Liquid Vapour Free Energies

Question

The free energy of a component of a liquid solution is equal to its free energy in the equilibrium vapour.

Partial molal free energies let us deal with the free energy of the components of a solution. We use these free energies, or simpler concentration terms to which they correspond, when we deal with a variety of solution equilibrium matters. Here we begin by seeing how the partial molal free energy of a component of a liquid solution can be deduced.

We cannot count on the assuming of ideal behavior when we deal with liquid solutions. The components interact with one another and generally produce free energy effects characteristic of the particular system. Thus, liquid mixtures contrast with gas mixtures for which the ideal solution results are often satisfactory. The strategy in dealing with liquid systems is to relate the free energies of the components to those of the more easily treated equilibrium vapour.

Consider a binary system that can consist of a liquid, a vapour, or a liquid and vapour in equilibrium with one another. In view of the relation illustrated the free energy of the entire system, with superscript l for liquid and v for vapour, can be expressed as:

G = n^l_A Gv_A + nlB G^l_B + n_A + n_B Gv_A

For this binary system 

n^l_A = nvA = nA    and     n^l_B + nvB = nB

Or

n^l_A = n_A - nv_A    and    n^l_B = n_B - nv_B

For equilibrium between the liquid and vapour, the free energy will be a minimum with respect to the fraction, or amount of the components in the vapour phase. We can form d/GdnnA and dG/dnvB and set these derivatives equal to zero to obtain

G^l_A = Gv_A    and    G^l_B = Gv_B

The partial molal free energy of a component in a liquid solution is equal to its partial molal free energy in the equilibrium vapour. This result can be used to relate the partial molal free energies of components in liquid solutions to be partial molal free energies of the components in the equilibrium vapour.

Example: the vapor pressure of benzene and toluene over benzene toluene solutions are shown as plotted points. What do these vapor pressures tell us about the benzene-toluene solutions?

Solution: the vapor pressures of the components are very nearly proportional to the mole fractions of the components. With the subscript B for benzene and T for toluene, this behavior can be described by the equations:

P_B = x_BP°_B and P_T = x_TP°_T

Or, P_B/P°_A = x_B and P_T/P°_T = x_T

When these relations are used, we obtain:

G^l_B = G°_B + RT In x_B and G^l_T = G°_T + RT In x_T

This is the component free-energy behavior that, according to characterizes ideal behavior.

Also the volume of a benzene-toluene solution is very nearly equal to the sum of the volumes of the separate components, and no appreciable enthalpy change accompanies the mixing process. Liquid benzene-toluene solutions confirm closely to ideal-solution behavior.