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A General Kinetic Theory of Liquids. VI. The Equation of State

A. E. Rodriguez

1949
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Proceedings of the Royal Society A
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73 that some boundaries etch very much more deeply than others, although no general relationship has been found between this property and the orientation; similar observations have been reported in connexion with other metals (Forsyth, King, Metcalfe & Chalmers 1946), from which it was concluded that the amount of precipitation and the extent of thermal etching are influenced by the relative orientations of the two crystals and the boundary. The present paper is concerned with the study of an
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... h the study of an approximate equation of state, first derived by Green, which covers both phases, liquid and gas, and can be used for numerical evaluation. It contains explicitly the roots of a certain transcendental equation. The differ ence between the liquid and the gas corresponds to the existence or non-existence of real roots of this equation. The solution of the transcendental equation has been reduced, by using a suitable expansion, to the solution of an algebraic equation. In this way explicit expressions for the equation of state are obtained which, however, are too involved to be generally discussed. The theory can be applied to argon using the Lennard-J ones potential for the interaction between argon atoms, and the results are shown in diagrams. The character of the singularity separating liquid and gas can be seen from these diagrams to lie at the lowest point of the isotherm in the unstable region. An estimate of the position of the critical point is made and found in fair agreement with the experimental data. I n t r o d u c t io n In the first parts of this series (Born & Green 1946; Green 1947; Born & Green 1947 a, 6) a satisfactorily general kinetic theory of liquids capable of describing the dynamical as well as the equilibrium properties of a statistical assembly of molecules has been developed. In the second of these papers (Green 1947) some attention has been devoted to the problem of condensation, and a new equation of state was obtained, valid for both phases, liquid and gas. The present work is particularly concerned with the study of that equation of state, with a view to adapting it for numerical evaluation. In the course of this investigation it has been found that some thermodynamical expressions, including that of the equation of state, as given in the original paper (Green 1947)are n°f quite correct. A new equation, which keeps the main features of the old one, has now been obtained, and is given in § 3. The final expressions, giving the internal and the free energy, and the equation of state itself, involved the solution of some integro-differential equations, in order to solve which several assumptions were made. It has been considered important to include a section discussing the extent to which these assumptions may influence the final range of validity of the thermodynamical expressions. Most of these results are contained in the paper by Green, mentioned above. Fuchs 1938; Kahn 1938) is confirmed by the present theory to the extent that a divergence of the cluster series is found, which is intimately associated with the process of condensation. On the exact point at which this divergence occurs detailed calculations do not confirm Mayer's contention that it happens at the density of the saturated vapour. The isotherm is continued into the metastable region above this density, rising to a maximum and falling to a subsequent minimum as the density is increased, much in the way foreshadowed by van der Waals over fifty years ago. The point of divergence is found to separate the metastable states of the liquid from those of the gas. This point, which from our standpoint may be considered as a sort of branch point, is identified with that at which one of the complex roots of the transcendental equation becomes real. In this way the thermodynamical properties are regulated through the nature of the roots of this equation. The equation of state here obtained is, of course, approximate; but it covers both phases, liquid and gas, this being the main feature of the theory. The difference between the liquid and the gas corresponds to the existence or non-existence of real roots of a certain transcendental equation. This equation, which in the original form contains an integral in the left-hand side, has now been reduced, by using a suitable expansion, to an algebraic, rapidly convergent series. In this form it is easy to formulate its analytic continuation without falling into ambiguities of any kind. The expansion also provides the way of finding the complex and real roots of the transcendental equation. The theory of condensation developed by Mayer (1937) and others (Born & The main object of the present paper is connected with the solution of this trans cendental equation. As has been said, the author has succeeded in reducing the problem to the solution of an algebraic equation which is of 2 + 1th degree, 4-1 being the number of terms used in the approximation. One of the sections is entirely devoted to a detailed exposition of the method in the most general form. In § 6, the equation of state is given in terms of an expansion, to show how the method permits the actual evaluation of some integrals involving the complex roots of the transcendental equation. 74 A. E. Rodriguez J 0 J -s -77% f j (s2 -t2)(t + r)a(t + r)dta'(s){l+f(s)}ds, (2-2) J 0 J -s where a(r) = e_^r)/fcr -1. (2*3) The left-hand side of (2-2) follows rigorously, the right-hand side is only approxi mate, since cubes and higher powers of/(r) have been neglected. This seems to be very reasonable. In fact, it may be inferred from experimental data that, even for rather low temperatures and high densities, the maximum absolute value of f(r) is of the order of 0*4, which implies that one is here neglecting terms that amount to 1% of the total value. * The proof is due to A. A. Sabry of th e D ep artm en t of M athem atical Physics of E dinburgh U niversity, whom I have to th a n k for kindly inform ing m e ab o u t it.

doi:10.1098/rspa.1949.0015
fatcat:6ujylsgswrhirazygfkqjvy3vy