Heat

.

LIBRARY UNIVERSITY OF CALIFORNIA. Deceived I

Accessions

No.U&*f&r

.

Class No.

LOWER

DIVISION

UNIVERSITY OF CALIFORNIA LIBRARY OF THE

PEPfrttTMENT 'OP

Pll'I'JieS

Received

\ccessions No..../...3..&..

Hi

Book No

HEAT.

HEAT

BY P. G.

,

M.A., SEC. R.S.E.

HONORARY FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH

Eonfcon

MACMILLAN AND

CO.

AND NEW YORK 1892 The Rights of Translation and Reproduction are Reserved

3

/rvPTtyc'

r

'

^

'OTi'T^

RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY.

First Edition printed, 1884.

Reprinted with corrections,

1892.

PREFACE. IT was with considerable hesitation that I sanctioned the

number of copies Some of my reasons

Publishers' proposal to print a very large

of the

first

edition of this work.

be found

will

in the

former Preface, reprinted below

especially those depending on the

fact that the

:

work was

written, in occasional intervals of comparative leisure, during

several very busy years. It

is,

ation

therefore, with proportionate pleasure that I

can reissue

find I

the

:

it

now

with a very moderate amount of alter-

only serious error discovered (besides other

pretty obvious typographical ones) being the unaccountable omission of some necessary words in the important quotation from Clerk-Maxwell given in

442.

But Teachers, who have used the book, have kindly pointed out to me passages in which they or their pupils

had found unexpected to

make I

these passages

difficulty

more

:

and

I

have endeavoured

clear.

have also rewritten the section which deals with the

isothermals of gases from the point of view of the kinetic

Here temperature, of which so much was said former Preface, reappears in an even more formidable

theory. in the

guise than before. P.

COLLEGE, EDINBURGH, November 2nd,

1891.

673242

G.

TAIT.

PREFACE TO THE FIRST EDITION. THIS work originated in an article which I contributed in 1876 to the Handbook to the Loan Collection of Scientific

As that article was Apparatus (at South Kensington). based upon the system of teaching the subject of Heat which, after many years' experience, I have found best adapted to the wants of intelligent students taking up the subject for the first time, Mr. Macmillan asked me to develop a

large

it

in the

form of an elementary

portion of

part of which

was put

in type,

Clerk- Maxwell's valuable

in

I

at the

;

In 1876 the

7

greater

and had the advantage of

Some

criticism.

when

chapters were utilised

an evening discourse

treatise.

work was written

the

had

of the earlier

to give at short notice

British

Association Meeting

1876.

Pressing work of a very different character, such as the

examination of the

leisure.

I

have

nearly as possible

at

Thermometers, interfered writing by occupying most of my

Challenger

from time to time with

my

last

on the

managed lines laid

to

finish

down

the book, as

at starting

;

but

it

cannot, under the circumstances, be expected to have the unity which written.

it

might have secured by being continuously

PREFACE. It

may be asked

subject which

*

vii

publish another text-book on a

Why

already thoroughly treated in the excellent (and strictly scientific) works of Clerk-Maxwell and Balfour Stewart ? The only answer, and it may be a sufficient one, is

is

Clerk-Maxwell's work

:

specially fitted for

the Physical

opening

Laboratory

for a

expression

;

that of Stewart

scientific

or experimental, are

more prominent

accurately the

facts

In addition to

phenomena. what is stated

the special arrangement and the present work, as to the perature,

been

way and

in

it

may be

By

that

who, without

career,

whether

desirous of knowing

yet

and theories of modern

science to such an extent as to give interest in physical

remains an

still

suitable for students

is

rather for

is

Lecture Room.

to the

any intention of entering on a theoretical

there

so that

;

work suited

mean a work

I

on the Theory of Heat, and

is

the Study

them an

in the

text

(

intelligent

n)

as to

of the subject in

division

well to say a few words here

which the

important subjects of temof absolute temperature, have

especially

treated.

Temperature is first introduced as a mere condition determining which of two bodies in contact shall part with heat to the other

(

In

6).

this sense

the pressure of the air in a receiver, the analogue of heat are

made

to

(

52).

is less.

And

this is altogether

quantities of air in the

In

57

it is

When

is

compared

air

two

is

itself

to

being

two such receivers

communicate with one another,

from that in which the pressure it

it

the

air

greater to that in

passes

which

independent of the relative

receivers.

pointed out that there

is

an absolute method

PREFACE.

viii

of defining temperature, which must therefore be the sole

method.

scientific

But

is

it

also stated that, in experi-

mental work, few quantities are directly measured in terms of the strictly scientific units in which they are ultimately to be expressed.

what

that

In

60 the reader

wanted, to prepare him

is

the

for

is

told

absolute

measurement of temperature, is "something which shall be at once easy to comprehend and easy to reproduce, and which shall afterwards require only very slight modifications to reduce it to the rigorous scale."

Thus

61) a temporary centigrade scale

(

means of mixtures of

And

this is stated to

ice-cold water

is

defined by

boiling

water.

" accord so closely with the absolute

that careful experiment

scale,

and

is

even to show

required

does not exactly coincide with it." that In terms of temperatures thus defined, the questions of Expansibility, Latent and Specific Heats, &c., are explained it

from the experimental point of view. But, before these are taken up, the

the whole

to the absolute

pointed out

(

option in the

and

that

is

ordinary air-thermometer. is

the

It is

leave a certain

still

so

possible

to

given at this stage) as to

close agreement between

meter temperatures

it

his

of absolute temperature,

definition

Thomson found

definition (which

and

measurement of temperature.

95) that Carnot's results

formal

rapid resume of

Thomson's application of

Reversibility, with

principle of

them

first

Carnot's Cycles,

subject introduces

new

scale

frame his

make a very

and

that of

an

Thus, the use of air-thermo-

justified

by

their close

to absolute temperatures, as well as

by

approximation

their comparative

PREFACE. simplicity

and

and convenience,

they are

thenceforth

the Indicator Diagram)

it

which follow

for the chapters

then shown, in

how

405,

;

employed (by means of has been fully pointed out what until

the scientific importance of absolute temperature.

is

ix

.

to

It is

compare absolute tempera-

by the air-thermometer.

ture with that given

All a priori notions as to strict logical order, however valuable, and in fact necessary, in a formal treatise, must

be

set

aside in an elementary work,

practice to

impede rather than

And

average student.

it

is

if

they be found in

assist the progress of the

solely

on

this

ground that

the above system of exposition has been adopted in the present work.

He who

expects to find this work, elementary as

it

is,

be deservedly disappointed. from real and great difficulties,

easy reading, will

everywhere No branch of science

even

is

either not

read at

free

Any one who

in its elements.

all,

or

thinks otherwise has

has confined

to

his reading

pseudo-science.

The

reader,

science of

who

wishes to

know more of

the general

Energy than was admissible into the present

work, will find a connected historical sketch of little

book on Thermodynamics.

There he

will

it

more of the purely analytical development of the specially from W. Thomson's point of view. P. G.

COLLEGE, EDINBURGH, December

isf,

1883.

in

my

also find subject,

TAIT.

CONTENTS. CHAPTER

I.

PAGE

FUNDAMENTAL PRINCIPLES

CHAPTER

I

II.

INTRODUCTORY

8

CHAPTER

III.

DIGRESSION ON FORCE AND ENERGY

CHAPTER

13

IV.

PRELIMINARY SKETCH OF THE SUBJECT

CHAPTER

21

V.

DILATATION OF SOLIDS

71

CHAPTER

VI.

DILATATION OF LIQUIDS AND GASES

CHAPTER

88

VII.

THERMOMETERS

103 b

CONTENTS.

CHAPTER

VIII. J'AC'.B

CHANGE OF MOLECULAR STATE.

MELTING AND SOLIDIFICATION II&

CHAPTER

VAPORISATION AND CON-

CHANGE OF MOLECULAR STATE. DENSATION

IX.

...................... CHAPTER

CHANGE OF TEMPERATURE.

THERMO-ELECTRICITY

OTHER EFFECTS OF HEAT

XI.

COMBINATION AND DISSOCIATION

2O

XV.

....................... CHAPTER

193

XIV.

................... CHAPTER

182

XIII.

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

CHAPTER

163

XII.

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

CHAPTER

CONVECTION

HEAT ......... 149

................... CHAPTER

CONDUCTION OF HEAT

X.

SPECIFIC

CHAPTER

131

230

XVI.

RADIATION

239

CHAPTER RADIATION AND ABSORPTION

XVII. 250

CONTENTS.

xii

CHAPTER

XVIII. I'AGE

RADIATION

276

CHAPTER

XIX.

UNITS AND DIMENSIONS

2QO

CHAPTER XX. WATT'S INDICATOR DIAGRAM

CHAPTER

298

XXI.

ELEMENTS OF THERMODYNAMICS

CHAPTER NATURE OF HEAT

324

XXII. 3$2

HEAT. CHAPTER

I.

FUNDAMENTAL PRINCIPLES. IN dealing with any branch of physical science it is absolutely necessary to keep well in view the fundamental i.

and all-important

principle that to the physical world save by or by mathematical deductions experiment,

Nothi?ig can be learned as observation

from data

and

so obtained.

On

such a text volumes might be written ; but they are unnecessary, for the student of physical science will feel at

each successive stage of his progress more and more profound conviction of its truth. He must receive it, at starting,

unanimous conclusion of all who have in a legitimate manner made true physical science the subject of their

as the

study only lute

;

the and, as he gradually gains knowledge by this method, he will see more and more clearly the abso-

impotence of

reasoning, to help

Man

all

him

has been

so-called

metaphysics, or a priori

to a single step in advance.

left entirely to himself as regards the of acquirement physical knowledge. But he has been gifted with various senses (without which he could not even know 2.

HEAT.

2

that the physical world exists)

and understand

to control

[CHAP.

and with reason

to enable

him

their indications.

Reason, unaided by the senses,

is

totally helpless in

such

The

matters.

preted by

indications given by the senses, unless interBut reason, are in general utterly unmeaning.

>wr\en reasGii ;and the senses

work harmoniously together

r l

His

6,

or

4'i5

per minute. Even the best mercury thermometers, when the column is rising or falling otherwise than very slowly, are found to

move

the

occasionally by sudden starts, the surface film a moment in the tube and thus subjecting

for

sticking

column

the bulb

Hence

to pressure or tension. recovers by starts from

It is possible also that its

state

of dilatation.

purpose of finding rates of cooling, as in the case above, it is sometimes better to make the observations of temperature at successive equal intervals taken as for the

the unit of time,

readings

and

to plot the differences

of the successive

the beginning and end of the In the figure the corresponding points

midway between

interval in question.

