Expressions for the chemical potentials of the components of gas mixtures and liquid solutions are obtained in terms of relatively simple integrals in the configuration spaces of molecular pairs. The molecular pair distribution functions appearing in these integrals are investigated in some detail, in their dependence upon the composition and density of the fluid. The equation of state of a real gas mixture is discussed, and an approximate molecular pair distribution function, typical of dense fluids, is calculated. Applications of the method to the theory of solutions will be the subject of a later article.

1.
E.
Wigner
,
Phys. Rev.
40
,
749
(
1932
);
Crossref
Search ADS
 
Uhlenbeck
 and
Gropper
,
Phys. Rev.
41
,
79
(
1932
);
Crossref
Search ADS
 
J. G.
Kirkwood
,
Phys. Rev.
44
,
31
(
1933
);
Crossref
Search ADS
 
J. G.
Kirkwood
,
J. Chem. Phys.
1
,
597
(
1933
).
Crossref
Search ADS
 
2.
L.
Onsager
,
Chem. Rev.
13
,
73
(
1933
);
Crossref
Search ADS
 
J. W. Gibbs, Elementary Principles in Statistical Mechanics, Collected‐Works II, Chap. VII (Longmans).
3.
We postulate the absence of coexistent phases containing any of the components of the fluid, since we wish to represent the system by a petit ensemble, each example of which contains the same number of molecules. Actually this is a trivial restriction, for Gibbs has shown that although an open system is represented by a grand ensemble, its equilibrium properties do not differ sensibly from those of a closed system represented by a petit ensemble, each example of which is made up of the average number of molecules of a system of the grand ensemble. In case coexistent phases exist, we simply regard υ as the volume of the homogeneous part in which we are interested.
4.
J. W. Gibbs, Elementary Principles in Statistical Mechanics, Collected Works II, p. 187, Longmans.
5.
It is to be remarked that the potential of Eq. (6) in no way excludes the simultaneous interaction of groups of more than two molecules. It simply states that in such a group, the mutual potential energy of any pair is independent of the presence of the other molecules. For example the electrostatic energy of a system of point charges is of this type. London has shown that the potential of the attractive van der Waals forces between chemically saturated molecules satisfies the requirements of Eq. (6). [
F.
London
,
Zeits. f. Physik. Chemie
11
,
222
(
1930
).] No such clear cut statement can be made regarding the repulsive forces, which operate at small distances, determining the effective collision diameters of molecules. However, it is plausible to assume that the potentials of these forces are approximately additive, and even if they are not exactly so, it is only necessary for our purposes that their narrow range of action is not sensibly displaced. In general, we should expect terms depending upon the relative coordinates of three or more molecules to be important only if the total intermolecular field is strong enough to produce a large perturbation in the electronic structure of a molecule in the fluid. Theoretical considerations suggest that not only in gases but also in liquids, this is not the case. Empirical arguments against any appreciable distortion of the electronic structure of molecules in most liquids are furnished by spectroscopic and optical evidence. For example, the molecular refractivity of most liquids is substantially the same as that of their vapor. There are to be sure some types of intermolecular potential which do not satisfy Eq. (6), for example those of chemical forces of the homeopolar type. Another example is furnished by the energy of polarization of a molecule by a group of ions.
6.
For a discussion of this point, see
L.
Onsager
,
Chem. Rev.
13
,
73
(
1933
).
Crossref
Search ADS
 
7.
Zernicke
 and
Prins
,
Zeits. f. Physik
41
,
184
(
1927
);
Crossref
Search ADS
 
Debye
 and
Mencke
,
Physik. Zeits.
31
,
797
(
1930
).
8.
Ursell
,
Proc. Camb. Phil. Soc.
23
,
685
(
1927
);
Crossref
Search ADS
 
R. H. Fowler, Statistical Mechanics, Cambridge University Press (1929), p. 173.
9.
After integration with respect to λi, the terms in μi involving 1/υ2 have the form
∑ k,s = 1Nk≠s≠i1υ2σkσs∫∫(1−e−βVks){VikVik+Vis(e−β(Vik+Vis)−1)−(e−βVik−1)}dωkdωs
, where the original summation indices, referring to the individual molecules, have been restored for clearness. For every term in the sum with the integrand
{VikVik+Vis(e−β(Vik+Vis)−1)−(e−βVik−1)}(1−e−βVks)
, corresponding to any specified pair of molecules k and s, there is also a term involving the same pair, having the form
{VisVis+Vik(e−β(Vik+Vis)−1)−(e−βVis−1)}(1−e−βVks)
. The sum of this pair of terms is readily seen to be
(1−e−βVik)(1−e−βVis)(1−e−βVks)
so that the double sum may be rearranged as follows:
12 ∑ k,s = 1Nk≠s≠i1υ2σkσs∫∫(1−e−βVks)(1−e−βVis)(1−e−βVik)dωkdωs
. This sum contains (N−1)(N−2) terms of similar magnitude. It differs from the sum over molecular types appearing in Eq. (55) by (3N−2) terms of the same magnitude. The aggregate of these terms bears a ratio of the order of 1/N to either sum and is consequently entirely negligible.
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© 1935 American Institute of Physics.
1935
American Institute of Physics
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