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On the Theory of the Virial Development of the Equation of State of Monoatomic Gases

dc.contributor.authorRiddell, R. J.en_US
dc.contributor.authorUhlenbeck, G. E.en_US
dc.date.accessioned2010-05-06T21:09:40Z
dc.date.available2010-05-06T21:09:40Z
dc.date.issued1953-11en_US
dc.identifier.citationRiddell, R. J.; Uhlenbeck, G. E. (1953). "On the Theory of the Virial Development of the Equation of State of Monoatomic Gases." The Journal of Chemical Physics 21(11): 2056-2064. <http://hdl.handle.net/2027.42/69794>en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/69794
dc.description.abstractThe 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 is1l!(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(52π)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.en_US
dc.format.extent3102 bytes
dc.format.extent637621 bytes
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dc.publisherThe American Institute of Physicsen_US
dc.rights© The American Institute of Physicsen_US
dc.titleOn the Theory of the Virial Development of the Equation of State of Monoatomic Gasesen_US
dc.typeArticleen_US
dc.subject.hlbsecondlevelPhysicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumH. M. Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michiganen_US
dc.contributor.affiliationotherDepartment of Physics and Radiation Laboratory, University of California, Berkeley, Californiaen_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/69794/2/JCPSA6-21-11-2056-1.pdf
dc.identifier.doi10.1063/1.1698742en_US
dc.identifier.sourceThe Journal of Chemical Physicsen_US
dc.identifier.citedreferenceWe 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).en_US
dc.identifier.citedreferenceOr better on the convergence of the series ∑blzb,∑blzb, where blbl are the Mayer cluster integrals from which the virial coefficients follow. See Sec. II.en_US
dc.identifier.citedreferenceJ. 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).en_US
dc.identifier.citedreferenceFor 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).en_US
dc.identifier.citedreferenceThanks 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).en_US
dc.identifier.citedreferenceFor a rigorous proof of these statements, see L. van Hove, Physica 15, 951 (1949).en_US
dc.identifier.citedreferenceFor the proof see B. Kahn and G. E. Uhlenbeck, reference 1.en_US
dc.identifier.citedreferenceTherefore it is also often called a “cutting point.”en_US
dc.identifier.citedreferenceFor 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).en_US
dc.identifier.citedreferenceK. Husimi, J. Chem. Phys. 18, 682 (1950).en_US
dc.identifier.citedreferenceA. 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).en_US
dc.identifier.citedreferenceR. Otter, Ann. Math 49, 583 (1948).en_US
dc.identifier.citedreferenceF. Harary and G. E. Uhlenbeck, Proc. Natl. Acad. U.S. 39, 315 (1953).en_US
dc.identifier.citedreferenceWe assume the convention that d(1,0) = 0.d(1,0) = 0.en_US
dc.identifier.citedreferenceSee 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,s(1,0) = 0, c(1,0) = 1;c(1,0) = 1; s(2,1) = c(2,1) = 1.s(2,1) = c(2,1) = 1.en_US
dc.identifier.citedreferenceWe 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.)en_US
dc.identifier.citedreferenceWe 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).en_US
dc.identifier.citedreferenceFor the proof, see R. J. Riddell, dissertation, University of Michigan, 1951, p. 65.en_US
dc.identifier.citedreferenceThe irreducible integrals in two dimensions have also been evaluated by Harris, Sells, and Guth (to be published).en_US
dc.identifier.citedreferenceUsing Eq. (13) one finds from Table I B5 = b4B5 = b4 in agreement with (42) For l = 4l = 4 the numbers s(l,k)s(l,k) are 3, 6, 1 and from (41) then follows B4 = b3,B4 = b3, again in agreement with (42).en_US
dc.identifier.citedreferenceThe 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)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.en_US
dc.owningcollnamePhysics, Department of


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