'

The time scale are put in and connected by dotted lines. is the same as before, but the degrees on the temperature

HEAT.

154

scale are ten times as long as before.

inserted

now

on the right

[CHAP. Its

numbers are

side of the temperature scale.

We

draw a smooth curve through the various points, but if we draw a smooth curve (the dotted one in the figure) which (perhaps not passing through any of them) shall show on the whole as much divergence of observed points from itself on one side as on the other, the find

it

impossible to

ordinates of this curve will be themselves the rates of cooling In required, and no drawing of tangents will be necessary.

the figure, the ordinate of this dotted curve, corresponding to 1 86, is 4'i2, thus agreeing within one per cent, with the result of the us, in

any

former method.

preferred.] 182. Another

Experience alone can guide is to be

which of these methods

case, as to

method of determining

specific

heat

is

supplied by the ice-calorimeter, in which the amount of heat lost by a hot body in cooling to o C. is measured by the

which

it

which the calorimeter

is

amount of

ice

The general principle on constructed will be obvious from

melts.

There are virtually three vessels, one within the figure. another, the two outermost containing ice and water at ordinary atmospheric pressure, and therefore always at o C. The middle vessel can part with no heat to the outer one unless all its

but this cannot be until its temperature changes; contained ice is melted ( 142). We have thus a

very simple and fairly effective method of preventing loss Into the interior vessel the substance whose of heat. is to be measured is dropped, at a known After a short is closed up. and the whole temperature, time the whole contents of the vessel are again at o C., but

specific heat

the heat disengaged ice

from

in the middle vessel.

the substance has melted some of the In the older and rougher forms of

the instrument, as designed by Lavoisier and Laplace (from

CHANGE OF TEMPERATURE.

X.]

155

one of which the sketch is taken), the amount of ice melted was given by the amount of water which drained away from Of course there was considerable error, the middle vessel. as

some of the water was

forces.

necessarily retained

by capillary

In the improved form, as designed by Bunsen and middle vessel is filled with solid ice to begin

others, the

with,

and the amount melted

is

calculated from the dimi-

nution of the volume ( 145) of the contents, as shown by an external gauge containing mercury in contact with the ice.

HEAT.

156

The

result of

[CHAP.

(151) the number by a known mass of a substance in

each experiment shows

of units of heat lost

By dividing passing from a known temperature to o C. the number of units of heat lost by the number of pounds of the substance and by the number of degrees of change of temperature, we obtain the Average Specific Ifeat of the

substance through the range of temperature employed. This method is very troublesome, and requires great care and skill; but it is much more trustworthy than any

when properly conducted. The only other method we, need describe is the common one called the Method of Mixture. Here we

other,

183.

specific heat of a substance by dropping hot, into a liquid (usually water) at a lower tem-

determine the it,

when

perature, acquires.

and observing the temperature which the mixture For rough determinations this process is sufficient,

and it has the additional advantage of being very But the corrections, which are absolutely necessary wish to obtain exact

results, are

easy. if

we

exceedingly troublesome

and require great experimental skill. These corrections are required chiefly on account of the loss of heat by radiation and evaporation, which do not affect the method last described.

M

Suppose pounds of one substance, at temperature C., to be mixed with ;;/ pounds of another substance at and suppose the mixture (all a lower temperature, t C. the have to corrections made) temperature T C., which must

T

;

T

Then the and /. obviously be intermediate between If the heat lost by the one substance has gone to the other. s respectively, two substances be *Sand heats of the specific the loss of the

first

(in units

of heat)

M S(T

T).

is

CHANGE OF TEMPERATURE.

x.]

The

gain of the other

is

(in the

same

y

/)

///

(T

'157

units)

Equating these quantities, we find at once

S s

m (r /) -~M(TT)'

which gives the ratio of the specific heats

;

or, if the

second

substance be water, the specific heat of the first substance. [It is to be observed that ^ is the average specific heat of

J/from Tto

T, s

It is well to if

no heat be

that of

m

from

r to /.]

observe that the temperature of the mixture,

lost

by radiation or evaporation,

MS T +

vi

is

sf

The quantities MS, MS, which appear in this expression, The water. are called Water-equivalents of the substances. equivalent of any mass is thus the number of pounds of water which would be altered in temperature to the same by the same number of units of heat, as the given mass.

extent,

[And we

see that the temperature of a mixture

is

given

in terms of the water-equivalents of the several substances

and

same

by a formula exactly the which gives the position of the centre of of a number of particles in one line, in terms of

their several temperatures,

as

inertia

that

masses and positions.] Another term which is sometimes of use, is the thermal This may be regarded as the capacity of a substance. water-equivalent of unit volume of the substance, and its their separate

numerical value

is

obviously the product of the density by

the specific heat of the substance.

(See

249).

HEAT.

153

[CHAP.

We

will now give some general notions as to relative But before doing so, of solids and liquids. heats specific it is well to remark that, in general, specific heat rises with

184.

temperature ; and that (as above stated) our experimental determinations give directly the average specific heat of a From substance through a certain range of temperature. the results

of such

however,

determinations,

sufficiently numerous and varied is not difficult to deduce the true

if

they be

in their circumstances,

it

specific heat as a function

of the temperature. at o C. requires

According to Regnault, a pound of water ioo'5 units of heat to raise raise

it

it

to

iooC., and 203*2

units to

to 200 C.

These are represented by the formula

H which /

+

/

0*000,02

/'-'

+

0*000,000,3

gives, for the specific heat of water at

^ 35

any temperature,

C., the expression i

+ o'ooo,o4

t

+ o'ooo,ooo,9

/2

.

Thus

the change for any ordinary ranges of temperature Refer again to 61. extremely small.

The

specific heat of ice

that of water.

at

oC.

is

is

almost exactly half it diminishes as

Regnault has shown that

the temperature is lowered. That of glass may be roughly

That of platinum

assumed

as

about

about 0-0355, and varies very

is

0*2.

slightly

With other metals, even for large ranges of temperature. such as copper, the increase of specific heat is (roughly) about io/. iron.

c.

for

iooC.

subject, as different results.

It appears to be rather more in no very exact knowledge on this specimens seem to give very different

But we have

CHANGE OF TEMPERATURE.

X.]

SPECIFIC

HEAT OF ELEMENTARY

Lead Tin

0x31

Copper

0-095

0-056

.... Iron Sodium .... Lithium .... The

.

.

.

.

.

.

.

.

.

118 63-5

0*293

56

23

.

...

0-941

7

59

SOLIDS.

207

...

0-114

i

.

.

.

... ... .

.

... .

.

.

6'4

6*6 6-0 6-4

67 6'6

column of the

table gives the specific* heat at second the atomic weight of the the ordinary temperatures, The numbers in the third column are the substance. first

first and second. It will be seen numbers in this third column are within about 10 per cent, of one another. Hence it is concluded that the specificheat of an elementary solid is inversely as its atomic weight. The small differences in the last column are attributed (in

products of those in the that the

part at least) to the known fact that the specific heat of each substance increases as the temperature rises, and

therefore that, as these specific heats are measured

the

same temperature, the

different bodies

all at

compared are much more

taken in different physical states, some being nearly at their melting-point than others.

It is probable that very important information as to the nature of matter will be obtained from this experimental fact. Perhaps its value may be more easily seen if (by

183)

we put

it

in the

form

:

The atomic water-equivalent elementary

A

is

nearly the same for all

solids.

similar law has

been found

to

hold for groups of

compound bodies of similar atomic composition. But the product of the specific heat and the atomic weight varies in general from group to group of such compounds. 185.

In general, as

we have

seen, the specific heat of a

HEAT.

160

substance in the liquid state

is

[CHAP. greater than in the solid

state.

SPECIFIC HEATS OF

ELEMENTARY LIQUIDS.

Lead Tin

0*040

400 C.

0-064

3

Mercury

0*033

C.

3C.

The second column gives the mean temperature of the range through which the specific heat is measured. The specific heat of Alcohol, which it is important to keep

in

1 86.

mind, is (at 30 C.) a little above 0*6. In the case of gases the problem of determination

of specific heat is not only more difficult, from the experimental point of view, than in the case of a solid or of a

but

liquid,

of view. gases,

also

more complex from a

theoretical point of the consequence great expansibility of have to specify the conditions under which the it is

For, in

we

is to be made. Thus the gas may be maintained at the same volume, or at the same pressure, throughAnd we therefore out the range of temperature employed.

measurement

speak of the

Specific

Heat at

Heat at

constant volume, or of the

constant pressure.

Thermo-dynamics gives us a simple relation between these quantities for the ideal perfect gas ( 126); and therefore for gases such as Air, Specific

Hydrogen, &c., one of the two

it

is

only necessary to determine directly This is a very happy cir-

specific heats.

cumstance, because the experimental difficulties of determining the specific heat at constant volume are extremely The mass of a gas at ordinary pressures is small great.

compared with that of the containing vessel, unless the and we cannot get over this difficulty by compressing the gas, for we must then

vessel be of unwieldy dimensions

make

the

proportion.

vessel

strong

(and

;

therefore

massive)

in

CHANGE OF TEMPERATURE.

x.]

161

The measurement

of the specific heat at constant pressure by passing the gas, at a uniform rate, through two spiral tubes in succession. In the first of these it is heated to a known temperature, and in the second it gives effected

is

its

up

heat to a mass of water in a calorimeter in which

From the volume of the gas is immersed. which has passed through the spirals, we can calculate rise of temperature in the its mass, and the observed that spiral

water of the calorimeter supplies the requisite additional datum.

This process was devised by De la Roche and Berard, its details were greatly improved by Regnault. Further remarks on this subject, especially from the

but

theoretical

of

point

view,

must

be

deferred

for

the

present.

showed that, for gases like tne specific heat of a given mass of the gas is independent of the temperature and pressure, and therefore 187. Regnault' s experiments

air,

124) of the volume. Hence the specific heat of a given volume of such a gas varies directly as the density. (

Equal volumes of different gases of same pressure, have approximately equal

this

class,

at the

specific heats.

SPECIFIC HEATS AT CONSTANT PRESSURE. '

Air Nitrogen It will

inverse

.

.

.

2 37

0-244

Oxygen Hydrogen

.... ....

0*217 3'4CQ

be seen that these numbers are very nearly ratio

of the

densities

of the various

in the

gases

(see

184).

Experiments on sound ( 397) have shown that the ratio of the two specific heats of air is about 1-408, whence we see that the specific heat of air at constant volume is o'i68. II

1

HEAT.

62

The dition

[CHAP. x.

specific heat of water-vapour is

0-48, a

little less

than that of

under the same conice

(

Hence

184).

water-substance has twice as great a specific heat in the liquid state as in the solid or in the vaporous state.

