The Chemical Potential and the Radial Distribution Function: General Formal Introduction of the Activity and of the Activity Coefficient

Abstract

The chemical potential μ plays a central part in the realms of physics, chemistry, and even biochemistry. It is related to the activity a of the species that it characterizes through a mathematical logarithmic relation. The latter can be formally written under only one kind of mathematical expression, whichever the type of activity is considered.

It is a well-known fact that, while relating the chemical potential of a perfect gas to molecular parameters to its number density is not endowed with any problem, it is not the case as soon as there exist interactions between the particles. In this case, the problem becomes, even, immensely complicated to solve exactly. This chapter mentions the setting up of general, but approximate, expressions, of the chemical potential of the components of a system, when such interactions exist. The first one links a decreasing exponential of the studied chemical potential to the difference of two other exponentials involving Helmholtz’ energies of the system. It is obtained within the framework of the canonical ensemble. The second relation is obtained from the previous one through the using of the pairwise additivity hypothesis. It is very interesting since it takes the form of the relation expressing the chemical potential of a perfect gas, but does possess a supplementary term. The latter only takes into account the mutual interactions of the particles and, hence, must be related to an activity coefficient. Finally, the chapter also mentions the setting up of theoretical relations between the chemical potential and the radial distribution function.