The Statistical Thermodynamics of Multicomponent Systems

Distribution functions, Fn(z, {n}), for multicomponent systems are defined proportional to the probability density that n molecules in an infinite isothermal system of fugacity set z will occupy the configurational coordinates symbolized by {n}. All thermodynamic functions may be obtained as certain sums of integrals of these distribution functions. These sums are always convergent, but impractically slow in convergence for numerical use without further transformation. In particular, the grand-partition function, exp [VP(z)/kT], may be expanded in a power series in the fugacities z with coefficients given by integrals of the distribution functions Fn(0, {n}) at the fugacity set 0. As has been previously demonstrated for one component systems, this is shown to be a special case of a more general relation permitting the calculation of the distribution functions (and therefore the thermodynamic functions) for one fugacity set from those at another set. The function —kT ln Fn(z, {n}) is the potential of average force, Wn(z, {n}), of n molecules at the fugacity set z; at zero fugacities, the potential of average force is simply the potential energy. When only short range forces are present, by employing the procedure previously used for the imperfect gas, the integrals of the distribution functions may be simplified to integrals of lower order which become the coefficients in such expressions as the power-series expansion of the pressure increment in terms of the fugacity increments. These series have the advantage of a much higher rate of convergence than those involving integrals of the distribution functions, themselves; however, they do not converge when the initial and final fugacity sets bridge a phase transition. The general equations are applied to the imperfect multicomponent gas, to isotopic gas mixtures, and to condensed systems with the variables converted to the usual activities and activity coefficients. A change of independent variable with a corresponding change in coefficients then permits the evaluation of the pressure increment in terms of the solute concentrations. The equations for osmotic pressure are developed and found to be entirely analogous to those for the pressure of an imperfect gas. Finally, the osmotic pressure in a binary system and its behavior near phase transitions are discussed with emphasis on the critical mixing phenomenon, for which the existence of a region of anomalous first-order transition is shown to be probable.