Resume of 178-187. Change of temperature cf mass by a given quantity of heat. Specific Heal. Method of Cooling. [Digression on Graphic Methods.] Method of Mixtures. Water EquivaIce Calorimeter. Thermal Capacity. Relation between Specific Heat lent. iSS.

unit

and Atomic Weight. constant volume, Water- vapour.

Specific

Heat of Gases

(b) at constant pressure.

:

Specific

(a)

at

Heat of

CHAPTER

XI.

THERMO-ELECTRICITY. 189.

We

have already stated,

49, the fundamental

in

by Seebeck ; and examine the subject more closely from For this purpose we must the experimental point of view. use the term Electro-motive Force. So far as we* require its as be enunciated follows properties they may

fact of Thermo-electricity as discovered

we now propose

to

:

The

strength of a current in a given circuit is directly proportional to the Electro-motive Force, and inversely

The energy of the current proportional to the resistance. is the product of the Electro-motive Force (contracted as E.M.F.} and the strength of the current. It will be advantageous to commence with the following statement of an experimental result Jf e be the E.M.F. in a thermo-electric circuit when :

and /, are the temperatures of the junctions ; and e tJie E.M.F. when the temperatures are / and /2 / then when the temperatures are t and /2 the E.M.F. is e + e'. /

x

[Consider this from the point of view given by the

lowing figure

Jour

;

where the

circuit is

supposed

to consist

of

two metals (say iron and copper). 49, where it is stated that the wires may be

wires, alternately of

See. again,

fol-

M

2

HEAT.

164

[CHAP.

soldered together at the junctions. No third substance, such as metal or solder, introduced into the solid circuit, is found to affect the result if

it

have the same temperature

at the

points where it meets each of the two metals. Let the temperatures of the four junctions be as in the and A together (if B and C had the same Then figure.

D

temperature

and

B

one another) would give E.M.F. e, and A had one common (if

with

and

C

D

together

temperature) would give E.M.F. is

Hence

e.

drawn, there would be total E.M.F.

the assumptions

made

(in brackets)

e

as the figure

+

e',

provided

above were superfluous.

Iron

Iron

But the E.M.F. may be looked upon as due entirely to C and D, for the tem-

the difference of temperatures of are equal, and peratures of

B

A

wire

AB

each metal to the

and

therefore the copper contribution of

Hence the whole E.M.F. of such a

produces no

effect.

circuit

depends

only on the temperatures of its ends, i.e. is independent of the temperature of the rest of the circuit.] 190. It follows from the experimental fact above, that interval of temperature, t to /?

we may break up any into ranges

E.M.F.

in

:

viz.

t

to

.,

,

a simple circuit

^ to 4 &c., /M to tn ; and the of two metals, when the junctions .x

THERMO-ELECTRICITY

XI.]

are at

t

and

tn will

be the sum of

the junctions are successively at /.,

its

/,

and

'165

separate values /,,

tl

and

/2,

when

&c, and

and 4.

We

shall now look upon these as successive equal ranges of temperature, and we shall define the thermo-electric power of a circuit of two metals, at mean temperature /, as the

E.M.F. which is produced when one junction is kept half a degree above, the other half a degree below, /. We thus get the means of representing in a diagram the relative thermo-electric positions of any two metals at different temperatures.

one degree in breadth, one of the metals.

The

Thus we may erect little rectangles below, upon a line representing

as.

area of each rectangle

is

the thermo-electric power

corresponding to the temperature at the middle of its base ; and, by taking the intervals of temperature small enough, it is obvious that the final upper boundary of the group of

This will reprerectangles will become a continuous line. sent at every point the relative thermo-electric situation of the second metal with regard to the first, as the first is represented by the line on which temperatures are measured. And the sum of any number of the rectangles (giving the

E.M.F.

for

any

t\vo

assigned temperatures

of

the June-

1

HEAT.

66

[CHAP.

is now represented by the area of the corresponding portion of the curve. 1 9 1. But there is a much more general experimental

tions)

result

than that in

sum of the p and y, is

At

every temperature the algebraic powers of metals a and /3, and

189.

thermo-electric

the thermo-electric power of a and y. of the iron wires in the diagram of 189 to be one [Suppose a third meial, say gold; and let the temperatures replaced by of the ends of one of the copper wires be tn and of the other /T .

Then

A

and

D

(if

B and C had

the

same temperature)

Iron,

'Cold

would give E. temperatures

J/. F. /

200

3pQ

177

^X.

40QC

HEAT.

178

The remark

[CHAP.

only line in the diagram which calls for special Its position and form are conthat of Nickel.

is

veniently studied

(as,

indeed,

is

obvious from the diagram),

by making a nickel-palladium circuit. When this is done it is found that in nickel the specific heat of electricity, negachanges sign somewhere about 200 C. and again about 320 C.

tive at ordinary temperatures,

199. Iron shows, but at much higher temperatures, the same peculiarity as nickel. Its line in the diagram intersects that of

N (one

of the iridium-platinum alloys above men-

tioned) at least three times below a low white-heat. In other 7 circuit has at least three neutral points. words, the iron-

A

If

we

"

raise the junctions of

such a circuit to the tem-

peratures corresponding to two of these points, the current produced is due entirely to the Thomson effect, i.e. to the

excess of the absorption of heat in one part of the iron, over the development of heat in the rest ; for (as we

N

have seen) has very

little, if

any,

Thomson

effect.

On the other hand, in the circuit of the two alloys, whatever the temperatures of its junctions, the current is due almost entirely to the Peltier effect. It is

probable that the above peculiarities of nickel and

iron are connected with the changes in their magnetic pro-

which take p!ace, as shown by Faraday, at about If there be peculiarities of the the same temperatures. same kind in cobalt, Faraday's results would lead us to look perties,

for

them

at a

still

higher temperature than even in iron.

200. In practice

it is very difficult to find the relative metals on the diagram for the purpose of two of position line of one the (when that of the other is assumed, drawing

or already determined) unless, either they have a neutral at point (intersection) within the limits of the diagram, or For least lie at no very great distance from one another.

THERMO-ELECTRICITY.

XT.]

179

they are far apart, the changes of E.M.F., however are usually small in comparison with

when

large in themselves,

the whole

quantity measured,

and may

thus to a con-

siderable extent be lost sight of in the inevitable errors of The lines shown in the diagram are very observation. irregularly scattered.

But what

it

is

is

between the lines

For This

by a very simple whose line of two given metals.

process, to procure

possible,

virtually a metal

shall lie

anywhere

purpose we use a double arc of these metals. made by soldering together at their ends a wire of

this

is

gold (suppose) and a wire of platinum laid side by side without contact ; and this combination will (when treated as

a single wire) have a line which passes through the neutral point of gold and platinum, and lies between them in Its actual position

thermo-electric position.

between them

determined by the relative electric resistances of the It would take us too separate wires of the double arc.

is

much

into pure electric science to give

so the following statement (from the Exam. 1875), which contains all that

the calculation

:

Camb. Math. Tripos

is necessary for the experimental application of the process, must suffice. Let wires of three metals A, B, C having resistances a, b, c, have their ends soldered together at two junctions y

which are maintained at

Then if

when

B

Ia

be

(different)

the current

when

is cut, the current in

C

A

constant temperatures. is cut, Tb the current

when

all

the

wires

are

continuous will be

a

(b

+ c) Ia + b(c+a] I

b

bc+ca + ab In practice the resistance c includes that of the galvanois usually large compared with either a or b;

meter, which

N

2

HEAT.

i8o

[CHAP.

and then the above formula takes the mate and exceedingly simple form

sufficiently approxi-

.

a+b

By

the use of this artifice

we can

fill

up

all

the gaps in

the diagram. 201. To conclude this part of the subject, we give a brief table of approximate thermo-electric data for some

These show how to draw the lines on the limits o and 350 C. Different specimens of the same metal do not agree very closely, so that the numbers are given on the same terms as are those in

common

metals.

diagram, between the

104, 153, &c.

NEUTRAL POINTS WITH LEAD.

\The numbei's in parentheses are

SPECIFIC

Fe. Pt

.

(soft)

Ir

.

.

.

.

Mg

.

.

Arg

.

.

Cd Zn

Ag Ru

-00247 -

'00056 'OOOOO

.

- -00048 - '00260

+ -00218 + -00122 + '00076 --00106

Sb

+

Co

--00585

The

of course, temperatures.,]

not,

HEAT OF ELECTRICITY.

'01140

Au

+ +

Cu Pb Sn Pd

+

-00052

-00048 -OOOGO -00028

- -00182 .--00260

Ni (to i/5C.) Ni(25o 3ioC.) .-f -01225 Ni (from 340 C.) - -coiSo - -00058 Rd Bi .

.

-0055

Al

.

.

+ '0002

quantities in this last table are to be multiplied by the The unit employed corresponds

absolute temperature C.

THERMO-ELECTRICITY.

XI.]

E.M.F.

to about io~5 of the is

because

adopted

181

of a single Grove's cell. This reader to form an

enables the

it

idea of the extreme feebleness of the thermo-electric cur-

The mode

rents in general.

a circuit of two metals

of calculating the E.M.F. of

be given

will

when we

later,

discuss

the thermo-dynamic theory of the circuit. 202. So far, we have considered the wires employed to

same

be each of the

[This

that differences of section in the

any parts of

circuit, or in

it,

had no

when

not necessarily true

is

and section throughout.

material

Magnus long ago showed

effect

on the E.M.F.

the variation of section

is

so abrupt that there can be finite difference of temperature within a range comparable with that of molecular force.]

But and

has been found that hammering, knotting, twisting, part of a circuit of one

it

stress in general, applied to one

metal only, makes rents

when

it

capable of giving thermo-electric curThe altered parts of the

irregularly heated.

unaltered parts as if they were of a In iron, nickel, &c., such effects can be produced by magnetising part of the wires. Also it has been found that if the two ends of a wire are at different

circuit

behave

to the

metal.

different

temperatures, and be brought into contact, a current (of very short duration) 203.

is

Resume of

produced.

189-202.

perties of thermo-electric electric

power

Thomson

denned.

Effect.

General experimental pro-

Consequences. Thermo-

circuits.

Peltier

Effect.

Neutral

Point.

Specific heat of Electricity in a metal.

Thermo-electric diagram. the Peltier and Thomson

How

it

effects.

represents E.M.F., and Mode of constructing

Anomalous behaviour of diagram experimentally. Nickel and Iron. Circuits in which there is (a) no Thomson

the

effect, (b)

of wires.

no

Peltier effect.

