Gibbs Free Energy

Gibbs Free Energy of Mixing

Recall that all spontaneous processes/reactions occur because of a decrease in Gibbs free energy. It should therefore not surprise you that the Gibbs free energy of mixing is always negative--otherwise mixing would not occur. The fact that _A < G°_A and _B < G°_B illustrates why compounds combine spontaneously--each compound is able to lower its free energy.
The above figure is hypothetical because we cannot measure or calculate the absolute Gibbs free energy of phases. For this reason, is always expressed as a difference from some standard state measurement, as , - °, or - G°.
The difference between the absolute Gibbs free energy G° per mole ° of a pure compound and the chemical potential per mole of dissolved compound is

• _A - G°_A = _A - °_A = RT lnX_A

This function has the following shape:
implying that when the mineral is pure (X = 1) then = 0, and when the mineral is infinitely dilute (X = 0) then the chemical potential is undefined. For example, in a two-component mineral if X_A = 0.4, at T = 298 K,

• _A - °A = 8.314 * 298 ln 0.4 = -2271 J
• _B - °B = 8.314 * 298 ln 0.6 = -1266 J

The equation of the G_mix line is the sum of the chemical potentials of the endmembers:

• G_mix = RT (X_A ln X_A + X_B ln X_B) or
• G_mix = RT (X_i ln X_i)

which looks like this for two components:
Actually, all this discussion has been predicated on the assumption that H_mix = 0. If this is not true, G_mix is not a simple function of composition, but has the general form:

Depending on the relative values of H_mix and -TS_mix, the free energy of mixing may be negative throughout the whole composition range if the entropic energy contribution outweighs the enthalpy increase; this is more likely at higher temperature.
The two free energy minima in the above figure indicate that minerals of intermediate compositions can reduce their free energy by unmixing into two phases. This explains the appearance and driving force for exsolution. Note that this can only be true if H_mix > 0, i.e., if 2_AB > _AA + _BB, which makes sense because it means that the A-B bonds have a higher free energy than the sum of the free energies of separate AA and BB bonds.