The Activity Coefficient

The Activity Coefficient
While we have defined fugacity coefficients of individual species in a liquid solution by 6.153, we may define yet another parameter called activity coefficient in order to describe the non-ideality of a liquid solution, especially at low to moderate system pressure.

We have by eqn. 6.155: 
And from eqn. 6.158:
Using the above equations:   
The left side of this equation is the partial excess Gibbs energy ; the dimensionless ratio appearing on the right is the activity coefficient of species i in solution, represented by the symbol  Thus, by definition:

On comparing the last two equations we conclude that γi is a partial molar property with respect to    Thus, we have:
It follows from the definition of activity coefficient  that for an ideal solution its value is unity for all species as GE = 0. For a non-ideal solution, however, it may be either greater or less than unity, the larger the departure from unity the greater the non-ideality of the solution. The derivatives of the activity coefficient with respect to pressure and temperature can be correlated to the partial molar excess volume and enthalpy respectively.

As we have seen in the last section that the value of the function  is relatively larger compared to 
one may conclude that the activity coefficients are far more sensitive to changes in temperature than to changes in pressure. For this reason for phase equilibria computations at low to moderate pressures, the activity coefficients are assumed invariant with respect to pressure.

Since the activity coefficients are partial molar properties, they are related by the Gibbs Duhem equation (at constant temperature and pressure) as follows:

The above equation may be used to validate or check the consistency of experimental data on isothermal activity coefficients for a binary system. The following equation may be derived from eqn. 6.177 for this purpose (by assuming negligible effect of pressure on the liquid phase properties):

Thus, if the function ln (γ1/ γ2) is plotted over the entire range of x 1, (fig. 6.8) the two areas above and below the x-axis in the resulting curve must add up to zero, if the activity coefficients are consistent.  Representative values and the
Figure 6.8 Thermodynamic consistency tests for activity coefficients in binary mixtures.
nature of variation in the magnitude of activity coefficients is shown in fig. 6.9; they correspond to the same systems for which excess property variations were depicted in fig. 6.5.