Null

effect of

Effects of irregular stress

and

unequal thickness

strain in a wire.

CHAPTER

XII.

OTHER EFFECTS OF HEAT. IN

204.

this

chapter

remarks on some of the

we propose

to

effects of heat,

collect a few

odd

the discussion of

which would have unduly interrupted the continuity of the

more important divisions of the subject already treated and to take advantage of the opportunity to introduce others which cannot as yet be strictly classified. We shall thus be able, for instance, to say a few words about some of the ;

chemical aspects of the subject, other than the heat of combination to which the next chapter is devoted. A curious example of mechanical motion, due 205. directly to expansion,

is

afforded by what

is

commonly called

A

the Trevelyan Experiment. piece of heated metal is laid on a cold block, usually of lead, so as to touch it at a place

which has been recently scraped or filed (to remove the If the heated mass be of such a form and so oxide). it can rock easily on its supports, the rocking (once started) becomes very much more rapid, so much so as to produce a quasi-musical sound which often continues

placed that

two bodies have acquired almost the same temperaof this phenomenon, given by Leslie and confirmed by Faraday, depends merely upon the

until the ture.

The explanation

CH.

OTHER EFFECTS OF HEAT.

xn]

fact that at the point

cold metal

'183

where the hot metal touches

it

the

so suddenly heated that it swells up, thus canting the rocker over to meet with similar treatment on the other side. Before it has come back again to its first is

position the swelling on the cold block has subsided, the heat having penetrated into the block by conduction, and

thus the process is repeated until the temperatures of the two bodies are nearly equal.

The

effectiveness

of

the

arrangement

is

much

oiten

increased by filing a notch in the lead, so that the rocker touches the block alternately on opposite sides of the notch.

The rapidity of the vibrations, and therefore the pitch of the note produced, is usually much altered by pressing the rocker more or less forcibly on the cold block ; and, when this process is skilfully conducted, sounds of the most extraordinary

character,

sometimes

almost

are

vocal,

produced. 206. An interesting example of the effect of heat, upon what are usually called "molecular" forces, is supplied by the modifications produced on capillary phenomena.

In general the curvature of the surface of water and other liquids in a capillary tube becomes less as the temperature is raised ; and at the same time the surface tension also height it

is

to

becomes

less

;

which the liquid

so that, on both accounts, the rises, or the depth to which

depressed, in a capillary tube,

temperature rises. The former phenomenon spection,

is

becomes

easily seen

and the proof of the

by

less

the

as

careful

latter is established

in-

by a

multitude of simple experiments. Thus if a thin layer of water of considerable surface be

placed on a large metal plate, and a small lamp flame be

UNIVERSITY OF CALIFORNIA DEPARTMENT OF PHYSICS

1

HEAT.

84

[CHAP.

applied under the middle of the plate, the effect of the lamp of exactly the same kind as that which it would produce

is

on a

The

tightly stretched sheet of India-rubber.

tension

of

water overcomes

colder

the

surface tension of

hotter

the

the same sort of result as

we

the

surface

diminished

and thus we have by putting a single drop

water,

get

of alcohol or ether on the middle of the water surface. the course of a sides

little

from the heated

time the water part, in the

is

one

In

dragged away on case, as

it

is

in

all

the

other case from the point at which the alcohol was applied. Thus we see why a drop of solder seems to be repelled

from the hotter to the colder parts of a soldering iron. 207. Another curious fact, closely connected with

this

part of the subject, is the effect of the curvature of a liquid surface upon the pressure of saturated vapour which is in

equilibrium in contact with it. When a number of narrow tubes, open at each end, dip into water in a receiver (free of air, let us suppose) the

water surface ture

is

surface

is

is

in

each raised in proportion as

Also the vapour pressure

greater.

more

raised.

bottom, with a

it.

its

curva-

less as

Hence, the more concave

the is

a

the pressure of saturated vapour If the tubes be now closed at the

water- surface, the less in contact with

is

is

water in each, evaporation or condenplace according as the water level is now

little

sation will take

higher or lower than before, and will proceed until the heights above the external water-surface become the same as

when

first

the

lower ends were open.

called attention to this

W. Thomson, who

phenomenon, considers

that

it

accounts for the great amount of water absorbed from damp air by bodies with fine pores or interstices, as cotton-wool,

Clerk-Maxwell pointed out that it also accounts for &c. the growth of the larger drops in a cloud at the expense of

OTHER EFFECTS OF HEAT.

xn.J

'185

the smaller ones ; and, at least in part, for that collapse of small bubbles of steam, which gives rise to the so-called " " singing of a kettle of water just before boiling commences. 208. This leads us to make a few more remarks on the

subject of boiling.

Water

boils at a lower temperature in a metallic vessel

than in one

made of

glass.

If water be carefully deprived of

air, it is

found that with

temperature may be raised in a smooth vessel many degrees above the boiling point corresponding to the pressure to which the water is subjected. But if some chips or care

its

of metal, or any solid with sharp angular points or It edges, be dropped in, it suddenly boils with violence. is possible that this property may have something to do filings

with boiler-explosions, at least when no fresh supply of water has been put in for some time before their occurrence.

it

These facts have not yet been thoroughly explained, but would appear that a nucleus of some kind is required for

boiling, just as

(

176)

it is

required for condensation.

A

very singular example of explosive boiling is furnished by the geysers, or volcanic hot springs, of Iceland. 209.

The phenomena

exhibited by these were carefully examined by Bunsen and Descloiseaux (Liebig's Annalen, 1847).

They determined the temperature of the water at different depths in the shaft of the Great Geyser, at various intervals before and after a grand eruption ; and on these observations Bunsen founded his theory of the action which takes place.

Between two eruptions it was found that the temperature of the water in the shaft increased from the surface downwards, but in no place rose so high as to reach the boilingpoint corresponding to the sum of the pressures of the atmoThe temperature sphere and ,of the superincumbent water.

HEAT.

186

[CHAP.

steam and very hot water being supplied from below and probably also at the sides of the shaft until a small upward displacement of the water column steadily rose at every point,

into

the basin

lowered the pressure to the boiling-point

one part of the shaft. This usually occurred at a depth of somewhere about seventy feet under the surface. Then at

there was violent boiling, blowing out the upper part of the column of water and thus relieving the lower part from ;

pressure, so that

it

also

Then

boiled explosively.

colder

water, from the geyser basin, ran back into the shaft, and the gradual heating from below recommenced. It is easy to exhibit the phenomena on a small scale, by artificially

and thus

producing the observed state of temperatures,

to fully verify the sufficiency of Bunsen's theory.

210. In 60, 134, while denning the fixed points of the Newtonian thermometer scale, we used the terms pure ice &c\&pure water. This implies, of course, that the freezing

and

of water (and

boiling points

affected

by impurities held

of other

in solution.

liquids),

are

[Mere mechanical

suspension of impurities, as in the case of produces in general no measurable effect.]

muddy

water,

The most important case, so far as the freezing point is concerned, is when water contains common salt in solution. found that the temperature of the freezing point

It is

is

Walker (in the Fox Expedition) notably lowered. found that the freezing temperature of ordinary sea-water is

then

below 28'5 F. salt is

He

observed, however, that a great deal of So he melted the ice, and reice.

extruded from the

hope of thus obtaining drinkable Although the freezing-point was raised by each operation, the water was still brackish after four repetitions of

froze

it,

several times in the

water.

the process.

The

lowest specific gravity of the liquid thus

obtained was from

1*0025 to

1*002.

He

alsb tried the

OTHER EFFECTS OF HEAT.

ty

converse process, removing in succession the various iceformed at lower and lower temperatures from a tub

crusts full

of sea water, in order to find

thus be procured. 211. Very extensive

and

how

careful

strong a brine could

measurements

have

been made of the boiling-points of aqueous solutions of various

In all of different degrees of concentration. is above 100 C., and the amount of

salts,

cases the boiling-point

approximately proportional to the percentage of the co-efficient of proportionality being

rise is

in solution

:

ferent for different salts.

Such

solutions,

when

salt dif-

boiling, give

almost pure water-vapour; and it is sometimes stated that, however high be the boiling-point of the solution,

off

comes off at the temperature corresponding (in There can be no doubt, Regnault's table) to the pressure.

the vapour

from special experiments made by Regnault and others, that a thermometer placed in the issuing vapour does indicate very nearly this temperature corresponding to the pressure.

But

just before

it

it is

to

be remarked that the aqueous vapour

leaves the liquid

is

certainly superheated

;

and

that superheated vapour leaving the liquid freely in a partially

closed vessel soon becomes saturated vapour, which on the bulb of the thermometer.

deposits a layer of water

The temperature water

is

of the bulb

is

therefore that at which pure

in equilibrium with water-vapour at the pressure to

which they are exposed, so that the method of observation is fallacious. We do not yet know, by any certain method, what is the exact temperature of vapour leaving a saline solution, boiling

under ordinary pressure

at

a temperature

above 100 C. 212.

solution

The is

rise of the boiling-point produced by salts in attributed to the molecular attraction between

the salt and the water.

Hence, when the circumstances of

1

HEAT.

88

[CHAP.

the experiment are altered, so that pure steam at iooC. is made to pass into an aqueous solution of a salt, condensaof steam

tion

point.

its

(with

latent heat) goes

on

until

accompanying disengagement of the mixture rises to

iooC. such

Thus, with steam at

its

solutions

boiling-

maybe

raised to temperatures very considerably higher. In Leslie's mode of freezing water, the greater part of the heat developed in the sulphuric acid is the latent heat of

In the next chapter the vapour absorbed. to account for the rest. 213.

What

is

shall see

called the spheroidal state of liquids

which, in

one

phenomenon been known from remote to test

we

whether a

how is

a

many forms, has a laundress wishes

at least of its

times.

flat-iron is

When

sufficiently hot, she dips her

water and allows a drop to the iron is not much above

gently on the iron.

finger in

fall

When

iooC.

in temperature, the drop spreads over the surface, and rapidly boils away. But if it be considerably hotter, the drop glides off from the

surface without wetting it, and without suffering much evapor[See Phil. Mag. 1850, I. 319, and 1832, II. 378.] The phenomenon is easily studied by dropping water

ation.

from a pipette into a shallow platinum dish heated from below by a Bunsen lamp. When the dish is to on it very much the water behave sufficiently hot, appears as it does on a cabbage-leaf or on an oiled or greasy surface. cautiously

The

water

is,

of course, not in contact with the metal. This

can be verified in many ways. For instance, if the dish be instead of a ray of light can be convex concave, slightly

made

to pass between it and the drop. Poggendorff connected one pole of a battery with the water and the other with the dish, and found that no current passed. The

temperature of water about 95 C. only.

in the spheroidal state

is

found to be

OTHER EFFECTS OF HEAT.

xn.J

i3y

The force required to support the drop is easily calculated. Thus, suppose it to be a square inch in lower surface, and a A water barometer stands at about quarter of an inch thick. and a half feet at the mean atmospheric pressure. the additional pressure required to support the drop is only about 1/33*5 X 12 x 4, or roughly, 1/1600 of the atmospheric pressure. This is supplied, as will be easily seen

thirty-three

Hence

when we

are dealing with the Kinetic theory of gases, by the

momentum come move

acquired by

air

and vapour

in directions

more

and

:

which have

On

leaving it they nearly perpendicular to the surface

than those in which they impinged cal

particles

in contact with the hot surface.

