Fluctuation Theory of Binary Solutions

Isothermal-isobaric binary solutions are by far the most common system of interest for FST. Since there are only two components, the number of simultaneous equations and the number terms in these equations are minimal, which means that matrices are unnecessary. Here, we derive the FST-based equations for binary mixtures as a simple illustration of the general approach. If one takes the derivative of Equation 1.48 (i = 2) with respect to the natural log of the number density of component 2 with p and T constant, then one immediately finds 

the derivative of the solute chemical potential (or activity) with respect to solute concentration can be expressed in terms of a combination of number densities and particle number fluctuations or KBIs. The ability to express thermodynamic properties in terms of KBIs is the major strength of FST. This has been achieved without approximation and the relationship holds for any stable binary solution at any composition involving any type of components. Derivatives of other chemical potentials can be obtained by application of the GD equation, or by a simple interchange of indices. The same approach can be applied to the second expression in Equation 1.48, with a subsequent application of Equation 1.27, to provide chemical potential derivatives with respect to other concentration scales. 

The most appropriate derivative of interest will depend on the ultimate application. Cabezas and O Connell (1993) show connections among derivatives with different properties held constant. 

After expressions for the chemical potential derivatives have been obtained, one can use them to determine corresponding expressions for the partial molar volumes and isothermal compressibility. Using Equation 1.50 in Equation 1.47 with i = k = 2 followed by some rearrangement using the GD expression provides 

The solvent partial molar volume is then provided by a simple index change. Using both partial molar volumes in Equation 1.46 and rearranging provides 

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