2-pn~paaol.A fractional distillation column diCl"ers from anordinary dislillation colu~nnin that the former is packed with glass heads. which provide a large surface area for the repeated condensation-evapora1ion process. propenies upon forming an ideal We can calculate the change irr thenlindyl~iim~u solution from its pure components. Let's lakc the Gibhs etlcrgy 3 s an cxi~mple.We clelinc the Gibbb energy nf mixing by --

I;;(T. P , n , ) - G i ( T , P . n,)

- ,l,)4; - IJ+;

= R T ( n , 111.ul -k n l In x,)

I

A,,,,% c; = G'I" ( T . P , n , . I [ , )

+

(10.20)

(10.11)

This quantity is always negative because x , and x, are less than one. In other uords. iin ideal solution will always f u n spclntaneuusly fro111 ~ t separale s components. '1.11~ entropy uf mixing of an ideal solutior~is giver1 by

Note that this rcsult for an ideal suli~tionis the same as Equation 6.26 fur lhe mix~ngol' ideal gases. This sin~ilarityis due to the fact that in both cases the molecule:, in rhe final sotutinn are randomly mixed. Neveflhzless. you should realize that an ideal solution and a mixture of i d u l gascs differ markcdly in the interactions involved. Althouglr thc molecu1c.s do not interact in a lnixlurc of ideal gikses, thry interact strongly i n an ideal solution. LII an ideal ~olution,the interaction:, in thc mixture and those in thu ~ ~ I I ' liquids are esser~tiailyidentical. 'I'he volu~ncchange u p n mixing of an ideal xoli~tiunis given by

and the enthalpy of mixing

IS

C

(see Eq~~atirjns lo,?1 and 10.22)

( I 0.?4) A rnrx H ' L11mA xG ~ ~ s T A ~ , , ~ s ' ~ = o

Therefore, there is nu volunle change upon ~nixing,nor i5 there any energy l i c , ~ ~ absorbed or evolved when an ideal solutjo~lis formed froin 11spure cornpunelits. Both Equations 10.23 and 10.24 result from the facts that the moleculex are roughly the same size and s h ~ p e(hcnce Aml1vtd= 0) and that the t ~ r i o u sinteraction energies are the cnmc (hencc A , n , 5 ~= ' d0). Equaticlna 10.23 and 10 21 are ~ndecdobserved to tK: true cxperi~iientallyfor idcal ~ o l ~ ~ t i uFor n s .most solutions, ~ O W C V C T ,A,,, H and A,,,,T I' clo not equal zero

Pure l i q u i d to cul)ection H a c k

.

Cr)lu~rlnp a t k e d w i l h glass rings

10-5. Most Solutions Are Not ldcal F ~ G U K E 10.6

A si~nplt.fraclior~aldiclillatiun column. Because r e ~ a t e duondensat~onand n.-evaporation wcur along the entire cululnn, lhe VRptlr bcconleh pwgresivcly ncher in the more vulahlc vulupmlent ac 11 mo\,es up i t ~ cuulurnn.

Ideal solutruns yrt: not very common Figures 10.7 and 10 8 show xfapor pressure diagram5 for carbon d~sulfidelditncthoxymcthdt~e [(CH,O),CH,] solutionh and tnchlornmethanelacctone solutions, rt.spectively. Thr behawor in Figure 10.7 5hows so-called positilc deviatic~nsfrom Rauult'x law hecnusc the llart~alrapor prcssures

F I G U R E 10.9

F I G U R E 10.7 T t ~ cv:twr , ortssure diacram ul-n carbon disultideldimethoxymethane solution at 25'C. Thi\ system chows pocitivc deviations from ideal, or Kauult's law, hehaviur. --

.

'I he vapor prescure diagrarn ot nlcol~ol/wdtrrsolutiuns i i s a f ~ ~ n c t l rol l n the riunjhcr o f carboll atvrns in the alcohols, showing increasing deviation from idcal behavior. The dashed line correspund~to ~netha~lul. Ihe dotted line 10 et11at1nI.and the dashed-dotted line la I-prrlpar~ol.

Figurc 10.9 sh(~wsplols of tllethanol. ethanol, :~ntlI-prc)panol vapor pressures i r ~ alcohollwlitcr 5olutions. Nole Iha~the positive dcviation from ideal hchaviur increases with thc size of the hyrlrocarbot~part of thc alcohol. Thih hehavior occurs because the water-hydrocarbon {repulsive) inieractions bccomc incrc:~singlvprevaler~las the s i ~ e of the hydl-rxnrhon ch;un Inureithes. There are sutne inipcl~Vantfcatul-es lo liotice i n Figures 10.7 and 10.17. I.ct's hlcus on cotltponerlt 1. Thc vapor pressure of uornponenr 1 approacl~csits Raoult's lam ~ a l ~ r r as x l approaches I. In an equation. we have (hat

F I G U R E 10.8 Thc vapor pressol-e(Iiagmln of a ~richlorun!ethanelacetonesolution a1 25°C. This system shows ~~cpalivt? deviakionq from ideal, or Raoult's Iaw, behavior.

of carbon di.sitlfide and dimcthoxyrncthane are grcatzr tha~lpredicted on the hahis of- R a o ~ ~ l tlit%. ' s Physically, positive devialions occur because carhon disulfided i r n e t h o ~ ~ ~ r ~ e tintertlctions h:~nr arc more rcpulsivc than either carbon disulfidexarbon

disulfide or dimcthoxytnctlia~ie4i1nethoxy1n~thane interactions. Ncgalive dcvia~iotls. or1 the other hand, like those .shown in Pigure 10.8 for a trichl(m>methane/acetone solution. are rluc to stronger unlike-njolccule ~ r ~ t e r ~ ~ c than t i o t ~likc-molecule s interactions. Problcm 10-36 asks you to stlow that if one component of a binary solution exhibits 1)ositivc deviations Crom ideal behavior. the11 the other component must do likewise.

10.25 trom Figi~res10.7 and 10.8. it is generally truc. Altho~rghu e deduced Equaticl~~ Physically. Lhis behavior lllay L x attributed tu the fact t h a ~there are so few compunei~t-2 molecules that 1r1os.t component- I ~nolcculevsecc~nlyother componetlt-l molect~lcs,co that the snlu~innbehaves ideally. Ranult'c law behavior is not observed forcomponcnt 1 as x , 4 0 in Figures 10.7 and 10.8, howcver. Althwgh not easily seen in Figures 10.7 and 10.8, thc vapor pressure of component I as x , -+0 is lincal. in x,, hut the slope is not equal to P; as in Equation 10.25. We emphasize this behavior by writing

,

,

In the special case of an ideal stllution. k,, = P;, but ordinarily k , S; P,' Equation 10.2h ls called Ilrtin~'sInti8, and k,, i'i called the Hrrrm's Itrw uonsrnnr c ~ t comporlent 1. As x, -. 0, the cunjpurient-1 ~noleculerare cntnplctclq si~rrt)unded by camponent-2 nlolecules, and thc value of k , , retierls the ~r~terrr~olecular i~~tcmctions hetueen the two co~nponcnts .4s u , + I , on the olher hand. the compnncnt- l

,

,

10-6. 1 hr Gil1t~5nuhem Equatiun

~noleculesarc completely surruundcd by curnponcnl- l molecules, and Pl 1s whal reflccts the interr~~olecular interactions ir~the pure liquid. Although wc havc focusscd

Kcldlt.s Ihr V,qilnr Pressure ut thc T ~ v oC o r i y r b ~ i ~ ~ i l ~

(Recall that thc argument orthe logarithm is actually this fr)mof I L , ( T , P) allows uq lo write

q / f i.where P - is unc bar.) Now

our disvucsion o n component I in Figures 10.7 and 10.8, the same situtation holds for cclnqx)ncnt 2. I!quat~ons 10.25 and 10.26can hc wrilten as and

Th~ls,in a vapor pressure diagram of a solutiutl of two volatilc liquids, thc vapor pressure of each colnponent approact~esHaoult's law as the mole fraction of that component approaches one and Henry's law as the ttlule fraction approaches zcm.

E X A M P L E 10-6 The v u p r preksure (in torrl oFcr)rnpo~~ent I over a bi~larysolution

B u l d x , = - - d x , (kuause .T, -tx, =

is givu11t q

Dcrenn~rie[he Lapor prehsure (P;)and ~ h t He~lry's . law constant (k,, , ) uf pul-e cornpunent I. 513L U I ION: In the limit a h 1,-. I . Thereture.

sa P; = 1x0 torr. As A , r',' k c a u c e r2 4 I ac I

that .r-, + I , the exponenlial factor

-, I hetauhe

1,

I), so Eq~~ation 10.28 hecnnica

which is another t'ornl of the Gihbs-nuhem equatio~~. If componetlt I obcys Raoull'h law as x , -+ I , then P, x , P ; and (8In P , / a x , ) , , , = I/x,#sothe left sideof Equation 10.29 becomes unity. Thus. we have the ccrnditic~n

+0

We now integrate this expression indefinitely to gel

-. 0. on I

Substitute these t w o expressions into the Gihhs-Uutlem equation to get

thc othcr hand, the exponential filctor apprvaches 4 0. Thu.c. we have

In P, = Inx,

+ constant

as

.rl

4

1 or .v: + 0

,

o ~ k,, ~ d = 807 torr

1 Wc will now show thar thc Henry's law hehavior of component 2 as .r, + O i \ a therrnorlyna~nicconsequence ot- the Raoult's law behavior of colnpunetlt 1 as r , 4 1. Tu prove this uunnec~~on. we will slart with rht: Gibbs-Duhetn equation (Eqi~;itioll10. I I )

+

I-,dp, ~ , d & = 0

(oonstnt~t1' and P)

Now. assuming that the vapor phase may be trcatetl as an idcal gas, both chemical potentials can be expressed as

Thus, we scc rhat ~ fcomponent ' 1 obey\ Raoull's law ;is x, + 1, then conlponcltt 2 must obey Henry's law as r, + 0. Pn>blem 10-32 has you prove the convcrae: 1 1 cumponent 7, oheys tlenry's law as x, + 0. then cofnponcnt 1 must obey K;in~~lt'.c law asx, + 1 .

10-6. The Cibbs-Duhem Equation Relates the Vapor Pressures o i the Two Components of a Volatile Binary Solution The ft>llowirlgexalnple shows that if rt7cknuw Lhe vapor pressure curve of one of tf~c compunenls over the entire cornpositiot~range, we can calculate the vapur pressure of the othcr coinprmerlt.

10-7. The Ccmtr,~l Thcrrr~c~dyriarriir Quarllily for Nnr~idpdlSnlutinnr Ir the hrtibilv

41 1

The exponential factor here accounts f o r the nonideality c ~ the f system. The chemical potential of component I i n this rase is given by

In Seclion 8-8. we introduced the idca o f fugncity to preqerve the I I ~ oI f lhe therniodynamic equations we had derived for idcill gascs. We will follow a similar procedure for solulions, uving an ideal solution as our standard. Tu carry over ihe l o r n of Equation 10.32to nonideal solutions. wc dcfinc a quantity called the nrririp by the equation 1 ~ : ' ' = 11: t RT Inn,

1 l C C f R E 10.13

( n l A tcmperaturc-compuc111~>ti di:~gram lllr thc system illustrated In I'igure 10.12. [hI A ~cnb~wrature-cnrrlpsition diagran~f o r n water/pher~oIsgslenr.

( 10.35)

/L; is the chemical potential, r)r the molar Gihbs energy. of the pure liquid. Equaliori 10.35 is the generalization of Equatiun 10.32 to nonideal s~)lulions.The firs! ol' Equations 10.27 says that $ = x,c'. as x - c 1 . If wc suhstitutc this resull inlo Equation 10.3 I. we obtain

where

Fig111-c10. I 3rr look x y mcthanc xt~lutic~ns at 35.2 I', and ~licscrintanrc plottcd i n Fipurc 10.15. Notir'c I h ~ hotli l curvcq approach Raoult'q law a< their ct>rrespontl~nglnrdc tr;~ctionsapprt~achunity. Thc di~shctllincs in the tigure rcprcxcnt Ilie 1inc;ir rcgiol~s;IS tlic currcsponding rnulc fractiotlk apl>rr~ncli7err). The s1npc.s of thcsc lincs givc tlic iicn~y'slaw constant folcnch component. Thc values come clut to hc k,, = I 1 30 torr s l ~ d k,, ,,,,,,rIl, = 1500 101-r. Wc call usc tlicsc valucs iind tlic valucs of thc vipcjl. prcssurc< of thc purc compuncnts to calculate activities and activity cocfticients based upon each standard statc. For example. Table 10.1 gives PC',,= 407.0 turr and Pd,rr,e,,t = 277.8 torr at rrs, = 0.6R27.

,

Thcrcforc.

,\, = 1 Pr, 407.0 torr = 0.701 1

(I(

P:,,

+ I, as x,

0, as can he seen by cumparing Equations 10.41 and 10.42. bhuatiot; 10.41 hecornrs equivnlcnt tcl Equation 10.35 if wc define u, by c,

P Itnrr -

f',,,,c,

0.8.1.35 0 9108 0.9554 I .OOIH)

?34.8 277 h 324.8 340.2 357.2 38 1.9 407.1) 424.3 442.3 458.1 4H I .K 501.0 514.5

37H.(1 7hO.X

342.2

313.3 277.8 250 1 217.4 184 9 124.2 65.1 O.(X)O

10-8

Actlvltic5 Must Hc CI~I~uldleCI w i ~ hR ~ q p ~tor tStandard St,~tcs

and

FlGlJRE

10.15

Vapor pressures of cmbon disullidc

dimcthoxyr~~ethnne over their sulutiuns at 35.2"C. The d i l l hlrilighl lines represent itleal behavior, and the dashed Ilnes represent rhe Henry's law hchavinr lor each crlnlponenl as the correspon~lingmole Iractions approach 7cro

where the superscript (H) simply e r n p h a s ~ ~(hat e s tlleae vaI tics arc hasetl upon a Henry 'b law, or solute, standard statc. Figure 10.1ha shows the Raoult's law, or solven~-based. a c t i v i ~ i ~and s , Figlire 10. l bh shows the Henry's law. or solutc-hiixed. aclivilies p l o ~ ~ e d ngaitlsr the tnole fraction ofc;uhun diculfide. M1c will scc in rhe ncxi ctlapter that i~ solute, or H c n ~ 's y law, slandi~rdstate is particularly appropriik filr ii substarlcc that doc< n o t exist as a liquid ai one bar and a t tlic temperature ul' the bulution under siudy. The activity cr~rfficjenlsbased upon thc Raoull's la^ stnrtdard sstatc (which i c the usual standard state fur miscible liquids) arc pluttcd in Figure 10.17. Nolicc thiit yr,l + 1 as A, --t I and that i t goea 10 2.2 as x, -- t 0. Ruth of these li~rlitingvalucr may he deducei horn the definition o f y ( ~ ~ u a t i c10.37) il

= ( 1 130 1orr/514.5 tori-) = 2.2, in iigreerrlent w it11 FigurclO.17. Thc activity coefficient of dinlcthnxymethane approaches 2.5 as ,i(,,,,,cil, 40 (.rL5, -+ I), i l l agreement with ydlmcrh kH,r,,mc,h/ Pimelh= (1500 torr/587.7 lorr) = 2.5.

k,,-,/T

-

with

w h e ~ thc c superscript ( R ) sirnyly cmphasizes that thcsc values are baked upon a Raoult's law, or solvent, standard state. .Siniilarly.

,,, , 407.0 rnrr acs2 = -= -= 0.3hO k , ,CSZ 1 130 torr

F I G U R E 10.16 (:I) The Kaoult's law irclir,iliec ui carbon disulfide ant1 dime~hr~xyr~>cthanc in carbon rliculfitlz/dimeth~~xy~nctl~ane solutions at 35.2C plnIted against the nlole fraction of-carbondisulfidr. ( 6 )'I'he Henry'\ law activities for the same sysrem.

10-'1.

Can Luristruc 1 .3 Molecular Mudel

Non-ideal Solut1un5

If we divide GE by the total number of moles n , G i h h errerg! nfnirirag,

cE:

I

and show 10.17

we ahkin the m d a r r:rrc.cs

I

E X A M P L E 10-9

Derlvc a forinula For

FIGURE

+ n,,

that

ckCOI

a binary sr,lulion In which the vapor pnlswrcs can he

cr is syn~nlctricahnu1 the 11111: -1, = 2 ; = 1/2

SC) L U 1 I(:) h': According to

Thc Kaouh's law activity culllicicnts 01- carbon tlisulfide (qolid line) and dirncthoxymcthane (dashed line) plo~rcdap~inct.r,,, for carbon disnlfideldrrncthoxy~nethane solutiur~sat 35.2'C.

the above t.xp~.cssionfor PI 2nd P).

Substitute these expws~ionsinto Equation 10.47 to obtain

+

We can calculale the Gihbs encrgy of mixing of binary solutions in terms of the ;rcrjviry cocRicients. Recall frorr~Equation 10.20 that Al),,x t i

= t l ] / ~+ ; ~tI2,1yn - n , , i ;

= r u x , x ~ irx,x:

- r12/l;

Rut, acconling to Equalions 10.35 nrtd 10.37.

which

II we divide Ami,G/K?' hy the - total n u n ~ b c rul' moles, C;~hhrojerxj nJ ~nixir?y.A,,,,xG.

11,

+ n,. we o b l a i ~the~ ntolrrr

ih

hyrrln~etricin I , and r , , and therefore, nhour thc lit~cc,

-- r,

= 1!2.

Many solr~tio~is can be desctl'hcd by the equ;~tion::in Exarnplt: I(&Y, and such solutior~s are callcd regulmr aulurio~s.Problems 10-37 through 1 0 - 4 5 i11volve rcgular solutiotls. e tn calculale U' . Wc can use yrS7 and ydlwllkthat we calculated tilr F i g ~ ~ r10.17 which is shown in Figure 10.18. Note that (he plot of G' vcrsus xCs is nut symmetric about x,.,. = l j 2 . This asymmetry ilnplics that 0 in the elnpiri:al vapor pressure formula ~isedirl Examplc 10-7, and that carbon disullide atid dimetboxyrnetha~ledo not form a regular solutio~lat 35.2'C.

+

'l'hc first ~ w tcrms o it1 Equation 10.44represen1 the Gibbs energy nfn~ixingof an ideal stduticln. To focux or1 thc elTect of nonideality, we definc an pxces.r tiihhs e n c v ? of 1 1 1 i . t - i j 1 ,GI-.: ~.

.

c;" 1% sce

fro111Equation 10.44 that

A . I; mil

-

A,,%G'~

( 10.46)

10-9. NIP Can Construct

d

M o l ~ c u l a rModcl of Non-ideal Solulions

I n this section, wc shall in~mducen simplc molecular rriudel for a non-ideal solution that qualitatively displays many uf the experi~nentalproperties of t~on-idealsululions. We shall assrlmc that the molecules uf components 1 and 2 al-c dislributed randomly

10-9. iVp Call CDIIS~~UCI d MOICLUI~I~ Model ut No111d1edl Scduti#~ri'l

We can locus on ihe nr)n-idciility ol'tllt' sc~lut~on hy introduuirlg rhc qt~;otIit):

Far an ideal solution. E , , = ( r , , 4-~--. , ) / 2 .and 50 vr) = 0. Thc ~nagriitudcof zu. ~hell,ib n measure of the deviation of' ~ h solutic~n e frclrn ideal behavior. Eqilation 10.48 1ahz.t on a fairly silnplc f ( ~ r mif w e clilliinate el, in l'ilvor of 111 using Eyrlatitm 10.49

F I G U R E 10.18 The molar exccss Gibhs energy of mixing of carbun d~sulfideldiructhoxy~~~etI~a~~c hc~lutionsat 35.2 C plottrd againft the mule lraction of carboll disulfide.

th~-i)~~ghout the solutiol], so that the entropy will bc the same as that of an ideitl solution. Therzt'ore, the dit'ference hetween the Gibbs ctlergy of our model sr>lutiolland that of an irlr,iil si~lutiuttwill he due to an energy tcl-m (actually a pnlenlial encrgy term). We exlwss rile potcntiiil energy t ~ the f solution in the form

where IV,, i $ the tiumhcr OK neighboring pair5 of molrcules of type i and j and where is tie interaction energy c~fan i and , j rnoleculc when they are nexl to each uther. Because we have assunled that the nlolecules are ~nndoinlydistributed, w e can dcrive qirnple exprz~aionsfor [he At,, . Lct's cottsidcr ,V,,. the rll~lrlhcrc~fneighhonng 1-1 pairs, first. We car] intcrprel the mole fraclion A , as thc probability that any neighbor of ;b ~nolcculei s a cr)mponerlt I mc~lecule.On dlc avcrage, thc tolal number of type I ncigIih(lrs of any given molecule, then, is givetl by x , , where 1 is the coordination riu~l~ber o f molecules around a central ~nolrcule.Typically z is around 6 to 10, hut we shall not reqt~irea precise vnluc. There arc N , corrlyrjnent 1 ~noleculesin ihe solutiun, so tlicrcfore the rlu1nhc.r of 1-1 ncighhoring pairs is (N1)(zx1)/2. where the l'xtor of 7, is inserted to avoid courlting each 1-1 pair twice. Similarly, we have h',> = :A? N 2 / 2 . Wr use the same value of z because wc assutile that molecular sizes are ktbuut thc s;inlz. The llurnhcr 01- 1-2 ncighhoring pairs is given by zx:Nl, or zx, N?. (he two expressions hzittg zyu~valent.Tlic total interaction energy in the solulian, then, 1s givcn hy

The last term here is equal t o zero when lr! = O and so reprcstrlts the non-irleiil hch;lv~tl~of the solution. Therefore, we can express the Gibhs energy uf the soluticm in the f o ~ m

We can express Fqualion 10.51 1n tertna o t nurrtber'; of iiioles rather thatl nurnbc.~. nf moleci~lesby dlv~dingN , arid N? by the Avogadru curlstant (:V,) to ohtaln

E,!

,VI 7

[ , ' = --

,,

i

is the chemical potentisl ol all idcal ~rll~rtinn [I!rlua

The term (;IC;,,r,,,,';$nl),,,, tion 10.17)

and the derivative in the second tenn in Equation 10.53 is equal to -

an,

,

I

-

II ,II,

01,+n2)'

>tT,;.i-,

22_c?; + nfrom ideal bch;~vior.For negative valucs of 11, the 1-2 intzractinnc are more fwt~rablethan the 1 - 1 and 2-2 inleractions. In this case, the mnolcculea mix rvcll. thus PI-nducingnegative dcvii~tiol~s f-roni idcal (Raoult's law) bchavior. -l.hc followil~gExample shows thal we can use Equations 10.55 and 10.56 to derive all expression for the mol;u Gibhs energy of mixing.

RUI

ux,xf

+ u;r?x: = u ~ , . ~ ~I .(. ~x, I , = u x , r 2

SO

A I,,II

c=R7'(,r,In~, t ~ , I n x , ) + u ~ , r ~

(10.57)

Similarly. Equation 10.47 givcs -F

G

- UX,.T?

(10.581

Prohlein 1 0 - 6 I has you show that

and

The entropy of mixing i q the .same as for an ideal soli~tionbecause we have assunled that the rnolecuies arc distribilted rantlomly thruughuut the soltilion. Urilikr at1 ideal salulion. AmlXH # 0 for a non-ideal solution. Equation 10.57 can k wrillen as

10.19 ,I plot uf P , / I',' rcrslh .r, given hy Equation 10.55 for u / K l = 1, 0.and - I . The valr~e u = 0 i c l d s an idcnl solution. Povitrvc val~~cs of u / H T yicld positive deviations fro111ideal (Knuolt's law) bchavior. and rregative salucs of ii:RT yield negalivc deviations from idcal FlGlJRE

hrha5,ior.

+

Figilre 10.20 shows plots of A , l , , x ~ for / r ~rcvcral values rlt R T / u . Note thal the xlopes of all the curves equal zero at the midpoint, r , = .w? = 1/2. Thr c u r t r tor K l ' / i i = 0 VJi~ qpecial in the \enst thal curves for values uf R T J u greater t l ~ 0.50 r ~ are concave upwards for all values of r , , whereas curves for valucs of R T / u Icss than 0.50 me voncale dowtlward at r , = I/?. In mathematical term$, ~ ' ( A , , ~ G 13.r; / I ~ )1~

F I G U R E 10.20

Plots of A,,,G/rr lr)r R T / u R T / l r = 0.40 (trip curve).

=

O.(a (hutturn curve).

RT,'u

posirive (a t t ~ i n i ~ n u m at) n, = w, = 1/2 for thc curves that lic betow the curve with R T . , u = 0.50.whereas a'(~~,G/cr)/ilx:is negative la maximum) at x, = .r, = 1/2 filr ..me> t h ~ li i t a h n t it TIK replan ahere h 2 ( l ..-> C!u):/h.l-' is negatlle r:prcwnr\ ail ,.,:>~~bl:rcgi,)n I Pr~>bl?rnl M 2 i and 15 31n1ilu IU dx l t ~ > p 01. > t k d:r \VuL ~ y i u r i ~ i,r l n ht R d l ~ ~ . h - f ; u o requatiirn g w k n T < I [ (Figurn 2 S). md 111 rh15 CLW ir,rrc.,p)nd 10 rt~iorlsIn u hiih thc. M-o liquid, ut not miu-ible. lhz critical I-alue RT . t = O.50 ~ , i~ hI lall I t . pmponion. ~t ternptratureb above T = 0.50l1,'H ;111d~rnrnihc~hlr at tzniptrature:. hclnn T = r i / R . Lrt'scr~n~it1c.r thc C L I T Vw~ ~ t hR l /I( = 0.10 ~ IFigure I 10.20.The 1 u . k ) ~~~iriirlla~,cprcsent two immiccihlz solutions in equil ihrium wirh each urhcr The oornpositiot~sof these Using Equation 10.62. two solution\ are given by the values of x , at each mirli~uun~. we

FIGURE

= 0.50 (middl: curvc). and

have

10.21

-

+

I

+

EXAMPLE I s 1 1 C x F ~ u i l ~ i o10.63 n w ~~alculatz thc compjsltiun oi r h ~rucj 1rr~irl1,cihI~. .olur~on, In equilibrium a ith each [*[herat a temperature given by R T / u = 0.41. SOL IJ 7 I ( 1 N , W e uukc l l ~ cNuu'tun-K;~pl~~l rnr~hncl~ h n ruc i~ltnduccdIn M;~thC'l~.lpter A. The functron {(XI oT Equation A . I ic

Equation A. 1 kcorrles

a s Lhe conditior~for the extrzn-ia O K AKc,,,G/u. Firvt 110tt:that .T, = 1 /2 solves Equalion 10.63 for any value of R T ! u , which accouiltq for the fact that all the curves in F~gurt.10.20 have eithcr a lnaxirnun) or a minitnum at .r, = 1/2. F3g plolting (XT/'la)llnx, - In(] - A-,)I ( 1 - 2 x , ) agiinsl .u, for various values ot' h ' l ' / ~ iyuu , can see that only .r, = 1/2 salislies Equation 10.63 h>r K T / u 1 0 . 5 0 . nhzreas two 0111er routs occur fur R T / u i0.50. The two rt~otsgive thc caml>o.silion ol- the two miscible solutions in c q u h i h ~ u r nw i ~ heach other. For (he case in which R T / u = 0.40, the two vill~~es of x, are 0.145 and 0.855. Figurc 110.21 shows the mule Iruclion of cornpoilent 1 in each o f the two imrriiscible solutions as a fil~ictiollof temperature ( H l ' / ~ i Note ). thiit Figure 10.2 1 i s sitt~ilrtrto Figure 10.13.

+

-

A lempraturc-colnposition diagram lur a hinary systcrn for which A,,,,lG!~r ( R T / u ) ( . Y ,In x, x2 Inx,) x,x2 (Equation 10.62). The curve gives the cnnlposltions o f the twu irnmisc~hlesnlut~o~ls as a function uf temperature. There is or~lyotle I>omugeneuuc phace i n the rcgion above the curve and there art. I w n inlrnisciblc solutiuns in equilthr~urrlwith each other in the regitjn heluu, thu curve.

with K I ' / I I

-

0.40. For one of the soIuticjl~s,we htart w i ~ hA , , = O.lo() u ~ l dgct

I

I b r thc other culution. wc st;jrr xith w,, n

,

= 0.9011 and get A

,

,

f'k)

10-4. Apply Euier'\ theorem

10

11 = I:( S . V , 1 7 ) . Do ?nu rccogllizc tllc resulting cqu:ltion'!

1U-5. Apply Eulcr's theorem to A = A ( / ' , I'. ~ r jVu . you rectlgnii.e Ihe rev~lringe q u a i ~ ( ~ n "

10-6. Apply Etller's theure~nto V = V ( 7 ' . P. n , . 1 1 , ) to cleriw Equalion 10.7. 10-7. The properties of many \olutionc are given au a iunc~ionof the mass Ixrccr\t of tlh: corllponclltq. If we let the Inass percent o f curnponen~-2he ti.. then tlcrixe ;I rzl;ilion het~seen,4, :inrl the n ~ u l efraclions. 1,and I ? .

Wc lrlust cmphasixc once again that (lie rcsults of this section are n result of the ai~llplt.t;l~lclomly-dislributrd nod el (fiat wc used. Although the model gives a nunlbcr of-qualitative res~ilts.i~ m u s t k borne in mind that it is just a n :ipproximale m o d e l that n c introduced to g ~ v erume rr~olccularor physical insight into the rlaturr of non-idei liquid binary solutior~s. blc will continue our rliscrlssic>rlof solutions in (tie next chapter, where we focus on hcllut ions ill which thc t w o colnpclnents iue not soluble in all proportiorls. In particular, Re u,ill rlircusk solutions of solid< in liquids, where the t e n n s solutr a n d solvent are ITI~~II>~II~CUI.

10-8. I'he CRL' Ho~rdl~ook of Uic7rni~lr?.(IIZI! Ptfvrrt.~givec Lhe d e r ~ s i t i eelf ~ tr\arly aqucouq sulurio~~s nq n function of the mass percentage of solute. Uure denntc (he dtnsity by p :~nrl the rn;r\s percenhge of cnrr~po~~t.r~t-2 hy A ,.the llrl~actbvokgivcs p = p ( A , ) (in g . lnL I ) . Show that thc quantity V = Or, M I 4 n , M , ~ / ~ ( A is, Ithe volume of the suluiicm crlntairlinp N(nv show that n , rrioles of cnnlponerit I and r t z lrlolcs o f con~pollcl~t-2.

and

I

I -

Problems

Show that

10-1. In ~ h ctext, wc aenr 11,onlEquation 10.5 to 10 h using a physical argumenl involving varying tflc cize of the ky~tcmwhilc keepir~g7' and P fixed. We could ; I I F ~have uscd a m;~!lic~l~ntical prucesf called EuIer's thrortm. Before we can learn about Euler's theorerr\, wc musl lirst delirlc a Ilr~mo~e.er~cnrr.r~rnction. A function j ( t , . z,. . . . . i , %is) said t be II(IIJ~O~CIICDIIS 11

in ;lgreeinenl with Ey11:ltion 10 7 10-9. Thc dcrlsity (ill g.mol I ) of a I-propanol-water solution at 2NC as il function oC ,I2. ~ h c mass percentage or I-pmpnnol, can he cxprcsscd n?

Argue that ex~cnsivct h c ~lnt~rlyrlnmicquantities are hurnogencr~u)I,show that the volume of the ~olutioncontaining rr, n ~ u l e so f c u n ~ p o n e ~I ~; IlI I ~

lnolcs uf comprlnent 2 i s givcr~hy 1' = ( r z , MI rliusc of coln1xmcnt j . Now shuw thal

11,

+ t ~ ~ M , ) j p < ~rvherr , ) , Mi

1\

tlh:

ruolm

10-16. Calculate thc relativc anlounts of llquid and vapur phacec a1 an overall cornpusition of U.5U for one of the pair of values, s: = 0.38 and s, = 0.57, that you obtained 111 Proljlcnl I(bI 4

10-17. In this prohlclrl, we w11l derlve arlalylic expreh+iorlxlrlr Ihe pres~~~rr-c.r~mpusi(lun cr~i-1r.h in Figure 10.4. The liquid (upper) curve is ,just

which is a straight linc, nc sccn i n Figurc I 0 4. Solvc the cyuaurjrl

Show that

for x, in terrrlq o i y; and cuhslilule intrl Equalion ( 1 I tr, ohl;~in

'

10-11. Thr dcrlc~ty( i r l g . t11ol ) of ;I I-propanollwuter solution (11 20 mde 1r;lction of 1-pr~,p;rnul. call he expre5hed as

r us a functiu~~ uf x2, the Plot t h ~ srcsull versus y, and show thal it gives ttlc vapor (1uwt.r) currr In Flgurr 10.1. 10-18. Prow t11at J: tally.