;

i.e.

thus, in the very thin layer

more nearly

verti-

between the water

and the metal, the gaseous medium exerts a somewhat greater

To pressure in a vertical than in a horizontal direction. in the a thicker or a similar result vapour layer gas produce mean free path (Chap. XXII.) increased in length. 214. This simple consideration gives the explanation of the chief phenomena shown by the Radiometer. This instru-

must be

may be

ment to

rarefied, so that the

much

consists essentially of a set of very light vanes attached

an

axle,

each vane side

is

When

about which they can freely turn. One side of blackened (so as to absorb heat), the other

is

polished.

The whole

is

fixed in a partial

vacuum.

exposed to radiation there is greater gaseous pressure on the blackened than on the polished sides of the 1 The first vanes, and the apparatus consequently rotates. it

is

experimental results which seem to have had a direct bearing this explanation are due to Fresnel, but they were left unnoticed.

on

215. With care water may be kept in the spheroidal state in But to insure success a glass vessel, such as a watch-glass. 1

Dewar and

Tait, Nature, xii. 217.

HEAT.

190

[CHAP.

must be nearly at its boilingWe may even have point when it is dropped on the glass. But a spheroid of water above the surface of very hot oil. here great caution is requisite, as explosions often occur,

in this experiment the water

scattering the hot oil in all directions.

Another

216.

striking

form of the same experiment is to silver heated almost to its

dip under water a solid ball of melting-point.

The

ball is seen

to

glow

in

the middle of

more rapidly when suddenly the water

the water for a few seconds (cooling, however,

than

would have done

it

comes

in contact with

it,

in air),

a slight explosion occurs, and

all

The experiment is considerably facilitated by previously adding some ammonia to the water. is

dark.

217. Brewster long ago discovered that in many specicrystals there are cavities, usually

mens of topaz and other

Sang microscopic, which are partially filled with liquid. observed that little bubbles of gas in the liquid in the cavities of Iceland spar seem to move towards the side of the cavity at which the temperature of the crystal is slightly If part of the walls of the bubble be formed by the raised.

explanation of this curious fact probably involves 206 above. But the capillary phenomena described in when the bubble is wholly surrounded by liquid, it would solid, the

seem

that the

motion

is

only apparent, the liquid

distilling

from the warmer side of the bubble and condensing on the As the liquid is usually carbonic acid, and under colder. considerable pressure, this explanation seems to accord with

known

facts.

Very

similar

phenomena can be produced on

a comparatively large scale by holding a warm body near one end of a small free bubble in a horizontal sealed tube

containing liquid sulphurous acid. 218. Davy's ingenious safety-lamp

lamp surrounded by wire-gauze.

is

He

merely an ordinary found by trial that

XII.]

OTHER EFFECTS OF HEAT.

what

is called flame cannot, except in extreme cases, pass If we lower a piece of wire-gauze, through such gauze.

whose meshes are not too wide, over the flame of a Bunsen lamp, the flame

is

arrested

by the gauze, although the un-

consumed gaseous mixture which passes through the meshes This may be proved by applying a is highly inflammable.

HEAT.

192

match

lighted unlit,

when

it

to will

it.

[CH. xii.

Or we may operate with

be found

that the

the lamp mixed gases can be

inflamed above the gauze without igniting the explosive It will be seen, in the next chapter, that mixture below it. the ignition of such a mixture begins at a definite temperaThe conducting and radiating powers of the gauze

ture.

(Chaps. XIV. and XVI. below) prevent the heated part

from being raised to

this temperature.

on flame will come more suitably 219. Resume of 204-218. Effects of heat

on molecular

curved surface of tions.

liquid.

bubbles.

further remarks

forces.

Trevelyan

experiment.

Vapour pressure over

Boiling under abnormal condi-

Freezing and boiling points of aqueous Radiometer. Motion of Spheroidal state.

Geysers.

solutions.

Our

in next chapter.

Safety-lamp.

CHAPTER

XIII.

COMBINATION AND DISSOCIATION.

As we have 7. 47, 51, 155 again to we know almost remarked nothing as to the ( 38), already We can measure, in state in which energy exists in a body. general, the amount which goes in, or which comes out ; and 220.

we can

REFER

tell in

what forms

it

does

so.

But we have seen that

molecular changes such as melting, solution, solidification, crystallisation, &c., are usually associated with absorption or evolution of heat. to the forces.

work done This

is,

however, as to

We

attribute these thermal

phenomena

against, or by, the so-called molecular

of course, merely an hypothesis ; so framed, in with the law of conservation of energy.

fit

The

actual nature of the process is wholly unknown to us. not to be expected, then, that in the present state of science we should have arrived at anything more satisIt is

factory than this in connection with the profound changes which take place in what is called chemical combination. Still, the little we do know is of great importance from the physical, as well as

from the more purely chemical, point of

And, from the purely practical point of view, it is one of the most important parts of our subject. For it is directly concerned with almost all our artificial processes for

view.

producing heat.

HEAT.

194

[CHAP.

221. It might at first sight appear, from some well-known experiments, that no very definite information is to be had on such subjects. For we can make the same two bodies

combine

in many different ways. Thus, to take a simple example, we may burn a jet of hydrogen in an atmosphere of oxygen, both originally at any one ordinary temperature ; or

we may mix and

together, once for

all,

in the

proper proportions,

same temperature, our oxygen and hydrogen, and apply a lighted match or an electric spark to the mixture. at the

The one

process

may be made

to

take place almost as

the other (in a slowly and as quietly as we please of moderate size) is practically instantaneous, 1 and ;

vessel is

ac-

the physical concomitants of a violent companied by if we consider the two processes in the But, explosion. of fundamental Carnot's principle ( 85), we see that light all

the water produced in each case is of the same amount, and has reached the same final state, the amounts In the case of of energy set free must also be the same.

when

slow combustion of heat

it

is

given out almost entirely in the form

in the explosion a great part appears at first as

;

sound and ordinary mechanical energy

become

:

but both of these

And

duced must be the same

the quantity of heat thus proin the two cases, when the products

have arrived at the same

final state.

ultimately

heat.

So

far, well.

But suppose

mixed

gases, just before the explosion, had been raised to a higher temperature than that at which the slow

that the

combustion took place, would the whole amount of energy involved in the explosion be the same as in the former case ?

We

simply mention this difficulty for the moment. beyond our imposed limits to

222. It would take us far 1

There is a definite rate at which a surface of combination runs along in each explosive mixture of gases, but it is usually considerable in comparison with the dimensions of any ordinary apparatus.

COMBINATION AND DISSOCIATION.

xiii.j

'195

give even a complete sketch, without details, of a subject like this, which has been developed in many directions by careful experimenters.

We

therefore content ourselves with

a mere mention (in addition to what has been said in 155, 156 above) of such facts as the following Heat is developed, often in considerable amount,

69,

:

when

such as chlorine, hydrochloric or hydriodic acid, Part of this heat is, of course, are dissolved in water. &c., due to the change to the liquid from the gaseous form. But

gases

the rest

is

be accounted

to

for

by the same process as are

the facts which follow.

When liquids, e.g. sulphuric acid of commerce, alcohol, &c., are diluted with water, there is generally evolution of heat accompanied by diminution of volume. But

it is

not always the case that there

is

evolution of

heat in the mere mixture of liquids. Thus, although when 27 parts (by weight) of water and 23 of alcohol are mixed

same ordinary temperature, the temperature of the rises about 8'3 C.; if we mix 31 parts (by weight) of bisulphide of carbon with 19 of alcohol at the same

at the

mixture

temperature, the temperature of the mixture is lou'ered by 5'9 C. Now it has been found by direct experiment that, in the former case, the water-equivalent ( 183) of the mixture

is

about

stituents

greater than the sum of those of the conin the latter case the excess is- only On the other hand, the volume of the first

\\\\

while

;

about TVth. mixture is 3-6 per cent, less than the sum of the component volumes with the second mixture it is 17 per cent. ;

greater.

The

first

of these supplementary facts would

tell

against the observed results, the second in favour of them. The subject is thus easily seen to be one of considerable

we cannot farther develop it here. mention, however, that experiments have been

complexity, and

We may

o

2

HEAT.

196 carried out with various

[CHAP.

mixtures of the same pair of liquids at initial

(common)

And

temperatures.

it

appears

probable (according to Berthelot) that the specific heats of mixtures vary as much with temperature as do those of simple liquids so much so, in fact, that the mixture of bisulphide of carbon and alcohol mentioned above would give rise to evolution of heat if both components were originally at a temperature lower than o C. 223.

The

application of the great laws of thermodynamics

37? 82 ) to chemical

(

combination

is,

in

its

elements at

The experimental least, a matter of no great difficulty. with the has been carried out of work great ability by part prominent among whom are Andrews, and (more recently) Berthelot, We must confine ourDeville, Thomsen, and Stohmann.

many

Favre

investigators,

and

selves to the

Silbermann,

more

salient features of the subject.

first law that, if any mixture undergo a definite chemical change without taking in or giving out work, the heat developed or absorbed depends solely upon

It follows

from the

no way upon one respect, a great gain ; for it much simplifies the reasoning from the results of experiment but it is also a great loss, inasmuch and final states of the system, the intermediate transformations. This the initial

and is,

in

in

;

prevents our obtaining (by this means alone) any information as to the nature and order of the successive

as

it

steps of a

complex

reaction.

Again, in any transformation which takes place without the application or the giving out of work, the heat developed is the equivalent of the excess of the original over the final potential

energy due to the chemical affinities involved. The reason here only of heat developed.

[We have spoken for

this

restriction

statement]

will

appear

at

once from

the

next

COMBINATION AND DISSOCIATION.

xin.]

its

197

In accordance with the second law of thermodynamics (or consequence, the degradation of energy), the final state

of every combination is that in which the potential energy of chemical affinity is a minimum ; or, what comes practically to the same thing, the reactions which take place

amount of heat. And, which in this way require are formed conversely, compounds for their decomposition a supply of energy from without, 224. Thus, to take a simple case, suppose equal weights are such as to develop the greatest

of hydrogen to enter into combination with oxygen no matter how. There are but two compounds which (so far

we know) can be formed, water and peroxide The heat of combination is found to be gen. as

of hydro-

nearly 50 per cent, greater for the former product than for the latter.

