The problem of the condensation of a gas is intimately related to the asymptotic behavior of the virial coefficients, Bm, as m→∞. The problem of the evaluation of the virial coefficients may be divided into two distinctly different ones. The first of these, which is purely combinatorial in nature and is independent of the intermolecular force law, is that of determining the number of a certain type of connected graphs of l points and k lines which are called ``stars.'' This problem is solved by means of generating functions, with the result that the total number of such stars is asymptotically equal to
(12l(l−1)k),
for almost all k. Arguments are also presented which indicate that the total number of topologically different stars is
1l!(12l(l−1)k).
With these results the combinatorial problem is essentially solved.
The second problem is that of evaluating certain integrals of functions which depend on the intermolecular potential. This problem is not so near to a solution. For a purely repulsive force, asymptotic expressions are obtained for k=l, and k=l+1. The partial contributions to the virial coefficient in these two cases are:
(−1)l·53(5)12(2b)l−1(l−1)l5/2,
and
(−1)l2·5324π3(2b)l−1,
respectively. Results for some simple one‐dimensional rigid lines are also given.
1.
We have especially in mind the theory of J. E. Mayer; for a summary see his book: Statistical Mechanics (John Wiley and Sons, Inc., New York, 1940), Chap. 13 Compare also
B.
Kahn
and
G. E.
Uhlenbeck
,
Physica
5
,
399
(
1938
);
B. Kahn, dissertation, University of Utrecht, 1938;
J. de Boer, dissertation, University of Amsterdam, 1940;
K.
Husimi
,
J. Chem. Phys.
18
,
686
(
1950
).
2.
Or better on the convergence of the series ∑blzb, where bl are the Mayer cluster integrals from which the virial coefficients follow. See Sec. II.
3.
J. G.
Kirkwood
and
E.
Monroe
,
J. Chem. Phys.
9
,
514
(
1941
);
Kirkwood
,
Maun
, and
Alder
,
J. Chem. Phys.
18
,
1040
(
1950
)
See also M. Born and H. S. Green, A General Kinetic Theory of Liquids (Cambridge University Press, Cambridge, 1949).
4.
For a criticism of this approximation with regard to the radial distribution function and the value of the fourth virial coefficient for a gas of elastic spheres see
B. R. A.
Nijboer
and
L.
van Hove
,
Phys. Rev.
85
,
777
(
1952
).
5.
Thanks to a communication of Professor G. Pólya the actual counting problems for finite l are almost completely Solved also. We will mention the results but omit the proofs, since the asymptotic behavior can be seen by more intuitive arguments. For a similar investigation of the number of different Feynman diagrams in various field theories, see
C. A.
Hurst
,
Proc. Roy. Soc. (London)
214
,
44
(
1952
);
also ,
R. J.
Riddell
, Jr.
,
Phys. Rev.
91
,
1243
(
1953
).
6.
For a rigorous proof of these statements, see
L.
van Hove
,
Physica
15
,
951
(
1949
).
7.
For the proof see B. Kahn and G. E. Uhlenbeck, reference 1.
8.
Therefore it is also often called a “cutting point.”
9.
For the general topological theory of linear graphs see the book of Koenig, Theorie der Endlichen und Unendlichen Graphen (Leipzig, 1936). Combinatorial questions are not discussed in this book. For these the basic reference is the paper of
G.
Pólya
,
Acta Math.
68
,
145
(
1938
).
10.
K.
Husimi
,
J. Chem. Phys.
18
,
682
(
1950
).
11.
A. Cayley, Collected Mathematical Papers (Cambridge 1889–1898) Vol. 13, p 26;
other proofs are given by Pólya, reference 9, and by
G.
Bol
,
Abhardl. Math. Seminar Hamburg
12
,
242
(
1938
).
12.
R.
Otter
,
Ann. Math
49
,
583
(
1948
).
13.
F.
Harary
and
G. E.
Uhlenbeck
,
Proc. Natl. Acad. U.S.
39
,
315
(
1953
).
14.
We assume the convention that d(1,0) = 0.
15.
See for instance Whittaker and Watson, Modern Analysis, Chap VII (Macmillan, New York, 1944). In order that Eq. (26) is valid for all l and k, we assume the following conventions: s(1,0) = 0,c(1,0) = 1;s(2,1) = c(2,1) = 1.
16.
We are greatly indebted to Professor G. Pólya for communicating his result to us. In this section we will omit the proofs, since they would take up too much space, and since we hope that they will appear elsewhere. They can be found in the dissertation of R. J. Riddell (Ann Arbor, 1951, p. 57.)
17.
We are indebted to Dr. P. Erdös for communicating to us various estimates for the range of k for which our asymptotic results are valid. With regard to Eq. (33) he thinks that k must lie in the range
k>(1+ε)l(logl)>12l(l−1)−k
, just as for Eqs. (18) and (27).
18.
For the proof, see R. J. Riddell, dissertation, University of Michigan, 1951, p. 65.
19.
The irreducible integrals in two dimensions have also been evaluated by Harris, Sells, and Guth (to be published).
20.
Using Eq. (13) one finds from Table I B5 = b4 in agreement with (42) For l = 4 the numbers s(l,k) are 3, 6, 1 and from (41) then follows B4 = b3, again in agreement with (42).
21.
The fact that for a single chain the integral (44) can always be reduced to a single integral by a Fourier transformation (since (44) is obtained by folding the function f(r) (n−1) times) was first noted by
E. W.
Montroll
and
J. E.
Mayer
[
J. Chem. Phys.
9
,
626
(
1941
)].Also Eq. (45) and the results for the ring and three chain integrals can be found essentially in this paper.
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