;.

.T:

i f P,* z P; and that

j1 .r

.r, i f P-: c P ; . Irrtrrpret t t ~ i rraull + phyw

where

a,,-- 0.49X23

lJae this expresclon to calculntc the values

10-19. Telrachlurumerhane and trichlomr~hylenrfornl ;In e ~ s ~ n ~ i ;idc,lI ~ I l y !,olu~iotl , ~ t40 C' .II all co~~ce~ltrations. Give11that the vapor prcssure of tetrachloror~~ctt~a~~t. and t n c h l ~ ~ r t ~ c h y lene a1 lo' C are ? I 4 lorr and 138 lurr. respectidy, plot the preshurc-unlnposiliunJi;ly;l~n for tl~i.;systcnl (scc l'rohlcrn 11L17).

u , = -0 171b3

I',,, atld ,

-

VI-l,lnp31,ula$ a filllction of

1,

10-20. The vapor prcssurcsot tctrachlornn~ctha~lc (1 alld lrictllon~.lhy Icl~c121 h c l ~ c c76.8 r~ and 87.2'C cw he expresced empirically by the formulas

r

accurtling 10 the equation in Prwblc~n1W10 10-12. Use the d;11;1 in Ihe TRI- I I ~ r r l l t ) o o kof C-/;~mir.tryc111r1 Physir.s to curvc tit the density ul i~ walcr/glyccml sulution tu a lillh-r~rderpolyllomial i n Ihe ~nolefraclion r l l gl ycernl. arid then detcrlnine the parliol rnnlnr volumcs of water and plycernl a + a tur~cliot~ of mols

and

Irucllorl. PI01 you1 1s,ult 10-111. Just h'fnr? Exairlplc I[)--?. we shuwrtl Ihar 11 orlc cunlpr>nrnl ut il I)I~,II).solulirln ot)cyc Knuult's I;IW r,iCer [he er~tirccolupusitiu~lrange, thc other crlmpjnerlt does also. 5 1, then 1 1 , = 1 1 ; i R T l r l 1, for Nou chou ttxit if / 1 , = 11 $ R T l r l .r: tor .r! ,blcln10 20 to collstnlct thc c1ltlr.c ~ C ~ I ~ ~ C I ~ ~ ~ ~ L I I ~ C - Cd1a~r.ilt11 ~~III~>OS~~~~II 01-a te~rachlo~nme~hanrltrichlororeth~lene suIuIiun. 10-22. Ttic ruprlr pres+ui-eso l hen~eneand ir)luene between 80 C: ant1 I 10 the Kelvin terngratrire are gih-en by the empirical forrnlllas

r a\

a functlr,ii

1,l-

Chapter 10 1 Sululinns 1: Lirluid Liquid Solr~tinns

and

70-30. Suppow the vapor pressurc of thc two conlponcnts of a binary solution arc givcl~by

In[ Fc:51,'torr) =

4514.6 K -

1-

+ 18.347

Assrlmirrg that kn/,enc 311dttlluenc form an ideal ~ ~ I u t i nUSE n . tllcse fonrlulas tu conqtruct :I t e ~ n ~ ~ ~ - ; i ! r ~ r t . - c ~diagram ~ t ~ ~ p ~ot ~ sthis i t icyktem ~ ~ ~ ~ at an at~~hienl pr,cssure nr 7hO turr.

10-23. C'onrtrucc tllc re~nperaturc-oompolc10.7, show that the vapor pressure of dimcthorymethanc is givt.11by Plot ttic?e data to dctcrn~il~e the I Ienry'~I:iw crlnctanr Ttlr elhanr~ltri watt.r or~rlfor 65:ttcr cttianrd 31 25 ('.

Vow p l u ~P, vtrsu?

.i,

a d coinpare thc rcault wit11 the data i n Tahlu 10.1. Do carbon

tliculfidc and dinletlwxyrnerhanr furln n rcgulnr solutioo at 35.2'r:) Plot tltc plot ~yinnieri-ic;11>c>iit n vcrticnl linc :I! r l = 1/2?

10-51. Usin? the data in Prohfem 1&50. plot [he activity c t d h c i c r ~ ~(hnscd ? upnll Knor~tt-s Ian) ol hrlltl t.ltianrd and wntcr against tllc lr~olefraction uf ethanol.

ELag:iinst s,.Ic

I0-48. A lnixtult. uf tricl~lnrcl~ncthilne ;mri ;icelonr N ilh xdcc, = 0.71 3 has a lotal vapor prcssure = 0.8IF; (il\eri of 220.5 trvr at ?X 2 (:, a r ~ dt t ~ ctnolc fraction of i~cetunein (tie vnpor is [hat thc vnynjrprcsstire uf pure trichlororr~ett~atlc nt 28.2':C is 221.8 l r r , cnlculatc the acctih-ity and the aclivily cWfticicnt (based upon a Rar)ult'~law standard state) of Irichlororncthanc in Ilie ntixture. ~ \ ~ < L Ithc [I~ yap01 c b ~ h i ~ itieally ~es

10-53. Surrw 5apor pressure dntn for a 2-prup;1not/benzenesolulitm

a!

25' C are

10-49. Cotisidcr o binal y solution form hich the vapor pressure (in torr)of one 01-thccornpone~lts

(bay corllporieut I l i + give11 cmpiricall!: h!:

Plrlt the activi~tecand the activity cocfficicl~tsof 2-propanol and benzene rzlatiw rr, a Kaoult's law ~t;~ndarrl (lare rercuc the rriule 1-ractiunt1[2-propn[1ol. r;~lculutc111c activity nlld the a h ity coellicien~of cornponcnt 1 ahen .r, = 0.25 hased on ;I rrdveril nllrl a aolutc sta~ldnldstmr.

10-54. Lluing the (lala it\ Fmhle~ri10-53. plot ?;'/KT vcrsus x ? .,~F.,,,~,l. ~~

10-55. .k~cts\fhermm~yntlrrr~c rlurrn/iticr are defined rclativc ~uthe values the rluantitics would have 11- the pure colrlponenls formed an ideal solu~ionat thc samc givcn terriprrature and pt'cssure For exi~~nple. wc haw [hiit [Equaticln 10 47)

at 1,= rl = I!?.

IEt h i ~result consistcnl wllh the graphs you ubtaincrl in P~,i>hlcln10 --ll'!

10-59. Use the data in T;ihle 10. I to plot Figures 10.15 through 10.18 10-b0. Use Equi~l~r,rllO.h2 to show t h ; ~lhc ~ slopec 01- all thu curveh in Flgure 10.2(1u c cqudl ru Lero when x , = u, = I/?.

Show Ihal

-

10-62. 111this problem. we will prove that ( a ' ~ , , , , ~ G j i l must x : ) bc greater ~ h a n,ern i l l a stal>lr. region of a binary solution. First choose - ~ o ~ point n c sy in Figure 10.20 and -.draw a strilighl line at x: tangen1 lu the curve of An,,%G against .TI. Now wgue (hat Aml,GI I I U S ~lic nhuvc thc tangent line fur the wglon around tu he spahlc, or that

Show that thc cquation for the tangent line is

-

Now expand A,t,,,Gin a Taylor series (hlathChapter C) n h x t thc point sj' and show that

for thc binmy ssolutivn to be stable. for d regular soluliun (see Roblcrr~l lL37) 10-57. Example Ifl 7 cxpressek I he vapor prcasure:, rd the two components of a hirlary avlution as

Shun lhal these

CXPICSSIOI~!,

are equiv;rlent to

~ s i r l gthcse expressions for 1t1c ac~irity ~ w t h c i c ~ ~Jtrive t \ . an exprebsio~lfur fiLI n tcrtlls ul'n and p , Show thai your s x p r c s s i o ~reduce, ~ lo that for kttor a regular \ol~~liun.

10-58. Pr0r.c thal Ihe maxima or n l ~ r ~ ~ufr ~A,,,,,C la dchr~edin Frr~hlcm10 -37 occur at x- = 1 !2 for :iny value 01- K1':u. NOW pnlre that

"

>0

a' A , , , , ~ E -i1.r;

- (I

for R T / u

;-

0.50

f u r R T I I I = 0.511

A-,

=

Solutions II: Sol id-Liquid Solutions

Peter Drbye (Icft) w;tu horn i n M:rilctrivht. the Nether1;tnds. on March 24. IHX4 and died i l l 1966. Dehye u a \ rli-igir~ally lruir~eda$ an electrical etlgincer but turncd his attelltion to physiclc tu presenl exact expressions fnl- the activi~iesand activity cucfticients in dilute solutions of electrolytrs. I n Sections 11-3 and I 1 -4, H e will discuss the colligative properties of solutions, such as osmotic pressure, as wcll its the depression of the freezing point and elevation of the boiling point of a solvtrlt by the adlditic~nof solute.

11-1. We Use a Raoult's I..aw Standard State for thu Solvent and a Henry's Law Standard State for the Snlute for Solutions of Solids Dissolved in Liquids 111 Scction 10-8, we considered sulutiona in which one of the compuncntq is only sparingly soluble irl the other. Tn cases such as ihese, we use thc tcrlns solurr fur the sparingly soluble component and sol~~errt for he cutrlporlerlt in cxccss. We cus~oninrily denote xolvent quatltities by a subscrip1 1 itnd solute quantities by a qubscrip~2. The activities w e defined for the solvent and st~luteart: such that 0 ,+ x , as x , + 1 and + x2 as +* 0. Recall that a , is dcfined with respect to n Raoult's law s t a n d ~ r d state (Equation 10.39)

a, =

11 ' -

p:

(Rauult's law standard state)

C h a p f ~ r1 1 ; Sulut~onrII: S n l i c l - L i c l t ~ r tSollrt~unr l

11 -1

-

p2

Rauult's a ~ ~I rlenry's l Standard Sldle

Ld\*is

We define the solute activity in ternis of molalit hy requirit~gthat

nt~dthat a , ic dcfiried with respcct to a Henry's law standard slate (Equation 10.43)

(L., - -

.

(1 1.2)

(Heriry's law standard slate)

kH1

whcrc the subscript .r enlphasi~esthat m 2 , and kl,,r ilrz bascd on a mole fraclion scale (P: = k , , xx,). Even if the soll~tcdoes nut h ~ v ae measurable vaporpressurc. defining the acl~vilyby Equation I 1.2 I Snevertheless crmvcnient because the ratio is still meaningful; evcn though 1': and k,,? may be exveedir~glysmall, the ratio P,/ k , is tini~e. Although w r havc defirlell the activities of the srdvcnt and solute iri terms of ~llole f~ractiuristhe use of mole fractions to exprehs the conce~~tration of a snlule in a dilute solutiur~is not numerically convcnicnt. A rtinre canvcniei~tunit is molrilifi (rn), which is delir~edas thc number ot.mt>lcsof s o h ~ t zper 1000 granis of solvent. In ill1 equation, we have

,

where the subscript In ert~phasizcsthal a?,"is bnscd on a molality scalc. \jie can exprcs5 Henry's law in terms of the molality rather than thc mole fraction by P: = k , , ) , , I I I . where once again [tie subscript ni eniphasizcs that k , ,,,, is basud 011 il ~ ~ ~ o l : ~sc;llc. l i t y In terms of k,, ",, the ac~ivityof the aulute i s defincd by

Another common concentration unit is trlolurity (r.), which ih the numbcr o l - n ~ r ~ l z s of solute per 1 OH) n1L of sdution. 111 an equation, n,

=

(11 Kl

.--

lO(l0 inL snlutic~n whciz 1 1 , i~ the number uf rnt~lesof solu~e(subscript 2). Note t h a ~the units of molality are ~nol.kg-I.We say that a solution containing 2.00 n~oleso f NaCl in 1.00 kg of water is 2.00 molal, or thal it is a 2.00 rtlul. kg-' NaCl(aq) solution. The relation between the mole fraction of snlutz (x,) and molality ( n t ) is

Note that lnolarily has units of rnu1.L I . Wc say that n solution containing 2.00 n1(>1e> of NaCl in 1.00 liter of solution i s n 1 . 0 h n o l i i 1solution, ~ or that it is u 2.W rriol-I. ' NaCl(nq) solution. We define the solute activity it) tenlls uf moli~rityby I-cquiring that a,,, + c.

wherz M I is the molar mass (g-r r i t > l - ' ) of thc solverll. The tern] 1000 g.kg I / M , is the number of moles (Tfbolvent ( n I ) in 1000 g of solvent and m , by definition, is lhe numbcr of moles ~ )solutc f in 1000 g of solvent. In tIiz case of water, I O(K) ~riol.kg-'/ M, is equal 10 55.506 mof .kg-', Erluatiorl 1 1.4 bccu~nes A-

nl -I -

55.506 mul-kg-'

+ In

Note h a t w, and nt are directly proportiot~alto each other if which is the case for dilute solutions.

1

tn

I' a 2.450 mol .L aqueous sucrose solutinn nt 20r is 1.3103 g .rn~.-I. 'Thus, thcl-e art' 838.6 g of sucrose. in 1000 1nL uf solutiun. wl~ichhas a total mass of 13 10.3 g . Of ihcse 13 10.3 g. 838.h g are due to sucrose. so 13 10.3 - 838.6 g = 471.7 g i ~ dilz c to water. 'Thc ~nolalitythe11i s given h!:

'

I

E X A M P L E 11-1

C'alculatc the mole fr:~ctionof-a 0.200 mu1 .kg-' (:I. ,H,,Ol, ( a q ) solution. . . 5 0 I U 1. 1 O N : The sulutiul~contarrls 0.200 mules ot wcrosr per 1000.0kg of water.

The iuolc fraciion of

sucre; is

I

E X A M P L E 11-2

The density

[ill

f - m ~ , - of l ) nrl nqueou~+ucroEI: c01t111011L A I ~hc cxpresbrri

ah

r h ~ p r e r1 1 I Snlutluns II: 50l1d-Llquid Sulutiuns

442

C,,I 1(ul;~te , thc mularitp of a 2.00-n10Ii11aqueous sucrose colulion.

Table 1 I . I sutnmarizcs the eqilations for the activities we have delined for the various conccntrution scales. In each casc, the acLivily coefficient y is dcfilied by dividing the activity by the appropriate concentration. Thus, lor example, ym = a , , " / ~ n Prob. lem 11-12 asks you to derive a ~ I a t i o nhctween the v a r i c ~ i ~solute s activity cncfticienls in 'Table 11 - 1 .

S (.) L U T I U N : A 2.IX)-molal ;Iqlleous sucrose svlution conpiins 2.W n~olzs(b84.bg) of sucrosc pcr 1000 p of H!O, or 2.(M nlolcs of sucrose in 1684.6 p o l solution. The dcrlcily nT 111c~ ~ ~ l u t is i ngivcri n by

11-2. Thc Activity of a Nonvolatile Solute Can Be Ohtaincd imrn I ~ F : Vapor Prcssurc of the Solvent T h e cqu:itions frlr the solute nctiviticx in Table 1 1. I are npplicnhlc. t o non\ol;~lilr as well a< volatile suluies. ?'he vijpcjr pl-cssure of ;I r ~ u n ~ o l a t i wlr~tc lc is so low, hourrer. tllat tllcsc cquaiic~nsare not practical tu use. Forlunately. thc Gibhq- Duhern q u a tion provides us with a way t o dctcrniine lhe activity of a nonvolatile s o l u ~ eli;clrrl a measurement o f the aclivity of the solvent. We will illustrate this prnccdurc using 2tr1

I hcrcfore. the n~olaritpof the solution is

t'rohlct~~1 1-5 asks you to dcrivc n gcncral rctatiol~between c ilntl m.

I

11.1 A sumrnary of thc equation.; fur the activities uccd for the varirlr~scr)ncentra~ionscales for dilute solutiuns. TABLE

I

E X A M P L E 11-3

Inw standard state

Solvent-Raoult's

.

-

11

( i ~ v c t l~~ cclct~sity1 0 ) ot tilt s01~ti1i11ill fi.n~L-I,dcrivc n gcllei-al rel;~lic>nhetween x, a r ~ dr,.

5 0 I U I 1 (.) '4: Cunhirler exactly a one liter sample uf the solu~iorl.111this casc, c = I : , , the nuniher- of n ~ o l eo f~ {rjll~tein flie one-liter sample T h e 1rinw of Ihc solution is yirrn by

Sr~lutc tlcllr-y's law standard stale

--

-. .

Molc tr~ctionscale mast

\(I

(IT the soluliun per liter = (IDCIO ml:~,-l)p

- ,

the rn;ls< o f the s o l ~ e n i\t In;].;\ of the colve~ltper liter

2=

nlass of the soiution - maw 01-[he solule

= (1000 n1L.L ' ) p - rM: wl~err.JW, is the

rr~olar~ r l i ~ \f he - ~ ~ 1 t > 1 - ' ) o i [he rulute. Therefom, n , ,thc llurnber of

inoleu rjl sulvcnt, i.;

(I2v,

= - p,--

Lrn

-;

YZ,,,

1(21.

l7:,"

4

3< 111 +

rn

0

P, . + k,, ",m a< m + O

(Henry's law)

Mrliarity scale n,

+ I-

ac

P2 3 k l l , ,c

r, + O RS

c +0

(Henry's law)

Chapter 1 1 / Solut~uns11: S n l ~ dLiqu~d Solutions

iIqueous solution of sucrose. According to a Raoult's law standard stare. the activiiy of' the water is giveti by P , / P , * .Now let's consider a diltlre solutiotl, irl which case a, : :x , . We now want to relate a , tu thc ~nolality of the solute, t n . For a dili~tesolution. III < < 55.506 mol-kg - I , LO we can neglcct m compared with 55.506 mol-kg-] in (he dcllolninatnr of Equi~~ion 1 1.5 and write

T A B L E 11.2

The wprlr pressurc of water (PI) in cquilihriur~lwith an arlueutls stlcrore ar)lullori 81 25°C as a fu~~ction uf molality (n~). Add~tiollaldata are the achviiy o f thc water r n , ) . the umotic crlctficic~~t ($), and the acti>ily coeffic~cnt( Y , , )~of the sucl-ace.

Therefore, for small concentrations.

where we have used (he tact that In(1 - .r,) 2 -x, for small values u f s , . Table 11.2 and Figure I 1.1 give experittientnl data tor the vapor pressure c~fwatcr iri equilibrium wilh an aqucui~ssucrose solution a1 25-C: as a funclion of molality and ttlolt. fratlion. rcspcctivcly. The eqi~ilihriumvapor pressure of pure water at 25-C is 23.756 torr, so u, = P , / P ; = P1/23.756 is gircn in thc third cululnn of Table 11.2. Equalion 1 1.12 rclotes ( 1 , to the rnolality w ( ( ~only r a d i l u ~ esolution. For cxa~nplc, Ltble 1 1.2 .chows that a, = 0.93276 at 3.00 molitl, whcl-eas Equalion 1 1 . I 2 gives It1 ( I , = -0.054048, 01. n , = 0.9471. To accuurlt lor this discrepancy, w e now define a qi~ari~ily 4, called thc nsntotir>roqfiricnt. by

Nut? that # = I if the sulution hehaves as arl ideal dilule solution. Thus, the deviation 01- 4 horn unity i s a 111easureof the ~lonidealilyol' Ihe scllution.

EXAMPLE 1 1 4 Using the d:bra 111Table 1 1 .?, calculale thc valr~eo f $ at l .On r r i o l kg-' S O L U T I(>N : We simply use Lquation 1 1.13 arid find that 'wager

" F I G U R E 11.1

Thc vapor ple\wre of wilter in cquillbriun~with dn aqucuus sucruse cnlution at 25 'C plotted again\( the ~rlolefractiorl nt water. Nore that Rao~llt'slaw (thc straight line i n thc figure) holds frrml I = 1.oU to about 0.97, hul [hilt devintiolls occur at lower valucs o f r s,,kr

in agreenicnt wlth the entry in Tahle 1 1.2.

11 -3. Culligativc I'ruperties Art. Solution Properlies

t~:,obtain

z:

where we have uscd the fact that 77: = A,,,,HIhr rllc port solvcnt. It' we intzgr=~tr: Equation 1 1.17 from pure solvent, where a , = 1, T = T,;,,to a solutiun with arh~lrarj values of a , and TtIII,me obtairl

F ~ C ~ U R 11.3 E

The lugarithlu of the aclivlty c u e f l i c ~ e ~(111 ~ ty,,,>) of sucrose in an aqueous sucruse aolution ill 25-C plotter1 against the molulity ( r r t ) .

11-3. Culligative Properties Are Solution Properties That Depend Only Upon the Number Density of Solute Particles A 11umbernf solution propelties, called c'uiiigriiz~,tp , q ~ r r t i e depe~ld, ~, at least in dilute soli~tion,upon only the number of scllute particles. and ntjl upon their kind. Colligative

properties include he lowering of the viipor prcssure of it solvent by (he addition of a soluie, the elevation of the boilill? point of a solu~ionby a nonvola~ilesolute. the dcpr-cssionof the freezing pomt of a solurion b y a atllute, and osrnotic pressure. We will discuss orlly frzrzing-point depression and osmotic pressure. A t the freezing point of solu~iorl,solid solve~ltis in equilrhriurn wilh 1t1e solvent in solution. 'The thcnnodyt~amiczorldition ol. this equilibrium 1 7 ~ t ~ t

\t71icrcas usual the subscript 1 clerjotcs sr,lvt.rjt iinri T,", is the freezing prjirlt of the solution. We use Equation 10.35 tor 1 1 , t o obtain

We have wntten p\ tor

~ 1 ;sllnply

Equation 11.18 can be uscd t o detcrrnine thc aclivity of the solvent in a solution {f'roblem 11-20). You may have calculated freezing-point depressions in gcncral chcmi\lry using [he formula AT,,,,= K,m

( 1 1.19)

where Ki i s D constant, called thefre~zitrg-/joitlrrlepressian coilstant, whose valuc ~ l c pends upon (he solvenl. We can derive Equation 1 1.19 from Equation 1 1.18 by making ;I few approxi~natinnsappropriate to dilute solutions. If the solution is sufficiently dilute. then Inn, = In x , = In( I - .r2) % -x,, and i f ' we assume that A,\,,His in depend en^ of ietnpcralure over the telnperiture range (?it,,,Ti,),wc obtain

- 7 ,:; iir'irirt

yt hy

Again. r~oiicethat the s u m of the exponent.s on both sides of Equations 1 1.38and 1 1.39 are thc samc. Given these delinitions, w e can now write Equation 1 1.37 as

E X A M P L E 11-8

Writc out Equatiorl I 1 40 c.rplicitly lor CilC1,. SC) L 1J TIC) N : In this tn)n

fit. hee

case, v+ = 1 and

that m , = III and rlr _

v-

= 2. Furlher~nore,accordillg to thc equa-

Mean ionic activity coefficients call bc determined experimenlally by the samc methtds used for the activity coefficients of nonelcctrt~lytcs.Wc will illustrale their determination froln the measurement of the vapor pressure of the solvent as wz did fix an aqueuus sucrorje solution i n Section 1 1-1. 111anslogy wilh Equatiun 1 1.13, we defiriz an osmotic coefficient for aqueous electrolyte solutions by

Notice that this equation differs from Equation 11.13 by thc inclusion of a fautc~rol' 1, here. Equation 1 1.41 reduces to Equation 1 1.13 for t~onelectrolytesr>lutionshecauw 11 = I in that casc. Problem 11-34 asks you to show that with this factor of 11, C$ -+ 1 as m + 0 for solutions of electrolytes or 11une1ectrolyte.s.Scarling with Equation 1 1.31 and the Gi hba-Uuhem equation, you can dcrivc ihe analog of Equalion 1 1.15 straighlfrjrwardl y:

= 2 m . Thur.

Table 11.4 gives the vapor pressure c ~ an f aqueous aulution of NaC1 as a functior~ of tnulality. Also included in the table are aclivilies of the water (calculated from Tllc rclntlons bctwcen Tahle I 1.3.

(I,.

m , i111r1 y, Inr uther ty~wsof elcctrolytc~are givcl~in T A B L E 11.4

The u p u r prersurt: ( P ), act~vityof thc water (aw),osrnoric coefficient (4). H ~ U and log~trlthn~ of the mean ionic activily coefficienl ( I n y,) ot thc NaCl in an aqumo"9 9olutlon of NaC1 at 25-C as a function or nithalily ( m ) . TABLE

11.3

lhe relations between the aclivi~y01- a tin cl!' is also l i ~ ~ cin a r,nl:' (Problem \ I - 5 ) . The quantity K in Equation 1 1.50is a cerl~ralyr~a~ltity in the Uebyc-Iluck~-1theory and has the following physical in~erpretation.Consider an inn with charge q, situated at the origin of a spherical coordinate syswn. According to Dehye atid Hijckcl (see a l ~ oPmble~n11-51), the net chargc in a spherical shell uf radius I- and thicknehs dr surrounding Itiis central ion is

If we inregrate 111;sexpression fro1110 to ,x.we ob~ain whcrl: r , is the molarity uf Lhe jth ionic s l ~ c i e sin , which case (Pmblcm 11-46)

This result simply say5 lhat the total charge surrounding an ion of charge q, is equal and of the opposite sign t c ~q r . h other it expresses thc eleclrooeutraiik): r)f llle E X A M P L E 11-11

First show rhat unillcss. as it

K huc

units

rll-

111

' ;~ndthcil shuw that

In y,

111

Equatior~1 1.-i9 is

be.

5 ( )I. I J T 10N : \Vc star1 % ~ t hLlualitl~~ I I ..iO Thc u ~ l i t c o q, f are C. s,, a r c ~ ' - \ ' , I : : I. m-', = kg.rl12.s-:-~--'. T are K,and N , / V are rK3.Therefow, thc unilc of A - ;ire

A,! :ue J - K - I

Using Equation 1 1.19 for h y*. F I G U R E 11.6 Values of In y, versus rn':' fur aqueous alkali halide +olutiuns at 2 5 C . Notc that even though the four curves are different, they all merge into one. the nrhye-Hiickel limiting law

(Equation 11.49) a1 small concentrilticlnc.

461

C:hsr~!cr 1 1 / Snlutiuns II: Colirl-Lrt(~~irlSolutiur~s

s~hutiun.Equation I 1 53,which IS plotted in Figure 11.7, xhows that there is a diffuw shell of net charge of opposilt: sign surround~tlgany given ion in solution. We say that Equation 1 1.54 describes an iunir utmosphere about the central ion. Furthermore, the In~utlrnilmin the curve in Figure 11.7 occurs at r = K so we say that K - ' , which Evaniple 11-1 I s11rlu.s tins units of rn, is ;I rneasure of the ihlch~casof the ionic atmosphere. For a 1-1 electnllyte in aqueous solution at 25'C, a handy formula for K is (Prohletn 11-51)

',

where r. is tbc molarity of'ttle solulion. The thickness of the iotlic atlnosphcre in a 0.01 0 tnoli~rsolution is approximately 3000 pm, or nh(~irtI0 times the s i ~ of e a typical ion. For an aqueous soiution at 25'C. Equation 1 1.49 hwomcs (Problem I 1-59)

According to Equalin11 11.52. [ related t c ~the ct~nuerltr;ition,but the relation itself rlcprrjds upon the type of clectmlyle. For exatnplc, for a I - 1 electrolyte, z+ = 1, : = - 1 , r _ = c , and r. = 1.. so I = (,. For a 1-2 electrolyte such as CaCII,, z+ = 2. : = -- I . r-+ = r.. and(, = 2r,so 4 = f (4r $ 2 c ) = 3r. Generally. I, is equal to sorrje i~u~ucrical tictor limes r.. whcrc the vitiue of the nun~ericalfactor depends upon the type of salt. Thercfure. Equation 1 1.56 says lhat a plot of In y, versus r'" should be a f straight line and thal the slope of the line should depend upon the t y p c ~ electrulyte. The slope will he - 1.173 for a 1-1 electrolyte and -(1.173)(2)(3':") = -4.06 for

a 1-2 elect~olyte.Figure I 1 8 s h o a~ plot ~ of In y, bersus rl,' lor NaCl(aq) and CaCl,(aq). Notice that the plots are indeed linear fr)r smaH concenirations and Ihar dcviatioris from llnear behavior occur at higher concentrations [ ~ l x' ~0 05 mol L-' ob c = 0.003 ~nol.l,-' Cor CaCl,(aq) and r'" 0.15 mol-id-' or c = 0.02 n ~ o l . ~ , for -' NaCl(aql1. The slopes of the two linear portions are in the ratin of 4.06 to 1.17

*

11-7. The Mean Spherical Approximation Is an Extension of thc Debye-Hiickel Theory to Higher Concentrations ?'he Dehye-tliickc.1 theory a?r~111es (hilt the ions arc simply point io11s ( 7 ~ 1 . 0riitlii) and that they inler8ct with a purely coulonlbic potential ( U ( r ) = ~+:-r'/4xt,f,rl. I n addition, the solvent i s considcrcd ;I co~~tinuous tnediuln with a u~liforn~ rebtivc permittivity F , (78.54 Il)r water at 25°C).Although the assumptions of point ions ;111d a continuulr! solvent may seem crude. they are quiie sntishv~orywhen the iau arc far :ippart from each othcr on the average, as they art. in \cry dilute solutions. C:onsequcntly, the Llebyc-liiickel expresqion for In y, given hy Equation 1 1.49 is exact in the limit of srnali corlccntr~tions.Thcre is no correspoitding theory for solutions of noncleclrolytes because, being ~ieulralspecics, nonelectrolylt. molecule^ do not intcract with each olher to any signilicarlt cxtent until they approach cacl~othcr ~rrlalivelyclosely. whcrc the solvent can hardly he assurned t o be a co~ltinuousmediuln. Figure 1 1.R e~nphasi~es that the Debye-Iliickrl theory iq a lirrliting law. It should not he considered a quantitative theory with which to calculare activity coet licierits exccpt at very Iow concentratiuns. Nevertheless. the Debye-Hiickcl theory has played ari i~lviiluahlerole as ;i strict limiting law that a!l electsolyte srdu~ionsobey. 111 addition. any theory that attempt^ io describe solutions at higher concentrations must reducc t o Equation 11.49 for s m a l l uoncentraticms. Many attempts have heen tnade tu construct

F I G U R E 11.7

F I G U R E 11.8

A plclr of the nel charge in a spherical shell of radius F and thickness dr s~lrroulldinga central i r r r ~rjf charge q:.Thi* plot illuctratcs the ionic atmusphere that su~rvuncl~ coch ion ill solutiun

A plot oCthc logarithm of the mean ionic activity coefficient (In y,) for NaCl(aq) and CaClltaq) at 25°C versus Note thal both curves approach the Debye-Hiickel limiting law (the ctrnight lines) as the moiarity prlcri to zero.