Hence, when hydrogen and oxygen act directly on one another, water alone

is

formed, and the excess of either And, by the first of the three

simply unacted on.

is

gas statements of

223,

if

we wished

to

cause water to be

peroxide of hydrogen, we should have to supply energy equal to the excess of the heat of combination of two atoms of hydrogen with one of oxygen, over oxidised into

that of

two atoms of hydrogen with two of oxygen. to find, as in fact we do

Hence we should expect that peroxide of

pound.

hydrogen

The energy

is

essentially

find,

an unstable com-

required for the formation of such

usually supplied from some direct reaction a stable (forming compound) which takes place at the same time. Thus peroxide of hydrogen is obtained during the

compounds

is

formation of chloride of barium from peroxide of barium

and

dilute hydrochloric acid.

We

have seen that a compound, which is formed by 225. the direct action of its components with the evolution of heat,

is

essentially stable.

To decompose

it,

external energy

HEAT.

198 is

peroxide of barium essentially stable, for heat is

Thus, in the case just

required.

[CHAP.

and hydrochloric acid are each

cited,

given out when either is formed directly from its elements. But a mixture of the two is not stable, for a rearrangement

of

elements

is

possible by which farther heat can be the energy required for the decom-

Here

developed.

positions is supplied by a chemical process. But there are many other ways in which supplied.

Thus heat

lime from limestone, is

effective

itself,

may be

it

can be

as in the formation of quickdirectly the agent.

under certain conditions, as

in

Light also the decom-

position of carbonic acid in the leaves of plants, or of salts of silver on a photographic plate. And the discovery of the

and earthy metals, Davy's grandest contribution to chemistry, was effected by the electric decomposition of

alkaline

their

hydrated oxides.

On

the other hand, compounds which are formed with absorption of heat are often liable to spontaneous decomThis is the case with many highly explosive position. bodies, such as chloride of nitrogen and the oxides of chlorine.

226. The displacement of one element by another from a compound gives an excellent instance of dissipation of Thus the heat of combination of iodine with energy.

hydrogen or other metal is less than that of chlorine with and chlorine has, in consequence, the the same metal

power of decomposing such iodides, setting the iodine free. But the whole of this part of the subject is much more easily studied

by measuring the energy, given out during a

transformation, in

heat

;

and

its

electrical

therefore, so far as

it

forms than in the form of is

not chemical,

it

belongs

more properly Although

to Electricity than to Heat. all combinations take place in

accordance with

COMBINATION AND DISSOCIATION.

XIIL]

the laws of energy,

laws alone do

these

to determine the result in

not enable

We

case.

any particular

1*99

us

must

have special data with reference to each pair of the substances involved. These, of course, can be found only by

As they belong much more directly to experiment. to Heat, we cannot enter minutely into than Chemistry what

is

known of them, but

general results of

given by

will quote, as

one branch of

a specimen, a few

this extensive

subject as

Berthelot.

Strong acids and strong

bases,

when made

to

combine

in

equivalent proportions, each being previously dissolved in a sufficient quantity of water, evolve nearly equal amounts of heat in the formation of stable neutral salts, whatever be the acid or the base.

The heat disengaged is but slightly altered by the presence of greater quantities of water, or of additional quantities either of the same, or of another, base. Examples of this are furnished by the neutral salts of soda, potash, baryta, &c., with hydrochloric, nitric, sulphuric, <Scc.,

acids.

Even in the formation of the alkaline salts of strong acids more heat is disengaged than in the combinations of feeble acids with the same alkaline bases.

The salts

feeble acids form, even with the strongest bases, which are decomposable by water to an extent in-

creasing with the amount of water, but greater excess of base or acid present.

less

as there

is

The

increase of decomposition goes on when water is added, without limit in the case of some feeble acids, but

tends to a definite limit with others. But there are instances in which a moderate amount of water wholly effects the decomposition, so that the effect on the thermometer is almost exactly the reverse of that due to the formation of the salt.

HEAT.

200

To

227.

remarks, values of

illustrate

we

a

give

[CHAP.

some of the above, and a few little

further,

of roughly approximate

table

HEAT OF COMBINATION.

H H

2

with

O

O2 C with O C with O 2 CO with O The

2

first

with

column

68,000

34,000

45,ooj

22,500

30,000

2,500

97>coo

8,100

67,000

2,400

indicates the substances combining,

and

Here H, O, C, the proportions in which they combine. stand for "atoms" (not "molecules") of hydrogen, oxygen, and carbon, in the chemical sense of the term ; so that their relative

masses are as

i,

16,

and

The second column

12.

gives the number of units of heat ( 179) produced when 2 Ibs. of hydrogen, 12 Ibs. of carbon, or 281bs of carbonic

oxide, respectively, enter into the oxygen combination indicated. The numbers in the third column give the units of

heat set free per

Ib.

of the substance to be oxidised.

The

by which these numbers were obtained were all essentially such as to reduce the compound to the same (ordinary) temperature as that of the components, and to measure the amount of heat given out during the reduction

processes

of temperature.

Thus, when a pound of hydrogen

is

oxidised into 9 Ibs.

of water, 34,000 units of heat are given out. directly peroxidised, only

Hence, to

effect the

If

it

could be

22,500 units would be given out.

farther oxidation of 9 Ibs of water,

energy equivalent to the difference of these, or 11,500 units of heat, must be supplied.

And we

number of units of heat set free by oxidising a quantity of carbon at once into carbonic acid, is the sum of those corresponding to the two stages also see that the

xni.]

COMBINATION AND DISSOCIATION.

formation of carbonic

oxide,

and

its

ioi

subsequent farther

oxidation. 228.

Many

combinations, even of the most vigorous kind, Even the exat ordinary temperatures.

do not take place

plosive mixture of one volume of oxygen with two of hydrogen requires a start, as it were. If the smallest portion of

the mixture be raised to the requisite temperature (as by a match or an electric spark) the heat developed by the

combination

suffices to raise other portions, in turn, to

the

not yet certainly known how a proper temperature. inflamed is of by contact (in air) with spongy jet hydrogen It

platinum.

is

It is possible that this

may be due

to the heat

developed by its sudden condensation, in which case this would be a phenomenon of the same class as that just mentioned. But it may be due to the mere approximation of oxygen and hydrogen particles by surface condensation. 229. We are now prepared to consider some of the chief of ordinary combustion. For the table, meagre how shows are the is, very large quantities of heat in the some of cases of combination. commonest developed When the gaseous products of a combination are raised to a sufficiently high temperature to become self-luminous,

phenomena as

it

they form what is called a flame. with a little detail. 230.

An

excellent

and

We

will

typical instance,

take one case

which has been

very carefully studied by Deville, is that of a blowpipe flame fed with a mixture of oxygen and carbonic oxide, escaping from an orifice of i^th of a square inch in area under a pressure of from half an inch to an inch of water. The flame proper consists of an exterior cone, three to four inches long, vividly blue in its lower parts and of a very faint yellow at the apex. Within this there is an interior cone, barely half

an inch long, which consists of as

HEAT.

202

[CHAP.

Just at the apex of this interior cone, yet unignited gas. where the gases have just begun to burn, the flame has the highest temperature, and it is there capable of readily melt-

The temperature of the ing even a stoutish platinum wire. flame falls off rapidly from this point upwards. [The velocity of propagation of the combining temperature (footnote to

221

) is

obviously that of the jet of gas at the apex

of the interior cone.]

To

find the composition of the luminous matter at

any

assigned point of the axis of the flame, Deville employed a f an mcn thin tube of silver, about T\t ns diameter,

m

which was kept cool by a current of cold water passing This tube was fixed transversely in the flame, through it. in

such

a

way

diameter) in

its

axis of the flame,

water was

that

a

small hole

(

T|^th of a inch

in

side occupied the assigned position in the

made

and faced the to pass

orifice

of the jet.

The

cold

so rapidly as to produce suction Thus a sample of the gases of the

through this small hole. flame was carried along in the current of water.

This was

escaping from the water, through caustic potash to remove the undissolved carbonic acid, and then carefully

led, after

analysed.

In order to determine from what amount of the

unignited

mixture

this

combustion

product

had

been

furnished, the experimenter had previously added to the This gas, mixture a known (small) percentage of nitrogen.

of course, passed almost entirely to the collecting vessel, the only loss being due to slight diffusion through the flame.

A this

great number of concordant determinations made by method showed that at the apex of the flame there is

practically nothing but carbonic acid (provided, of course, that the original combustibles were mixed in the proper proBut, from the apex downwards, the proportion of portion).

COMBINATION AND DISSOCIATION.

xiii.J

263

to combined gases steadily rose, till at the apex of the interior cone (where the temperature is highest) the uncombined gases formed about ^rd of the whole. This

uncombined

'result is, at first sight,

tion

not far to seek.

is

sociation 231. to

(

On

47) due to the high temperature. the theoretical aspects of dissociation

make some remarks

assimilate

somewhat surprising, but its explanaWe must take account of the dis-

its

later.

laws to those of evaporation.

that a liquid, at

any

definite

temperature,

equilibrium at its free surface with definite pressure

(

we intend

Meanwhile we may, roughly,

its

162), so that, at the

We is

have seen in

kinetic

own vapour

common

at a

boundary,

there is constant condensation accompanied by an equal amount of evaporation, with equal continuous disengagement and absorption of latent heat.

Thus, in a compound gas, at least at any one temperature within certain limits (which now probably depend on the there is kinetic equilibrium maintained by a constant amount of dissociation, accompanied by an equal amount of recombination, and also with equal continuous pressure),

disengagement and absorption of heat, the heat of combinaThe percentage of the whole, which is dissociated, tion. from zero at the lower limit of temperature (at, and below, which there is no dissociation) faster and faster till about half way, and then slower and slower till the upper rises

limit

(at,

and above, which the whole

is

dissociated)

is

This half-way temperature, at which one half of the gas is dissociated, is called the temperature of disreached.

sociation.

When begin to

the mixed gases are gradually heated, they cannot combine till a certain temperature, depending on

the constituents and their relative proportions (in a way not is reached. Thus we see how the explosive

yet ascertained),

HEAT.

204

mixture

(228)

requires a "start."

[CH. xill.

