Activity coefficients and solution standard states

In this section, we derive PDT expressions for activity coefficients and standard state chemical potentials that are conventional in physical chemistry and chemical engineering thermodynamics. We assume here a single homogeneous solution phase composed of several components, and write the following conventional expression for the chemical potential of component a in this multicomponent solution  

Recognizing that = x p where p is the total number density of all molecules in the volume V, we can rewrite Eq. (3.56) as 

The first term on the right side of this equation must be equal to j8p-° +ln7o from Eq. (3.55). Since p,° and are defined with respect to one another, we can multiply and divide the term in brackets by a factor that defines the standard state, and then separate the numerator and denominator into factors j8p,° and In Our first choice for a standard state is the pure fluid, and therefore, the factor of interest is the average Boltzmann factor for coupling a distinguished molecule to its pure fluid  

Multiplying and dividing by this factor appropriately in Eq. (3.57) gives 

The Boltzmann factor in the denominator of this equation corresponds to coupling a distinguished molecule of component a to the solution. This result is reminiscent of local composition free energy models that are widely used to calculate fluid-phase equilibria for multicomponent mixtures of nonelectrolytes. We note that 1 corresponds to less favorable interactions in the mixtures, and 1 as 1. 

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