Chapter 1 1 / Sulut~uns11- Solirl L~quidSolutions

Most of these theories that have h c c ~dcvclopcd ~ rcquirc nu~tlericalsolutiot~s10 lairly complicated equations, but one is notable in that it prrlvidcl analytic exprecqiclns lor the variuu:, thennodynamic propediec r)l- elec!rolyle solutions. The nun~t.o f this theory, the mean spherical approxitr~ation(MSA), derives liorri i ~ oiiginal \ formula ti or^, and the theory can be viewed as a Debye-Hiickel tlieury in which the linitt. (11onr.m)) size of the ions is accounted for in a fiirly rigon)ux n3annt.l. A central rchult of rhc mcxn spherical iippruximation is that

thcrjrizs h ~ more r ctlr~uen~rated electrnlyte solutions, but nlosl have mct with only linli~edauccess. One early attempt is called the Extended Debye-Hiickel theory, in which Equation 11.49 is ~nodificdto be

Thi:, expression becomes Equation 1 1.49 in Ihe litnit of small concentrations because !.I1' bttcornes negligible colr~paredwith unity in he denominator of Equation 11 3 7 in this limit. where In y2' is an electrostatic (coulomh~c)contlihutiun to In y+ ant1 In ytiS is u hardsphere (finite-size) contribution. Fur solutions of 1-1 clcctrulytcs. In y:' is givcn by E X A M P L E 11-12 Use Equariun 1 1.57 10 calculate 111 y + for 0.050 molar I.iCl(uq), and cornparc the result with that obtained from Equalinn 1 1.49. The accepted expt.rimenta1value is -1). 191.

where p is thc number density of chargcd palticlcs, d 1.; thc sum uf thc r a d i ~ rof a calion and an anion, and x = ~ dwhere , K i . given ~ by Eqiration 1 1.53. Although it is oot obvious by casual inspection. Equatiot~1 1.59 reduces to the Dchye-Hiickrl l i tnitil~g law+ Equation 11.49, in the limit of small col~centrations(Prohle~nI I (A)). The hard sphere cr~ntributionto In y, is given hy

S O L 1J T I O N: For a 1.- 1 salt such ax LrCl. I' = c., so

lny = -

l . 173(0.050)]"~

= -0.214

1 + (0.050)~'~

Although Equation 1 1.57 provides some irnprtrvenlerlt over the Debye-Auckel limiting law, st is not very accurate even at 1).050 ~uolar.At 0.200 molar, Equatior~11.57 gives -0.362 for In y, vcrsus the expzrimenlal value of -0.274. I

i

Another .cemienjpirical expression lor In y , lhat ha^ been widely used to fit experi ~ ~ i e ~data ~ t aisl

In y- = -

I.I7B[z+z I(I,/inol,L

I)'"

1 -t (I\./ r n o l . ~ - - ' ) " ~

+ C'm

where C is a parameter whose value depends upon the electrolyte. Although €quation 11.58 can he used to ht experimental In y, data Up to one molar or so, C is still strictly an adjustable parameter. 111the 1970s, significant advi~nceswere rnade in rhc theory c~felectrolyte solutions. Most of the work on thehe theories is based on a model called the pri~tritivemodel, in ivl~ichthc inns are considered I~nrdsphcrcs with charges 211 their centers and the solvent is considered a cgntirlunua rnedlu~nwith a unifur~r~ irlntivc p r ~ n i t t ~ v i t yIn. spite of the obvious deficiencies ul' (his ~rlodzl,it addresses the long-range uoulumbic interactions bctwccn the ions and their short-rangc repulsion. These turn out to be ttlajor cor~siderations,and as we will see. the primitive model can give quilt salisfactory agreement with expzrimer~taldata over a lairly large concentration range.

whcre y = n p d 3 / 6 . It1 spite of the fact that Equations I I .hO and 1 1.61 are sotnewhat lengthy, they arc. easy tu use becauhc once d has been chasen, they give In yf in tzrlns ot' (he mcll;~~-ity rm. Figure 11.9 shows experi~nentalvalues of 111y, for NaCl(aq) at 25'C and 111y, ;ls calculated frorri kqi~ation1 1.59 with d = 320 prn. Given essentlall y one atljuslable parameter (the sum (>I' the ionic ~ a d i i )the . agreement is secn to he quite gurld. We alsr) show the results fr)r the more ct~rnlt~only seen Equation 11.57 in Piguru 1 1.9.

Problems 1 1 - 1 . The de~witpof ;I glyccn,l/water solutiun thal is 40.0% glycerrll hy mas\ i\ I . 101 g - m l . ' :)I 20 C. Calculu~ethe niolalily urld the r~lolarityuiglycerol in the \r)lulion a1 20'C Calculaie the rrlr)lality ;IL U C. 11-2. (:onccnlratcd ccllfuric acid is sold as a st>luliorithat i > YX.O!'r su1t~11.i~ ncld :111d2.0'c wntcr hy Inass. Given that the density is 1.84 p-~nl:~, cillculi~tethe rlialariry clf crlncenlralrd sulfuric wid.

11-8. Derive a relaticm hetween thc 1uo1e fraclion c ~ lhe l snlkent and the mulalily o l a holutlon. 11-9. The volume uf ari aqueous sodium chlol-ide srllu~ic~n a1 25 C call he eepre\\ed as

V/mI, = 1Onl 70

+ (17.298 kg mol

')mi

+ (0.Y777

kg'.mol

')rl12

0 5 nt 5 6 mol-kg

wllcre rrr chloride.

F I G U R E 11.9 h ct>n~parisonof In y, from the mcan sphericai approximalion (tquatiun 11.59) with cnpel-imental data for N a ~ l l a q )at 25-C. The line labelled EDH is the extended Debye--Hiickel thet~iyrcsult. E q u i t ~ i ~ 1l r1.57. ~ The value of d , the sutn of the radii ul-the cation and anion, ic

t n k c ~to~ k 320 pnl.

I 1 -3. Cr)ncentratcd phouphoric acid is sultf as a solution thal is H5% phusphoric acid and 15% u.;ltcr by tnnqs. Give11 lhar thc lnolarily is 15 r11o1.L I . calculate the dznsily rllconcel~tratetl phnsphnric acid. 2 1 -4. Calcul:~[etilt 111olc fraclion (IT flucove in an aqueous solutirln thnt i~ 0 . 5 N rnrllal irt

glucose

iq

the

'

molality. Colculntc the mnlarity r>T n soluliur~thnt is 3.00m r h r in s o d ~ u r r ~

11-1 0. I K x T , ?a". nnd c" are the mole Iract~ot~, nlolnlitv. onrl mtllarity, respectively, of a solute at infinite dilulic>n,show that

where MI is the molar mass (g,mol-I) and p, is the densily ( g . r n ~ - ] )of the solvent. Note that n101e frttvtiun, molality, ntld 1no1:rl-it?are all rlirec~ly~jroportional tu each other at low cwnuclllrill~r?ns 11-11. Consider two '.;nlutionu whosc solute activities are and cry. ~vferredto the xime standard statc. Show thal the dilferencc itt the che~nicalpolentiall., o f tl~csctwo solr~tiuns is indeprrlde~ltof tllc standard slate m d d e p n d s only upon thc ralio u i l u ; . Now choose une o f these soll~tionsto he at an arbitrary co~lcc~ltration and the olher at o vcty dilute cr>ncentrnrion(csscntially infinitely dilutc) and argrle that

11-5. Shou that the relalion hctwcenmt~larityand nlolalily ior asolutiwn with a singlc solute is 11 -1 2. Uhe Fquatioos 11.4. 1 1.1 I , and the r z < o l to~f tttc previo~lstwo problems to show that where r , is t l x rncllarity, r n is the mulality, p is the density of the solulirln in g , ~ n L',- and M, i u the rrtoliu Inass (p mtjl-'1 of the solutc. 11-6. The L'KC Hanrfbook of C/~emi.v~ry rrrtd P&sic,s has tablcs of "'conuentrativc properties 01aqueous solutions" for many solutiunri. S o ~ n centries for CfCl(s) are

where 6) iic thc de11sity of the s u l ~ ~ t i oThus. n. we qcc that the three tliffrrent activity coeftciel~ts;(re related 10 nnc at~otl~cr.

11 -1 3. Use Equations 11.4, 1 1.1 1, and the reuultf of Prt>blcln 11 -l? to derive

Given that h t: density nf an aqueous citric acid (IU, = 192.12 g . In01

')

solulion st 20-C is

given by

where A i.; the rrlass pclccnt nf the solute, p is the density of the srjlutiorl. and r is the ~ilolarity.L.\irig Il!csc d a h . calcl~lnrethe ~rlrjlnlityn! cach concerrtr;~titi~!. 11-7. Uct-ive a rrlatiot~ between the Inass percenvage ( A ) of a svlutc in a solution and its rrlulality (m). Calcul;ltt the rnulillity of an aqueous silurofe solution that is 18% sucrosc by Illasc.

plot y,,,,/yl, versus r.. IJp to what concentratir~n

yz),,and y2( differ h) 2 W ?

Chaptcr 1 I 1 Solutions H:

hulirl-l

irlllial

Solutions

'1 1-24. Calculate the valuc of the freezing puinl depression tonstam lor ni~rr)bcnzene,whobe free~ingpoint is 5.7'C and whwe molar cnthalpy uf Cusion I$ 1 1.59 !d.ni(T I.

11 - 1 4 . Ihe C'RCH(~ndbmkr,SC'hemrst~onrlPhysrrs gives a table of mass pt.rcenI rlf sucrose in all aquerrus solution and its corresponding rnolanty at 25'C. Use lhese data to plot mulal~ty rcr,sus molaritg fi)r an nqrlcous sucrrlhe solution.

11-25. IJst all argument sin~ilar10 illz nnc wc useti derive Equat~ons1 1.24 and 1 1.25.

11-15. llhit~gthe d;~t;l in Tahle 11.2. c;ilculale thc octikity cueUflcic.llt of wdtrr (on a rllolc fraclinn basis) at a sucrose conccritratiu~~ of 3 (XI molal.

t l l rlr

Equotior~s11 2 2 and 1 1.23

to

11-26. Calcuklte the hoilrr~gpoint clcvatio~~ consIan1 I l ) r cyclubexi~nsgiien ~ l ~ T%&,' a t = -434li

11-1 6. llsinp tile data in Table 1 1.2, plot the aclirily uocfficirnl r)l wattr against the mule frilctiun oC wa1t.r. 11-1 7. Gsing the d a h 111 Table 1 1.2, calculntc the value or$ at e;lch value tri?tire I I -2

I O derive

and

AvaI,H = 29.47 kJ .nlol-l.

11-27. A solut1011containing 1.470g nfdichloruben~cnein 50.00 g of k n t e n r hnil+ul X0.hO C' at a prescure of 1.00 bar The bnlling point uf purt. h e t ~ z e ~is~8U.W e C'. and thc ~nuliu enlhalpy of vaporizativi~vf pure ben~cricis 32.0 kl mu1 ' I . Dctcrlnine the n~olecularIila.;s nf dichlurohen~enefrotn thcsc data.

~ 1 1 I-epruduce d

11-18. l i t the data fur Ihe ocmotic coefiicierit o f sucrose in Tablc 1 1.2 to a 41h-degree polyou1nh1and calculate the value of y,)" fnr a 1.00-rnulal solution. Compare your result nilh the clrjc obt:~inedin Exarrjplc 11 5.

11-28. C'onsider Iht Iollowtn~phnsc di;~pramlrlr a typlcnl pure si~hh~ur~cc. L:lbel the rrgiljn ctlrrcsponding to each pha~c.Illustrate how this diagrnrn chailges k)r u dllutc solutiu~lu l ' ; ~ nonvuliitile solute.

11-19. l!hir~g the data fur sucmcc given in Tahlc 1 I .2,d'tcrnline the vnluc of In y,,,' a1 3.00 ~nolal h plotting (9 - I)/m versus m and dcternlining the area under the curvc by numerical inltgration (MnthC'l~aplrrA ] rnthcr than hy cutvc fitting 4 linl. Colupare ycwr r e ~ u l twith the value g~vcrli r ~I'ahle 1 1.2. 11-2U. Lrluatiun I 1 18 ciul br uhsrl 10 dctcr~ninethe aclir ily of the solker11at its freczlng poi111

A s w n i n g that AC; i* lndcpcl~denloi~tenlpcrature.show lhat

nlierc AlUbIi(l;,) i, (tie rilolnr enthalpy rlf tusion at {he ireezing point of the purc solvent ( T i > )and ij, the difference i n thc nlular heal capacities of liquid and solid solvent. llcirlg Equation 1 I . I X, show that

~r~

',c:

'

11-21. T u k e A l u , ~ ( ~ ; {= L )6.01 k h r l ~ o l = 75.2J.K mol-',andC'; = 37.hI.K-'.1nol ti) h o w that the equation for - In (dl in the previous problem becomes

lor an aqueous solril~on.The frcclllng point drprescirlr~of n 1.95-molal aqucou~sucrose Cornpare your rehuit with srhutinn is 4.45'C. Calculate [tie value of cr, at h i s conccntra~ir)~~. the villue in Tablc 1 1.2. The valuc you calcula~zdill this prvhlern ir. for 0 C, whrreu\ thc u l u c in 'l'i~blc1 I . ? i, for, 25 C, hu! ~ h diffei-ence c ~c tairly srnall because u , docs nut vary 4 11. grwlly with tcrnpsleture (Prtlhlcm 1 1 11-22. I'hc I r e e ~ i r ~pgi n t of ;I 5 0-liiolal ;\clrlto calculate ;~n(lplc~tIri y- as "unction

uf tn':'

1 1 -37. Clse Cquatic~n1 1 43 IU calculate In y, for N a C l ( q ) at 25°C a=, a fuitction nC molaiity nt~dplot it verwc !TI'". Currjparv your rewltc with lhosc in Table 11.4. 11-30. 111 PI wble~li1 1-1 9, you detcrmi~~ed In y?m Frlr sucrose by calculating the area under the curve of - I versus III. Whcn dcaling with st>lutinnso f electrolytes, it is hetternumeriually to plnt (,$ -- I):JIII" versus ml!' hecauct. (yi the natural dependence I C 9 on ml". Show that

+

Show that thi.; detinition mudilies Equation 11.53 fi)r dilute solutions ro k

For an aqurclus solution at 2 5 C , where p is the density of the sulvenl ( ~ rg.mL i I).

11-47. Show that

fur all aquevps culution at 25-C, whcrc I _ i s the i o n ~ cstrength expressed in terrris of ~nolality.T&c e, tu he 78.54 and ~ h derl\ity c at water trl he 0.99707 g ~ I I . - ' .

11-39. I!\+? rllc data in 'Table 1 1 -4 lo calculntc 111y, for NaCl(aq) at 25 C by plotting ($ I ) j l l l 1 ; ? agnrmt rlrl." and determine the arcn under the curve hy nurrjerical irltegration (M;ithChap~crA). C n t n p m your values or It) y, with those you obtained in kobIem 11-37 whcrc you cnlcul:~tedIn y+ iron\ a curve-tit expression ol 4 as a p l y n o ~ n i n in l m':'.

11-48. Use the Uebye-Huckel thcory tu calculnl~the value uf In y, for a 0.010-nlolar NnCI(:lq) solution at 115 C. The experirnc~ltalvalue uf y* i i ~0.902. Take F, = 78.54 for H?Oll) at 25.0'C'.

4 72

(haptcr 1 I !Snlurinns 1 1 : 5ul1d-1lqtrirl Solutions

11-49. Derive the general cquatiun

(Hin!: Sce the derivat~onill Problein 1 1-35.} Use this result to sitow that

Show that Equation 3 can be wnttcn

d!

ah

1r $ - ( r ) ] = ~'[rt,b,!rjl

Now shu-, that the only sulution for tlr,(r)lhat is Gnitc fol- largc ~alue!,of I.

for the Dehyc-Hiickel thcury.

11-50. In Ltie Dcbgr. lliickel Iheurq.. the ivnc are rl~ndclcdas pr>in~lrlnc, and the sulvcn~is lut)dt.lecl n ct,nrinuous i~it'dium( n o ctruclurt) with H relarive perrnlrtivity F , . Consider '111 iun or typc r (I = u cution or at) anion) ait~ratcdat the origin 01- n spheric:~lconrdinatc cyclclli. Thc ~ I C ~ C I ~01 C Itli'r Z ion at lllr orlgili &ill attracl ion> rlr opposite charge and rcpcl ) the ~iuntlrerof ions (IT rqpt j ( j = a caric~nor an Ions u i ttjc same charge. 1x1 ~ y , ( rhe dniun) aituatcd at a dislance r troln Ihe cenlrill ion of tppt. i (a cntlorj or a n ,~nirm)We can irsr a Bultztnann hctrir to say that

whcrc N,jV is the hulk n u l n k r tlenxity o f ( ious and n:,,(r) is the interaclinn cnergy uf an i i'rrn with a j ion. This interaction energy will be electroslatic in 01-igin, so lei I I : , ( r ) = 4, t//, tr). wt~ercy, is the charge or1 the ion or type j and $, ( r ) is the t.lectrnstntic porcnl~aldue to the central ion of type I . A f~u~idilmmphyhics that rslaleh a cpllcrically ay~nmeri-ir, zlcctrostatic pnrari~iolt;',( r ) to a \phencally symmelnc charge densiry p, (r) is Poisson's equa~irln

I d

-

-

r'dr(

.d@ f--

-

d r )

Fx

P, 0.)

i~llerca, is the relalive pernjittivity ol Itle crllvcllt. In oui- c;r\e, p , ( r ) is ths c h a ~ g edenslty 31-c~und the central inn. First, show that

IS

where A i s a conshIlt. Use the fact that ~fthe cnllcentratiol~i \ \try small, then $,( I . ) is just Coulolnh'h law and so A = q,/.lne,,c,and

the Equation 6 ic n central resiilt of the nehyc-tfuchcl theclry. 'The iictor uC t'. Y'niod~ti~lt.h resulting Cuultlntbic putcntial, so Equation 6 is called a rr-rt.~ 10 shuw thal [he llel chiup< 111 3 y h r ic.11 ~ shell uf radius r surround~nga central ion tlt type i is

11 -52. Use the recult of the previous prublern to bhow th;it khc inwt prr>bahlc valr~cnf I - is 1:'~.

11-53. Show that

ahere c is the rnolarity of an aqueout \elution rrf a 1 -1 clcctrcdytc at 25-C. Takc e, = 78.54 for H,0(1) at 115-C. wheru C 1s the bulk nu~ribcrdun.;i~yof s p x i c s j (C', = G'1. Ll~lenrizerlie rxporientinl Icrm url;usc the ct~nililitmol~clcct~oneutral~t!: to shdw t l ~ a ~

Nrlw +uhctitutop , ( r J into Poicst>tt's equation

to

gut

for a O 50-rnr;nr aqueous sulution ufa 1-1 elcct~,olpteat 25'C. Take E , at ZSC.

= 78

54 fur H.O(l)

11 -55. How d w s the t l ~ i c k n e sof~ the ionic atrr~osphrrc.curriparc for a 1-1 clcclrulytc ,mri 2-2 electrulytu?

&i

C h ~ p l 1~1r / Solulinnr ll, Solid I iquitl Solutinn?

474

11-56. I n this problenl. we will calculale the tolal electrostalic energy of an clectroIyte solution in the Dehye-Hiickel theory. Use the equations in Problem 11-50 tu show that the number of ionh OK t y j ~111 ns"p\erical ~llcllof radii r and r dr about a central ion of type i is

Use the rcsult you go1 lor A" in the prevui~usproblem to show that

+

U Ethc ~ formula

The tolal Coulonlhic irlleraction k t w c c n thc ccntral ion of typc i rind thc ions of type j in Ihe sphci-ical {hell i u ,V, [ r ) u ,(*)37rr'rlr/ , I'wherc rt, (r) = q,q,/4rts,srr. To dctcrlninc tllc clectrostntic inter,jctibn energy u i all the ions in t i e ccolution wilh the renlral ion (01typc i ) . L',",sum ,$I '? ( r ) u'I ( r ) / lJ over all types r)f itmv in a sphericaI shell and then integrate o w r :ill \pliericul shells 10 get IJfe thc clectruneu~ralitycondition v + q ,

Cse electroneutrality to show thut

+ v-q-

= 0 tcl rewrite In yr as

11-59. Dcrive Equation 1 1.56 from Equalion 1 I .49. 11-60. Show thnt Equdion 1 1.GO reduces ro F q a t i o n 11.49 for small concentralions

NOR. using Equatiun h of Problenl 11-51), ur~tingthe eriergy

o!herwise7 11-57. Wr derived an expression Tor U" irl rht. prcvir~l~c prohlerl~.Use the Gihbs-Helmhull/. cquntiori fur 11 (Prr)hlerri 8-23) to show that

11-61. In this prohlem, we will inveqtigatc the temperature dependence (11. activities. Starling with thc cquation p , = p; RT In a , , show that

+

z,

1 f -62. I Icnlg's law h:~ysIII:II the prcwul-col-ng;~\in erluilihriutll with ;I 1lonclrctioly1cxolutirln of the ga\ i r ~n liquid is proportic~nalto the mulalily of lllc gas in the solutior~fur suAicienlly dilute solulions. What form du you think Henry'u law tiikes on for a gas such as HCl(g) dissolved in watcr'! Use the fr)llowing data fur HCl(g) at 25' C to tehl your prediction.

P,,, j 10-I bar 0.147

0.238 11-58. If we assume that the electrostatic interactions are the sole cause of the nonidealiw of an electrolyte solutiol~.thcn wc can say that

z,

is its panla1 molar where H1 is the rnular enthalpy of the pure solvent (at one bar) and enthalpy in the solution. The difference bctwwn H: and is slnall for dilutc solutions, so n , 1s fairly independen1 of tclnperature.

0.443 O.bh3 0.85 1 1.08 1.62 1.93 2.08

C1icipf~r12

478

I r h f r n ~ c ~Equilibrium ll

( \',)rles from 0 to some niaxrrnuln value dictated by thc stoichiome~ryof the reaction. l'cor cxamplc, I I 11, and n,,,, in Equalions 12.1 are equal 10 v, nloler and v, moles, respecti! ely, then 5 mill \ary frotn O to one mole. Differentiation of Equations 12.1

We shall denote the right side of Equatiot~12.5 by A,G. scr that

gi\,es

Thc ncgative signs indicate that the reactants are disappearing and thc positive signs intllvatr thi~tt t ~ eproducts are being forlnctl as the reaction progrcsqcs from reactants to products. Now let's corjsider a system cotitaining reactants and products atcowtant T and 1'. l'he Gibbs cncrxy lor this multicumponent system is a function uf T, P, n, * I

where the subscript n , in the f rst two partial derivatives stands for n,, n,, n,, and n,. Using Equation< 8.3 1 lor ( a U / i l T),,ni and (aG/a P),,n,. dG becorries

and

with siniilar expressions for pI,, 1 ~ and ~ p.,. For a reaction that takes place at constant 7' and P, dG beco~r~cs

The quantity b r G ' ( T ) is the change in s~andardGthhs energy for tlrc rcactlon between unmixed reactants In thclr ctandard states at temperature I' a~id a pressure of one bar to form unmixed products in their standard sbalrts at the same temperature T and a pressure of one bar. Recause the standard pressure P" in Equation 12.9 is taken to bc one bar, the P"s are usually not displayed. It must he remenhered, however. that all the pressures art termed to onc bar, and h a t Q conrcquentl) is unitless When the reactiun system is in equilibrium, the Gibbs energy must be a minimum with reqpect to any displacemet~tof the reaction from its equilibrium position, and rj t at cquilibrium, and the prexcurc P .

S O L U T 119 N : At cquillbri~tm,lherc will he 2(I uT K , ( g ) . The lottal nr1rnht.r of ~ncrleswill be (2 spccicc will be

'

'

cq) mule%of K(g) and tq moles

cq).

The piutial prcssure u l each E X A M P L E 12-3 The valuc of K , , ( T ) (based upor] a sundarrl state uf one har) for the reaction descrihd by

NIll(g)

+ !Ii2(g) * i ~ ~ ( g )

'

1.36 x 10 at 298.15 K. Deter~rw~e thc corrcspond~rrgkdluc of K = 2 11101 whcn ( = I mol. l b c Gibhs energy of thereastion mixture i s givcnby

,

-

(2?(AfG' l N 0 2 ( g ) l )-

K, . nr

l)(AfGr[N204(g)l) -

(8.3135 J - K I ,mol 1)(298.15 K)

If (he reaction i s uarricd out at a constant Iota1 pressure of clnc bar, then -

and

The total nunikr of tnoles in the reaction mixture is (1 have

,

P , 2 = X ~ ~ , . R ~

- 5 ) + 2 = I + 6 , and so wc

rdb. the reic~ionp r u ~ r c d sfrotrl 1ef1 to right as written. In ternlc ot' S , l i , it' L),, -= K,.. (her) A,G < 0, inclicati~igthat the reaction I Ssprjntaneou~frum 1t.R to riglil :I\ u r i ~ t c ~ ! . Conversely, if Q , , > K,, then Q must rlecreaxe as the reactirm proccetlh lo ey~~ilib~-i~irr~ and su the przshures of the products 111ust decreast. a r ~ t fth(7s.e of the rcirutanta rtiu;;~ increase. In tel-ms of A;G, il-Q , > K , , thcn ArC; > 0, indica~ingthat thc ~ractior)i s spontaneous from r i g h ~to left as wrilleri.

,

or tCq = 0.1892: in agrecrnenl W I I ~Figure 12.2. Problems 12- 18 through 12-2 1 ask you to carry out a similar analysis fur two other gas-phase reactions.

1 12-5. The Ratio ot' the Reaction Quotient to the Equilibrium Constant Determines the Direction in Which a ~eactionWill Proceed

I

EXAMPLE 12-5

'lhr cql~ilihriulnconsl;inr lur ttic runttion dchcrilhcd hy

15

K,.

= IO

at 960 K. Cnlculat~.I!.,(;

procecd spnraneously

and

lildiv:~tcin n l i ~ c clirtc~iam l~ lht. I-citctiot~ will

Cor

2SO,(1.0 x 10-'bar)+O2(020lharl ;= 2 SO;(] 0 x 1 0 'bur)

Chapter 12 / Chcri~icalFquilibrium

S V 1. I 1 T I O h ' : U!t. l j ~ v lc;ilculnte Ihn rrfiction qunticnt under co~ding111 Equahon 12.15.

the disswiation of N,O,(g) takcs place spontancously. The partial pressure of NIC), (g) NU,(g) incrca~csuntil equilibriun~is reachecl. 'fhc cquilibri(tr~t state is dctcrmined by the condition A f G = 0, at which point Q , = K,. Thus. initially A,G has a large negative valtte and increases lo zero a< the reautiuti goes to

these condition'. Ac-

decreases mid that of

equilibrium. We should point uut here that even h u g h A,G 0, the reaction may not uccur at a detectahlc rate. For examplc, consider the reaction given by

Note that l h e ~ equantit~csare u~litlesshccausc the pressures arc taken relative to one bar Using Equation 12.26, we have

= ( ~ . 3 1 4 ~ -mol ~ - ')(4130 ' K)In

5.11 x

)I)-'

Yhe lhct that A,(; -:O implies that the rractiun will prvcccd from left to right as aritten. This nray also hc seen from the fnct that Q , < K,.

1

12-7. Thc L'.~ri.zlicnnt an Equ~l~hrium constant with Tt.nl[>~ratbreI 5 tiivcn by llw Van't Hoff Eql~ativrr

I

12-6. The Sign nf A r c And Not That of A,.Go Determines the Direction of Rcactjon Spontaneity It i c ilnpndant to ilppreciate the diffkrence bctween A,G and ArGn.'l'hc superscript ' on AtG ett~phasizcsthat this is the valur 01- AlG whcn all the reactants aud products are unnlixed at partial pressures equal to OIIC bar: A,G" is the sranrlclrd Gibbs energ; changr.If A:G < 0, thcn K, > 1,meaningthilt thereaction wi1lproceedfrt)mreactants t o products il. all the species are mixed at one bar pdrtial pressures. IC A r c " > 0. then K , < 1, n~eallingthat 11ic rcaction will proceed frr~mproducts to reactiints if all the species arc rnixcd at nnc bar p a r ~ a lprcssurcs. The fact that A,G" > 0 does no! mean that ihe rcaction will not proceed from reactants to products if the species are niixcd under all condilions. For exarnplc, consider the reaction described by

for which Arc' = 4.729 kJ.mol-' at 298.15 K. The corresponding valuc of K , ( T ) i s 0.148. The fact that A, G" = -t4.72!, kJ.mnl-' docs no[ mean that no N,O,(g) will dissociate when we pl;tce some ot-it in a rcaclion vessel at 298.15 K. The value of A,G l o r rhe dissociation 01 N,O,(g) i s given by

Let's say lhat we fill a container with N,O,(g) and nu NO,(g). Initidly then, the logarithtn term arid A, U in Equation 12 27 will be essentially negative infiriity, Therefore,

The valur uf A,(;' at 25'C f-or this reaction i s -237 kJ per mole of H,O(I) for~ncd. Consequently, HzO{I) at onc bar and 25°C is much more stablc than a rnixturc uk H,(g) and(.ll(g) under those conditio~ls.k t . a mixture rjf Fl,(g) and 0,(g) can be kept indefinitclq. I1 a spark or a catalysl is introduced into this mixture. however, then the reaclion occurs explosively. Thiq uhservation scrves t o illustratc an impurlant point: Thr "no" o~therniorl~narr~ics is emphatic. ii'thermo+namics says that a certain process will not occur spontancously, then it wili not occur. The "yes" of thermodynamics, on the other hand, is actually a "maybe". The fnct that a process will occur spontaneously does not i l ~ ~ pthat l y it will nrcesfarily occur at a delectahIe rate.

12-7. The Variation of an Equilibrium Constant with Temperature Is Gi\:cn Ily the Van't Hoff Equation We can uae the Gibbs-Hclmoltz equation (Fq~tatior~ 8.61) AH'

to derive an equation for the tenlpernhtre dependence of K, (T). Substitute AG' ( T ) = - RT In K,.r7',l into Equatirjn 12.28 to ohtain

Note that ~f A, H " > 0 (end(~ttlermtcrcaction), the11 K , ( T ) illcreases with temperature, and if A r t { i0 (exother~ntcreaciion), the11 K , ( T ) decreases with Incrraslng tempraturc T h s is another example of Le ChPtclier'.; principle. Equation 12.29 can be intcgraied to give

12-7. The Variation uf 'In Equilibrium Constant n ~ t h Tcrnp'rdlurr

If lhc tenlperiiturr: range is s~nallenough that we can considcr A r H to he a corlstaut. then w e cat1 rvritc .

Equation 12.3 1 buggests that a plot uf It] K,.( T )versus 1/ T should he;] straight line with a slope of - ArH"/H over a sultiuier~tlysmall temperature range. Figure 12.3 sh(1u7s such I: plot for Lhc reaction H,(g) CO:(g) 7CO(g) tl,O(g) rwcr the temperature range hi10 C to 900'C.

+

+

Is Cdver by the Vdn't

t t r l l l Frludlloll

In Section 5-12 rve discl~ssedthe Lernperature variation of h r H ' .In particula~.u:c derived the equatiun

where AC; is the dlfferencc bctwcen the heal capacities uf thc products and react;]nts. Exper~rnentalheal capacity data over temperature ranges arc often presen~edas p l y nomials In the temperature. and if t h ~ sIS Lhe case. then A r l f ( T ) z,ln k expressed In the tomi (hee Example 5-1 3)

If this form for A, l l ' ( T ) is suhstitu~edinto Equation 12.29, and but11 sides itltcpl-ai~11t'd indefinitely, then wc find Lhat

The constants a lhrnugh 6 arc knuwn from Eq~~ation 12.33 and A i s an integration constant that can be evaluated from a knowledge uf K , ( T ) a1 some particular tenlperalure. We could also have intcgratzd Eqi~ation12.29 from some tznjperature TI at which Lhe vlilue of K,(?') is known ru an arbitrary temnperilturc T tu ohlain F I G L I K E 12.3

A plur of

It1

lempcrntllrt.

+

K , I T ) versus l j T furthc reaction H,(g) ( : O l ( g )+ CO(g) -tH,O(g) over the rarlge 601l'C: lo 9(H)"C.Thc circles rcpresenr cxperinlentul data

I

EXAMPLE 12-6 Givert Ihat A, H ' has arl axeriigc valur. ot - 69.8 kJ~rnul rrver the 500 K tu 7(W) K for the 1-eactiundescribeti hg

S (ILCJT 1 0N : We

IIW Equa~ir,~~ 12.3 1

ttlnperatllrc i-;lngc

with the a h w c xalucs

Equations 12.34 and 12.35 are gci~cralimtionsof Equation 11L.3I to the case whel,tl the tclnperature dependence of A,FIr ih not ignored. Equarion 12.34 shows that if In K,,( T ) is plotted against I / T, then the slope is not cunstanl, hul has n slight curvature. Figure 12.4 shows It1 K , ( T ) plotted versus I / T for thc ammonia sytltfiesis rciictir~n. Nutc thal In K,, ( I ' ) does not vary linearly with 1 / 1'. showing Illat A, H ' is tcmpcriittlrr: dclxndent.

I

E X A M P L E 12-7

Coll~ldcrthe rrilction dcccrihed by

The mular heal capncities of N,(g), I I:(:),

and NH,(g)

call bc cxpl-e\sed in Lhe 1urr11

Nutc that (he 1,cactionis exothcrr~licand so K, ( 7 ' =70n K) is le+stha11 K , ( T = i O U K).

r; [NH , ( g ) J / ~K -.I ,

lnol

' = 25.93 + 32 58 x

10-'?'

