Basic Statistical Mechanics

Abstract

In this chapter we introduce distribution functions for molecular momenta and positions. All equilibrium properties of the system can be calculated if both the intermolecular potential energy and the distribution functions are known. Throughout, we shall make use of the ‘rigid molecule’ and classical approximations. In the rigid molecule approximation the system intermolecular potential energy u(rNωN ) depends only on the positions of the centres of mass rN ≡ r1 . . . rN for the N molecules and on their molecular orientations ωN ≡ ω1 . . . ωN; any dependence on vibrational or internal rotational coordinates is neglected. In the classical approximation the translational and rotational motions of the molecules are assumed to be classical. These assumptions should be quite realistic for many fluids composed of simple molecules, e.g. N2 , CO, CO2 , SO2 CF4 , etc. They are discussed in detail in §§ 1.2.1 and 1.2.2; quantum corrections to the partition function are discussed in §§ 1.2.2 and 6.9, and in Appendix 3D. In considering fluids in equilibrium we can distinguish three principal cases: (a) isotropic, homogeneous fluids (e.g. liquid or compressed gas states of N2 , O2 , etc. in the absence of an external field), (b) anisotropic, homogeneous fluids (e.g. a polyatomic fluid in the presence of a uniform electric field, nematic liquid crystals), and (c) inhomogeneous fluids (e.g. the interfacial region). These fluid states have been listed in order of increasing complexity; thus, more independent variables are involved in cases (b) and (c), and consequently the evaluation of the necessary distribution functions is more difficult. For molecular fluids it is convenient to introduce several types of distribution functions, correlation functions, and related quantities: (a) The angular pair correlation function g(r1r2 ω1 ω2). This gives complete information about the pair of molecules, and arises in expressions for the equilibrium properties for a general potential.