And,

if the

constituents be

such as to combine directly with evolution of heat, this commencement of combination is sufficient of itself to raise the temperature, with the effect of producing farther and This goes on (provided the gas lose farther combination. no energy to external bodies) until kinetic equilibrium is established, is

in the

i.e.

until the percentage of the mixture

combined

which

state corresponds to the temperature

reached by the whole in consequence of the heat of combination set free. If, however, the gas be permitted to lose contact with a cold vessel or by expanding as by energy,

and doing work, the process of combination will go on farther and may become complete, in which case the final temperature must be under the lower limit. Similar consequences may be traced if we suppose that first individually heated above the higher then mixed, and the temperature gradually reduced. Different modes of com220-231. 232. Resume of bination. Heat or cold produced by mixing. Direct con-

the gases be limit,

Stability of sequences of the Laws of Thermodynamics. Heat of combination. Comcompounds. Displacement.

bustion.

Composition of flame.

Dissociation.

CHAPTER

XIV.

CONDUCTION OF HEAT. We have already seen that 4, 73. 233. REFER to bodies in contact ultimately acquire the same temperature. This experimental fact is the necessary basis of almost all our methods of measuring temperature. The same must, of course, be true of contiguous parts of the same body. This

can only occur by transference of heat from part to part of the body. For the present we need not inquire whether this equalisation is effected by the mere passage of heat from the warmer to the colder parts, or whether there is a mutual interchange in which the warmer part gives more to the

So far, indeed, as the colder part than it receives from it. ultimate result is concerned, it does not matter which of the correct view, for we cannot identify any portion ; though the question is theoretically of great interest and importance, as we shall find when we come to

these

is

of energy

To this latter head also we will refer all cases which the transfer of heat in a body appears to take place directly between parts at a finite distance from one another. 234. That bodies differ very greatly in conductingRadiation. in

power

for heat,

is

matter of

common

observation.

We

hold without inconvenience one end of a short piece of

HEAT.

20b glass,

flame.

even when the other end

No

one would

try the

[CHAP. is

melting in a blowpipe

same experiment with a

copper or even of iron. The pieces of or bone which are inserted between the handle and

similar piece of

wood

the body of a metal tea-pot owe their presence to a recognition of the same fact. So does the packing of the interior

of a Norwegian cooking-stove. Two of the ordinary forms of lecture experiment also

may

be mentioned here.

First, that of

Ingenhouz, in which a metallic trough, from number of rods of the same diameter,

the front of which a

but of different materials, project.

These rods are covered

with a thin film of bees-wax, which melts whenever

it

is

raised to the proper temperature. Suppose the whole to be at the temperature of the air, and hot water to be suddenly

poured into the trough. By watching the melted wax we can judge of the relative

limits of the

thermometric

conducting powers of the various materials. A less objectionable form of the experiment consists in placing end to end two bars of iron and copper, of equal sectional area, having small bullets attached by bees-wax to their lower sides,

and placing a lamp flame

The temperature

at

sufficiently to let the bullets fall

is

between them.

impartially

which the wax softens found to travel

much

CONDUCTION OF HEAT.

xiv] faster

along the copper than along the iron.

does really indicate superior conducting

107

And

here

it

power on the part

of the copper, for the thermal capacities ( 183) of iron and copper are not very different 235. Some creditable work was done on this subject But the first to give a thoroughly last century by Lambert, satisfactory definition of (not, in

Fourier.

English at

The

conducting power, or conductivity

least, conductibility

entire subject

or conducibility) was

of the present chapter, with

but its later experimental developments, may be said to have been created by his Theorie Analytique de la Chaleur, first published in a complete form in 1822 ; and these later all

developments may be said to have been suggested, or rendered possible, by Fourier's work. Its exquisitely original methods have been the source of inspiration of some of the

and the mere application of one ; simplest portions, to the conduction of electricity, And more, in this famous. has made the name of

greatest mathematicians

of

its

Ohm

work of

Fourier's,

attention

was

first

specially called to

the extremely important subject of Dimensions of physical [See quantities in terms of the fundamental units.

Chapter XIX.] 236. Conducting power must,

ceteris paribus, be meaWe must of which the heat passes. quantity by therefore find how to estimate the amount of heat passing

sured

HEAT.

208

[CHAP.

from part to part of any one body, and what are the circumstances which have influence on it. We can then place two different bodies under precisely the same determining circumstances, and thus compare their conductivities by comparing the quantities of heat transferred. 237. The simplest mode of passage is that in which the transfer of heat takes place, throughout a body, in parallel lines. Thus, for instance, when a lake is covered with a uniform sheet of ice, the ice grows gradually thicker if the

temperature of the air be below the freezing point. This is caused by the passage of heat through the ice from the water immediately below it, the upper surface of the ice being kept colder than the lower. Thus the layer of water next the ice freezes. And as the temperature is the same throughout any horizontal layer, whether of water or ice, the transference of heat i.e.

layers,

it is

Hence we choose, solid,

is

wholly perpendicular to such

vertically upwards.

a plate of

as our

typical

uniform thickness

form of conducting

and of

whose

(practically) sides are kept permanently at two

different temperatures.

After a lapse of time, theoretically

infinite surface,

infinite

but in general practically short, a permanent state

of distribution of temperature slab, it

arrived at throughout the layer parts with as much heat as under these circumstances, we measure the is

and therefore every

receives.

If,

quantity of heat which passes through a square inch, foot, or yard of any layer of the plate in a minute, we may take this as representing numerically the

To

conducting power.

compare plates of different substances, we must of course take them all of the same thickness, and use the same pair of temperatures for their sides. Thus we are led to define as follows ductivity

of a body at any temperature

is

:

the

The thermal

number of

con-

units

CONDUCTION OF HEAT.

xiv.J

269

of heat which pass, per unit of time, per unit of surface, through an

infinite plate (or layer}

of the substance, of unit

thickness, w/ien its sides are kept

at temperatures respectively half a degree above, and half a degree below, that temperature. The units which we employ are, as hitherto, the foot, the the

minute,

pound, and

chapter we

special

will

the degree centigrade.

show how

to pass to

In a

any other

[It will be observed that conductivity the above definition goes, depend upon may, the temperature of the body. This will be considered

system of units. so

as

far

later.]

238. The methods chiefly employed for measuring thermal conductivity depend ultimately upon observations of temperature of the conducting body at different parts of its mass. Thus the amount of heat which passes across any

surface in the body, one of the quantities in terms of which the conductivity is defined, is deduced from the change of

produces in the body itself. Now the a given quantity of heat are inversely temperature as the thermal capacity ( 183) of the body. Hence what temperature which

it

effects of

we

directly

ductivity capacity.

deduce from such experiments

itself,

it

and of

not the con

to the thermal

is called by Maxwell the Thermometric and by Thomson the Thermal Diffusiuity^\

follows that the determination of conductivity re-

quires that these set of

is

its ratio

[This ratio

Conductivity,

Thus

as defined above, but

methods be supplemented by a separate

experiments for the determination of specific gravity specific heat.

now think of the state of things in the interior of the slab while a steady passage of heat is going on. The only reason why it goes on is that every layer differs in 239. Let us

And, because it temperature from those adjacent to it. on same amount across each to the goes (thin) layer, the p

HEAT.

210

[CHAP.

difference of temperatures ot the sides of such layer must be is proportional to its thickness. [This easily seen by supposing the layer to be divided into a great number of sub-

and then considering two, three, Thus the amount of heat passing across any layer depends ceteris paribus on

layers of equal thickness,

or

more adjacent ones

as a single layer.]

the gradient of temperature across that layer. This may usefully be substituted for the difference of temperatures of the sides of a layer of unit thickness. For we thus have the means of calculating the amount of heat which passes across any layer, even when there is a variable state of as for instance when one side of the slab is temperature at a constant kept temperature while the other is alternately :

heated and cooled. 240. All experimental results hitherto obtained are consistent with the assumption, for small gradients, that the

which must evidently be true amount of heat which passes

across any layer, kept at a given temperature, is directly proportional to the gradient of temperature perpendicular to that layer. is necessary to point out that the exten[It sion to high gradients is an assumption found to be consistent

with

facts,

because conductivity

is

generally measured under

circumstances in which the gradient is very much greater than iC. per foot, which is that implied in our definition.

And we cannot yet positively assert that the quantity of heat which would pass through a slab a millionth of a foot in thickness is exactly a million times as great as that passing through a slab one foot thick, when the surface temperatures are the same, even if the conductivity of the material were the same at all temperatures. The corresponding proposition in electric conduction (which is called Ohm's Law,

though directly based on the splendid work of Fourier) involves the same assumption, but here we can much more

CONDUCTION OF HEAT.

xiv.] easily verify

and the

it,

verification

extreme lengths by Maxwell and

211

has been carried to

Chrystal.~|

realise experimentally 241. It is practically impossible to the simple conditions which are assumed in the above Hence, in order to measure definition of conductivity.

the value of this quantity in different substances, experimental arrangements of a somewhat more complex character

must be adopted.

Two

chief principles have

The

been employed as the bases

of

requires a steady state of temperaexperiment. The second ture, and a consequent steady flow of heat. first

requires a variable, but essentially periodic, state of temperature, maintained by external causes in one part of a

body and thence propagated

In

as a species of wave.

both we calculate, in terms of the (known) gradients of temperature and the (unknown) conductivity, how much more heat enters an assigned portion of a body by conIf the temduction than leaves it by the same process. perature remain constant, this part of the body must lose heat otherwise than by conduction ; if it do not so lose

In the first method, as heat, its temperature must rise. each part of the body maintains a steady temperature, we must measure by some independent process how much heat the assigned portion loses otherwise than by conducIn the second, we must equate the gain of heat hy conduction to the quantity of heat indicated by the thermal

tion.

capacity of the element and its change of temperature. A third method of some celebrity involves both of these considerations. 242.

The

first

method was employed by Lambert,

in

an

but was afterwards greatly improved by imperfect form Forbes, who obtained by means of it the first absolute :

determination

of conductivity of any

real

value.

p 2

The

HEAT.

212

[CHAP.

of the method is extremely simple, but the experimental work and the details of the calculation from it are both very tedious.

principle

A to,

long bar of uniform cross section has one end raised and maintained at, a definite high temperature, while

is exposed directly to the air of the laboratory. Small holes are bored in the bar at regular intervals these (when the bar is of any other metal than iron) are lined

the rest

;

with thin iron shells, and in them the cylindrical bulbs of

accurate ;thermometers are inserted, each surrounded by a calculation based few drops of mercury. [It is found by on the results of the experiment itself, that these holes do not, to any perceptible extent, interfere with the transference

of heat along the bar.] second bar, of the same material and of equal cross section, but of much smaller length, also provided with an

A

inserted thermometer,

The

is

placed near to the

first.

us say, in the have been left night, then be of the will no heat being applied. They laboratory, same temperature throughout, viz., that of the air, and all bars

all

let

CONDUCTION OF HEAT.