-

3.036

x 1 O-."T'

to 1500 K. Give11 that A, H' INH,(g)l = -4h.l l kJ m n l - ' at 3(W) K and that K , = 6.55 x 10-' at 725 K, derive a gener:il cxpressiun For ~ h vmiaticln c o f K , . ( T ) with ternpt.mturc in the form n i Equation 12.34. rwcr rhc temperatu~c range 3[H) K

with 1; = 300 K and A, H 'CT, -300 K) = -46. I I kJ.lnol

' and I

ono K 1 T

F I G U R E 12.4

A plot uf In K , f 7 ' ) versus 1/T for the ammonia sgllthcsi~reaction, NH,(g).

H Z ( g )+ N ~ (= ~ )

12-8. Wc Can Calculate fquilihrium Constants in Terms of I'artition Functions or

'

ArH (Tl/J,~nul =

38.10 x 10'

- 31.17T + 13.44 x

IO-~T'

Now wc uke Equatir)~~ 12.35 ~ i l TI h = 725 K and K,(T=725

-

1.965 x IO-'T'

K) = 6.55 x lo-'.

An Ilnporlant chemical application of stadstlcal thertnodynamics is the valculat~onot equilibrium constants In tarn.; u t ~nolccularparameters Consider the general homogeneous gax-phase c h e m i c ~ lreactiun

il,,A(g)

+ ~s,B(g)+-

11,

Y (g)

+ v,Z(g)

in a reaction vessel at fixed vulutne and Lemperaturt.. In this casc we have (cf.Equation

9.26)

+

d A = /lAnn,,

~

-t p > d~r l y

~

+ p,rln, (

~

(conslant T ~and V ) l

~

instead of Equation 1 2.3. Introducing the extent or reaction through Equations 12.2, however, leads to the same cortditiori for cheniical equilhriu~nas in Section 12-1,

-1his erluatiur~WRS used tu ger~eratcFigure 12.4. A1 M)O K, 111K, = -3 21. or 0.U40, in exct.llent agrcemenl with thc cxperi~nenlalvalue o f O.Wl.

K,

=

1: i s intcrzsling to cornpare the results of thia section 10 those of Scctic~n4 4 . where we dcrivctl (lie (:lausius-dapcyron eq~tation,Equation 9.13. Note that Fquntions 12.31 and 9.13 are rwentially the same because thc vaporization of n liquid can he represented hy thc "chernical equiition''

We now irltroduce statislical Ihermodynamic< thmugh the relation betweerl the che11lical potential and a pwirtitiot~funclion. I n a mixture of ideal gases, the species are independent, and so thc partiti011 function of the mixture is a pnlduct o f the partitirm funutioris of he individiral components. Thus

12-8

The vtiern~caipurer~tialof cach species is given by a11 rqi~atiorlsuch as (Prohlcrn 12-33)

where Stirling's approximiitiun lias been used for N,!. The Nl subscript un the panial derivalive indieales illat the numbers uf particles of the other specie5 arc held fixed. Equation 12.37 simply says that the clic~nicalpotential of one species of an ideal gas nli.rturt: is calculated as if the other species were not prcscnt. Thi.s, c ~ course, f is the vast for an ideal gas mixturc. 11' w e substitute Equation 12.37 inlo Equatiori 12.36, then we gct

Can Calc~ll,~teE q u ~ l ~ b r ~ l(I(mct,>nl< lrri ~n lrrnlr or P;lrt 1t.m runcr~r,n+

Using F,qu:ltiol~4.39 for the molecular parlit on funcrioi~sg i ~ c s

where we have replaced DF in Equatiun 4.39 tiy D, - t h11/2 (Figure 4.2). All the necessary parameters are given in Table 4.2. Table 12.2 gives the nurneric;il valiics of K , , ( T ) and Figure 12.5 shows In 'A plotted vercus I j T . Frotn ttie slope of the linc in Figure 12.5 we get A,H = - 12.9 kJ. mol-' colnpared lo iht: experi~tientnlvalue of ,- 13.4 k J . l ~ ~ u .The l discrepancy is due to the inadequacy of the rigid rotalor-har~t~o~liL. oscillator approximation at these 1ernperatLlreP.

For an ideal gas, tile molccula~-parlitiorl I'utictiuri is of the form , f ' ( T ) V (Section M ) so that q / V is a fiinction of temperutilre only. If wc dividc cach factc~ro n both sides of Equation 12.38 hy C"', i~nddenote thc n1111ibe1density N l / V hy p,, then w e havc

Note that K c is n function of temperature only. Reciill that K,(T) and K c( T Iare related hy (Equation 12.17)

F I G U R E 12.5

By means of Eqi~at~un 12 17 and Erlurttton 12.39, alung wrth the recults of Chapter 4. wc can calcula~eequilibr~umconstants in lerrns of njolecular paramctcn. Thia is best illustrated by means oT ex;~~nples.

The logarithm of the equilibrium constant verhuh I / 7 lor the w e c t i o ~H,(g) ~ + I:(g) + 2 1Il((g. The line is calculnted from Equatiun 12.41 and the circles are tlh: c.rpc;imental values.

T A B L E 12.2

A.

A Chemical React~onInvolving Diatomic Molecules

The valucs of K,,(?') for the reaclir~ndzrcribcd hy H2(g) calculated according to Equalion 12.41. TIK

firm 5Oi) K to I (XI0 K. The equilihr-i111rlconstan1 i b given by I

K,,(Tj

+ I?(g) e 2 21Il(g)

InK,(T)

U.

,4 Keaction Irlvolving Polyatomic Molecules

T A R L E 12.3

The Irjgarilhnl of the tquilibrium constant for the reactinn H:Ig) Ollg) + IlzO ($1

+i

As an example o f a reaction involvillg a polyutomic molccule, consider the reaction

TIK

111K,(calcj

In K,.(exp)

wllose equilibriutn constant is given by

It is allnost as corivenient to c;~lculateeach pLutiticlnfunclion separately as to substitute them into K c first. The necessary parameters are given in Tables 4.2 and 4.4. At 1500 K, thc three partition functions are (Equations 4.39 and 4.60)

Tahlc 12.3 compares the calculated vntucs or In K,,with exy~rjmenlaldata. Although thc agreement is fairly g o d . the agreement can be considerably i~npruvedby using 111oresophistjcated spectruscopic models. At high temperatures, the rotational energies of thc molecules are high enough to warrant centrifugal distortion effccts and other extennsof the simple ripid rr~rator-hartnnnicoscillalor approximation.

12-9. Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated

and

The factor 01.3 rxcLr1.s in yU2/ I/ hecausc the degeneracy of the ground electronic state ofO2 is 3. (Tablc 1.31. Nuticc that cach of the above q ( T . V ) / V has units of m-?. This ells us that the rztercncr state in thU (~~~ulccular) case is a concentrations of one nlolccule pcr cubic rneter, ut. that r,' = one molecule.~n-'. tlsing the values of U t ,from Table 4.2 and 4.4, the value of K, at I300 K is K c = 2.34 x 10 '. To convert to K,, we divide K r by (

~

)

l

NAPr

"

_

[!l&)(8.3145 i.mol-'.K ')(1500 K j (h.022 x 10" mol-')(I@ Pa)

to obtain K, = 5.14 x lo5, hased upon a orje bar standard state.

In the prcvious section we have seen hat the rigid rotator-hamlonic orcillator appruximaiion can hc uqed to calculatc cquilihrium constants in reasonably good agreenjerit with expclimeni. and because uf the simplicity o f the model, the calculatiorjs itlvolved are not extensive. If greater accuracy is desired, ht~wever.one must ii~cludt. corrections to the rigid rotator-harmonic oretical calculations. The thermodynamic lables that wc are about to discuss in this section, then, represent a collection of the thermody~lamicandor statistical thermodynamic properties ol' many substanccs. One of the niust extensive tabulation< of the thermochemical properties of substances is an American Chemical Suciety publication. Joumul of Phpaicnl Cfietnicrtl H c f e r ~ n rTjatrr. ~ volume 14, supplement 1, 1985, usually referred to as the JANAF Goint, ormy. ~ravy,trir force) tables. Each species listed has about n full page of thermodynan~iclspectrosct,picdata, and Table 12.4 is a replica of the entry for ammonia. Nolc [hat the fourth and fifth columns r)f thermodynamic dara are headed by - ( G ' - H ( < ) ) / 1 ' and II" - H'(7;). Recall that the value of an energy rimst be refcrrcd to some fixed reference pr~irit (such as a zero of energy). The reference point used in the JANAF tables is the standard molar enthalpy at 2'38.15 K. C:onsequently, G ( T ) and H ' ( T ) are expressed relative tu that value, as expl-cssed

12-9. T A B L E 12.4

A replica oiiht. page o1-NH,(g) data in thc IANAF tables J M A F THERMOCHEY1CAL TABLES

1598

Molecular Part~tiunFunctions and Kel.ll~dThermdynarrtir

Data Arc Extcnsivt.ly

T~hulatcd

501

by ihe headings - { G 3 ( T )- H"(298.15 K ) J / T and H - ( T ) - H'(298.15 K). FAble 12.4 gives - { G '(T) - H'(298.15 K)}/'I' for ammonia at a n111nbcr of temllcratures. The ratio ( G (1')- H"(298.15 K ) J / T rather than I G 0 ( T )- / I ' (198.15 K ) J is given because { G - ( T )- H"(298.I.S K ) J / T vurizs rtlure slow,ly with tempcrature. and hence the tables ;Ire easicr to inlerpoliitc. It is ~iotneccasary lr) specify a reference point for thc heat capacity or the entropy, as indicatcd hy lhe headings to the second arid third ctllumns. The .sixth and scvcnth uolur~~ns give ~ a l u z s of A,H' and A,G at various temperatures. We learned in Scction 12-3 that thesr data can he used to calculate valucs of A, H ' . ArG', and equilibr~urnconsta~~ts of reaction&. Because C (T) and H " (T) are expressed rclaiive 10 H ' (29K. 15 K ) in Tablc 12.4, wc must express the molecular partition function q ( V , 7.1relative t c ~a zero of energy. Recall thal in Section 12-5 we wrote q ( V. 7') as

where q U ( V ,T ) i s a molecular pwiitiun function in which the ground state cnclgy taken to be zcro. If we substitute Equation 12.46 into Equation 3.41, thcn we obtaitl

For one mole of an ideal gas,

= H r { T )=

af

PV =

is

t KT, and so Equ;ition

11.47 becomes

where H," = N , * E Because ~. y O ( ~T) , is thc molecular pa~litronfurlction in whizh l h c ground state encrgy is taken to be zero, q " ~ ,T ) is givcn by either Equation 4.57 or 4.60, without the factors of e-'~,h,,'?' ant1 e f ) ~ ' ~which ' , represent the ground state ol thc ~nolecule.Using either Equation 4.57 or J.bn, Equation 12.48 btcomec

2

J

- I

(I~ne'lrtuolccule)

( 12.4911)

12-9. Mnlerular Panition Functions and Rclatcd T h c r r r l u d y n , ~ mLl,~ta ~ ~ Are t x l ~ ~ n ~ i v Ir3t>uldlrd c~lv

Ch~pler12 1 Thpmical Equilibrium

The vi~lueobtained from Table 12.4 is more accurale than the valuc calculated from Equation 12.49b. .4t 1030 K,the ammonia moleculr ic exciled enough that the rigid rotator-bmnooic oscillator apprvximativn beginc lo become unsatiflhctory.

=~RT+Z R(-)\i~l.j

,(,

,iT -

1

(no~~linear molcuule)

(1 2.49h)

Nr11r t l tlie~i. ~ ~;II-c n o tcrms involving 0%,, , / 2 T or De/k,T in Equations 12.49 as tllcre arc in Fqu,~~ionr 4.58 and 4.61 hecauqe we have taken the energy ot the grclunri vibratrunal stale to be zero. Wc c a n u w Equation 12.49b and the parameters in Table4.4 tocalculate H"(298.15 K) If" for alrunon1:l

l ' t ~ ?very first cntly i n he liflh column i n Tablc 12.4 is - 10.045 kl.mol-'. 'Ilus value represents I J (0 K ) - H' (298.15 K), which is thc ncgative ot H (298.15 K ) H' (0 K) that we just calculated because H,' = H' (0 K). Thus, the value given by t q ~ ~ a t i n12.49h tj and the value given in Table 12.4 are in cxccllent agl.etment.

E X A M P L E 12-8 IJse Equation 12.49h and the parameters in Table 4.4 lo calculate H L ( T )NH3(g)at 1000 K and m e b x . Co~npareyour result to Tdble 12.4.

We can alst~use the data in Table 12.4 to calculate the value o f q n ( ~T. ) for amrnunia. Recall from Section 9-5 that we derived the equation (Equation 9.36)

-

-

where E, = N,,E, = Ii; and P ' I bar = 10' Pa. Equahn 12.50 is valid only for an ideal gas, and recall that q ( V , T ) j V , (M-q " ( ~ T. ) / V , ir, a l'unc~iurlul'terripxalure only for ail ideal gas. Equation 12.50 clearly displays the faw that the chemical potential is calculated relative to some Lem of energy. Because C;, = p' for a pure substance, wc can writc Equation 12.50 as

Il i s easy lo s l ~ u wthat G" .-• H i as T -+ 0 (hccause T In T H," i s also he standard Gibbs energy at 0 K.

-t

0 as T

0). arld so

According to Equation 12.5 1

H,"for

S 01.C! I l O K : Equatiorl 12.4% gives

'

H ' ( I O ( H I K ) - If,; = 42.290kl~mol Tatllc 12.4 gives

= 10' I'a. The fourth colurnn in Tablc 1 2 4 gives - {G - FJ (298. I S K ) } i T instead of -(G" - I-I,')/T, hut the first entry o f the fifth column gives H,; If'(298 I S K). ThereFtlre, h e cuponenl~alIn Equation 12.52a cat] be obtalt~edfrom

where I'

-

(G" - H i ) T

15 K)) ( H i - HU(29R.15 K)) -- (Go- H"(20X. +

e x ~ t l e n in t Equatio~l12.52a

If we sublract Equalion I from Equation 2. then

we obtain

T

T

founh column m Table 12.4

hrst entry of lifth column in Table 12.4 d i v ~ d dby T

(12.52b)

Let's use Equa~ior~s 12.51tocakcula~ey i ' ( ~ T) , for amnionia at SOU K. Suhqlilu~irlg 111edata in Table 12.4 inlo Quation 12.52A gives

The entries ~nrhc .lANAF tables for H(g) and N(g) givc 3 ,H (I)K ) = 2 16.035 L l .rnol-' and 470.82 k l . mol - I , rzqpectrvely. T h e ~ c\alum conespond to the equations

and If we suhslitute tthi~\:slue inlo Equation 12.57~1,then we obtain

If we subtrack Equation 1 from the suttl of Equaticln 3 and thrcc times Eqilatiot~2, thcn we obtain tquation 4.60 gives (Probleni 1 2 4 8 )

Thc v d u e given by Equations 12.52 is thc Inore accurate becausc Equation 4.60 is hased on the rigid rotator-harmonic oscillator approximation.

The value given in Table 4.4 is I 158 kJ.mol I. EXAMPLE

12-9

I

The JANAF tables give - I(; - H (298.15 K ) ] / T = 231.002 J mul-' K-' and H,, - f l (298 I5 K) = -8 683 ~l.rnth-' for O ? ( g )at 1500 K. IJse these data and Eq~d~r>r~\ 11.52 to calculate y"( V, 7')/1' fur O , ( g ) ut 1500 K

E X A M P L E 12-10 The JANAI: lahlcs glve A , I I - ( 0 K)lilr Hl(g). ki(&), and I(g) 10 k 28.5535 kJ,rnol- I. 216.035 kJ. lnol ', and 107. lh !d.mol-'.respectlvcly. Calculate [he value of I),, For HI@).

I

SO I I) TI C) N : The above data cml be presented 3s

and Equation 12.52~1gives

y1'(V.T I 1'

-

(h.022 x.-10'' nlol-'1(105Pa)e12!51w3 (8.313J-mul- I . ~ - ' ) ( 1 5 1 ? i ~ )

,K--~

L1!6114

,,,,(>,

I E;-:

If we suhtracl Equat~on1 from the aurn of tkiclatic)ns 2 and 3,then w e oht;~ln

The talue givcn in Table 4.2 is 294.7 kJ-nirjl-'. Labtly, the t h e r n l o d y n ~ r ~data i c in lht JANAF tahlcs can also be used to calculate halues of U,, fur molecules. Tahlc 12.3 gives A, H ' ( 0 K) = -3K.907 kJ,mol ' for Nt I, (g). 'I'he chetnicill equalion (hat rcprcscnts thih process is

f H,@)

+

N,(g)

+ NH,(g]

A , H ((I K) = -38.907 1;J.mtli-I

(I}

The thermodynanlic tables contain a great deal of therrntdy nnrnic andlor slntisticnl thermodynamic data. Their use requires a little practice, but it i s well worth the d'lilr~. Problems 1 2 3 5 through 12-58 are menrir to supply this practice.

1 2 10. Erl!~ilibr~umCurislants for

12-1 0. Equilibrium Constants for Real Gases Are Expressed in Terms of Partial Fugacities Up to ttii\ poit~tin illis chapter. w e have discussed equilibria in systetns of ideal gnsts only. In this sectirln. wc st~alldiscuss cquilihriil in systems of nonidcal gases. In Section X-8 we introduced the idca of fugacily through the equaticln

where iC(7') is the chcmical poential or thc corresponding ideal gas at onc bar. Once again to simplify the ontation we shall not display the f " in the rest of this chnptcr. Therefore. Equation 12.53can he writtcn in tlic fornl

ror~scqi~ently. we must kecp irl mind that f is taken relative to its qtandard state. In a ~~tisftll.e ( ~ ~f ; I S C S ,we would hake

Because h e ~nrhec~ilcs in n mixturc or gnscs in which the gdses do not behave ideally are IIOI iridcpendent c ~ onc f another, the parlial fugiciry of each gas generally depends tlpnn the concentrations of 311 thc other gases in the mixture. Now let's consider the general gas-phase reaclion

Ke,~lCasirj Arc: ~xpresscdin Terms of I'anial Fugacitiw

where the equilibrium constant K , is given by

Once again notice that the equilihriu~~l constant is a fullctiun of tc~npzratureonly, as dicl;~tedby Equation 12.57. The equilibrium constant defined by Equation 12.57 is called a rltt,r7nndynri1nilcquilibriuni ronrtrml. Equation 12.57, which relates Kf to A,GPis enact. being valid for real gases ax well as ideal gases. At low preshures wc can replace the partial fugitcitics by panial pressures to obtain K,, but we should e x p t thih approximalion to fail at high pressures. Thc formulas t c ~calculate partial fugacities t n ~ mequation-of-statc data are extensintjs of the ft~rmulitsin Section 8-8 where we calculated fugaciiies for pure gdses. In urder to obtain the parlial fugacities to use in Equation 12.58 we need rather extensive pressure-volume data fur the mixture of reacting gases, These data are available for the important industl-ial reactiun

Table 12.5 shows hoth K , and K j as a Kuriction of the total pressure of the rrac~iun rrlixture. Nore thnl K, is ncll a constant, b u t that K, is fairly constant with increasing total pressure. The results sliown in Table 12.5 emphasi7.e that we must use fugacities and not pressures whcn dealing with systems at high pressures.

T A B L E 12.5

The change in Gibbs energy upon cclnverring the reactants at arbitrary partial pressures to prodllcts at arbitrary parlial pressurcs is

Vdlucs of K, and K t as a fut~ctionof total prescure for the arnrr~oniasynthesis equilibrium at 450'C.

total pressurelbilr

K,/tO-'

K c /IW3

If we substitute Equation 12.55 into this equalion, then we get

Notc thd Equalion I?.% is the gcnrrulizatiun of Equation 12.24 to a system of nonide;ll p i w s . Rci~lizcthat thc values of 1he fugacities at lhis point arc arbitrap, and not necessarily equilibriutn values. If the reaction system is in equilibrium, then A,G = O ;~ndall the fugacitjcs lake on their cquilihriu~nvitlucl;. Equation 12.56 becomes

E X A M P L P 12-11

The cqu~librium constants K,, and X, can be relalt'd by a quantity K y ,such that K, = K I K , and K y has the form of an equilibrium conslallt. h u ~involving activity coefficients, y,. First derive an exprescjon for K y and then evaluate it a1 lhe variuuh pressures given in Table 12.5.

I

Chapter 12

/

c'heni~cal Equilibrium

12-1 1 . Thermodyndmic

Equilthrium Curr5l;inrs Arc txpresscd r r l lrrlns ui 4.1t ~ v t t ~ o

Now let's consider the general reactiun

5OL U TION: The relaliort kta'eer~pressure and fugaciiy is gwerl hg

11-we aubstit~~te 1h1+cxprzssirm into Equalir)n 12.58, then we tblairl

The change in Gibbs energy for converling A and B in arbitrary states k) Y and Z

iri

arhilrary states is given by A,G = v y p y

+ I I ~ J --L ~

1

1

~

1

~~

~

~

,

~

p

~

If we substitute Equation 12.59 into this equation, then wc ohtiiin where w e have used k t stmldartl\tatc ~ f' = P

--

L'\

+

I bar. Us~ngthe data in Table 12.5,

ArG = A r c ' , K I ' In

we see that

mvz

+k

QA"%

where K
Also shrw that

=

1hcr;(2

-

cYj2/27(1 - CCu)''= 0.I(I

12-21. Supposc that wc have n lrlixturc of f l ~ egases Hz(&+C02(g), CO(p), and AIO(g) a1 1 2 h I ) K , ~ i t hP =0.55 bar, Pr,, = O 2 0 bar, P,,= 1.25 bar,and I',:,=O10 bm.1~ H. the rcaction d c ~ c n b c dby the equat:otl

at equilibriuln undcr thcsc conditiorl~:lI r ~ i ~int ,w l ~ direclinn t w ~ l the l reactiun prtweed tu attain cquilibriun~l and that K, = 3trt1/(1 - -

t,)'

=

5Y

12-22. Given ttint K,,= 2.21 x 10' at 2S'C

[ ( ~ rthe

equa(inn

12-1 9. Cottcider thc reaction dcscribcd hy

at 401)0 K aljd a iota1 pressure of urlc bar Suppclsc t h a ~we start with two moles of ti,O(g) and no II,[g) or 0.1;). 5huw thnl

prcdict thc direction in which a reaction mixlure fix which PC., 0. l o bar, and PC,, = 11.0050 bar procecd~to allain equilibrium.

,, = 10.0 ha^;

=

K, at 500 K fur a gas-phase reaction doubles when thc tcnlpcrature is increased f r m 300 K to 400 K at a fixcd prcqsure. What is the value of- A r H ' lor this reaction'?

12-23. The value or

12-24. 'The value of A t t i c is 34.78 kJ.nlol-' at 1(HX) K

for the rentlion desurikd hy

Gi>en thal the valuc O F K,, is 0.236 at 800 K . eslinlate the value of K , , at 1200 K, nss~lming that h,ll is independent vf temperature.

12-25. 'l'hc ~ l l u of e A, tl is

- 12

97 bJ . lrml

12-29. The knlprratu~-edependence of the eqrlilibriuin conqtant k',, li)r Ihe rc;~vticrodcscril>cd

bv

at MI0 K fur ic given by the equation

Assuming that h , H IE it~depcndentof tempzriiture. calculate K,, at 700 K given that K, = 29.1 at 1000 K. Calcula~eIhe values of A, ti", A r H - ,and .INSfor this reaction at 52.3

12-26. The etluil~hriur~~ ~or~htaril to1 tllr: rractiun described by

12-30. At 2000 K and one bar. water vdpclr is 0.53'8 dissociatcd. At 2 1 tH) K and one bar. ~t I \ 0.88% dissociatcd. Calculate the valrie of ArII for the dih.crjcialiun of water at oor~cbar. assurning that the tnthalpy of reaction IE corIaparit o w r the range from ?(H)O K to 2 100 K.

can he expre5hed b) tllc cnlpiric;il Crlr~nula Irl

E:

=

-6.375 r 11.64 15 III(T/KI

K

-1 1790 K T

12-31. The follou lng tablc g i r ~ cthr \t:~nri;lrtl ~ n n l ; ~C;lht)\ r c r ~ ~ ~(11'g Ir>ri11:11 y irw of C'I(g) :II rhree rlil'ftrr~lttenlpcrilturch.

Ilcc this form111o to liclsrmine ArII as a furlctiun of temperature. Calculate A r H - at 2 5 C a r ~ dcomparc your result to the one you ~rhtainf r ~ m 'Table 5.2.

Use these dala to determint the value uu K , a1 each telrlpcraturc lor thc reuclirln dcsu1-ibed hy

A s s u m i ~ ~that g ArH' i s temperature indcpcnderir, determine the wlue of A, H rnlrrl lhekr data. Cumbine your results to determine A S S at cach tclnperature. Inlerpret your resultc

12-32. The Kolluu ing rxperi~ncntaldata were rlrrl 3n rl~r!rochcm~c,~l Ccll

Reduction takes place 3t the right electrrde, and the equatiun for the reaction is

to wite the ~ q u a ~ i of (n~ the r cell reaction unalnhigously. For the abovc cell diagram, ther~.wc have

Ztlls)

+ Zn"

Cu:-(aq) t 2

(aq) P

+ 2 P-

=+Cu(s)

(oxidation a1 the IcCt-hand electrode)

The equatiun Ibr the nct cell reaclion i s

(reduction at the right-hand cleclrodc)

+

Z l ~ ( s l 2 I l ' (aq) .-+ Z1i2' (aq)

+ H,(g)

This convenrion is easy 10 rcrnembcr because reduction and right both bcgin with r . Thc cherrlicn[ cquatiorl ctil-rcspo~ldingto t h e cell rcaction is the sum o f t tlc t*o e l e c ~ o d c

I

E X A M P L E 13-3 Writc the erluation~li>rthc eIectl-ode rt.:~c.tior!sand the net cell reaction fur the elecrrt).

I

cherr~~cnl cell

E X A M P L E 13-2

Writc ~ t i c c q ~ n t i o ntor r thc elcctrutle rcnc~irln~ nlld the net cell reaction for the electroche!t>icolccll

Nnle that this cell (irlec not Rnr~ca \all bridge hcf~useit ha.; orily

ottc

clcctrul!liu

solutirrn S O L U T l O h': Oxidation lakes place n t the left electrdc. Oxidation of Cd{s) yicfds Cd2+{nq).so we write Cd(s)

--t

Cd" (aq)

+ ? e-

(An ilIristr:ttiorl of tl~ic1.~11is show11in Figure 13.4.) The (reduction) reactiun at thc right eleclrude is 5 0 1 LlTlOh

Oxid~tiont a k c ~place nt the left electrode. T h u ~ the , yuativn for the reactinn at the letr clectrade is -

l'he neL cell rzacrlnn is

The cell ~ ! c s c r i k dtrcw is very i m i l a r 10 thc M'c,.irt)r; .~trrr~,l VL,,au thc cell voltage. -1V , is pusi tivc. A key pruprrly of a cell [hat allr~wsus tu apply the~,modynnrrlicsto elzctruchemist~y i n the ~ ~ 1 c r t r r ) n 1 c j ~ i 1 ~ (emf), c ~ f i r t ~t. ~ cwhich ~ is defined by

If a ccll ih reversible, the chemical reactiun clccw~ingwilhin the cell car1 ptoueull in either direction, depending upon the flow of current, and at the null point, the driviilg force of thc I-ci~ctiunis exactly balanced by the emf of thc putcntic~mt.ter. Becausc E is associated with a balance point of the reaction, whcrc it will gr~oibz direction or thc other with small chi~ngesabout E , one might wspect that E is relaled to thc change in Eiihhs energy for the reaction. Furthrrniort.. if- E > 0, the cell reaction p r c c c d s spontaneaurly and if E < 0. the revcrsc reacticln d t ~ sThis . concept follows from F4uation.c 13.3 and 13.4. bur the cell diaprum given by Equi~~iuri 13.2, t' :, 0. Clearly, then, rht: relation hctween E and A G ~riuslinvolvc a negative sign heciu~\e A G < 0 for the reaction associated with the ccli diagram. I ~ t ' see s just whal the relation bctv,,ccn E and A(; is. Consider a cell ill which n molcs o f electrons ;Ire (ransferred from thc left electrode to the right electrodcc. The magnitude the ckargz of one nlole of clcctrons is given by the product of tllc magnitude of'the charge on arl electron (1 -6022 x I O - ' ~C) atjd the Avngadro constant (6.0721 x 10'' mol-I), and is equal to 96 485 ~ : . r n o l - ' .'1-his quantity is callsd the Funitlr~ycortstirnr and is denuted by F.'Thus. if 11 molcs of clcctnlnh pacs ~ t ~ n i u gthc h extrrnal circilit, then the total clectric charge passing ihrougl~thc cxtcra;il cil-c~~il i.; II I.'. If AV 3 0, then the electroris will ffow from left t c ~righl (by our c o n v e ~ ~ t band ~ ) ,an amount of work equal 10 n f - AV will be done by thc ccll. II' this work is done revetsihl y, A V is replaced by E. and the electrical work done by thc ccll will he n F E . Note that iC E > 0, the cell docs work or is able to do work. Atcordin? to F.quatic111X . Ih

E q ~ ~ ~ i 13.1 i o n says (hat h e errif of a cell is (he poteritial difference of the cell rneasured i~ndzrthe ct~nditirlnt)i'rIu flow of electric curretlt.

13-4, The Gibhs Energy Change ot' a Cell Reaction Is Directly Kelated to rhe Electromotive Force of the Cell Si) f ~ r we , have no1 in~ruduvedthzrlnody rii~~nics into our discussion of electrochzrnical cclls. Such an intrriductiorj njilsl be done with some thought because the equations of rhenuodynarr~icr,a5 we have developed thcm. iipply only tu systems it1 eqililibriunl or 10 ~ v r r s i h l zpi-ligi~tlqwhen the current changes Irom onc riirectlcln to thc othcr, the ccll

which is the desired rclation hetween A G atid E , and is thc central equ;~tir>n of the thrtmodynamics of clcctrochemical ce1l.c. As we will sec. cicctrovhe~niui~l nieasurerrlcnts can be used to dctcm~inetherrnt~dynamicdata of ionic rcactionq thai occur i11cells. l'he relation AG = -n F E tells us [ha11the emf of il ccll tlcpeltds upon l l ~ econceritrations. or lnvrc prcciaely, upon the activities, of tlic reactant.; and 11roluctsin tlic cell reactiorj, For the general chcmical ~rcacliondescribed by

A G is given by*(~quation12.60)

537

11-5. The Standard Emf uf an E I e c t r w h ~ ' m ~Cell ~ d l Can Be Fuurld br txtrdpolalior~

Here A G = ~',p; + u , l ~ ; - v,p; - v,&, where the p; are the chemical potenlials in some appropriate and conve~~ient slandard stale Now because A G = - n F E ,

and the overall reaction associated with the above ccll diagram is

The Nernst equation for thc cmf of the ccll i s Equatior~13 7 1s called the Nertts; pqlmtroil and shon7show the emf r>t a cell dcpcnds upun Lhe autw~tics,or the concentrations, of thc qpecies participating in the cell reaction. Thc quimtity. E ' , cilllcd the standard emf of the cell, is the emf of thc ccll when the ; I C ~ Iities L nf thr products itrid rcacttlnts are equal to unity

1 3-5. The Standard Ernf of a n Electrochemical Cell Can Be Found b y Extr.~polation Bcc:iu';e rc-e ran calculate thc value of I7 for arbitrary activities once a e know E' for n ccll. it is ir~~y)urtar~t I'or 11stu be able to determine the value of E' . The following calculation illustrates a standard procedure for daing this. Consider thc cell pictured in Pipure 13.6. The cell diagram Cur this ccll is

l%ec;iuscAg(s) : I I I ~t\gCl(s) arc stdids, u-csel irAP :111d(I,,~,., et111;llto ~ u ~ i tiny l i ~ 1 1 1 : i tion 13.10. I r l addition. we regulate the prcssure of t11c hydrogen gas over the plaliuul~~ electnde such that u,,> = I. which aluounts to sctting its fugxity equal to unity. Fur a gas xuch as hydrugkn .nl ordirla1,y pressures. the fugacity is cswntially equal to the presxure. s o we can set the pressure equal to unity (one bar). The number of clec~rons transferred in the overall reaction as written above is one, so n = 1. in Fquatirjn 13.10. Finally, then, Equation 13.10 becomes

From Table 11.3, a+a- = u i and a , = y,tn, su Fquation 13. i 1 becomes Tile Ictt electrode i c a Ilydr~genelectrode, and the right electrode is a silver-silver chloride clcctrode. ,iccordit~gto our convention that oxidation occurs at the left electrode in the ccll diagram, the two elcctrode reactions are

In our discussirjl~of the Dehye-Iltick~Itheory, however, we saw that (Problem 1 1 4 7 )

A.g(s) f(>il covercd with AgCl(s)

for an aqueous solution of a 1-1 electrolyte such as HCl(aq) at 298.15 K. so we can write Equation 13.12 as

, FIGURE

13.0

:In rlrctrochctl~ic~al cell co~~sisting ol'a t~ydrogenelectrdc and n silver-silver chloride electrode.

