On the Statistical Mechanics of Change of State

Abstract

The nature of the pressure against density curve at constant temperature is investigated by the methods of the grand canonical ensemble for a system of identical molecules between which only short range forces act. It is shown that, except for surface effects, the total energy, entropy, etc., are proportional to the number of molecules in the system. The pressure is proven to be a non-decreasing function of the density which may have regions of constancy as a function of the density. This shows rather clearly the impossibility for any analytic equation of the Van der Waals* type being able to give a satisfactory account of condensation without ad hoc assumptions. I T should be possible to understand the isothermal change of a vapor into a liquid in terms of the forces acting between the molecules. Since this change takes place for nearly every system of molecules, one would expect that the detailed nature of the forces is not important. Only such general considerations should enter, as whether the forces are attractive or repulsive, and their behavior for large distances of separation of the molecules.We shall show for a system of molecules of one species between which only short range forces act that: (a) the pressure is a monotonically increasing function of the density at constant temperature, although there may be regions in which the pressure is independent of the density; (b) the energy, entropy, free energy, etc., are approximately proportional to the number of molecules present. It is only in the asymptotic approximation for a large number of molecules and where the dimensions of the space containing the molecules are large compared to the range of the molecular forces that (a) and (b) hold. The neglected terms may be regarded as giving rise to surface effects. It should not be implied that short range forces are necessary for (a) and (b), although we have been able to show that (a) and (b) are true only for this case.The grand canonical ensemble will be used for the statistical description of the system because of its elegance and convenience. We will furthermore consider the system to obey the laws of classical mechanics. This has been shown by Kahn and Uhlenbeck 1 not to be an essential restriction. The appearance of Planck's constant in our equation is for dimensional purposes. It has no quantum mechanical implications.The partition function of a classical grand canonical ensemble with generic phases isIn terms of the function, \f/> the pressure, average number of particles in the system, and the mean square fluctuations in the number of particles are:<[iV-(i\0 Av ]%= -fid**/da*.Equations (2) and (3) do not insure the existence of an equation of state. For a general \p the pressure is not necessarily a function of the temperature and particle density only. To investigate the conditions under which an equation of state exists we write 0 = ln V. (5) 1 B. Kahn and G. E. Uhlenbeck, Physica 5, 399 (1938). 63