Calculation of Liquid-Phase Activity Coefficients

The composition of the liquid phase can have a profound effect on the K value for a given component of a mixture. In fact, experimental evidence shows that large variations in the ICs result from changes in the liquid compositions at pressures so low that the vapor not only forms an ideal solution but may also behave as a perfect gas mixture. Thus, for many systems, Eq. (14-37) reduces to 

In spite of the simplicity of this relationship, the expressions for computing the activity coefficients for multicomponent mixtures are quite cumbersome. Since the equations of Wohl,76 Van Laar,68 Margules,43 Scatchard and Hamer,57 Redlich and Kister,52 and others are well documented in the literature, a restatement of these equations is not presented. Instead, a brief introduction to the newer methods including the Wilson equation, the NRTL, UNIQUAC, ASOG, and the UNIFAC methods is presented. 

There follows a presentation of certain fundamental relationships needed in the prediction of activity coefficients for the liquid phase. 

Thermodynamic Relationships for Fugacities and Activity Coefficients for the Liquid Phase 

This section is divided into three parts. First, condensable components and then noncondensable components are considered. Then the use of the excess free energy function in the calculation of the activity coefficient is presented. 

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A distillation column is separating 100 mol/s of a 30 mol% acetone, 70 mol% methanol mixture at atmospheric pressure. The feed enters as a saturated liquid. The column has a total condenser and a partial reboiler. We desire a distillate with an acetone content of 72 mol%, and a bottoms product with 99.9 mol% methanol. A reflux ratio of 1.25 the minimum will be used. Calculate the number of ideal stages required and the optimum feed location. VLE for this system is described by the modified Raoult s law, with the NRTL equation for calculation of liquid-phase activity coefficients, and the Antoine equation for estimation of the vapor pressures. 

In an attempt to place calculations of liquid-phase activity coefficients on a simpler, yet more theoretical basis, Abrams and Prausnitz used statistical mechanics to derive a new expression for excess free energy. Their model, called UNIQUAC (universal qua si-chemical), generalizes a previous analysis by Guggenheim and extends it to mixtures of molecules that differ appreciably in size and shape. As in the Wilson and NRTL equations, local concentrations are used. However, rather than local volume fractions or local mole fractions, UNIQUAC uses the local area fraction 0,j as the primary concentration variable. 

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