T t ~ ccell diapra~nir)r Illis cell i s Pt(r~. so the procedul-c is to plot the left side c ~ Equation f 13. I 3 versus mu'?attd extrapolate to m = 0. The intercept, thcn, is E ' . Table 13.1 gives data for E versus m at 298.15 K for the cell pictured in Figure 13.6. The data in Table 13.1 are used to plot the left qidc of Equatiorl 13.13 against m'!' in Figure 13.7. The linear portion at small values of rn cxtrapt)lateq to E-' = 0.222 V.

13-5. Thr St.~r!ddrtl En11nt an E l e c l r ~ h e m ~ c Cell al

340 T A B L E 13.1

l'ht c~nf rer\u\ I I I at 298.15 K fur he cell whosc cell diagram is Ptt$)lH,(I har~~IiCl(trr~lAgC1(s)lAg(sjI

( , I ~ l5e I

1-ound by F x t r . i ~ ~ r r l a t i l ~ n

Table 13.1. Note l t ~ a the plot is less curved than the urlr in Figure 13.7, mak~ngrl~e required exIrapolation much easier. Thc plot shown in Figure 13.X i b cilllcd a Hirr,hcocl: plot and should be used [or an accuratc determination of thr value of E - .

I

E X A M P L E 13-4 Use the Iollt>wingdata to detern~lncthe v;~lue01 E' for thc cell who+c.ccll diagrani i,

S O L U 1- t 0X : Using our convent~onthat oxirlatirm takcs place ar the lcft electrc~lein a cell diagrdni, we write the two clecuode reaction\ a8

and

F I G U R E 13.7 A plrlt cjt E + ( 2 K I ' l k ' ) Inm ngalnst mi.'' at 1'38.15 K Tor the ccll whose cell d~agra~n is given by Fyua~iun13.8. The extl-apolation

tllc value of E

'.

The net wactlon is

illustrated i n Figure 13.h and or this plot to m + 0 yields

the +~andardclnf of the cell.

Although the plot in Figure 13.7 can bc extrapolated to m = 0 withrwi tnn much difficulty, we can obtain a more reliable method by using an extension of the DehyeHiiukel expression for In 6 to higher concer~tratinns. For example, we can use thc sen~ienipiricalexpres.cion given by Equal~ r j n1 1.58

Thisequiitinn is analoguus to Equatio1l13.9,so wccan~rseEquation13.13, u r k t t c r yet, Equation 13.1 6. The resulting plots using Equatiun 13.13 (plotting the lcft side against I;>)or Equation 13.1 h (plotting the left cide npainsl m l arc shown 111 F i ~ u r c13.8. Notice thal the Hitchcock plot 15 much easier 10extrapolate to m = 0, yielding a valuc. 01- E' = 0.0730 V.

where C: is an adjuh~ilbleparanw tcl: Equation 1 3.1 4 becomes

fol.

a 1-1 electrr,lyle such as HCl(nq). 11- w e substitute Equatiun 13.I5 into EquaK, wc ohlain

lion 13.12 at 298.15

Now if wc plot the leR side uf Equatiun 13.16 against ni, we can obtain the value of E" by extrapolating t c ~m = 0. Figure 13.8 shows such a plot [or the data givcn in

F I G U R E 13.8

The detcrminntiun of E by means of a Hitchcock plrlt for the data givcn in 'lhble 13. I

for a reaction in which the electnlns on each side cancel (see Plnblc~n13-57). I t we apply Equ~tion13.22 I(> the cell diagram giver1 by Equation 1.3.17. w e have

Note thal we have indicated that rcductit~nt;ikcs place at (he rinu electrode by wl.iting E , I Z I I ~(aq)/Zn(s) + 1 and that oxidatiun takes place at the hydrogenelectrode by writ ill: E 'I1l?(g)/HA(aq)J.By coupling other clcctrodcs with i i hydrogen electrode, we will obtain results similar to Equation 13.21. Fur example. if we use a AgCI(sbA?(s) electrode instead of a zinc elcctrodc, we would obtain F ~ G U R E13.9 T\\rj pltrlq

illustl-;]ling~ h cdctern~inatin~~ r)f the value of E - at 208.15 K f o ~!he cell whose crH diagr.n!nis PI(\) lH:(l har)lHRr(m)[AgBr(c)ws) from the data given i n Exanlpte 1 3 4 (;)I Thc left side ill Equation 13.13 plotted ilgalnst m I;'. (b) The left side a f F~pation13.16 ~>lr)tted njiainsl m. Both plo~scan he extrapolated to 0.0730 V, but the extrapolation is more

it i s riot pcwsihle to tileasure the voltage of a singlc electrode: only the difference irj voltngc butwecn two electrodes can be ri~easuted.If, howcvcr. wc agree tc! choose a numer~c:ll value for the standard voltage of some particular electrode, then wc can assign standard voltages tn single electrodes If we takc E" for the hydrogen eleurrode to hc zcm, we can write

13-6. We Can Assign Values of E" to Single Electrodes I'cmsider the cell whose ccll dii~grami s

and

The value of F for the corresponding ccll i s -0.723 Ihr electrode rcactiuns alc

V at Z(18.15 K. The equatiotis for

Hl(g) 4 2 ilA(aq) + 2 r -

(uxidaticln at the left clectrode)

(13.18)

~ n ' .( n q )

(reduction at the right electrode)

(1 3.19)

2 P-

-

Zn(.s)

Brcai~aeE - implies that a11 the reactants involved are in t h e ~ standard r states, wr could write Equdlion 13.27 more explicitly as

n ~ the d equatior~for the overall reaction is

If we let ArxI;-'and Arc,,(;' he the standud Gibbs energy change fur Equations 13.18 and 13.19. then A r c ' for Equation 13.20 is

Because AG' and E d are directly related by Equation 13.7. we can use Equatian 13.21 to write

E:ell = E,,

and cotltinue the proccqs by ctlupling other electrodes with a hydrogen clectrode. Thus, we will fortnally write

+ EFt,

to emphasize thal H,(g) must be at ini it fugacity (which i s essentially equivalent to P = 1 bar) and that the stmng acid corresponding to H ' (aql [fur example, HCl(aq)] must he at unit activity. Equations 19.25 and 13.26 are exan~plesof the standard reduction potentials 1ha1 we can axhign to?lectrode reactions ('ljble 13.2). We can use the entries in Table 13.2 to calculate the value of E' for a cell whose cell reaction is the sum of the two electrotle reactiotis. According to Equation 19.22

ulicre reduction rakcs place at the nght elcctlade and oxidailon t;~kciplace at the lett c1ectrt)dc We can emphasize this r~ght-lcftcunventfon hy writing Equation 13 29 as

TABLE

13.2

Standard reductinn poterr~ialsin watcr a1 2 j C. Elccll-c>dereaction

But, fi,r a pnnictilar electrode-rraulion cquatiorl, oxidatic~nis thc reverse 01' wductioi~,

K i a q j + L,-

- --tK(s)

~ a (oq) "

t,-

+2

+

SO

I?&

.--+ Cays)

--

hs+(aql r' + Na(s) E8>k,l

Llsirlg lhe Fact that

=d ; ' -

A ~ ' ~ (+a3~c i- . i\l(s~ 2 H;O!ll T 2 e - -- H,(g) r 2 OH (aq)

I

E:,= - El;,, Equation 13.30 can be written as

20'' (aq! A 2 e Zn(s) A g I S ( s , - 2 ~ -+?A@(sj+S2-(aq) FeL'(aq~+ 2 e -+ Fefs)

Tl~usthe emf rjl' the cell is the standard reduction poterltiat of the right elcct~udein the cell diagram m i n u ~thc standard reduclion potentiill of the left electrode. Consider the cell whose cell rliagr~rni s

C? [q) 7 P - -.+ CrZf(aq) ~ : d " (aq) -1 ? c -

Phil(s)

+ 2 L.

PhSO_(s) t 2 rPbBr,(sj 2 e

+

The overall cell reiiction cquntio~li s 2 Ag(.,

$

CLI" ( 1 00 rn)

+ 1 Ag' (0.100 nil + C:u(s)

(13.32)

a ttd

C'U"(~,)

+ 2 c == Cu(s)

Wc. bee from T ~ h l r13 2 that

Ag+jaq)

+ r = Ag(s) +

C L ~ (xi) " 2 e-

+ l'u(s)

E = +0.799 V

E' = t0.337 V

Pb(hj SO:-(aq) 2 Br-(aql Pb(s)

-0.364

-03583 -0 274 -{1.2hb

-0.255

Nits)

-0.30 -0.15 I

4

~ n ' iaq) ' T 2 r- --4 Sn[s) ~ h "( a q ~I- 2 u- - + I%($) Lig,ll(kj t- 2 e -+ 2 Hg(1) t 2 1-(ay) FZ' (aql + 3 1,- -+ I;c(sl 1 H - { ; q ) i 2 e . -+ H,(g) AgBr(5) t c --• 4 g ( s ) t B r - ( a q )

-0.03b O.I)(lO i0.0732

HglRrlis) $ 2 e -* 2 I Ig(1) f 2 Ur-(atll S~i'+(arl) 2 P ~n'l (aq)

+(I. t 396 +O I5

+ i\gCl!?i) + u- + A&(" + Cl-(atl) IlglC:II(>~ + 2 -+ 2 Hgll) + 2 CI-cay) Cu' ' iaql + 2 rCurs)

---P-

--

l>(sl 2 c

2 1- (aq) i l g l S O - ( > ~ 2 e --,2 Iig(1) + Sui-(aq)

Notc tha~the Ag(s)lAgl (aq) half-cell emf- is rror multiplied by 2. Thc cnlf of an electmche~nicalcell is independent of the size of the cell: emf i h an inre~lril,r,property. (See also Prohletns 13-54 thmugh 13-58.) The cmf of the cell wtiu.se cell rcuction is described hy F.quation 13.32 i s given by the N z ~ n s cql~nrion t ( ~ ~ u a t i o 3.7) n * ~with rr = 2:

-0.440 -0.408 -0.403

V?-(aq)

--.A & ( s ) + l (aq)

Agl:sb k e

and the two electnde rzaclions arc

+ Ph(s) + + -+ 2 C1 (nq) + Pb(sl 1 1-(aq)

-4

-

PhU12~.h) 1 2 L.v'I {aq)+ tr~ i : " i a -~2) P-

Cd(s)

-4

--•

-2 '125

-2.866 2.711 - 1-66 - 0 828 -0.763 - 0.7051

-

A g L S 0 4 ~ s+ lZ F- --+

---

~ r " f o r ~ , + r .---

Ilgjiaql

Fez-(aq)

+ 3 r-

Ag' i a q ) -I s

~ d ? (:q) ' -h? r

2 Ag(s) t SO:-{uq)

2 Hi(])

Ag(s)

Pd(s)

+

C)ZIg) 1 3 Ii+iaqj 4 e - -+ 2 H,0(1) Cl,igl t 2 e --,2 C1-paq)

C O " ( J ~ I+ r - + ~ u ~ ' ( a ~ )

-0.1.lh 0.126 -0.0405

+O.222,1 +0.2hX f0.337 4-0.53 15

+O.hIS5 +O.b53 +0.771 f 0.7% +-0.799 f 0.9K7

+ l.22q I 1 361

+ 1. X 1

Chaptcr 1 1 /

Therrnndvnarrlir~o i Ell~rlrochcnlir~l Cclls

:

Settillg the activities (>I.~ g ( s )and Cu(s) equal to unity and using the relati(lns I - y:ml ::y:4m' (Table 11.3)- w e =a. U , k p ~ l ~ =, N : = V I M ; = y f ~ n ' ando pet 7

13-7.

Cleclrnrllrrr~~~al Cells Tan RP l l ~ r kd1 DCterm~l~~ At tivilj

Cwfii~ient~

We set the activity of the Cu(s) and Zn(s) q i ~ a to l u~~ity and write

According to Tahlc 1 1 3, a,n,ll = y:m; = 4y:m' and u,,,,,, = y:m: -- 4y:rn'. Using the fact that y, = 0.396 fnr %11C1?(0.500 111) anti y- = 0.-119 Cur TuC'1,t I .MI In). the Nemrt equntion gives

Givcn that y , , , ,

= 0,721 at 13.100m and that y,,u,+

j3,1

= 0.456 at 1.00 m,wc have

.I.l~enegative sign iridiciiles that Equation 13.32 will be spontaneous fmm right to left ti~ldcrttie given conditions.

f X A M P L E 13-5 Use tic data in Table I ?.2to calculate thc value of the emf of a ccll whoke ccll diagram

We should point out n lirnilaticln of Equation 13.31 M o r e we go o n . It trlrns out thai 1ht.r~is always a dilferencc in electrical pjtential acrosq the interfacc of two solutions unless they have an idelltical composition. This poten~iali s called a liquid junclion porrmiul. Fr~rexample, consider the salt bridge connecting the two solutions in Figure 13.1. There is a liquid junction potential at each salt bridge-solution interface, so Q u a ~ i o n13.31 in this case shr>uldbc written as

and

where E l ' , is the sutn of (he two liquid jitnclion potentials. The magnitude of the liquid junction potential causcd by n salt bridge depcnds upon the rclativc itiobili~ies of the cation and anion that uanstitute the salt hridgc. Potassium chlrjridc is used because a pvtassiutn ion and a chloride ion are about equally ttinhile in aqtreous solution. su the toti11 liquid junction potential due tc~a potassium chloride s:~ltbridge is quite small (around a millivolt or less) and can be neglected. Generally, in designing electroche~l~ical cells with a liquid juncticln, we must take care to rnini~riizethe liquid junction potential. We will discuss liquid junction potentials in Chapter 14.

,uldthe ccll reaction i\ given by

13-7. Electrochemical Cells Can Be Used to Determine

Zn(s)IZ11C1,(0.500m)llC'i1C12(1.00 m)lCu(s) Take y,-

=

0.396 for 0.50Um 2nC$(aq) and y, = 0.419 for 1.00 nr CuCI,Iaq)

5 0 1 I! T I O N . The two clectrodc reactions are

Zn(s)

7.rl(s)

~n"(0.500m) + 2 @ -

4

+ Cucl2(1.I)() m) -

We first caiculatc {he value rlr

(left)

Activity Coefficients -t

Cu(s) -tZnCI,(0.500

from Equation 13.31,

and then uw lllc Nernst cquatirjn with n

= 2 (Equntio~l13 7)

m) In Chapter I I , we learned that activity coefficients can be deterttlined from vapor pressure measurements and from lreezing point measurements. One of the muqt convenient and accurate mcthods for determining the activity cocficients of solutio~xof electrolytes involves electroche~rlicalvelIs. I ~ t ' sgo back to Equation 13.13

which we derived ti>r a cell whose cell diagram i s Pt(s)JH,( I atn~)IHCl(aq)IAgCl(s)lAg(s). We detennincd the value OC E ' to be 0.222 V in Section 1.3-3 by plotting the lcft side of Equaticln 13.13 against nl"' and extrapolating

Chapter 1 3 !Thermuriyn,~m~r s oi tlpc trlr~lien~ic.al Cells

t o In = 0. Pirlw tha~we k n o ~the value of L ' , either by extrapolation ut B versus nr data or, rnorz conveniently, from a table o f standard reduc~ionpotentials, we can use the sainc E verrus nl data (Tablc 13.1) and Equation 13.12 tn dererrnine y , venub tn fur HI:l(aq) Sulving Equatiun 13.12 for In yL and using the first sct of data ~n Tiiblc 13.1. we get

13-7. Electmchemic~lrells Cdn Be Us~d to Det~rnmineAr tivitv Crrt.lhcicnt3

5Cl L 1J T I O R : The rzactiw at tllc Icft elec~rtale(oxirlarion) is

arid thal at the righl electrode (reductrot\) i:,

The equatioll for the net ccll reactlon i c

{ H1(Ihar) + AgRr(x)

-

1-1' (aq)

+ BrC(aq) t Ag(s)

The corresponding Nerrlkt equalion i s 01- I/+ = 0.738 a1 in = 0 . 1 2 3 8 ~rirh-kg-'.'lhe other values obtained from the data in Tablc 13.1 are plotted In Figure 13.10.

?

7

If w e set u , ~d.l q R R , . and aH? e 4ual to u n i t arid uce [he rrl;~tiunsr ~ , . uBr = t r : = y;rn(Tablc 1 i .3), w e obpain

Solving this equation for In y, glves

For the tirst set of data i n thc above table.

13.10 'I llc rucan loriic ncctivity coefficient of HCl(aq) ohluinrd lrmn Equatlutl 13.12 and the data in Tahlc 13.1 p1ut1t.d agalnkt ~nIi'. FIGURE

E X A M P L E 13-6 Show 1ha1thc Nernct cquation for a cell whose cell diagram i s Pi (s) lH,(I bar]lHBr(aq)IAgBr(s)lAg(s) can be written as

or y+ = 0.970. The other datlr are plutted in Figurc 13.1 1

Buth calculittions done in this aec~ionhave invcllvcd 1-1 electml yies. Lei's now du an cxample ir~vol\:ingZnCI,(aq), a 2-1 electr.olyte. Corisider LI cc.11 whoac icH diagram i s

The reaction at (he lefi electrode (oxidatiotl) is where y-rn 1s thc activity ofHhr(aq). Use thr. 1-esk111 uf Example 13-3 and Ihe Ir)llowing cell data ru calculate and plot y- versus , ~ r ' : ~ . A

and thal at (he right clcctrode (reduction) i s

13-8. Elcrtruchcm~c~~l Mca,~rrenler~tsCon Rr I ! ? ~ rIn l Determ~neValucs ui A r H and A,S ui Cell Rr.actlun5

The following values of E at 298.15 K versus rn can he used t o calculate the value LIT y, as a function of nl (Probleni 13-14):

'These results are plotted in Figure 13.12

FIGURE

13.11

']'he mean ionic iwkivity coeriicient of HUr(aq) plotted arainst m i i 2 .The value< rlf y-- are

obtained accordin2 lo Example 13-5.

The equation for tllc overall cell reactiorl can he written as Zn(s) t- 2 AgCl(sl - +

ZIT"!^^) f 2 C1 (aq) + 2 Ap(s)

F I G U R E 13.12 The mean iioric activity coefficient of ZnCl, (aq) plotted against m'.".

Recause the activities of the three solid phases can be set q ~ l a to l unity, the emf of the cell is given by

13-8. Electrochemical Measurements Can Be Used to Determine Values of A,H and A,S of Cell Reactions The 2 in the denclminator of the 111+~:+a:,- term results because tlvvr, electrons are trrlnsferred in the cell reaction. as we have written it. Using Table 11.3, we write ~,~~+rr= : , 4m3y:, so Equation 13.33 bccomes

Equation 13.6 serves as the bridge hetween thermndynamics and elecmxhemis~ry.We can readily express A H and A S in tertns of E by using Fquation 8 . 3 t a

and then the rclatirln A G = A H G ~ b e nE versus m dala at cuffiuientlj !ow conccnlratjons, we ccli~ldplot E f ( R T / 2 F ) ln4m3 versus m 1 and then extrapolate to m 4 O to determine E", as we did in Section 13-5 for thr cell whose cell diagram is given by Fquaticln 13.8. Altematmely, we can use the data in Tdblc 13.2 lo ohtain

-

7'AS to obtain

We can usc Fguations 13.115 and 13.36 10 calculate the values of A, H and ArS f ( ~ any r cell reaction. Fur example, the temperature dependence of E' for the cell Pt(s)lH,(g) IHCl(aqjlAgCl(s) IAg(s)

is given by

Solving Equation 13.34for 111 y, gives

55 1

1 1--9. Sulubilr~vPrtrrlurts Can Be

from 0 C to 50'0'. The equation for the cell reaurion is

thc value or 4, S

Using Equation 13.35, we have

U e ~ r r m ~ n LVIII~ c d kl~ctruchcmical( ell,

is

=

(2){96485 C.rno1-~)(1.45x lo-'' V - K

= 28.0 J . K - -111ul

I)

'

and the valac of A r H ' is

a1 75 U. arid using Equa~ion13.36. wc have

13-9. Solubility Products Can Be Determined with Electrochcrnical Cells Because AC; = - I I F L : = 0 a1 equilib~.iuni,we can use Equdlion 13.7 tu derive. ;I rrlatlon between the standard errll' and the thernicxlyna~nlcequilibrluln conatant 01 llie equallcln assoc~arzdwrfh the cell reactlcln Setring E = 0 in kqu,~~ioii13.7 giieh

E X A M P L E 13-7 l'lic cell whohc cull di;lgl-iiin is

hats a stalldart1 erni (11-0.5359V i ~ n da lerr\pcrature cr,cffuient uf 1.35 x 10 ' V K ?'>K. IS K. Ucterlninc the cell rcaction and the vulucs rlf A,(; A,H' and A, S .

.

al

.4n important upplicstion c ~ fthis equation I S ru the determir~atior~01 the solubili~y yrodl~ctso f sparingly soluhlc salts. .As a collcrre exarnplc, let's consider thc detrrminatiu~~ of the soluh~lity proclucr ol' AgCl[s). 'She t~lualiul~ far the dissolution of Agl?l(s) is

.

i~ndthat at the right cluctrodc (reduction1 i s

+

H ~ ~ C l ~ i 2s r.)

;IIIJ the cell reaction

2 Hg(1)

+ 2 CI

(aq)

IE

Ph(s) Thr value elf

P

+ Hg:Cl,(s)

4

Wc :an const]-uct ~ I ulcctrochernical I cell uhuse net ccll cqualion is E~lr~;irior~ 13.38 by i~singaAg(s)(AgCltsllCl (aq) electrode and a Ag(s) 1 Ag ( a q )elec~roclc.Thc reduclion elezlrtdc-rzaction eqtlatiolls corresponding to thebe IWO clcctrodes ~ I I - t .

'

PtK1 ,(F) -r 1 IIg(1)

w

i.i

A K G= , -,IF E' = -(2)(96 485 C>mn1-~)(0.5359V) =

-103.4 k j rnt,l-l

and Ap+(aq)

+E

-+ Ag(.c)

13-9. Snlul>ililyI'ruductr Tan Be Uctcrnm~ned~virhFlr~rrnchemiralCells

To obtain the equation for the dissolution of AgCl(s), wc subtract Equatio~l13.41 from Eqi~ation13.40. Therefore, the left eIectrode (the electrode at which oxidation takes pl:ice) it1 the cell diagntn i s lhe Ag(s) IAg- (aq) electrode. and the cell diagram is

555

The preceding two calculations of the solubility product uf a sparingly soluble s a l t illustrate the general method. Let the sparingly soluble salt l x My4A,_(S I , su that the cquation for its dissolution is M,, A , (s) I

.

+ 11, M'+ (nq) + u..

AL-(aq)

(13.42)

Notice !hat this cell ha.: no liquid jurlctin~~ and herlcc has no liquid-junction potetitial. 1-hevalue of E at 298.15 K for h i s ccll i.s given by

and its solubiIity-prod~~ct expressiorl is

According to Eqi~ation13.37.

We can conwuct an electrochemical cell whose [let cell equatioti is Equat~on13.42 hy using an M(s) lM, A,, (%)/A'-(aq) electrodc and at) M(s) IW- (aq) electrode. The net chemical equation+of the reactiutl wuurrlng in a cell w ~ t hthe cell diagram

:II

298. I5 K.Thc ~tnndardstate hcrc is a orle-molal solutiun.

is Equation 13.42, so the valuc of E' of this ccll will give the sofubilily pmduct of

M,, A"- (sJ 1

E X A M P L E 13-8 U ~ the C data in T~hlc13.2 to calcula~etllc value r)f thc suluhilitj~producl ot' Pbl,(s).

S(31 (!TI O \ . C'sing the above calculation as a guide, we use a Pb(s)lPhl,(~)II-(aq) electnxle and a ~ b ( s l l ~ b ~ +electrode, (a~) whose curresponding reduction electrode-

rcaction equations arc

E X A M P L E 13-9 Dcvise an electrochcmica1 cell (without a liquid junction) that can ht used to d c t e m i ~ ~ e the solubility product o i Hg,Cl,(s). I!se lhe data in Table 13.2 to cdculale the valuc r)l- lhc solubility prcduct uf Hg,UI,(s) at 298.15 K .

5 0 1 CJ T I U N : T t ~ eequation for the rli\uulrition ol- Hg,Cl, (s) is

Tho electrode rcactiorls that give this n e l equation are To obtai~~ thz cquatinn Ibr tile diusolu~ionof PhIz(s), we subtract Equalitln 2 from E u u a t l o ~I ~ . Therefore, h e lc,ft electrr)de(the one at which vxirlatio~~ takes placc) is the PhlsjlPb" [aq) eltctrode. and the cell diagram i\

'Ihe valuc of E -~IL 298.15 K for this cell is E

= b,',

E;

=

-0.364 V

-

(-0.126 V) = -0.238 V

Hg,CI,(s)

+ 2 e-

2 Hg(l1

-4

f 2

CI (aq)

and

We place the oxidation electrodc reaction on thc 1el1 and the reduction electrodc reaction on the right, and so wc ha\e the cell diagram

* Therefore, accardiny rn Equation 13.37 According to Tahlc 13.2,

Ch,~l)trr1 1

1

1 h~rmodyndrnicluf Elec

~rr~rlien~ical Cells

1 I - 1 U. The Uisswi.~tion(:unstantr vf Wcak ,4crds Can Be Determined vvirh

If we solve Equation 13.46 for tioti 13.45, we ub~ain

The solubilily prduct is given by Equatloll 13.34with t! = 2

CI~crrncl~~~l~ical Tells

u,,, = mH.y H and suhstitutc this rcsult into

Equn-

We clin remdrlge this q u a l i o n into the hlnn

13-1 0. The Dissociation Constants of Weak Acids Can Be Determined with Electrochemical Cells

where

Co~isidcrIhe electrcxhcmical cell

where IiA is a weak acid and NuA is its sodium s a l ~'File . equntinll.; t i ~ the r electrode reaclions are

AgCI(s) -tt, + Ag(sj t-CI ( n q )

(right)

and the equation I i j r (he net ccll rcaclion is

The procedure is 10 vary rn,,,, m,, , and m,. and to evaluatc thc left sidc o f Equ;jtion 13.47 as il function of ionic strenglh

This procedure givcs In K i as a function of lorlic strength. We now plor ( f ' I K 7 ) ( E E") ln(m,,,nr,, / m A ) against ionic strength, and thc extrapolation to zcro ionic stretigth gives 111 Ku because yl,,,y,, /y,, + 1 as I,,,+ 0. F~gurt:13.13 shows such a plot using the follouitig data for a c e t ~ cacid [HAc(aq)J,IL 298.15 K:

+

Ciiveli t h a ~the fugacily (rsse~~tially the pressure) of thc H,(g) is orit. bar. lhc cmf uf h i s crll is given by

= 0.2224 V i s thc standi~rdreduction pr,tential of thc silver-sil\er chloride electrode. Using the rclatic~no,,.ci,,. = y,. n l , , , y,, m,,-(Table 1 1.3)-we have

whcre E'

Because acetic isid is a fairly weak acid, we have neglected the a111i1llcontributiol~11ial H (aq) and Ac ( i ~ qmake ) to Im in Figure 13.13. The extrapolation of the best linear tit to the data ploued in Figure 13.13 gives - In K,,= 10.958, or K', = 1.74 x 10- '. If the H' laq) and Ac (aq) frorn the dissocintior~of HAc(aq) arc taken into nccourll, you o h t ; ~ i rK,( ~ ='I .75 x 10 ', in agrccmcnt wit11 thc ":rccptcd" vi~lucof h',' o n rnolality scale. PI-ohlem 13-41 treats the acid dissociation of- f-ormic acid. whic.11 1s about ten limes stronger 111arlace~icacid. In this case the H'(aq) 2nd .4c-[iiq) i'rorn ttlz dissociation of thc for~nicacid cannor bc ncglec~etl.

557

13-1 1. \,Ve Can Arslp Th~r~norlvnam~c Vd1ut.s In InOividual lun5 i r ~Sr~lu~ion

where T = 298.15 K , E' = 0.2224 V, and nr = I I I , , = mN3, = t r z T h e irlnlu strength is equal to Im= m,*, im = hi.Figure 13.14 i s o plot of - In Ki agnit~stCz.f5xlrapolutir)r1of a linear curre lil gives all intercepl of 11.227, or Kc,= 1.33 x 10-'.

Ionic strength I r n o l , k g - l F I C U R F 13.13 A plot 01.thc [eft side

of Equatiot) 13.47 wilh E' = 0.2224 V against ionic strength using the supplied data for the cell R(s)lH,(l bar)lHA(m,), NaA(m,). NrlCl(rn,)lAgCl(c)lAg(s).The e~tm~ulalion ol the hest lincm fit to thefc data gives - In KO = 10.958, or Kc,= 1.73 % 10 for t~icticacid.

' Iunic strcngth 1 rn01.k~-I

E X A M P L E 13-10 The value of KO fnr propat~uic acid can be dcternjined using the cell whose cell dii~pran~ iu

F I G U R E 13.14 - In K,13 given by the exprussiull in Example 11-10 for prupanoic acid, against i t ~ l i c stwngth for the cell WE)IH2(p)lHP(aq).N W a q ) , N~rlPaq)lAgCl(s)lAg(s). A lincm f i t g i w s an intercept of I 1.227, or K,,= 1.33 x lW5.

A plot of

IJw the inllnwlng data tu dctennine the value of KO at 225'C.

