On the theory of grand partition functions

Abstract

ABSTRACTThe grand partition function is derived by averaging over all partitions and numbers of molecules of a grand ensemble. The statistical parameters that arise are interpreted thermodynamically. A formula for the mean fluctuation of the numbers of molecules is deduced, and it is shown that the latter can become very great for singularities of the partial potentials. In that case, the system becomes statistically indeterminate, and separates out into several phases. Conversely, in the statistics of actual systems, the search for singularities, at which the partial potentials become independent of the composition, leads to the conditions under which a phase-change takes place. It is shown that, on certain hypotheses, the grand partition function is the generating function of the ordinary partition function, and that the latter can always be represented in product-form. The generating function of a factor in this product-form is called a reduced grand partition function. The significance of its parameters is discussed, and thoir use in the statistical thermodynamics of liquid mixtures is reviewed.