Thermodynamic Fluctuation Theory for Multicomponent Systems

In a multicomponent system the mean square value of the fluctuation in refractive index (zl )2, which determines the total turbidity, is a function of the concentrations, and refractive index increments, of each of the components of the system. It also involves cross terms, involving the correlation between the fluctuations of the concentrations of different components. For a given pair of components, i and /, the cross term is zero if the chemical potential of i is unaffected by a variation of the mass of j in the system but if the chemical potentials are not independent in this manner the cross terms do not vanish. 

The fundamental conceptions involved in the application of fluctuation theory to such systems were first advanced by Zernike (1915, 1918). Recently they have been further developed independently by workers in three different laboratories namely, Brinkman and Hermans (1949), Kirkwood and Goldberg (1950), and Stockmayer (1950). The fundamental equations of all these authors lead to the same results. 

The general equation, for the turbidity, t, of systems at constant pressure and temperature, may be written 

The summation is tdken over all but one of the components. Generally it is most convenient to omit the solvent from the summation to compensate for this omission the complete equation should include a term for the turbidity of the pure solvent, arising from density fluctuations in it. This term is generally small, in systems containing large molecules, and for brevity it is omitted from (27). However, in practice, we have generally determined the turbidity of the pure solvent and subtracted it from that of the solution. IP, denotes the molar refractive increment of component i that is An per mol of solute per liter of solution (or per kg. solvent). The terms in the determinant ,j denote the coefficients 

Our equation (27) differs from Stockmayer s (1950) in that he defined the terms as dfj, i/Sm, where /r denotes chemical potential whereas we have employed activities. Thus, his expression differs from ours by a factor RT in the denominator. This difference has been taken care of in the formulation given here, so that our working equations are essentially identical with those of Stockmayer. Also the volume factor V, appearing in Stockmayer s equations, does not appear explicitly in equation (27), since the concentrations are here expressed in volume units. 

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