13-1 1 . We Can Assign Thermodynamic Values to Individual Ions in Solution The thermodynamic properties of singk ionic species are tiot measureable quantities, but we can still set up a table of single-ion therlnodynamic properties in the fnllawing manner. Consider the reaclion Z n ( h ) t.2 H ' (aq) + zn2' (nq)

+ HZ(g)

The standard emf for the associatcd elec~rochetnicalcell is 0.763 V at 298.1 5 K. I f all the species arc in their standard states

S O L ( , T I O N . We plot ~ h cleh side of Equation 13.47 against ionic strength and eukrapulatc to /.cru ionic ctwngth. Neglecting [he small concentriltions 01 H' (ilq) and P-(aql due to the rlissocialioo o f HP(:lql, the expression for - 111 Kc:is

Tile standard Gibbs energies of formation of I$(g) and Zn(s) are equal to 7cro by convention. and so wc can write

13-1 1 . We Cat1 Assign Therrr~udyna~nic V ~ l u c stn Ind~rldualIons in hulutlon

We can continue this pruccss and determir~evalues of A , F for various ions relative to h,C;-ltI+(aq)].For exalnple, consider (he I-eaction

5hl

These results suggest that we choose to set S'[Ht(aq)l = 0 ai 79K.15 K, md wc wriit:

from Equation 13.54 and

The blandnrd einffr)r the associated electrocherriical cell is 1.360 V at 298.15 K. Thus, we can wnle

from Equation 13.55. We can calculate A, H - in each case frun~the equatiun

For Equation 13.4'1, wc have Notice that the combitladun of znL+(aq)and H (aq) is rlectncally neu~ral(zn2+ 211 ) in Equation 13.50 arid [tiat the cornb~nat~on of Cl (aq) and H+(aq) is electrically neutral (CI- H + ) in Equi111on 11.52. This resi~ltalong with tnany other sini~larresults shows that H+(aq) %ill cancel fr)r any neutral salts. Consequently, we art. 211 I~hertjtu sct t112 value n t ' h , G ~ I I ~ + ( a qarl>~rrarily )l and we choose to h e 1 A,F'IllL(aq)] = O at 298. I5 K. Accepting this convention, we call wrllc

+

from which we ohtain A, H '[2n2+(aq)]

- 2A, H [H' (aq)] = - 154 k l . i i ~ c d

and fork.qudtior~13.51, wc have A - H - = -1h7 1;J.niol-' or frcrin Equation 13.50 and

We set A,H [HI (ar])] = 0 at 298.15 K. and s o we have fro111Equation 13.52. W e can detcrmi~~c A , H ' and AfS' liom lhz temperilturc variation of E u for the appropriate cell. For the cell aascxi~te(1with Equation 13 49, ( i l k - / i ) T ) , = -0 119inV.K I , s o u t ' c a n ante

and

The standard molar. entropies of the neutral species H,(g) arld L n ( s ) are 130.08 J . K .moi -' arid 41.h3 J . K - ' .mol I. rcspectively, at 298.15 K. st] Fquation 13.53 hecomes

Values of the thertnodynamic propertics of single ions are tiihulatcd in a numbcr of placcs, but one particularly valuable source is n joint Amcricarl Chemical Society and American Institute of Physics publication [Jor~rnolc!fPlljsir.rzI rrvd Chv'hrrnil.rr1RP/>T~,IIC.P Daru, vol. 11, suppl.2 (1982)], callcd the NBS Tables uj C./rc+~rr;.rllI'/~lr.rrl!o(i~"rm,nic Pn~pr,rti~s. Srlrutr.rl Vulucs{ur Inoqnnir. rind C', and C2Olgrrnir Sr t b b 1rmrr.5 it1 .S1L'i~irs. Figilrt 13.15 sl~oasa typical page of these tables. Unlike the JAKAF tahlc., (Scutiorl 1 2(I).which give data over a wide range of tcIrlperaiurzs for ahour IS00 suhs~anccs.the ahovc tables givc data only at 298.15 K, hut for several thousanrl s~~bstn~lues. LVc car1 usc the data in Figure 13.15 along with A,GLIC1-(aq)]= - 13 1.228 kl.rnr,l (frorn page 2--47 of the tables) to calculate the value of the sofithility product t ~ TICl(s) f at 298.15 K. The equation for the dissolution of TICl(s) is

'

'

The uorrespunding rcsult for Equation 13.5 1 lurns uut to be

TICl(s)

-

Tl' (aq)

f

Cl (aqj

13-1 1 , We Can Arsigl) Thcrmdvnamic Lalucs tu Indlvirlua I Ion, in Solution

and so

and so

I

TI' ' TI ' TI"

1 I,{) T>rl>a 1

E X A M P L E 13-11

Lis~rlg ttle \slues of A,G'[Ag (nqbj = 77 107 kJ.~nol-', A,C'[Br (aq)J = Oh kJ 11io1', arid A,G [ApUr(s)J = 96 40 !d. r n o l - l at 298.15 K. calculait. the value of Kqp01 AgRrO) at 298.15 K - 10'4

SO1 11 1 I0 N: Thc equation for the dissoluiion of AgBr(s) 1s

AgBr(s)

4

A g ' ( q ) 4,Rr- (aq)

and

and K I>

I

TII:l TIi.1" TICI,' I lN:l, TII'I.

-

= e - l , C - / n T - eup

-70.M7 x -

10' J,rnul

'

~ X . J IJ.K-I J ~ .t ~ l ~ ~ - ~ ) ( 2 9 s . i %

We can i11st1use tabulated values of Gibbs energics of fonr~ationof ions to calculate acid-dissocia~ionconstants. For examplc. the NB.7 Toh1p.r c!fC'lae??~iral Thrrrnorl~~im~~rir Properries give

TICI,

and

11:1:19 TICIO. 1 111

F I G U R E 13.15

I'ngc2 135 fioril t l ~ cNRS Table.; uf Chernicol 'rhern~odldytlamic Propertics IJ. Phy.~.Uhtm. Rrf 1)rrtrr.

vol. I I . \11ppl,2( I t ) K 2 j ]

at 298.15

K. Tlie equatic~nfor the dissociation of fonnic acid car1 be written an

Ch,i~>lrr1 1 i Thermodyndmirh oi

klrrlmrhem~calCells

13-12. Batteries dnd Fuel Cells Are Devices That Use Chemical Reactions to Produce Electric Currcnts

and .;(I

K = p-lrl;

!RT

= exp

[

-

21.3 x LO' ~ . m o l (8,3145 i.mol-I K-')(298.15

KT]

E X A M P L E 13-12 Given the tollowing d;~lafrr,rr~ the IVBS 7itl)lesc!f'Thr*~rtic.rrl Thrrmnd~nnnzicPrryr~*rrics, c;llculak thr: value o f rhe prrltonation cullstunl nr an~intl~rlethane at 2'18.15 K.

Batteries are devices that are cunlmonly used to provide energy for numerous ilppl~cijtiuns. Batteries are classiticd as primary if they are not rechargeable and as secundary if thcy arc chargeable. Examples of primary hawries ai-e alkaline hattcrics, commo~il y used in flashlights and toys; lithium hatterics, uscd in wristwatchrs; ~IIILIIlierclIry l>.~iteriea, used in heart paccrnaken, hcaring aids, and computerh. all of which requil-e ~ h s constunt vultagc this type o f battery pluvides. txamplea r>fsecondiryhattcrics arc lsad storage batteries used in automobilzs, and nickel-cadmium (NiCad) battcrics used ill co~dlesstools, shavers, toothbrushes, and many other device';. The cell diagrani of a mercury battery can be descrihcd by

with a sirnplitied version of lhe v t ~ r r e s p c ~ i d itell ~ ~ grc:~ctinnbeing given by

Norive that all the sp'cies in the abovc ccll rcacrion are iri cunde~isedphases. Constquenlly. there is no change in reaction concentration during discharge. which nleatls tttat the voltage (1.35 V) stays constint during discharge. This ctlnstiailt volragc, along with a long shclf lifc. is unc of thc imyurtiiot distinguishing features of a mercury battc~y. Its low atunliz miIss and slrong reducing polt.ntial givc lirhium many advantagcx iix a rttaterial lo use in batteries. For a given size, lithiu~nbilttcrica have the highest powc~,to-mass ratio of all batteries. They can also be designed t o t ~ a v elong dl-;tin r a b dnd lung lifelimes, excellent characteris~icsti)r an application such as cartliac pilcc~nakcrs arid wrislwatchcs. A lithium battery can power a wristti~;itchfor marc t l l ; ~five ~ ~ YCII[.S. The classic exarnple of a secortdary battery is a lead rlcrraye hat~cry,wt~oszccl t diagram is

5 13 1 U 'I I Cj h : The equarir)~~ tor Ihc protolraritln uf CH ,NH2(aq) is

where PbO,(s). PbSO,(s) denotes a hc~en~grrieous mixlure ot thc two stllids. U'hcn fully charged. this cell pruduces about ~ w uuolts. The 12-V hi~tteriesconin~only11acd in autt~~nobilcs coli\ist of six CCIIS in scr ics. Thc overall ccII rci~ctionupon dihchalp is

= c-L

0 ,Kl

--

--

exp

[

14.26 x 10' J mu1 --

'

I

~ s . ~ 1 a s r . m o i - ~ . ~ - ~ j ~ ~ ~ 8 . 1 ~ ~ - ~ j

During discharge, leu1 sulhte is funned at both clcctrudcs and si~lluricacid is used up. In the process of k i n g charged, the above reaction is reversed.

Chaptcr 1 1 1 Thermodynamics u;

13-12

tlpt t r r ~ c h r n i c a Lclls l

ll,{g) i 2 OH (aq) -+ 7- H20(1)

+2 t-

(13.56)

'i'he OH-(aq) i < cupplied by the concentrated NaOH(aq) or KOH(aq) si~ll~tion bathing 111~two electrodes. The oxygen is reduced at the othcr electrode according to

567

Note that the overall reaction is the combustion of hydrogen and oxygen to f u m wnter

Anuthercomniunly used rechargeable battery is a nickel-cadmium inicad) battcry. A sealed nickcl-cadmiurn battery is more stahle than a lead storage hattery and can he left jn;lclive for long pr.dods. The rwerall charge-discharge reaction can hc written as

In alrr~ostall hattcries. a lnetal in the form c~fan electrode is oxidized by an oxidizing iigcnt. Although batteries have the uor~venienceof being portablc and self-contained, they arc not huitahle for very large scale energy production. A promising il1tcrnatii;e IS a file,] ccll, which is an electrrxhemical device that utilizes the oxidation of a file1 t o produce electricity. Fucl cells d~CCerfrom batteries in thal the tucl and uxidizcr art: fcd rorltinr~ouclyinto the cell, so that its operating life i<essentially unlimited. The H: ( ; ) - 0 2 ( g )tuel cell, which has been a power source on many manned space flights, is show11schematically in Figure 13.16. Hydrogen is fed into a porous, ~ne~alljc (often ~ i i c b e l )clcctrode, whecc iil i s clxidizrd to fomm wnter acct~rdingt c ~

Rattcrics ,\rid f u ~Cells l Are Lle~,ictlsThat Use C h ~ m ~ rRcaclinrlr al in Prnducc t lrclrir Cu-rents

The currerlt tlltouph the cell electrolyte is cnrricd by the Na- (sq) and OH-.( a q ) , s t ) h~gli cr~nuentrationsare used to favm a low intcrnrtl cell resistance. The mqcu appeal ol~fuelcells is thar they are not limited by ihc inherent cfticicncy o f heat engincs. Recall from Section 6-7 that the maximum ei'liciency that we could obtdin if we burned hydrogcn tu ruri a heat cngine is givcn by (Equation 6 - 3 5 )

N; Thc cornhristion reaction of one molc of

methane i q given hy

F I G I J R E 13.16 cchcmar~cill~~rrratinu of a H , ( g ) - O : ( c )fuel cell. ?'he H,rg) is p;r?;sc,dover n porous electrode, wl~crcoxidalioll takes place according to Equalior~13.56. The recullir~gelectrons pas%through rhe uxtcrn;il circuit, and (lie oxidizcr [usjrrg O,(g)J is reduced according to Equation 13.57. The current is carried through thc cell hy 011 (aq).

In Equation 17.58, the oxidatirln Etate of the carbnrl atom gws firm -4 to +4. and that (11- each of the four oxygcn atoms goes from 0 to -22.Thus, n = X irr Equation 13-58.

1

Chapter 1 I / Thermodynamics ui Elcutrucbr.m~t~~l Lells

13-31. The temperalure dzpenderlce of thc standard crnf of a cell whobe cell d~agramIS Pt(sjIH,I 1 bar)]HCl(a = 1.00)1AgCl(s)~Ag(s) is given by

13-39. The value of the acid-dissuciation cvnclant of propanoic acid call be determined u4llp the cell

1132 the Iollr>wing data at ] ( P C to detrrnmine the vi~lueof K,, frlr propanoic acid at I0 C.

u h e r e Lhz tcrnpcrattlrz is in degrees Celsiuk. Write tht: as.cociated cell rcwtiun, and dztermule the slantlad change in Ciibbs energy. the cnthalpy, and the entropy at 298 I 5 K.

13-32. -rhc rzmvlaturr dcprndel~ccuC the slurrdard errlf nt n ccll Prl OIH2il bar)lHUrrlr= l.OO)llIglBr~~s)IHg(l) is giwn by

whose

T:ke L' fur {he cell to be 0.23 142 V at IWC.

cell diagram is

wllsrr I ic the Ccl91us tcrnpcrsturu. Dctvrtl~inc the chilnge in the rlarld~lrcl Ciibhs c.ilcrFy, tlic srarlcli~rdcnlhalpy. ant1 the sta~ltiarderllrupy at 298.15 K . Givcn that h I li,(g) l = 13[l 68.1 I ~ K - ~ ~ ~ o I - 5 - ~[lIg!Hr,(s)] . = 21R O l - ~ - ~ . r r l u l - ' , and 5' [llg(l)l = i~re 7h.ll2~.k-l.rl,r>l- , ~ t2~)8.15K. c;llcul;ite Ihe value of S lor Br-(aq). C u ~ ~ ~ pyour an\wer W I I ~~ h cratuc in the ,\'.8S 1i1l)lrs(IJ Chrrl~rrtrll%rrttir~cI~~ruttzrt. P~-o(,er-ficr.

13-33. 1Jse he data ~ I Tablc I 13.2 to c,llculate the value of Ihc \r>luhll~ty product of H&Cl?(s)

1 3 4 0 . The value the cell

lhe acid-dissociation constant vf acetic ;icitl cun he dctcrnlirlud us111g

at 198.15 K .

13-34. Ilse tic data III 1:1ble 13.2 to vdlculatc the value oi the soluhili~yproduct r d A&SO,isi at 298.15 K .

13-35. IJce the data at 298.15 K.

111

table 13.2 trj c;~lculatsthe v;ilue of lhe suluhility pnxluct n t PbSO,(sj

13-36. Givelithat dlcstanrlard rzdl~ctionpc)tential oi 2 D+(aq) ~ a l c u l a t ethc valuc of the equilibrium cvnstanl uf

+ 2 t-

-

Use the i'ullowing data at O'C to dcternlillc the value WE K,, Cur acetic acid : ~ t (PC. ' h b c L fm the ccll to be 0.23655 V at 0 C.

L),(g) is -U.OG34'

13-41. The \ a l u ~01 lhc acid-dissociatlo~constant c~fft>rr~ucacid can Lw dctcrn~ined~ h i n g the cell

ClO-(aql -t H,0[1)

+ 2 c-

+ U1 (aq)

+ 2 OtI

(aq)

E' = (1.40 V

colculatr. the vnlr~enl' the acid-discncialinn constant of HUIIXaq). Hint: k'rx~n [lie fact that h',&= I MI x 10 ".

d to u x

Determi~wt ~ l ccquation for the uverall reactiorl nt this ccll and write thc corrz$pt>ndir~g Memst cqwalrnn Now use tilt rclation

13-38. Givrrl the rrriu~:lir,~~ ele~~rt~Clt'-rt;~ctiun diitii at 298.15 K to show that = calculate the balue ct'thc acid-dissocintion constant of 13C102(aq).

~HF~YH,,"~cI-YcI-

-

F

YF~

k'

Ill

K,,

Chapter 13 / Therrn~d~nam~cs ul Clectru~h~mical Cells

where E' ic the stat~dardentf of the cell

Show that this t q ~ ~ o t i ocan n be writt.cn a?

Ws)lH,(g) IHCNaq) IAgCl(s)lAg(s)

Using rhe relation Llescribc hum tllc value of

% call be detcrlninctl by plotting show that

sg:iins~ionic strcr~glhillid cxtl-;~pc>lating to /.crt>.U~tlikcthe cusr uf prop:lnr)lc acid prcsentcd in Exmrlyle 13-10, the HI (aql and FoC(aq) frvm thc diswciation o f Ht'o(aql cannot he ipntrred. Thr way to prrlcced is as follows. First, calcuhk tlje values of In Ki and ncglecling the H7(arl) and Fo (;iq) fr.olu the dissociatiotl of formic acid. Plot In K: against I",, and ubtnir~a preliminary value uT KL>by extmptdaling to zero ionic strcl~gth.Now use Ihe preliminary \aluc of K O I r ) calculate nt,,, using

uhtl-e mHIvand mGdS,>ilrz thc stoichiometric concentrations 01- HI:u(aq) and NaFdaq). Realizing t h a ~m,,,will be fairly small, ncglecr m , , , with rcspecl 10 nr,,, and mN3, and

wrile

'1'11e valut. of ; . Ican k eestim;~te~l usin? Erluotion 1 1-56 on

;I

rnolnlity rcsle

Using !he above prucrdure and the lollowing dntn at 25'C. calcul:~t' the value of Crjr tornlic acid at 25 C.

K,

13-42. T t ~ value e c ~the f discnciatiun cor~stautOF water. Kw, can be determined u ~ i n g the cell w

P I ( E ) ~ ~ ~ : ( ~ ) ~ K OKCI(aq)lAgCl(s)lAg(s) H(R(~I~ Show that thc emf rll. this cell is given by

(1)

Usin: the fi~lluwingdala fi)r t t ~ cccll givcl~by Equation I with I I I , , , = 0.01W)rnol,kg-', determine thc value of Kw at each ternpemture by plotting thc left side of the abuve equalirm again5t ionic strcngth and extrapolating to zero. Take E" 10 k given by

Now plot In hry against 1/ T , curve f t ywu~rcsult to

where

vl,,,

15the

partial molar volume of the HCl(aq) at mnlality m. -kt,, i a [he p,ir.rl:ll

md TdGCI are the molar vnlumej of Agl s) molar volume of H?(g) at pressure I>. and and AgCI(s). k g u c that at rel~tivclylow prehxlires

;111duse Bqualitm 12.29 to dctcr~ninethe value o f A r H ' for the dissociatirln of water as a quadratic polynonliai in T. The experimental value of A,H" at 2S'C is 55.9 H .nlol-I. 13-43. L'cc ~ h data s in [ablc 13.2 tu ualculatc tire valr~eof thc frrluhility produci r \ l PbCl,(s) a1 298.15 li,and compare your rcsult wilh the value that you obtain uslng the data in the bL3S lirhIes uf Chrmicrrl Th~r~nt)d?.lrurnic Propertirs [ A Phj.s. C h ~ l nDohi, . vol. If, suppl. 2 ( 1982ij.

13-44. U.rc t l ~ tiah c in'l'ahlc 13.2 to calculate the kaluc uf the solubllity pruduct of HgzSO,(s) XI 298.15 K, and compare your rewlt with the vnlue that yo11 o b t a i ~using ~ the data in the NBS 7ii blcr r!f r ! i ~ r r ~ iThcr~(~tyi~clnnlr c~~l Pro~~c,rfirr [J. Ptiys. Ulwm. ncrtr~.vol. 11. suppl. 2

where Ej is the emf of the cell at pressure P,

13-53. Consider lhe cell whose cell dingrim is

-

Show that A rV x - V , , ,. Using a vlrial equation (11' state throi~ghthc wcond virial ccotlilicient (Equation 2.23). shuw that (see Prubler~~ 13-52)

{1982\(.

13-45. G I V Cthat ~ A,C; [ ~ h ~ + ( a q=) ]-21 43 kJ rr~ol I , A,G' [~O:-(a~)l = - 7 4 53 kJ mol ' ar~dA , G [PbS04(s)J= -81 3.4 k) mol-' at 298 15 K, calculate the valuc o f the mlubility h ollc obvulned in P r o b l c t ~17-35. ~ p r o d u ~ clf t PbSO,($) Cornpare your rcsul~& ~ t the 13-46. Clse the dam in Figure 13. IS along with A I G L [ R .taql] r = - lI13.96 kJ ~ n o l - 'tocalculatc thr value of the solubilily product of ttlalllum(1) bl-on~ideat 298,IS K.

-

Now use thc data in Table 2.7 and Figure 2.15 to cstitnate the value of B2,( T I B , , ( T ) / R T fur H,(g) at 2S5'C.[The vulue of B;,(I") from numerical tables is 0.4 l h.1 Pillally, use tllc lilllnwing data at 25'C Tor the above cell to calculatc the value ot E a\ a function of pressure, and cornparc the result o f yrlur calculatio~lwitti the c-upcri~r~ent;il -cult5 graphically.

1 3-47. I!5s t hc data rn kKS 7i1blr.5 r!l ( 'hr+izic.ulY henno~t~ntlt~nr Yroprr-fi~,~ 1.1 l'lfys. Clrrt~~. il,ttr~,vol. 11. ~ u p p l .2 (1982)J lu c,ulcuIate thc value of the solubility product of barium sulfatc at 2921.15 K.

-

13-48. Given that AIC; [HSO;(aq)l -755.9 1 kJ-mu1 and thal A,G' [ ~ O : - ( a ~ )= l -744.53 kl.nlnl-', calculate thc value of the second acid-dissociation conslant of sulrui-ic acid a1 298.15

K.

13-49. l l s e the data ill ,VRS T,lbl~x1 , f C'l~rrnir~cll Il%ermorl'~ntil~ii Pml~rr-liraIJ. Pliys. C'hcm. UCIIU, vol. 11. suppl. 2 (1982)] to calculatc the u l u e of the first acid-dissociation constant d a r c e n i c acid. H,AsO,(aq) at 298.15 K.

1 3-50. IJse thc dava in NBS Yirbks rd Ctierrricul T/ler~noclrnurnic Prr,l,rrfirs [J. I Chrm. Zlufu, vnl. 11. siippl. 2 (198?)] to calculate the value of thc f i r ~ base-protonatinn t cr~nshnt of hydrilzillc at 298.15 K. 13-51. Cr)rupalc tllu idcaliied alnnullt ut work avail;iblz w11ct1one rnc~lrof prrlpanc is burned ill ii turl cell at 2'18 K and in a heat cl~ginewith 1 , = %)(I K and 7: = 300 K. Calculate the valuc 01- E ' for this tuel cell. 13-52, Dcrive a g e ~ ~ u rrcl;ilir~n al for rllc prussure rt~.pcnderlr.ucher~lical

The next jive prob1rm.r involve ~~olrultrrrrfg !he vc~lur( B E -fora rlt'~.fr~dr W U C I ~ U thilr ~ ~ IS (I rombinutinrl cg'urhrr el eat rod^, renrritms. 13-54. Somerimes II rri~ghthe nccecsary to calculate thc viluc of E' ior a reduction electrode reaction that is a coinbination uf othcr reduction electrode rcnctionc. For cxample. cons~clrr the lwo reduc1ir)n electrode renctirlns l . K H 1 ( a t l ) + M r ~ O , r a q ) + 5 r -4H,Cl(l)+Mn'1faq)

+

2. 4 H+(aq) + MnO, (s) 2 e- + 2 HLOtl) + Mn2+(aq) We can use these ddla to calculate the raluc (]I' F' for 3. 4 H ' (aq) MnOj (aq) + 3 e - + MnO,(s) + 2 H,O(I)

+

F.]~l49lV E; = 1.20K V

W e hrst ~lntt.that Equaliu~l3 resulls from subtracting Equation 2 frnm Equation 1. It ir important I(>reali~ethut E , i~ not equal to Ey - E;. howwcr. because E" is an i ~ t r ~ n a i vproperly. r The standnrd Cibhc energy change for Equatiot~3 is given by AG; = AG; - A[;,. howevcr. Using this fact, uhow thnt

18-55. Given l t h t

calculatt. the raluc of E ' for

13-56. Ciiten t h ; ~ ~ ~;'(nq)

+ 2 e-

-

Cr(s)

B' = -0.9 1 V

calculate ihe value uf E' for

13-57. 'Ihe p r e ~ i o u sthree problems rlevc!op the idea that you must use value< of AG' In cnlculntc v;lluzs of 8 - ul- reduction electrode reactions from the val~lcsof E irotn other

clcclrrxlc 1,cactiuns. When the cambination is such that the electrons or1 each side of the rquutiu~~ c,~ncel. hnwevrr, we do nth have to use AG" as an intermediate quantity and can write E Y ~= ,, k ,, 1 E , , dirzc~lyas we did 111 muation 13.22. In this p~oblern,we derivc Equation 13.22. Consider the ~ w reduction u clectrude reactions

1.

A+tiIY

2. B

+ n, r

-

X

-+Y

Ei,md

Ei.A~

Show that the comhrnntion in which the e l e c w n s cancel

3.

~r~,\+?l,

Y + n, B + n , X

is

13-58. Crmsirler thc reduction electrode reactiot~

Using the fact that the dissociation constant for H,O(I) is K y = I .(XI8 x 10 calculate the value nf E" for

l4

af 29X. 15 K1

Nonequilibrium Thermodynamics

To this point, we have discussed the thenuodynamics only OF systems in cquilibrii~~n.

Lars Onsager wah born i n Oslo, Norway or1 Novr~nher27, 1'103, and dlcd in 1076. Hc received a dcpree in chemical crigincering I-iun~the Nurweg~anlnstitule nf 'I'echnology in '1.1-undheimin 1025, altcr which ht: wcnt to Zuricl~to discusc with Peter Debye a flaw he had discor*eredin the Dchyc-liiickel theory. Oncagcr was a graduate ~tutlentwith Dchye fro111192h lo 1'128, dunng which time he dsxeloped the Onsagcr linliting law for the crrnductivily ot dilute ~ v l u ~ i of n~~s electl,olytes. In IY28, he c m i ~ r ~ t ctod he llnitell Sla~ehand spenl livc years all dssoc.ialz in chcmi$trg ut Brown University. where he developed tlh: Onsager rcciprucal relations, which ttw Norwegian Ir! p,, meaning that rnattcr will fluw rrrrn~con]partment I to compartment 2. as e x ~ c t e d . We call relate Equation 14.19to Equation 14.18 in the following way. Because the V = V, 1', is fixed, we can write

+

for the two-compartment systcm, and so

Substituting Equation 14.15into Equalion 14.13gives

(constant T and V )

To make the complete connectiun heiween Equations 14.18 and 14.1'1, substitute Quation 14.19into the above equation to obtain

Thc first tern) in Equation 14.16is essentially Equation 14.12. I ~ t ' look s at the second term. Fquation 14.16 says lhat

I ' C I S ~=, ~- d A = - d n , ( p 1

-

p2)

>O

in agreement with Equation 14.18.

if 1; = Tz and 11, = f i 2 .Thus, d V , > 0 if PI > P,, which means t h a ~the flexible wall t+-illmove to the right in Figure 14.2i f P, > P,, which is what wc should expect. The third tcmi in Equation 14.16 has to dn with matcrial flow, ur diffusion. If we lake T, = T, = T and PI = P2 for simplicity, then Equaticln 14.16 i s

I

E X A M P L E 14-1 Suppsc a two-compartment syslem is encloscd by flexible. diathcrmal walls and is immersed in a heat bath at temperature T and pwssure P.Dcrive the analog rlt Equation 14.21 forthis system. Assunie that the wall sepwatjng the w o compartnlcnts allows only rrlatter to flow from une compartment In the other.

Kecall that (replacing activity by concentration) SOLUTION: $Vestartwith

lijr a solution of conccnlration c, or [hat y depends iogarithnlically on the cc~ncentration. 'rhos. Quatior1 14.I H says thar d n , < 0 if p , > p,, which means that matler will flow frt~rnuorrlpartment I to compartment 2 if the concer~trationof the dilfusing species i~ grr:itel- i n cunlpartlnent 1 than in cornparlrrlent 2, as we know from experience. S o far w e have becn treating an ixolated (two-compartment) system in this section. Suppose instzad that the ~wo-cn~npartment system is in contact with a heat bath at ternperature T, so that T, = 1; = T. Furthermore, for simplicity, let the wall separating the two compartments he rigid so that V , and V, are fixed. In this case. the twu-compartment systcm is at constant I ' and constant V l and V,. The appropriate thermodynamic state functic~nunder thexe conditions is the Helmholv, energy, A = A ( T , V = V , V,, n), aod the governing equatin~risd A = pdn at constant T and 1'.Thus, we have for the Iao-cornparttrle~ltsystem,

+

dC = - S d T

+ V d P + pdn

= pdn

(ton\tani T and P)

Becawe the pressure is held comstant, we can wrile (see Equurioti 14.20)

d.Spd =

TdS-rlH T

-

dG

- --

T

(consrant T and P )

potential dimerencc is called elertroo.~mosis.For a given system and diaphragm, the elecrmo~moticpressure (EOP) is delincd as

where .I,, i s the volunle flow across the diaphragm. The electroostno~icpressure is the ~in~l,ortionnlity constant relating A P to A @ at the point whcn thc volulrle Row has ccaqed.

Now coucidcl. rhe opposite cxpcrin~entwith thc same system in which the electrodes arc sllort circuiteti so that A111 = 0 and the solution is forced thrrlitgh the diaphragm by n pistun (Figurc 14.5). I n this expc-riinent, therc is an eleclric current through the diaphragni us well as a volume Ril\v, J , . It turns out that the magnitude of the electric current is directly proporlional to the volume flow. We define a quantity called the srrerrmi~lgcllrreni. SC, by the relation

F I G U R E 14.5 A schem;itic diagrnnl of an apparatus to meawre ~trcamingcurrelit Each comparime;lt containc idenlical solutions r ) f an electrvlyte and the two conlpartments art. separated by a pllrnur diaphragm. .4nammeter nieasurcs the eleciric current that accr~nipatliesthe flu~dHnw uar~setl by the piston.

with a pressurc rlifmtlce A P , a ~ c ~ l ~ diffcrence age results. If this voltage difference is measured with a potet~tiomcterso Lhilt I = 0, then A$ turns OUL to be proportiunal to A P. This fact allows us to dcfine a strr,nming pnicniiul, SP, by

The stiranling cu~rent1s the pn~por[ionah(yuunstant relating the current 1 In the vrllume flow J , . It is found empirically that the tnagnitude of the electroosmotic pressure differs only in sign fron~the streaming current, so that

EOP = -SC

Similarly, &hen a potential rlifferencc i v applied across the diaphragm under conditions in which A P = 0 (horizt)ntal tubc), there is a volume flow. J,, as well as a current flow,I. It rurns out that J,, is proportiunal to 1 under these cot~ditions,and vie dchne a quantily ~lrurmo~rmotiujlow, EOF, by

We will see below that Equation 14.36 results from the Onsager reciyrucal relatiutls.

E X A M P L E 15-3 Stlow that the clestroosn~olicprclsure and the streaming current have the same unils. SOLUTl(1h: The S1 units of A P arc ~ , r n . =J,m-' ' and thosc uf A $ J-C-', so that the units o f EOP are

The S1 units (if I are C - s - I . and thost. o f

JV

UC

The streaming ptential and elecltoostnotic tluw are relatcd to each other by

V=

are m3.5 ', so that the units of SC are

fhus the llnitr nE EOP and SC are the same.

Then: ale other pairs of eltctrnkinetic quantities that are relatcd to each other. Fur example. if the fluid is forced through the diaphragm shown in Figure 14.4 or 14.5

Equations 14.36 and 14.39 are just two of a number or relations hetheen various electrokinetic cluantitier We will 11uwshou how Equations 14 36 and 1 4 . 9 result frum the On~agerreciprocal relations. If h e apparatus illustrated in Figure 14.4 is at a fixed temperature and pressure. then the appri~priatethermodynamic equations are

Chdp~er14 / Nonequilihri~lrnThermodynamics

i +6

The Dnsa;r+ K e ~ i p r u c Kclationr ~l Are Haserl url thc Princ~pleot II~td~lrd Hdldn~c.

a h e r e p.,, i s the chemical potet~tiitlof the cation, / I , - i s the cl~ertiioalpcltcntial of the anion, p,, i s the chendcal potential of the water, and is the electrical potential in compartment I . 'The Ii~sttwo terms in Equ;ltion 13.40 rcpresmt the electrical work involved in changing the nurtlher c ~ fmoles of cation hy d r l , . - and the nutnber of moles of-aniot~by drr,-. if we use an equation similar 10 Equation 14.40 l i l t compartment 2,

The correspnrading linear fl ux-force x1arirm.j arc

thcn

We can riow use Equations 14.46 lo pro\c F q ~ a t i o n s14 3h and 11.39. The elzctruoslnotlc presaure (Equation 14.34) i:, obbainetl by letting J, = 0 in Equat~on14.16~ and solwng fur A P / h @ to obiain

+,

IJqing the facts that dn,+ = - d n ,

Jn,- = -dn, , and dn,, = - d ~ 1 ~gives _ The streaming current (Equallon 14 35) is obta~nedby setullg A $ = 0 in t q i 1 ~ 1 1 1 o n ~ 14.46 and dividing Equation 14.4hh by 14 46a lo obtdrn SC = --Ll,,

Exatnple 14- 1 showcd tl~atrlSiIK,= - d C / ? - for a system a t constant 7 and P, s o Equation 14.41 van hc writtell as

(14.4Xj

Lvr Erlualions 14.47 and 14.48 show that EOP = -SC bccause L,, = L,,; lhus ploying the validity of Equation 14.36.

w h c ~ cA F , =I"+ - - - p I , , = p:- .--J L , , A/L* = pZu- p n , and A$ = $,. Now divide Equation 14.42 by dr to get

l?r, EXAMPLE 1 4 4 Pruve that tilt srrcar~lingputenlid. SF! is lhc ncgnti\c of elec~ronsint~tic Row. EOt:

whcrc. .II = d n , , / d r , . I = d n , . : d t . and Jw = dtrla/ d t . The flow of the neutral 1-1 elc.ctroly~eacross the diaphragm i s ,l>= J+ = J the clcctric current across it is I = r J+ - e./ , and the chertdcal potcntiid of a I -- 1 rlectrolyt:: is p. = !I I p - . Suhhtituting these lhrec rclirtiol~s~ I I ~ Equiitior~ (I 14.43 gives

.

