Quantum Dynamical Correction for the Equation of State of Real Gases

56

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PROC. N. A. S.

University of Kansas, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The writer is indebted to Dr. D. McL. Purdy for suggestions and criticism. 1 Hecht, These PROCEEDINGS, 13, 1927 (569-574); J. Gen. Physiol., Baltimore, 11, 1928 (255-281). 2 Houstoun, R. A., Phil. Mag., London, 8, 1929 (520-529); Houstoun, R. A., and Shearer, J. F., Ibid., 10, 1930 (433-450). 3 Volkmann, A. W., Physio!ogische Untersuchungen im Gebiete der Optik, Leipzig, 1863.

QUANTUM DYNAMICAL CORRECTION FOR THE EQUATION OF STATE OF REAL GASES By HENRY MARGENAU SLOANE PHYSICS LABORATORY, YALE UNIVERSITY

Read before the Academy, Wednesday, November 18, 1931

Recent progress in the understanding of intermolecular forces has revived theoretical interest in the equation of state of real gases. Several investigations2 make it appear very probable that the forces which cause the deviations from the perfect gas law are due to the interactions of rapid electronic motions within the molecules; and calculations of the second virial coefficient B, based on these concepts, have produced reasonable agreement with experimental findings. It was only in the case of very light gases, such as H2 and He, that satisfactory agreement could not be obtained. Here the discrepancy proved to be such that for low temperatures the calculated values of B were much smaller (algebraically) than the observed data. A possible explanation of this difficulty was first pointed out by London,' who suggested that the existence of zero point energy associated with the vibratory motion of molecules in quantized collision states might render the attractive Van der Waals' forces partially ineffective. However, a detailed quantitative investigation of this phenomenon and its effects upon the equation of state has not yet been presented. Such an investigation will be the object of this paper. We shall begin by assuming-in accord with the results of papers listed under 1)-that the forces between any 2 molecules have a potential energy e which if plotted against the distance of separation r of the two molecules, yields a curve similar to figure 1. If now these molecules are regarded as mass points and their mutual potential energy E(r) is substituted in the Schroedinger equation, this will lead to solutions corresponding to discrete vibrational energy states El... E,, of negative total energy, as well as to continuous positive states. It is not difficult to show that the relative spacing of the discrete states,

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and the position of the lowest possible state E1 depend on the depth of the minimum of e and on the masses of the molecules. In particular, for light gases like H2 and He the lowest energy state lies very high, indeed not much below the 0-axis. Moreover, it is likely that for He there exists but this one vibrational state.3 This situation implies that all pairs of molecules bound together by their -potential energy, i.e., having a negative total energy, cannot exist in that part of phase space which corresponds to the shaded region unless they be in one of the states E1 ... .p while there is no limitation on the states of molecules which have a positive total energy. In order to obtain a correct equation of state it is necessary tomodify the usual calculations by excludng the inaccessible region of phase space, and considering instead the set of discrete levels El. . .E^.

FIGURE 1

The pressure of a gas may be obtained from the thermodynamical relation p = kT

log Z,

(1)

where Z is the complete partition function ("Zustandssumme" for the discrete states, phase integral for the continuous ones). If the gas cqnsists of N free mass points (e = 0)

(2) =h3N J.... -WkT d++N)2vd V for dv,. ..dVN5, dV for dxl... dzN. (2) is easily evaluated

Z

e

dv is written and leads to the law for ideal gases pV = NkT, if substituted in (1). If the molecules, instead of being free, are considered to have a mutual potential energy e(r) and the existence of quantized states is neglected, Z differs from (2) only by the appearance of the total potential energy of the gas in the exponent of the integrand; that is,

PHYSICS: H. MARGENA U

58

[2(v'

= M Nf. .J'k

+

*--

PROC. N. A. S.

DN) + E(ri-n) +... f(rN

-rN)1d,JV (3)

Here the integration extends over all velocity components from - co to + c, and over all space. By (ri - rj), the argument of e, we designate the relative distance between molecule i and molecule j. A calculation of (3) may be found in textbooks on Statistical Mechanics.4 The result is approximately

/(27rmkT 3N/2 VN(1 + t)N

Z =

2irN

c

e(r)

(e kT - l)r2dr. yields the usual equation of state: with

=

V

pV = NkT

1 -

-V

(4)

This expression, inserted in (1),

(e kT

-

l)r2dr}.

(5)

It is evident that this equation ignores the effects of quantized collisions. A better approximation to the actual state of affairs may be obtained as follows: We consider the gas as a mixture of n pairs of molecules in states of -mutual quantization and neglect their interaction with similar pairs and with all other molecules. N is the number of molecules in continuous energy states. The total partition function now consists of 3 parts: (1) the sum of state resulting from the vibratory motion of the

pairs (e-kT

(2) the phase integral due to the classical motion of F(47r-mkT)3/2V1

and (3) Zp, an translation of these pairs, which reduces to [ h3 integral like (3) but extended only over the accessible part of phase space.

(Ve

B,.i ) (4irmk~T)3/21f T

(6)

Substitution in (1) shows that the first part of (6) contributes nothing nkT to p, the second part only the partial pressure ,VX which may be neglected since, by ordinary statistical reasoning, n may be shown to be very much smaller than N at all temperatures of interest. The problem is then to evaluate Zp. The reasoning which leads from (3) to (4) involves the assumption that only "binary encounters" between the molecules can occur. This same restriction will here be imposed. The integrand of Zp may be transformed by the identity

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VOL. 18, 1932

59

m

2;a; e= 1

Htm(Ca-

1) +

,m-llm-i(e~ $1) + Em-21m-2(ei- 1) +

*

+ 1.

Hr indicates the product of r terms (ea_- 1) with different indices i and selected from the total number of m such terms, while 2 denotes the sum over all possible products (m) in number of r terms. If now we write e, for e(r, - rj), etc., allowing i to take on values from 1 to N(N - 1) mt 2 and W for the total kinetic energy of the N molecules,

ZP

= ()

Xf

kT f e {kT(e -l) +

Em-lm 1(e

***+le kT

-

kr-

1) + 1 }dvdV. (7)

The limits of integration p are defined as follows: If one molecule X is so far from all others that e(rx - r,,) practically vanishes, where j, may be any other molecule of the assembly, then the velocity components of X are to be integrated from- co to + 00; however, if e(r