A Statistical Expression of the Activity of a Species: A Relation Between It and the Corresponding Concentration in the Case of an Imperfect Gas

Abstract

This chapter brings some elements of answer to one of the most significant questions regarding the notion of activity, i.e., how are, mathematically related to each other, the value of an activity and that of the corresponding “concentration” of a species when the latter, no longer, tends toward zero? Recall, indeed, that G.N. Lewis, when he introduced it, defined the notion of activity by the following sentence:

a quantity which is “an active density number which bears the same relation to the chemical potential μ at any density that N/V does as N → 0.”

The results mentioned in this chapter constitute a first mark of the fact that statistical thermodynamics permits, at least in part, to answer the question. The content of this chapter shows that the setting up of the expression relating the activity of a gas to its corresponding concentration stems from a reasoning which, at the onset, requires the definition of the activity in terms of statistical parameters. It also shows that the obtained relation involves terms which are related to the virial coefficients. According to the theory, an activity z of a compound can be identified to the product of its absolute activity λ and of the second canonical function of the grand ensemble Q ₁(N, V, T) (that is to say that corresponding to the presence of only one particle in the system), product divided by the volume V of the system. The relation also shows that z exhibits all the properties of Lewis’ activity. It has the form of a series development of z in density ρ, the coefficients of which can be, in principle, calculated from the experimental values of the virial relation.