Mixing Helmholtz energy change

An important excess property is the excess Gibbs energy GE. Many models have been developed to describe and predict GE from the properties of the molecules in the mixture and their mutual interactions. GE models often refer to the condensed state, the solid and liquid phases. In case significant changes in the volume take place upon mixing, or separation, the Helmholtz energy A, defined as... 

Note that the term in brackets is not the change in Helmholtz energy on mixing A, because the terms that define A must be at a common pressure, but the terms in (E4.1.5) are at a common extensive volume. Substituting the ideal-gas expression (4.1.45) for A j in (4.1.49), we obtain... 

There is another process in which mixing is involved and in which both the Helmholtz energy and the entropy do change. Consider process II in Fig. 2.2. We start with two pure components A and B as in the previous experiment. We let the two components mix in a volume 2 F, instead of F as in the previous experiment. This can be achieved by removing a partition separating the two compartments. 

Now, as documented below in Section 1.12, A = A T,V,n is supposed to involve temperature, volume, and composition as the control variables for the Helmholtz energy, whereas Eq. (1.10.3a) involves a mix of quantities. To achieve the desired change, we express the entropy in terms of the appropriate control variables by setting S = S T,V,rij). This leads to the appropriate differential form... 

If mixing is carried out at constant T and V, the analogous criterion for miscilibity is that the change in Helmholtz free energy, AA ... 

The mean-field lattice gas model (MFLG) represents a pure component by a lattice, the sites of which are randomly occupied by i moles of molecules. The molecules are allowed to occupy rrii lattice sites each the volume per lattice site, vq, is kept constant. The total number of sites equals q- - iWi(= where 0 is the amount of vacant sites in moles. Pressure and temperature changes affect the density of the system and can be dealt with by appropriate variations of Hq. In one of the more elaborate versions of the model (see, e.g., ref [53]), the Helmholtz free energy, AA, of mixing Hq vacant sites with i moles of molecules is... 

In an analogous manner, the changes in the internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy upon ideal mixing are found to be as follows ... 

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