313

the thermometers will show that temperature. Now suppose heat to be applied to one end of the long bar. [This is or in done it a of melted bath solder, usually by inserting

which is maintained by a proper gas-regulator at a steady temperature.] After a short time the thermometer nearest to the source of heat begins to rise, then rises lead,

faster

and

faster.

Meanwhile, the next thermometer, in And so on. In Forbes's iron bar,

turn, begins to rise.

which

is 8 feet long by about i inch square section, the steady state of temperatures was not reached till after a When this steady state lapse of from six to eight hours.

final

was reached (i.e. when, for half an hour or more, no change of reading was observed in any of the thermometers) it was found that even the thermometer farthest from the source indicated a temperature perceptibly higher than that of the of the room, as shown by the thermometer in the short

air

bar.

ing

The others showed each a rise of temperature, increasmore and more rapidly as they were placed nearer the

[The conductivity of copper is so much greater than that of iron, that the farther end of the eight-foot bar has to be kept cool by immersion in a vessel, into which a source.

steady supply of water

is

experiment, repeated tures of the source, it

if is

admitted from below.] From this* necessary with different tempera-

temperature along the axis Fourier's methods,

we can obtain a very permanent distribution of of the bar. [And calculation, by

obvious that

accurate determination of

the

shows that the temperature

is

practically

the same throughout any cross section of the bar, unless the section be very great, or the conductivity of the material

very small.] 243.

only

:

So

far,

the investigation has dealt with temperatures consider heat. The quantity of heat

we must now

which passes, per minute, across any transverse section of

HEAT.

214

[CHAP.

the bar, must obviously be the product of the area of the section, the conductivity ( 237), and the gradient of

temperature course, be

(

The gradient can, of 239) at that section. from the observed distribution of

calculated

Hence, temperature, and the area of the section is known. the heat passing is expressed by a definite multiple of the unknown conductivity. But that heat does not raise the temperature of the part of the bar to which it passes, for we have supposed that the stationary state has been reached.

Hence must

the rest of the bar, beyond the section in question, by cooling in the air (and, in the case of copper,

lose

the water- bath at the end) precisely as much as it gains by conduction. It is only necessary, then, to find what amount of heat is thus lost, and we have at once the

determination of the conductivity of the bar at the temperature of the particular cross section considered. [The words in italics are of special importance, as they indicate one of the

most valuable consequences of the improvements due Forbes, which we now give.]

to

244. Starting afresh, we once for all heat the short bar to a high temperature, insert its thermometer, and leave it to cool in the neighbourhood of the long bar (which is, in

the temperature of the air). The is now read at exactly equal intervals of time, say every half minute while it is very hot and cooling fast, then every minute, every five minutes,

this

experiment,

left at

thermometer in the short bar

and

finally every half hour, till it has acquired practically Some of the thermometers in the temperature of the air. the long bar are read at intervals during this process.

From what we have from heat

this is

said

above

it

will

be obvious

form of experiment, we can calculate lost

by the short

length, at each temperature

that,

how much

per minute, per unit of within the range employed ;

bar,

CONDUCTION OF HEAT.

xiv.J

and hence we can

calculate

what was

^15

former

lost in the

any assigned part of the long bar, since we know what was the stationary distribution of temperature at

experiment along

it.

The

loss of heat by cooling depends mainly upon of the excess temperature of the bar above that of the for is the purpose of finding this that we use it and air,

245.

But it also the two bars in each part of the experiment. some the actual to extent, upon temperature and depends,

and thus the two parts of the ment should be conducted as nearly as possible same air-temperature and pressure. (See 336.) pressure of the air

246. is

is

The

experi-

;

at the

unit of heat, in terms of which the conductivity

242, 244, given by direct comparison of the results of of course that which raises by i C. the temperature of

unit

volume of the substance of the

one of Forbes's experiments on are as follows

The

bar.

iron, in

results of

terms of this

unit,

:

THERMOMETRIC CONDUCTIVITY OF IRON. o

Temperature C.

0*01506

Thus

would

it

at

first

100 .

sight

.

.

0*01140

200* .

appear that

.

.

0*00987

the

conduc-

of iron for heat, like its electric conductivity, is diminished by rise of temperature. Forbes had expected

tivity

to find

it

so,

having remarked that the order of the metals is the same as their order as electric

as conductors of heat

conductors, while the electric conductivity of every metal is diminished by rise of temperature. This remark was fully

confirmed by the well-planned experiments of Wiedemann and Franz, but Forbes's farther expectation was not, as we shall see, realised.

The

following results, in terms of the same or similar

HEAT.

216 units,

[CHAP.

were obtained by Tait in a repetition and extension The iron bar employed was that

of Forbes's experiments. of Forbes.

THERMOMETRIC CONDUCTIVITY. o

100

0*0149

o'oi28

Temperature C. Iron

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

Copper, electrically good ...0-076 Copper, electrically bad ...... 0*054

German

In in

Silver

this

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

list,

all

...0-079 ...0-057

o "0088

...

300

0-0114 ... 0*0105 0*082 ...0-085 0-060 ...0-063 0-0092

0*009

...

52-9(1+0-0014*)

53-4(1+0-000880 52-5(1+0-0009*)

0*0094

46-6(1+0-0009*)

seem

the substances but iron

therm ometric conductivity by

To

200 ...

to improve

of temperature. convert these numbers into thermal conductivities rise

above ( 237) they must each be multiplied by the numerical value of the water equivalent of a cubic foot

as defined

of the corresponding substance, i.e. by its thermal capacity These water equivalents are given in the last 183). (

column of the

The temperature changes

table.

indicated

by the factors in brackets are very uncertain. The following table contains the results of Forbes, with

some more recent determinations.

Mitchell employed Tait's

bars, nickelized to prevent oxidation.

THERMAL CONDUCTIVITY OF IRON AND COPPER. To foot, minute, and degree C. Iron.

Copper.

0-835 (l-:O*COI47/)

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

Forbes.

'

0796

(,

-o.oo.870.

0-666(1-0-000230. 0-677 (I-0-002/)

.

.

0-6,9 (,+0-00070

.

.

.*,

.

.2-88(1+0-000040. 2-04 (1+0-00570 .

.

-

.

.

.

Lorenz. Kirchhoff.

.Mitche,,.

is to be noted, however, that in deducing his numbers from the data of experiment, Forbes did

247. It final

not allow for the

increase of specific heat with rise of

CONDUCTION OF HEAT.

xiv.]

^17

temperature ( 184). Angstrom unfortunately says expressly that this cause can affect the change of conductivity with temperature only to an inconsiderable extent, so that his numBut the specific bers also are not corrected for this cause.

heat of iron increases by about i per cent, for every 7 C., and the introduction of this consideration would reduce to about -|th of their amount the changes of conductivity given by Forbes, and they would then be (in a matter of such delicacy and difficulty) within the limits of experimental This change of specific heat has been taken into error. account by the other experimenters whose results are given above. Thus, it would appear that, though the order of the metals as heat conductors is practically the same as their

order

in

electric

conductivity,

sought by Forbes does not

exist.

the farther analogy of elec-

The diminution

conductivity by rise of temperature holds for all metals, and nearly to the same extent in all. On the contrary, the thermal conductivity seems (at least in the majority of

tric

metals) to improve with rise of temperature. But it has been shown that copper of good electric quality has higher thermal conductivity than that of bad electric quality.

The whole

subject, so far as experimental details

are concerned, is still in a very crude state, as may be judged from the preceding tables. Of course, a good deal of the discrepancy arises from the fact that the materials

operated on differed but also physically,

not only in chemical constitution, i.e.

having been

cast, rolled,

drawn,

annealed, &c. in

248. Angstrom's process is the mixed method referred to It consists in alternately heating and cooling, 241.

and to a fixed amount, one end of a bar, which need not be nearly so long as that employed in Forbes's method. This periodic change is steadily carried for fixed periods

HEAT.

2i8

[CHAP.

out until the indications of each of the thermometers at different parts of the bar

have also become

or at least practically so.

strictly periodic,

Fourier's method, applied

to

problem, shows that (at least if the conductivity and the conductivity specific heat do not vary with temperature) diminution of of from the rate can be calculated directly this

ranges of the successive thermometers, and the postponement of their dates of maximum temperature, per unit of length along the bar ; altogether independently of the rate of loss of heat by the surface (provided this loss be everywhere proportional to the excess of temperature over that of the

air).

The

details of the calculation

cannot be given

here, but a general idea of its nature will be gleaned from the solution of a more restricted problem to be given in the

next section, the mathematical part of which may be omitted by any reader without interruption of the continuity of our exposition.

Two

extremely important practical questions connected with this part of our subject are 249.

How

does the internal heat of the earth affect (by conduction) the temperature near the surface ? How far, and according to what law, do the (2.) fluctuations of surface temperature, from day to night or (i.)

from summer to winter, penetrate into the crust of the earth

?

The

second of these questions obviously involves a problem somewhat similar to that presented by Angstrom's method which we have just discussed. And both questions afford

us

very

simple

instances

of the

application

of

Fourier's beautiful method.

So long as we confine ourselves to the upper strata of the we may treat them as planes, the temperature throughout depending on the depth only, and the passage

earth's crust,

CONDUCTION OF HEAT.

xiv.J

of heat being therefore in parallel greatly simplifies the investigation.

219

(vertical) lines.

Let v be the temperature at a depth

x under

This

the earth's

Then

the temperature gradient is dvjdx. Thus the rate at which heat passes upwards through a horizontal surface.

area of one square foot, at depth x, is kdv/dx, where k is the conductivity at temperature v. For a depth x + So-

under the surface

this

k

becomes

d dx

The

+

dx

excess of the latter over the former denotes the rate at

which heat has been (on the whole) communicated to a slab of crust a square foot in surface and of thickness

The

rate at

which

its

temperature

S_v.

rises is dv-dt\

whence,

183) of the crust, we have as ( another expression for the rate at which heat is communicated to the slab, per square foot of surface, and thickif

c be the thermal capacity

ness 8x, ;

^

-

\.:'v

:

:.'.

.,,

Equating these independently obtained expressions for the heat, we have finally

same quantity of

dr

d

i

r (x

=R

=

)=(!--* f(*)=(l--* a

we have

).

-

at

a'.

Through the points P",

P'", &c., draw adiabatics meeting

Then

the

successive additional points /"', P'" heat units are taken in from P' to /"', /"' to

&c., as well as from

,

in

mark

i