+

h l o s ~electrokinetic experiments arc carried out such that the crtnccntratiar~*)I' the solution is the samc in botl~ .partments. In this casc, there is n o conceritratinn dspcnder~ceof A / l , or A p *. s r ) (lquation 9.9)

*-here i sbands for s sr w. Suhstituring A l l l = 1'(A P into Equatiutl 14.43gives

The tcm1 in parentheses is sirxtply thc total \'r)lumt- Row, J , , acnls, rhc diaphragm, su we finally havc our basic erlli-opy production equation for eleutroklnetic phencbmena:

SCI L U T I Cl h : 4~.corrlingtu Guatioll 11.37, SP = (Arl,; LIP),.^,. Lct I = 0 irt Equatioo 13.4hhl~~gt15P = - L i b ~L,,.ElcctronrmolicHowisgivcn hy EOF = (.i,.!II,,=,, (Equalion 14.38).Let AP = (1 ~ I Equalion5 I 14.46and diwdc orlc t)y the ollier lo ubtain EOF = ( J , : ' I ) = I.,,iI Therefure, SP = -EOF heci~uwI .,,, = L , .

,,,-,,

.,,.

,

Problerris 14.. 11 to 14-15 have yo11 derivz relaiic~nshctwccr~other electrokinetic quantities.

14-6. The Onsager Reciprocal Relations Are Bascd on the Principle of Detailed Balance 111this section. ~2 will usc a chcmical kinetic ccheme to gain some insight to the r)ripi11 of the Onsager reciprocal relations. OE (he lniany applications of nonequilibriun~the,,to i~ariousprtlccsacs. the applicatin11 i o cliemicul kincrics is exceptionally rnrxiy~arni:~ Ijmitcd in practice because of thc requirement that the Ilux-forcc rcl;~tionsbe fincar. This lineari t j requtrtment mcans that chemically I-eilcting >ystems rnilht be ulosr. 10 eq~litibriun)Tttr he f~wmalismof r~oriequilihriumthcl-mudyr~atr~ics 10 he LIPPIICL~~IC. Nevrrthzlrrs, wc nil1 sce in this section how the 0n.cagc1.rcciprociil rcli~lit)nsariw hg studyi[q 'L u ~ n p l cchcmical reactiotl.

14-6. TheOnsag~rReclprucnt Relations 4rc H,lfcd nn thc Pririrrple of neta~lcdH d l , ~ n r ~

Let's start with a reversible first-ordcr elementary chemical reaction described hy kxv

X+. Y

+

where we ha\ e used the fact that [XI [Y]= constant. Equation 14.53 is our fundamental entropy production equation. We cat1 write Equation 14.53 as

k,, 'rllis reaction i s tcw simple to involve the Onwger reciprocal relatjon~,but we can use it to introduce sonle not;itiotl. The ratc oT this reaction is given by

where J i~

where [XI and [Y1 are concentra~ions.We can define the flow of !hi%reactirlrl fmm left lo ri$t hy

ih called thc nj'jinig. Thc linrir Rux-fnrcc relalion is

At cquilibriu~n.J = (1 and Equation 14.49 givcs

]XItq

and

+

A giwn in Equation 14.ih is rafid onl? for small values of A/ R where K i~ the molar gas conqtant. This mcans that [XI and IY] n~usthc cloqe to their equilibrium values because { i , = pI at equilibrium. koblern 14-18 shnwi that

u,=WI-~Y[,~

(14.50)

+

Note that a, +a, = O because [XI [Y]= [XIcq [Yjq. Using t h ~ sresuit, Equatiorl 14.49 cat1 he rxpresred in terms of' a, by (Problem 14-16)

Now let's apply r~u~~cqi~ilihrium thcr~nodynamicsto this rc:+clion. For cnwsriience only (scc Problcln 1 6 1 7 ) -wc take the reaction system lo he isolated. Fmm the quation

Re

Equation 14.49 and v.hcrc

EuperimentalI>, the linear relation betwccn J and

Now define

n,=~~I--

~ I V C I hy I

if L Y ~ ~ Lns LIII i t ~ 1w0 t sirlcs of thc r n c l n h ~ ~ ~This n c . potellti;ll difference is often c;illcd lhe ,Vvrn.\t oolc.trtir;l, especially i n biophysics. A number of biological mernbranea are ~ n u c hnlorc pcrmcablc to one particular ion than to others. For example, squid axon nerve cell nienihranes in their resting state are altnust zxclusively permeable lo pot~ssiumiclns. Tlic concentrations of the potassium ions on the two sides ol' the membrane art ahour

Poruu+ d-aphragtrl

where thc electron. e-, is in the Ag(s) electrode. The reaction that takes place at the lctt elzctrodc(oxidaticm) is described by

Now if we let the electrical potential that an ion experiences i n electrode campartmen1 Or I j be $" or $&.then (Equation 11.73)

'I'hc equatiot~h ~ the r nvcrall cell I-eactinn is given by the sum of these two cquations

Solving for A$! and using Equation 14.85 gives us

'I'hus, thr clrlving iorce ol the cell is due oniy to an activity chnngc, or a conccntratiun change. Such a cell is ualled a mncenrmtint~cell. If wc apply Equa~ml14,77 to the cquations for the two electrode rcnutions given hy abovc. we Ilnvc

In Equatiorh 14.86, Aft i s the liquid junction potential and E is the emf o f the ccll. We w7iII low develop the noncquilihrium thennndynnniic cquntionsfor this (isothermal) systelrl. Following thc rcasoning that wc used in previous secticms, we may wri~t:

and

where a and B dcnrlte thc solution clcotrode compartments. Using the fact that the B electrodes are sirnib, we can write p,*,, = @ ,! and pi,, = p,, so if we subtrucl f;.rlu;uion 14.8 1 T~.omEquation 14.80, we have

where J (.I 1 is the llow oT cations (anions) attd Jy is the flow of w:atcr ~no~cculcs. M'e will fir->I [I-anstimnEquaiion 14.87 into one involvitlg the Aow of neutral snlr, Jluine flow is givcn hy

'v_

Rut >> V,, especially in a dilute solution. so J , z Equntin~l14.90 in the form

about M)S: of the ionic curretit. 'The fraction of the electric current carried by an ion of !ype j, I ; / I. is called the trrmsport n~tmbrra i d is denort.d by

+

~ ~Thus,7we can ~ write .

Experirncntally, a concentru~ioncell such as that depicted ill Figure 14.6 is run such that J,,, the vc>lume flow from one cotnpartment to the other, is zcro, so uur hasic eritropy production equation ir

For a binary salt. t+ t- = 1. Tmnspon numbers can be deterniithed by experimental methods (see Problem 14-28). U'e can wntc Equation 14.97 in terms oFt- = I+,/I.

Sulving Equation 14-99ror

r+

varlzty of

gives

The afsociated phenomenological equations itre

- "-L

Ll2-

L,,

The emt of the cell is determined untlcr cunditions of zero curxnl flow. so if we sci I = 0 in Equations 14.94. wc find that

il

v + z ,F

IT we substitute Equation 14.100 inlo E q ~ ~ a t i c14.95 ~ n and substitute t h a ~rebult inlo Equation 14.H6, we ohwin

+

The rntio L , J L , , has a nice physical interpretation. To see what this is, f i r s t divide 1, by 1 u~Equations 14.94 with All, = 0 to get

w h e ~ ewe havc used the O n s ~ g e reciprocal r relation in going from thc second to the third ratio. But using the relation J.+ = rJ- J, (the electrodes react rwersihty with the anirli~l.w e her

'1-I,c

I ,it it) I I ,'I in ~ ~ u iEtl rt~14.97 i s thc fructir~nof the total inni; cLlrrrnt thal is hy the cetinr. S o ~ ~ t r i l tu r y what you ~lliphlhave thought, the cntinns and anions do no1 neressarily c;irry tht s;lmc alnoilnr of currrnt heuousz they do no1 rhrcessarily tnmc .iI thc same speed. For exun~plz,the chloride ions in n NaCl(ay 1 rnlutit~nw r y

cal.1 led

Last, we use b p s = IJ+ A,?+ u- Afi-, electroneuual~~y ( v , + V _ Z _ = 0), and the tact that r, -t r . = I ( 1 , and t are thc fractln~~s of current camed by the uatlona and anions) tu pct (Probletn 14-35)

If we use the rzlatic~nsA p , = R T I n ( ~ + , ~ / u)+and , , A p - = HT In(u , , / m Equation 14-101 becon~es

,

,), thcn

Let's apply Equation 33.102 tci n 1-1 rlcctmly~e.If w e replace the aur~vitic.: by rimcentratinns ill Equation 14.102, we habe m , - ? = r . , -l = t.,, c r - , ? = r . - , = (,., .*milrr = (. ,, = r > , , and so t i + , = (,+, , =

,

,

14-9. Tbe L~quid]unction Pofential Is kvell Approxirr~dledby thc Hedwsun Equat~on

Equation 14.10.1for this cell is

Note that Equatiorl 14.103 says that thc magnitude of the liquid junction potential deper~ds.upon r - I + in this case. It turns our that t,+ a t , , . , so we expect that the liquid ;unction potential should k quite small for a concentration cell whose cell diagrain is -

+

whcrc we have uSBd the fact that I,, I,,-= I in going from thesecrlnd line to the third Line. If we assume that fM+ is constani and integrate Equation 14.105 from electrode compartment I to electrode compartmenl2, we gct

EXAMPLE 14-8 Llsc Equation 14 101 and the data k l o a to c~timatethe liquid junction pulential at 25 C for the cells

.4f(c)lAgCl(h)lh1CI(rl) ! MCl(r,)lA~Clfs)lAp(s). I,,

SOLUTION:

u,/rnol .L-'

M ' = Na'

0.392

0.010

M' = K+

0.4'4)

M+ = H+

0.H25

0.010 0.010

For Na',f,x+ =0.392,so!,,-

u2/rnul~~-' 0.0050 0.WSO

0.0050

where we have used h e relalion a: = a , ~ a , , - in going from the first line to the second line. Before we can use Equation 14.106, we must deal with the (nonexperimental) ratio a,,- ,!ti, . According to the Debye-Hiickel theory, or its simple extensio~i(see Equation 11-57),

= 11.608. Therefore,

,,

(8.313 ~ . m d - ' . Kl)(298.15 K ) 0.0050 InA$ = (O.608 - 0.392) 96 4 ~ c5 mol-' ' 0.0 10

.

Iny = Thc values fur K' and Ht are -0.36 m V and + I 1 .h rnV,'.respectivcly.Theexperimental values are es~rmaterlto be -3.68 ntV, -0.33 mV, and + I 1.13 mV. r e s p t i % e l y .Note Ihaf the magnitude of A$ for KCl(aq) i s aboul a factor of 10 smaller than the others.

I

1.173i;?(Zr/r n ~ l , L - ~ ) ~ ' * I

+ ( l r / ~ n o l . I .I)"'

Note that rhis equ~tionsays that y,,, = y,,-, so n , + = a,- in each compartment (Problem 14-29). Jf we accept this result, then u7ecan substituteac1- = (U,,U,~ )"' = (a:)'!' = a , into the second term on thc right side of Equation 14.106 to obtain

Equation 14.101 i s oftcn written in the differential form

Equation 14.104 is a fundalnental equation for the liquid junction pr~tenddWe will apply Equation 14.104 lo a number of special cases in the next section.

14-9. The tiquid Junction Potential Is Well Approximated by the Henderson Equjtion Once again, we comider a cell whose ccll diagram is

We used Equat~on14.107 (with activities replaced by coricentralions) in Example 1+X to calculate A @ fur several concentration cells. We can also use Qualion 14.104 to calculate rhe values of the emfs of the concentration cells in Example 1 4 8 . We start with the first line of Equation 14.105 and use = 1 - t,,. tn obtain

!,-

14-9. The I i q u ~ dlunctiun Porentiai

+

whcrc we hale used E q u a ~ i o11.32, ~ ~ which says that p, = v+jr I v p . Because the electrodes react reversibly with the C1 (aq) ions. we now use Equatior~14.84 in the differential form

Is Well hpprux~mdtcdby the Hendrrmn Eq~ral~orl

613

ion in a dilute gaq will bc accelerated by an electric ficld, but an ion in a sr,lut~on~ u c h as an aqueous sulutic~nwill quickly (of ihe order of 10 s, see Problem 1439) curue to a constant vclncitp as the viscous drag on thc ion balances the torce arising flnm (he elcxtric field. If wc let v, be the (cljnstant) drift \-clocity r d t h e jlh ion, thcn thc mobility of the j th ion, it,, is defined by the equalion

''

to write Equarion 14.108 as

If we iIssikme thar 1,- is constant and integrate from electrode cornparlment I to eleclrorle cnmpanment 2, we obtain

whcrc we have used the fact that u = u:. Recall that the electrodes are rci~crhibll:wilh respect to C1 (aq); it they were reversible with respect to the cation, (hen t,,, would bc replaced by -1, ,. tn Equahon 14. I09 (Problem 1+36)

I

E X A M P L E 14-9 Use Equaliori I1.1(W and the d;~t;~ in Example 14-8 to calculnte the vallle nl' E ior

I

each cell in Example 1 6 8 .

where & is the electric field strength. Thus. the mobilily is the drift velocity of an ion in a unit electric field. Because the units of v, are mes-' and thuse o f f are V . ~ n - l ,(he SI units of mobility, u,, art: m2.v-'.s-'. For a 1-1 electrldytc such as NaCI, the current density, j. is given hy

Note that the utiits of j are (mnl.dm 3 ) ( ~ . m o l - ' ) ( r r i - s - 1=) A .m-', whcrc A stnr~rls for amperes. Thc first term of the right side uf Equation 14.1 11 detiotes a current Hux of cations, and thc second term d e n o ~ e as currrnt Rux of anions. The ca~ionsand anions are moving in opposite directions under thc influence of E. but they have opposite signmp;lKtnent to Ihc other. In lhc vocabulary ill-nnr~cquilihrium ~hern~odyr~arniva, we say that such systems arc discontinuous. \Ve can irnaginc systclns such 3%a tnetnllic bar in which the rzrnperature vane< slnuothly from onc cl~td to the other. a>r a aolution in which the concentr;ltiol~varies so~oothlq'h-rlrn rlnc rcgiur~ lo another. We say that systems like these arc continuous xystclns. 111 [his cectiun. we will show how llonequilihr~umthenriodynamics can be forlnulnted to treat co~ltinuous systems. For simplicity only, we will consider systemc in which the pmpcrties v;lry in only one direction. which we take to hc along ihe .w axis. We now assumc that we can subdivide the system into slices of thickncss Ax = .k, - A , (Figure 14.8). which i., large enough tlrat therrriodynamic ~ i l r i i t h lsuch ~ ~ as 1.. P, and p llave well-dcfitictl local values hut small enough that these local valucs are the '.;amcveryuhere within thc rcgion of th~vknessA x . We furthcrrrlnrt: assume thai heh he Inca1 tlicr~nodyr~u~i~ic quantitie:, satisfy tliz same theniindynarriic equatior~athat we havc idcrived t i ~ requi11bnuin systerns. These two assumptions are formali7cd by the Posrularr r$ Lorul Eqrrilibrhm, 1-his pustulate obviously places restrictions on the systems we ca11trt.;lt. hut eaperie~lceshows that it is a good approairnaiio~~ for rliarly systems. Most syuetus r l f chcmical interest vary rather smoothly on a rr~checuIarscale. with nntable exceptions bcing systems involving phenumcnn such as shuck waves or exploaior~s. Ler's consider Eq~ratiun13.23,

(I;) = c,,. (.r). I f we substitute Equation 14.114 into Equation 11.1M and integrate, then we obtain (Prt~hlem1441)

because c,,

ill agreement with Equn~ion14.107. Nnw let's consider a liquid junction such as HCl(uf. KCl(r). I n this caiz, h e Hcndcrson equation yields (Prohlcm 1 M 2 )

I

E X A M P L E 14-11 Ljsc the data in Thlr 14.1 lo calci~l.lte the valt~cof A $ a1 25 1: for thc liquid jt~ncliun

SO 1.U I'IO N , IJclr~gtilt

mobilities in Table

13.1. Equatio~~ 14.117gives us

in g o d agrutnlent with the expcrirrlerlv~lcaluc of 31.2 V. w

1

There art: other integratirm schemes for Equation 14.104, but Henderson's is relatively simple and gives satisfactory agreement with exycri~nentalvalues.

F I G U R E 14.8 The geo~netrjused to dcrive Equation 13.119. The pusitively dirccrrd fluxr5 J,, and J,: are rhvwn poiuting iri the pusitive .Y dircction,

14-lil. Th? Flux -Force Relations for Cor~tinuousSystems

which ;~pplicsto the discontinuous system shown in Figure 14.3. We now apply this equation to any subrrgio~~ of volume A Ax and divide Spnvlby IJ = rl Ax to write

)

n ACjijT) + -1 ( d1 -AX

11

A

cdt

whcrr A ( I , / l ' ) = I/TZ - l / T l and A ( p / T ) = P,/T: sn~all,wc write Equation 14.1 1 K as

AX

- /L,/T,.

(14. 1 181

-

K e a l i ~ i nthat ~ AX is

61 9

E X A M P L E f 4-1 2 Discuss the unla irr Fquat~on14 123. S O L U T I U Y : Thc utjitsof o = $,,Xn ~ ~ict allowcd to ;~djuhttn its steady-state value.

s~,,,

X ,is

Let's hee huw S7TG,varies with X m kccping

X,!fixcd. To do this ~nathematically,

w e differentiate Sprndgiven hy Quation 14.135 with respect to

X nto obtain

But thc right side here is equal to 2J" by Equation 14. I B h . so wc have

uscd the fact that Jn = 0 at the steady slate. A second derivative of with recpecl ru Jin gives ( a ' ~ ~ ~ \ , / a=x2Lnn : ) > O, so w e see that the rate of cntropy production i a a minimum at a steady state. Equation 14.136 has a nice physical interpretation. At steady state, thc unrcstrained li~rcewill adjust itself so that the rate of etltropg pmduction is a minimu~n. Equation I I . 1 3 h is an exarnplc of the I'ri~lciple of Minimum Entropy Prcxluctiun. An equilibriirm stare i s n statc uf zcrn entropy production; a steady sllite is n xtatc of ~ n i ~ ~ l r n uentropy ni production. In a sense, a steady state plays the same role in nunequilihriun~thennodynamics as an eq~~ilibriurr~ slate plays in classical (cquilihriun~)thcrmodqnarnivs. For tl~zilca~e trea~edin Example ]&Id, the rate of entropy production is a minimuni. Thus, if we hold 2; and T, conswnt and let cncrgy flux artd niaterial flux take place until a steady start: is reached. then s will decrease to its minimum value vr~iislstcnt p-' with the fixed ( n u n ~ ~ rthermrdynainic i) force X , .If we were then t o let X,, v ~ n i s hby removirig he contact uf the cumpartments with their respective heat baths, the system xvnuld come tn equilibriu~n,when Sprtd= 0. where we havc

We can rllustrak these results pictorially 111 Figure 14.10, which shows mSprTl~rrrrtr>rlvn;brni# r~u~li s

626

If the tetnpertlture i s fixed 31 the twu ends of thc bar so that a ( l / T ) / a t = 0 a1 x = O and x = I, then the lirsi term on the right side of Fquation 14.142 vanishes and we have

Wk obtain the inequality because the integrand in Equation 14.147 is always positive (see Problem 340 or 8-58),

I

E X A M P L E 14-15

Prow the inequality in Equation 14.141 for the case cyf one-d~mensionalisothermal diffus~onin a binary st)Iution.

We can writc a J U / 3 x in terms of 3 7 ' / 2 t by referring tu Figure 14. II. The difference between the flux of energy at x Ax and that at x , J , (x iA x ) - Ju (x), is equal to the change in energy it) the volume A A x . If we let u he the energy density. then we

SO L UTI 0 N I In this case, we see fi'orl~Equation 14. I24 that

+

have ;uld EO

there is o~ilyone flux and ivrce, with J, = ~ , datid X, = a ( - ~ ~ , / T ) / a = x S~rbstitutc.Il and ,Yl into Equation 14.141 and intcprak by part5 to

i-. 1 :Ticd/i.;d.r). get

'l'he ~ ~ e g a ~ai vi ~e noccurs hccaurrihi!~at~ons of floxcs and force:, and $till presene the Unsilgzr reciprocal relations. In man) applicalior~, ,>I ~ ~ o n e q u i l i b ~ ithrrmudynamics, tin~ certain lincar flux-force rrlatir!ns ;Ire rimre c r m \ z ~ ~ i c [ ~ t lhan olllers, ~ r l dthc ~ c s u l tof this prohlsm say\ lhst wc can urc ;my convenient linc.,lr cotnbinations that we wanr. W e wilt prnvc tluf rcsult for r>oly a special ~ d s e hut , the ~ . c h u l t is general First sti~rtwilh S,Tr, = .I, X, + .I,X, with

+ l,lzx: J. = L,, X I + L,,XL

Jl

= Ll,,Y1

To keep the algebra tu a rlnnrttlutn. dcfine ilrv fluxes by

W ~ ~ C T r~ C ' mid 11 a r t constants. No* stdvc rhcsc two equation.: f o r .I, and .I: and s ~ ~ b s t ~ t ~ r r c them into S to obtain

Problems 14-1. Cunsicltr a two-co~npart~nunt systcm it1 colltact with a hcnt bath at tcrllpcratbrc T, so rtiat 7,= 7,= T. Let the two-cunipartr~le~lt sysleln be surmunded hy rigid, tnipernieahte wall.;, hut let rhc will1 separiiling rhc rwo companmcnts bt. permei~hlci ~ n dRcnible. Shuw thi~td A = rlrr (11 . - - ;I ..) - rl C', ( P, -- P, ) 5 0 N r m sliuw thal

which szrlea to tlefirle X ; and Xi. Now define Ihe phe~ln~ncr~olngical coeffizirnla !W,, tq

,

Conwrt thcsc equstions into the h r m ut Equatio~~s 1 , nlld show thui ,PI,, = ,I.1:, follrrr\s from L , , = L , , . 14-2. n~\cussthe physical meaning

t d

each term

on the right side uf Equatioll 14.12

14-3. Ehtenri Equalion 13 21 11, include a flexible wall between ths two comp,~~-trnzots. 14-4. Show I ha1 the 1uo Iernl!,

c)ri

[he right hide

r>l- Sprlrl =

J I , XI, + J,, X,, have I he 58me units

ah

14-5. Show thul Fick't law in the iurrr~J,: :x - 11

, IS small.

14-7. I t thc

\ ~ C ~ Z I iInI

/I

11,

- / d l can

he wrilren

ah

.I,,

t,,

-

&he11

14-12. The didgunal Irrrr~hi ~ Equation i 14.36 v;in be directly r c l a ~ ~tod expcrilncrllally rllr~iconducmrtrr. ( J , / A P),rL,) is c q u d to L , . surahle quant~ties.Show that the ~nrr~horiir~ut Shoiv hat the rirmr fr-icol ( ' o r d u c t i ~ t 1~J~, ~~,~ I / J ) ~is~ ~. -q ~u ,ato l L,:.

,.

14-1 3. 1-ih: r r r :nr~tl~lerrrr~o.smmti~~,flo~i~ i r dutinzd as ( J , / A I ) ),,,._,,.and tlw clrrn,rit is delined a!, ( I / A P ) , $ o . Show lhilt lhew quantitlus iuc sqrlal.

w t otlil

.rrrt',r~r~i~rc

14-1 4. lr~stcado f writing the fluxec as lirlzar.cr~nlbi~~atiuns uf I he Irlrceh. wv car) wrirc the force\ For twu fwccs and turofiuxe5. w e t~avc a:, linear ctrnlbtnarlnns of the

F~gurc14.3 tlac a Hexihle *all helween [he two crlrnpurlmenlc. then Sl> , ( ,

-

J[,X,:

+ .(,XmZ+ J , X , .

0

J , = d Vl ! O r iind X,. = ( f', / T , ) - t I'?/ T1).Write uut the linear flux-f(,tcerrlatiuns fm this system. Show that L , , , and L,.,. h a w ttlc E ~ ~ I units. I~C where

14-8. P l o w that L , , L : ? > L ; : ~ 14-9. PIOYCthat L2:=- 0 and that LC,L,! >

Lf!if .kpm, =

XI I.,, X,X,.

14-1 0. Provc that the largest phcnnrr~er~oiogical ct~erf~cienl musi he one of the diagonal ones.

arhe~c h = L , I L ,,, - - L l l L 2 1 Nole . rocdlrzhllhrn& L , , = L ? , .

that

R I i = I(,, as a consequence

r)r tllc Onsafer 1rr.ip-

14-15 . Shri* 1 hat Ihc lsvo electrokinetic qualllilics ( A $ !.I, ), ,, (.\rrorid ihcufr why the rcilctinn cchemc

If wc ignore any small transport of water mrrlecules, the conditloll for equilibrium ~ c r o c s the mctnbrane is 14-23. 'l'l~isprr>hlrni illuctratcs , u ~ alternate derivation nf the Orlsager reciprucnl relations for the triangular kinetic ~cIrelncd i s c ~ ~ s s eind Section 1 4 6 . First show that A, given by Brluatio~i14.hO car! bc writtcn as

Now

Lice

Equnt~orls14.68 and the fact thal A, = - ( A ,

Show that if we replace activilies by cunccntrations. then

these two conditions

give us

+ A:] to derive The pofe[itial,A ylr, is called the Drmnrm pofcnfiuL Show that c,+,,r ,,-,, = c., Now show that electroneutrality girea r

and that 11-21. Cnlculale the value of the Irankn~enlhranepotential at 298,15 K ol a membrane thal is pcrnmcable cmly lo potossiu~nions if the solution on the twn qidcs of thc tneinbranz are

+

+

=c

and

cK+,?= crl-,? = ('

,?L?~

,,

= r2.

Uce ~hewequaliunslo verliy the entries in the last four c o l u m ~ in ~ sthe follou~ngtable (all concentrations me mol L and the temperatulr is 298.15 K).

'

c ~ f4g+(aq) must have migrated uul c~ithcclcctrode cr~rnparl~rieril. Ttlc fraction ul'~hzt r ~ r a l current flow (0.0 I00 faradays) carried by the Ag+(aq) ions is

Similarly. at Ihe otherelectrode,the drcrrilhr o l llh: mnoullt of Ag- Pay) is(O.I(KH) - 0.0947) mules = O.iKl53 moles. But in the ahser~ceof ~uigratiun,thc passags of 0.0100 taradiljc would tesult in a decwnw of 0.0100 mules. \(I O.MW7 moles mu>[ haw rn~gratedi n t o the elwtnide conlpartnlent. Su w e cee oncc again t h a l

I t is

oftende\irahle 10 suppress the Dwn~raneffect and the abo\,c table shows rhal this can

tx done by adding relatively high conccr~tratinr~a of ~ a l t . 14-27. Considcr tllc ccll whew ccll d~agrarr~ I!,

Wrilt lhe equalinns fur the electrude reactions and the overall rcact~ono l this ccll. Would yc?u call thic a concrntrfition cell? 14-28. 'Fhi5 problem illustri~tesa melhod for dercrrlulung transpo1.r ~ l u ~ n b c cnl)c.r~rr~er~tally. rc Corlsidzr rlle schematic diagram below.

The tuhe i i filled with AgNO,(aq) and here are silver electrcdes at each end of the t u k . When a current is passed through the ccll (cluctrolysjs), the reaction< at tIie twtl elrztmdes arc deacrihed hy A:(>) -* Ag'(uq) e - and Ag+(aq) e - + Ag(s). Thus, if olle mide of chiirgc ronc faraday) is passcd th~nugllthe solul~rln,then one rnnle ul .4r+iaq) iivr mill Irrr~r~ a1 ulie clcclrutlc and one molt will be removed at the other. 'Thc current between the elrctl-des is carricd by Ag+(aqj ions movirjg tn orlu direction and NO;(aq) ion:, nioving In thc other dircctiurr. Btcauac thc .Ag+ (q) ionndr) not carry all Ihe current, they r l o nrlt inure away t h r ~ the j elrctnjdr a1 n h i c h they are h r m e t l as t'asl as lhcy form Situjlarlj. thcy do not a l ~ i v eat [fie r>t!wr slcctrcxie a5 fasi as they are rrmove-ed.'l'he~cfarrs.Ag4!arl) ions ilccu~nulatc~lrok~o(l the electrode at which thcy arc produccd and me dc~,lr.tdaraurld rll' clectside at which thc) iirc rclrlc)vcd. ~~tjc:, Cr)nuidr~-~ l duc c ~ r o l p i hcell ~lictui-ctlall>rnr.1 1 ) be divided I I I I ~two ~ l ~ ~ ~ i c~)unes par,lrnentr, and cuppuce lhal each electrode cunlpartlnent initially coorain, 0.1000 r ~ ~ o l of ApNO,(atl j . Now silppose that 0.0100 fwadnys arc pn~scdthrough the A&NO;(aq) xolulion, and we find that thcrc are O.IU53 n~oIu!,ot AgNO,(aq) i11 the electrtlde c;)mpsrtment in uhich Agl (nq) is producd arld 0.0947 rrlolts in t t ~ ecrlirlpartrnent in which 4gttaq) is removed. If there Rere no rriigralic)n of .lg ' ratlj ions fi-om one electl.xIc c a ~ ~ ~ p a n : n e n l trl the olher. the pkdshagc uf (1.0100 fiiradoys would rccult in an irlurca~edl- 0.0100 mnlc!: A g ' ( ; ~ q )in thc clcctrodc culrjpoi,rnlt.nt in ullich ~ ~ + ( isaprt>rluccd. ~ ) The ob+er:ed incrcasc. howevrr, is rmly 0.0053 mules: thei-efore, (0.0100 - O.(H153) nlolcu = [I O O l i IIIOICS

+-

Suppose now that 638 Care passed through n AgNO,(aq) sr)lutlon and that 3.312 g id' AgNO,(aq) are found in une electrode curnparlmetlt and 4.602 g in the cjther. Calculate the transprl number uf Ag+(aq) in AgPiO,(aq). Aasurtlc that the initial unlc>urltof AgNO,(arl) in the rwcl cornpillirnenl ic ttlc same.

14-29. Show [hat thc lkhye-Huckel thtory says that

u , . = u,

fnr ;i 1-1 0, howevar, dJ2 tnust lwvc the same sign as SX,. Now argue that if 6 X , > 0.then 6 J2 > 0 means that X, is made smaller, thus causing X 1 to g back toward its steady-state value. to

14-41. Use the Henderson intqration scheme tu derive Equation 14.107 (with a c t i ~ i l i ere~ placed by conccntratiuns) fur a liquid junction hlUl(c,) ! MCl(c,).

14-42. Use the Hcndeaot~integration scheme to derive Equation 14.1 17 fur a llquid junction

N w argue 1 1 ~the t same is true if JX, i0.

Chapter 1.1

: Nonequ~l~hrium TIternmnd\,nam~cs

14-51. In this problclri, wc will develop a gcncral prnnf nbuul thc rtahility ot t11c stcad) state for a system near erluilibrium. Let XI, X : , . . . , X , be tixed furczs and X , - , , . . . . Xn be variable fnrcc~,so that . I , , = I,, I = . . . = J), = 0. Nnu, let o ~ l uof thc bariahle fnrccs, X,,. change siighllp by d X n E ,s o that X S + X: 6 Xm,with L 1 5 ) I I C: n. Show that S J,,, = L ,q,,,b X ,,z. Arguc as wc did in thc prcviuus pr,oblcrn that thc sign of 4 JSjcwill always tie such that [he flun .Irfi reduces X,,, hack to its cteady--calltddiffusirw cqun11011

14-52. 111 lhih prnbleln. we prow that

.

rltl,ii~lsa lninimum value with respect tu each of the nnnfixed forces, X,,, . ... X,,,in a steady state The mathematical condition thrit be a nliniinum iand not a n1;txiinum ur at) lnf ccticm poitit) i? thal

untl

Nuw show that

The solutions trl this equation give the ct)ncc~~tration of ;l diffusing \ulntance at any prrinl and tiine. starting trom some intitial dihtr~butionof crmcentrution. l'lls ~nlutioniu lllc diffusi4,n equatloli if all the diffuring substance is inillally lncntcd at the uripin ic

where 4 1s ttw total cuncerltration, initially Incated at the origin. F i r r ~show that ( , ( I , r > does indeed hat~kt!,the diffusion equation. Plot c ( u , t)/r;, versus x for inclrasing vnluc\ of Dr and interpret the result physically. What do you chink tlie follow inttgml equals?

