Derivation of the potential distribution theorem

5 Once you have the chemical potential for the van der Waals model of the previous exercise, find the equation of state by integrating the Gibbs-Duhem relation. Compare your result with the equation of state obtained from the approximate partition function using 

Consider a macroscopic solution consisting of the solute molecules of interest and other species. Focus attention on a macroscopic subsystem of this solution. We will describe this subsystem on the basis of the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, volume, and the chemical potentials or, equivalently, absolute activities for all solution species. The species of interest will be identified with a subscript index a, so the average number of solute molecules in this subsystem is 

The average displayed in Eq. (3.14) is particularly relevant to our argument here. The summand factor n annuls terms with = 0 and permits the sum to start with = 1. The latter feature means that the overall result will involve an explicit leading factor of We are then motivated to examine the coefficient multiplying Of course, a determination of establishes the thermodynamic property [Xa that we seek. To that end, we bring forward the explicit extra factor 

Notice that we haven t written the of Eq. (3.14) explicitly in the summand of Eq. (3.15). It has been absorbed in the , but its presence is reflected in the fact that the population is enhanced by one in the numerator that appears in the summand, rii. indicates the population after one molecule of 

We ve positioned this factor out front in writing Eq. (3.15) so that the leading = 0 term in the sum will be one (1) as suggested by the usual partition function 

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