14-56. We can lllterpret c ~ A r- ,) d r / c , ,in the pl-etm~ous problztn as the pruhabil~tythat a diffir\irig particle is lucated k t w e e n x and x -td.x at time t (hlatt)C:hapter B). Firs1 $how t t l ~ t O and that idp,/,Jvr, I,,, = ~ - 1 ( i ~ ~ (2, 7 )/, ,~ >c 0~ )

'

Given th.1 U = 10 crn2.s-' for all ion tliltusing in walcr, calculate tllz viiluc rjl lo-'' s, 10-' s, and 10 s. ~ ~ S C U your E S result%.

I,",,a1

14-59. Cirnsidzr a clowd circuit consi\tirlg of two dirferer~t~nctalliccon~luctorsu-heie the junctions be4wot.n the dissimilar metals are tnaintained a1 uniI~rcnt teinpcl-alures, Ti ~ I I L I 7:. It turn\ OUI that a voltage wilh a resulting electric c u n r l ~ twill he g c r ~ e ~ a l eilld this circuil. This eticct was firs! r)b.cerved by T.J.Sezhrckin 1822 ;utd 1s now knuwn ,I> lhr .Sc~ht,rI. c f l d ~ rThe . ~ i ~ r r ~ h p ~ l l clnf d i l l in p lhc circuit ( m c ~ s u wlth d a potrntiumrter) dcpcnds U ~ I I rhc tctnperatilres al the two ,juoctiuns arid upon the Iwo n~ctals.'Thus, !f unz junctlon 1s liked ar a kn4,wn tel~~peri~lure, tt~cnrhc ternpt.rdlllre at tllc other ltlnctic>r>can he J e t r ~ - ~ ~ ~ i r ~ t . ~ l by mca~uringthe emf. Such a device is called a tlr~~r.ttrot.o~tplc. ~ltldis uhed rxtrnlr~vclyto nltasurr Ivrnprature. [,el's now cons~dzlthe inverse ~ l i f t ~whcl-e t, lhe trro jul~ztiuiicart. at Ihc snme temprrature a ~ t dd n e l ~ h currcnt c is mainlained across the junctirr~~s bctween the lwo nwtals. 111 this caw, thc :empcralure at nnc junction will increase and the ternperaturc at the other w ill decrr-we,ui>lts1K/I q:1m Y.803 at 5!HHI

+,

K

4-23. 2. I. 12, 24.2. 1

,

MathChapter C C-1. 1.25 :-:lOK'%,197 x I 0 ' 5 . O 467'A

C-2.0?498,U4?9%.

.,492S

4-24. t-lvl = 2-1-11 K : Ov,,, = 3016 K; i4v , , , A = 1 0 3 K; I-),,,,, = 3765 K; C,:R = 6.21

4-25. 2: 1.JbO r H)-" kg.ln2; 1.7U2 K ; 2XJZ K:3649 K;4715 K; IU49 K; Rh3 3 K;4.33R

4-28.I = h.74t x 11U (:Im = 0.547K.

kg.rn2.

4-29. see 'I'ablc 15.4: 5.30R Chapter 3 3-7. -hyB1.I)

4-39. (a) 3K!2 (b) 7 K / ? IC) 6 K !dl 13K/2 (c) 121I

,,= 4480 K; O ,,,,,,,= 5464K; U

4-34. U,, (4

?,,,,,

n,,HD

= 42.6 K; = 64.4 K

Chapter 5 5-1. KE = 9.8U kl; 11 = W.3 rn.s I; 27.1 C 5-2. 15 bar; 3000 J 5-3. 28.8 bar; 3 60 J 5-4. 4.01 kJ 5-5. - 1.73kl 5-6. I I ..IkJ 5-7.+325 J: +309 J ; they d ~ f f tbccause r 711 i:, il path fl~nction 5-9. -3.93 kJ.mo1 5-10. -3.92 kJ-invl 5-12. V,=11.351.; V2=22.70i-. T 2 = lO90K: AII = 1 0 . ? k l - ~ n o l - ~A; H = 17 0 k ~ . m o l - I ;q = 13.6 kJ; u. = -1.40 kJ 5-13. 418 J 5-19. T, = 226 K. 11: = -898 J 5-20. 519 K 5.21. 421 K 5-22. q , = 122.9 kJ-mol-I. AH = 122.9 kJ-mol I . AU = 113 2 k~-rr~all-'. u: = +9.77 k ~ - m r l l - l . q , = 113.2 k ~ - m u l - I , At1 = 112.9kl-mu1 I, AIJ = 113.2 k ~ . r u o l - ' w , =0

'

Answers tu the

'

5-23, h,U = 288 3 kJ in01 5-24. 74.6 kg 5-25. 2rl5 K 5-26. 1340 kJ 5-35. A , H = 416 hJ 5-36. A, I I = 521 6 kJ 5-37. A r l l = +2 9 kJ 5-38. A , H"[iructose] = +1244.3 kl,rnol 5-39. rnt-thnnol. -22 7 k~.g-'. N,H,(I) = - 19.4 k 1 . g 5-40. 32 5 kl 5-41. (a) -44 14 k l , cxolhermic ( b l -429.87 kJ, cnotherm~c 5-42. 47 X kJ mol-' , 44 0 kl mol-' from 'I',~ble 19.2 5-43, l J h '31d kJ.mol '

Kurr~eric-a1 Prvblems

6-9. 19. I 3.~-' ; positive because the gas is expanding

-

6-1 0. qwU= Pl(V,

Answ~rslo :kc Numcr~calProblems

7-10. Suhslanct:

-

I',1;

Pentane Hrxme Hcptilr~t.

' 6-1 3. AS = 37.4 J K-' 6.14.

AS = 30.6 J K

'

mol-'

'

Lthylcllc nsidc Reiisz~le Dl ethy l ether Z;.trnct~lnrnrr~elhane Mercury Bromine

-

h-17. A S can ht. posithe or ncgative for an ~ ~ o t h e r mpmcesq. al AS = -5 76 1 K-' 6-10. A S = 217.9 1 K

' 58.7

Penrdne Iiehane

6-19, A S = 4 4 O J . K

Hzptanz

Ethylcnc chide Bcnzc~~c Uicthyl cdler Tctmchlnr~~r~~erhane

73.5 77.6 32.0 35.7 4h.3 13

Mercury Brorriinc

7-13. 1'12 05 J K '-mu1 7-16. 223 2 J , K - ' ,rnr,l- ' compared tu 223. I l - K 1no1-' 7-1 8. 23? R J li- mol 7-20. 19h.7 J ,K-' ,rnl>l-' 7-21. 139.3J.K-' ,anal-' 7-22. 272.6 J - K - -ulcl-' 7-23. 27'4.3 J - K - -n~oL-' 7-24. 1 5 1 7 3 K .mol-'; residual entropy 7-2 5. 185.6 J . K- .t ~ ~ a l - I 7-30. 212.8 J , K- . mnl-I 7-31. 159.9 5.1(-' .rnul-I: residual entropy

E-5. Each urnhe her in a row is the sutn of the two number\ above i t . E-6. K4 E-7. 1.12 x 10. vs. 0.0194 in Table J.1

'

6-41. 191.6J.K '.niol-' 6-42. 2 13 , 8 J - K - ~,mul-~ 6-43. 193.1 J . K-' ,mol-'

6-45. 21% at 1 atm:

Chapter 6

6-2. d :!y

+

i:

6-6. ym, = l : ' ~ , , ( T ) ~ i l

-L

.

...

at 25 atm

' ' '

'

C,(T)~T

V P,~~Y:AS=KITI-~ -

VI

6-8. 5-76 J .K '; positive kcause h e gas is cxp;~nrling

Chapter 7

7-2. 37.5 J.K-' 7-3. 192.6 J.K-' 7-4. 3 . 7 5 J-K-' 7-5. 44.51 J . K - '

7-32. 153.1 J - K '-mol 7-33. 3 5 . 4 J.K ' .mol- ' 7-34. 113.: I . K-': mnl-' 7-35. S i H , , 29s t 5 K j = 130 7 J - K ',mrll I : S'(D2,298.l5K) = I43 9 ~ ~ K - ' . m n l - ' : S3(H1>, 29XK) = 143.5 J.K-' ,mcll-'

' '

'

Chapter 8 8-1.

'

E-4. b

7-36. 253.5 1 . K ,rnol-'. The experimental value is 253.7 J.K-' ,rnr,I-'. 7-37. 234.3 J - K-' ,niril-' The ex~t.rirnental value is 210.1 J,K-' .rnol-'. The dil-fewnce i h due Ir) rt..;idual mtrr)py. 7-38.- 172.7 J . K - ' . 1no1-' 7-39. -49.6 5 K -mul-' 7-40. (a) CC), (h) CH,CH,CH, ( c ) (:H,CH,CH2CH,CH, 7-41. (a) D 2 0 (b) CH,CH:OH rc) CH,CH,UH?CH,NH, 7-42. (d) I. la) > (bl > ( c ) 7-43. ( c ) > (h) % (d) (a) 7-44. ~ranrlmionaltin hr~th 7-45. 239.5 J - K-' -mu1 7-46. 188.1, J K -mu1 7-47. ( a l 2 . 8 6 J . K '-lnol (h) -242.9 J - K - '.mril-' (c) -112.01.K-' , ~ n o l - ' 7-48. (a) -332.3 I . ~ - ' . n l n l - ' Ib) 252.66 1. K - ' .nlnl-' ( c ) - 173.0 J - ~ - ' - r n u lI

'

09'C) = 0;

A , , a p c ~ 70°C) 5 = 0 33 kJ-mol Avmpc(~ O5C ) =

-':

-0.44 kJ-mu1

'

8-2. A w , , I ~ ~ ( ~ 0 . 0= 9 '0; C)

~ ~ ~ ~ ~ ( = 7 435.9 5 . 0 J.rnol ' ~ )- I ; ~ ~ ~ C ( 8 5= . 0-429.8 ' ~ ) J.rnol-I; no 8-5. PV = RT and P(V - b ) = R7' 8-7. - 0.050R kJ.rntll8-8. R 8-9. 7.87 x I O - ~d n ~ ' . h u - ' . ~ - '= 0.7H7J . K - ' .rnql-' 8-1 2. fmrn data: 155.6 bar; from van der Waals: -4.80 har; from Redlich-Kwong: 161.1 bar 8-13. -0.M52 kl . ~ n o l 8-16. ( i + F p i 2 ~ = )T 4.47 x lV4 dm3.rnol-'.K-'; 2, = 25.27 J - K I-mol 8-17. 138.8 J-mrll 8-19. V and U 8-20. 0.01561.K ' . ~ n o l

'

-'

'

'

'

Anrvvers tu tht~Numerical

Chapter 13 13-1. (a)

+ 2 HI(aqj

Prublems

-

PbJ,(c) I H2(ri ib) Cu(s) + 2 A g U l O , ( q ) + 2 Ag(9) + Cu(ClO,),(aq) (c) 2 ln(s) 3 C,d2+(aq) + 7Cd[l Pruhlerns 13-22. Znisl lZnCl,(aq)l IHCl(aq)lH,(g); +0.7h3 V 13-23. -0.21 V 13-24. 0 41 9 \.:; yes 13-2.3. -O.?P? V: no 13-26. 1.326 V 13-27. 0.7 12 V 13.28. 1."+9V 13-29. HZ(p)lH,SO,(aq)IPhS04(~)IPb(s); -0.3s3 v 13-30. (1.0732V. U.981, 0.975. 0969.0.963. 0.955 13-31. H,(gI ? AgCI{r) + 2 4g(s) 2 tICI(aq); E ' = 0.2224 V; A , f T = -42.92 hJ-mrll I : A,S' = -124.6 J . I I I ~ I - - ' . ': K 4 , H - = -80.0: k ~mnl-I . 13-32. Ill(g) - lig,Ur?(s) + 2 Hg(1) -1- 2 HBrrrrq); ArG0= -26.96 kJ.nlol-'; A,S' = , 29 7 I - m o l - ' . K - [ : A . f f ' = -33.8 kJ-mu1 I; S 1Br-(nq)] = 83.5 J-mol .K-I 13-33. 1.21 x 13-34. 1.16 x lo-' 13-35. 1.40 x I U - R 13-36. 1.30 13-37. 8.R x 10 13-38. 9.4 x IU-' 13-39. I .?3 x lo-' 13-10. l.h5 x 10 ' 13-41. 1.78 x lo-' 13-42. Thz values of Kw at O"C. t OC. 20-C, 30'C. W C . T O T , and 6U'C me 1.15 x 1u-IS, 2 94 x 10 1 5 , 6.85 x lo-". I .45 x 10- 14, 2 92 x 10-1; ,445 1 o 14. and 9.60 x 1 O-.'",respectively. The correspondirhg balues uf A r H nare (in kJ.rnul I) we 63.88, h0.33, 57.26, 54.67. 52.58. 50.97. and 49.81.The value nf K* at 25°C crnrl-s pur to be 1.01 x lo-' and the value of A,H' comes out to be 55.W k J mul-'

+

+

"

13-43. 1.85 x 13-44. 7.90 x 13-45. 1.64 x 13-46. 3.70 x 13-47. 1.08 x 13-48. 1.01 x

lC1-I and 1.71 x 10 lo-' and 7.99 x 10 10 I O-' 1 (I-"'

"

-'

'

1K2 13-49. 5.65 x 10 13-5U. 9.85 x 10-' 13-51. 148U kl versus 2 1 O X k t ; F,'" = 1 .OY V 13-53. B?,. = 6.1 x 10 atm-' (estrnlatcd). or 5.39 x I fl a m (using tnblualcd

'

'

values)

13-55. 0.852 V 13-56. -U.74 V Chapter 14 14-24. 59 nlV 14-25. 12.7 kJ.mol-' 14-27. H 2 ( P lI + Hz(P?); the cell ir a concentration cell 14-28. 0.47 14-3U. -3.95 mV; -2.31 mV; -0.356 mV; -0.208 mV; f I 1.6 mV 14-37. 13.93 m k 4 1 . 7 8 mV: 17.45 mV: 52.36mV: 29.42 mV; 88.26 m V 14-39. TAe the radius o f the Na+(aq) ion to be 100pm. Then / - = 1.7 x 1 0 - I k g - s I. Take the mass of the NP-(aq) iori tn be 100 amu. Theri J!nt = 1 x 1013s I . 14-43. 26.9 ntV. 26.9 mV; 31.2 ~ I V -4.37 ; rnk -4.37 mV: -5.01 1nV 14-45. p2,/j~?I= -0.013 for c, = r , = 0.0010 M; -0.043 frlr r , = r, = 0.010 M; -0.15 for r , = c, =0.10 M 14-46. For an ideal solution. 14-47. R T L , , / M ~ c ~ ~=~-9.9 s x lo-' and R T L , , / M . c ~ ~ . s -= ' -9.5 x lo-' 14-48. K ~ ' L , , / M . C ~ ~=. ~-1.5 - ~ x 1N6 and RTL,,/M.cm2.s-' = -1.5 x 14-57. The integral i s equal to zero. 14-58. ( x ) = 0; xn,, u ~ r uvonr m dt. Gtsch~edznis >an de X.tuur~~ete115chapp"n~ I,e~dcri. (:oulrcsy oi AIP Elnillo Scgrc Visual Archi\?,. Chdpter K: Photr>:rdph by I+.l.ange, courtesy of AIP Enlillo Scgri Viaudl Archives. Chapler 9:Ccurtehy c:f ALP Kmil~oScgre Visual Arurle+yrrf The Huncroft I,ih:;lry, Un1etl51ty(11 Califr>rniu,Berkeley C'halwr I I : i l k h y t ) Pnaograph by Francis S~mon.ctjllncsv :\IP Er~~iliu Segrk Viauul Archive!, l'hapter l?: Courlc5r. of :ZIP Et~lrlluScgri. 15su::l ~ k ~ h i v eW> , Cddy Cullecliun Chiper 13 t<eprintcii with perm~usiontmni the Jr>~iu~iirl :t rhrmk,crl Ei!,rr.rr!iun, Vul, 61, Nn I . ilDli4l. C ' ~ ! p y r i ~I?) t ~ I1984 by lunn nr T11cl~iicalF ~ I I C U ~ I ~Oc I. I. . AIT,:~JC~II ~ I I c ~ I . I LS,~; II! C I C I ~ , Ctiupl-r :1 C o ~ r l c +~i y l"il< Uniwrjily L ~br'iry.\l,i:.~r,c;ipIs .rnd Archives.

TEXT FIGLIKES Fig. ?, 10: W ~ t hpcr~riirsior~ Irorl~C~uug-.lenSu. Indu.rrrra1and D~gf~lreri+lg I'l~mristrc Rr~rrin-h38. 803 (1936). Cupyripht 194h Americim C'hemic;~lSocie~p Chapler 5 quote: With permihhion of Open ('curt '1.1-adeB Academi~Honkh. a divhitin ot Caws R~bli\l~ing Crhml~aliy.P c n ~ IL, . tr.c)rllrllI>r,rrEin'insrri~rPhili).>ophcrSr'jnretl!$ it11 l>crrni,hii~nt ~ n ~Hr .lH . Nzwron, birlii\rrrtrl clod Er~girrrrr-lvg (71vnli\lrj R lur ~ h c Nii~ir)nalI i ~ ~ t i ~ru~( Science tf . i~nd Teclu~ology. Fig. 13.15.Wrlh pcrrr~iaaiu~i Urmn Joclr~~t~ul r$ I'l~yvi~-ol a~lclI'l~entrrnlH ~ f i , ~ , ( ! Dutcl mt.~ 1 I. supplelnrr~t2 (19811 Copyright 1982 Amcricerl C hcrrjicul Sucic~yand Ihe Anicricun Insrilu~cuE iJhyaic9 f r ~ihc. K;lriuri;~lI r ~ ~ l i l ~illr tScicncc c ar~d Tcchnulopy.

l ndex

A absolute entropy. 279 activity, 4 I 0 if.4431. 455R con(lensmi phase, 5 10 dilute uolutiun. 443# tclllyeralurt. dependence, 475 iictivitp coeffiiicnt. 41 I activity quolienl, 504 adiabatic cumpressibility, 344 adiabatic exyansiun. l96fi, 222 adiabatic fli~tnetctnperature, 225 adiabatic prwess, 194 Ictnpcritrure change. 196 affinity, 599 a~tlplitudc.10 at~gularv e l t ~ i ~ 19 y, a!iharmr>n!c terrnq. 33 anharmonicit) cojlstarkt, 14 anodc. 5 3 1 a\vmmetric top. 27, 1 6 5 8 atmt>sphere.atm (unil), 5 1 average. 96

azeotropc. 407

B Ral rner Furmula, 4 BaIn~crscrics, 4 bar iun it). 5 1 battery, 5 h5 Berthclot equation. 335 binary ~ i l u t i v n 38s . binonlial meuicient, 230 binrmnisl expansion, 230 binomial series, 230 B A r Irequcncy condition. 3 Bohr rad~us.3 1 bojIir~cp i n t elcr.~tint~ constant, 450 boiling point elevatiotl. 450 Bolr~mannconstant, 7 3 Boltz~~latln tac~or.105

Boltzmann, 1,urfwig (bio). 1W Roltzmann statistics, 123 criterion. 1241 boson, I21 Ruyle temperature. 7 1 Boyle's law. 197

C calcite-aragonite equiIibrium. 527 canwnical ensemble, 106, 107J C'arnot cycle, 2 5 8 8 cathude, 530 cell diagram. 533 Celsius scale. 52 centrifugal distortion conrtant, 36 chen~icalaffinity, 599 chenlical potcntid. 361f, 3938 partition [unction, 369fl Clapeyrun equation, 362 classical harmonic oscillator, 32 cIas';icnl thermndynamics, 2 Cluusius4lapeyron equation, 365 8 Clausius, Rudol T (hio), 236 Clauaiuc, statement of second law, 248,583 cwfficient of thennal expansion. 182, 3 11 coexistctice curve, 61, 35U, 4UY cdIigative properties, 4.188 compressibilily, adiabatic, 344 isothermal, I81 con~pressibilityfactor, 54 con cent ratio^^ ccH. WS,606 consulate temperature, 408 cunvergencc tcst, 136 corrcction for nonideaiity. 320 correspondi~~g statcs, law of, 67#. fugacity. 330 Rcdlich-Kwong, 68 second virial coefficient, 75 van der Waals equation. 68

coup1111gcoefticient, 590 critical consrantn,400 D ~ I I I Lcffcrt. ~ I 63 1 I)uni~anpntcntjnl. 63 1 d~,oplct.vapor precsurc. 381 8 nulong and Pctit. 116. 130

E L~llcteinmodel rd otorl~iccrystal, 115. 130, 140 Llllstein temperalurc. 14 I elcct~icalwork. 602ff elec~rocheri~icnl cclls. 529 a l ~ ddicswi;ilic>n corlsrits, 55bJ aiui holuhjllty product. 5 5 3 8 clcc~r~>chernlcuI p>tcotial. 603 elrctrkjtl. 530 clcvt~ubinrlicphrnrlrnclla. 593 /j elcvtro~n;lpneticspsctrum. 1[) ~ I E c ~ ~ ~ I I I I oforce. ~ I ~ ' ~ 5% elr;ir\ln r,olt, cV, dcfincd. 3 1

clcctronic energy, 8 electronic partit~onfunction, 143,# electronic temperalure, 171 elec~rr)rlarrlosis.594 . clccrrous~nr~lic lluw. 595 clcct~'oosmoticprzscurt., 593 rrrii, i 3 h and tnllwlpy, 55 1 anrl rnlropy, 551 anri Ciihhc erlergq, 537 pressure deprdericc, 575.576 cndr,thermic. 207 cncrgy

elzclrtinic. 8 8 trans1;itiulnal. 7 vthralio~lai.13 >irial expanciol~.339 cnsuntble, I06 en,errlhlz a v c r q e . 112 cr~thnlpy,200 precsu1.c cizpeodsnce, 31 5 temperarure dependence, 21 7 virial expansion, 339 enthalpy ar~dcruf. 551 entropy. 240 and di+ordcr,249 8. 288,u. altd emf. 55 1 arid phnsc transitiori. 277 anrl \tillis~icalthcr~nudynamicr,?491J: corl-scticln tor ~~orlrdc,~lity, 31 9 8 1~i1.tltio11 func~ion.2hIl.jl., ZX I ,g v~rlalc x ~ p l ~ s i o339 n. enlrulpy cxclla~~ge. ?hh, 107. 581 cnlropy r l t rllixir~:. 252. 255 fl cntrupy product lor^. 147. 266. 3 7 . 5x2 fl and disur.dei. 2 4 3 8 . 2KX I!. arid h ~ d t(low. 2hb and partition runcliurl, 28 1 and ijrlal cxpat~siun.339 rate of. 588 etll~ilihriur~~ n. 4x1 ~,rjr~vcrurc.321 -6 nnn-i'- V w-r~rli.? 106 3 virial expa~~sion. 3 39 Gihhs-Duhen~ equurion. 3 9 0 8 Gibbs c n e g p of mjxing. 41 8 Ciibbs-Hcltd~oltz :quaIio~i.323, 335 Chhbs, J MJlllard ihiwl, 348 Gihhs phase rule, .Y8 Glansdori-Prigoqce ineyualiry, 6 2 5 8 gntiicnt. thcrniody~larnrc,61 K g r ~ p h i ~ c d i s m n - rquilihium. ld 511

tiurd sphere pulentlal, 82 second virial coefficient. 83 harmonic oscillator. 11 heat. 186 r n o l c c u l ~irltcrprctatio~j.14%11. tleot capacity, I 14. 202 fi Uehye thetlrq, 139. 278 Eirlateln, I 15. 130, 140 prehsure deperidence, 3 34 vrnal exI,arlsiorl. 339 volume dependence. 333 heat CIIEIIIC, 150 elriciencg. 259 heat of cumbustion, 107 hcat of fusiuri, 205 heat uf reactiull, 210 hcliun~-ncon lnscr, 3 1 Hvlrnl~oll~ cncrgy, 302 virial expansion, 340 Ilelmholtz. IIerlnann von (bio), 3W Her~deranricqualion, h 13jf tlellry's law, 403 Hcnry's law constarit. -103 Henry's law ctarldard \tale. 4 14.440

hertz. 30 Hcss's In% ZOX, 2 15 Hildehrand. Jr1t.1 [hro),3x8 Hitchcr)ck ptrjt. 541 homogeneous functicln, 426 Ilooke's law, 10 hydrogen atomic \pectruln. 4 hydrogen elc~,trl,~lr. 532 H~ickel.Ench (hlo), 438

1 ideal gas eq~t;~tiun, dl) ideal rubkt bmd. 342 ideal solution, 395 8.401 inequalily c > i C'ldusiu\. 24H. 583 iticxijct dlffer~er~t ial, IBO, 192 i r i tiurzd active. 24 infrai-ed illactive. 24 lrr~eg~atirlg factor, 240 intensive quilntily, 50 i~~trrmolrcrilar interacrion, 7 3 8 ir~tci~lal pressure. 333 i u ~ n calniospt~erc,462 ionic niuhi tity, 61 2,@ ionic strength. 460 lonizatiorl etlctgy, 3 1

1sol:ited 9ystem. 245 isutherm, 60 isothermal comprrssibili~y.181. 22 1 . 312 iteraliot~,39

I JANAF tablev. 449 Joulc. James (hit>), 184 ~ o I I I ~ J? ( U I I ~ ~ 2) . Jr~ulcThorncon cuefhcienl. 227, 3.10 fl JouloThomson cxpeiimcnt, 226fi Joule-Tllomsc>n ~ n v c ~ s i otclnpilrature, n 311 K, kelvin. K (unit). 5 1 l'H8pital'c rulc. 139 law of cnrreuporidii~gstatelume 79. H[lt postulate of local equilibrium, 617 putentiornett% 537 pressure-cornpsition diagram. 396 8-429. 130 pressure tinil-. 5 1 f primitiw model. W principle r>Cdttaild balance, 601 principle of minimum entmpy production. h22 ,JJ prubability. detinrd. 95 pure heat transfer. 168

u?;ldatic>n-reduction clectrrde, 532

R

P parlinl deriwtive, 175 partial ~nolarqunntite.;, 3#7# pafiial lnolilr tolurne. 7848 partial pressure, 86 partilio~~ funclinn. 1098 and chelnical pntctltial, 365'8 arid energy, 1 10 and cntropy, 2 M # and equilibrium constanl, 4958 atomic. 282 diatu~nicmolccule, 159, 282 electronic, 144$, 28 l ff: independent, distinguishable particles, 120

Raoult's law. 1Wff Raoult's law. negative deviations, 402 Raoult's larv puqitive deuiarinn. 402 Kaoult'b law $tandad slate, 414.439 rate of entrap? production, 588 ratio tcst, 136 rrac1it)n quolletlr, .LEN !( rcciprvcal relation\. 591 j 1 rcctilinem diameters, 375 Rzdl~cLKtl.otrg?onstants. 581 RedlicbKwunp equatir~n,5 7 reduwd mass. I I. 19, 3: redwed parameters. h7 regular soultion. 419. 4 3 2 8

molecular Lheory, 41 Y f l residual entrnpy? 289 reverse osmusis. 453 reversible cell, 536 reversible e l e c t d e , 531 revcrsible process. 189 reversible wurk, 184 rigid relator. 1R rotatio~~al constant, 20, 1 5 , 25t rwtational &@es of freedom, 23 rotational panition function, 1 5 6 8 nltatirmal temperature, 151r, 156, 1 6 3 rubber band. 342 rubber elasticity. 342 S ~ a lbridge. t 510 Schriirlinger equalion, h screened coulomhic polential, 473 Seconrl r a w r ~ Thermndynamics, f 248, 249 Kelvin's statement. 260 second cclitral Inolnent, 97 second eIectrut!~moticRow, 629 secrlnd electnlc>smolicpressure, 629 second moment. 97 sccond streaming current, 629 second streaming potential, 629 second virial coefficient, 70 Seeheck effect, 637 selection rule, 14, 19 anharmonic oscillator, 35 harmunic oscillator, 14 npid rotator, 19 semipermeable membrane, 45 1 silver-silver chloride eleclr&, 531 Simon cquation, 374 Simpcon'f rule, 44 s i n g l e 4 e c h r d e pntenlialh, 542x Sl units. 2, 50 solid-gas coexistence curve, 350 solubility product, 5 1 4 , 5 5 3 8 and elecclmchernical cell, 5533 solute, 439 soluliun activities, 4 4 3 salvent. 439 spherical top. 27, 165 spontaneous process. 301 g square well potential, 83 second virial coefficient, 83 stability. 343 fi slahility conditions, 3 4 3 8 , 385 standard boiling point, 352 standard deviation, 97

Index

\t;tndard emf, 538 standard ttntrrlpy of reaction. 289,R ctandard Gibhs energy oi'fornation, 484jf..

4HSr stariclard mcliing polnt. 35 1 standard riiol,v er~ihalpyoi for~rlalion,2 1 I , 114t sti~ntlardn~oldrentl-opy. 2 7 9 8 , ?X4t, 2 x 5 8 standard reaction enthalpy, 2 10 statidartl wductiori potentialc, 544, 5451 standard state, Iicnry's Inw. 414, 440 standard slale of a gas. 3 1 9 8 alanrlard ctate o f a solution. 41 3 ~ 3 hrandard stotc. RaouIt'h law, 114,434 stntc lunctlon. 1x9 station;try stute. 3 ?i~i~tis[icul ttlem~udyria~nioa, 2 stzildy \[sic. 622J. Stirlirtg's approximalir)n. 232 Srokc'h law, 634 strea~nir~g curmnl. 594 alrearr~ingpotential, 595 suhlirnarion, 353 surrounding*, 1% syln~r~ulric top. 27, 165 splrimetry n u m k r , 15'1, I h?r system, 186 T

Ihylra expanaiun, 33,141 tzmpcrature-composition diagram, 3 4 8 8 , 429. 430 rhcnnocl~crni\try,207 thcrrnacr>uple,h37 thertr~crdyn~mic energy v~rlnlexpansirln. 334 \-olume dc.pendence. 310 thcr~ric~tiynnm~c rqullihriuui constant, 507, 5 10 thcrn~tdynamic1i)rcr. 589 ttrer~l~r)d) namics of ~rseversiblrprocch.\es, 581 Th~rriLAWof Ctier~nodynan~ica. 275 fl, tic IIIW.39h tom (rlnir), 5 1 lotal derivative. 178 Iraycctrjry. 33 lru~~hlarior~al degrees of freedom. f3 Ir:~njlillirmalcnelgy, 7

[ransport number, expcrimtntal determination. 631 irdpczoidal approximnt~on.43 tl-iplc point, 350 I'roulun's rule, 142

v van dcr Wlnls cnnhtants, 55, 5ht and cri~icalcrmstants. van der Waala eiluatiun. 55, 1 18 vnll der Wralh. Johannea (biu), 48 vm't t4ofirqu:1don. equilibrium constant, 401 ff.

viin'~11011'. Ji~ct~hus (blo), 476 vapul p c c w r e and external prcscure. 38 1 vilprr? pruacure. droplet, 381 J. \arlancc. 97 ~ ~ h ~ a t ~ tdegrcex l n u l of frcedo~u,23 vihratlunal C I I F T ~ Y .I3 vihr,~~ional parlition function, 152 fl, r i h ~ ; ~ t ~ i ltrnlpcralure, nal 15 I!. 152. 1621 vi~ialctwlliclent. 70 virlal equulion 01-state, 70 v ~ ~ iexp;lnhlun al encrgy. 339 enthnlpy. 330 cnlropy, 339 Glhbc energy. 334 hcal c;rpacity. 339 H r l m h o l ~cncrgy. ~ 340

W walr, 31 wave furictiut~,6 u,avcrlumber, 5 Wcalon standilrd cell. 535 wrjrh. 18h clc~,lrlval,h02J. rnuximurrl available, 303 111c>lrclkirinterpretation. 198.ff: nun-Y-V? 3Uh Z zcrr I-poirlt energy. 13

Molecular Thermodynamics

Molecular Thermodynamics of Complex Systems

Molecular thermodynamics of complex systems