alt="equation"/>
Notice the similarity to the Helmholtz free energy; in that case we subtracted the TS term from the internal energy; in this case we subtracted the TS term from the enthalpy. The Gibbs free energy is the energy available for nonPV work (such as chemical work). It has two other important properties: its independent variables are T and P, generally the ones in which we are most interested in geochemistry, and it contains the entropy term (as does the Helmholtz free energy), and hence can be used as an indication of the direction in which spontaneous reactions will occur.
2.11.2.2 Gibbs free energy change in reactions
For a finite change at constant temperature, the Gibbs free energy change is:
(2.125)
The free energy change of formation,
(2.126)
Like other properties of state, the Gibbs free energy is additive. Therefore:
(2.127)
In other words, we can use Hess's law to calculate the free energy change of reaction. Values for
2.11.3 Criteria for equilibrium and spontaneity
The Gibbs free energy is perhaps the single most important thermodynamic variable in geochemistry because it provides this criterion for recognizing equilibrium. This criterion is:
Products and reactants are in equilibrium when their Gibbs free energies are equal.
Another important quality of the Gibbs free energy is closely related:
At fixed temperature and pressure, a chemical reaction will proceed in the direction of lower Gibbs free energy (i.e., ΔG r <0).
The reverse is also true: a reaction will not proceed if it produces an increase in the Gibbs free energy.
On an intuitive level, we can understand the Gibbs free energy as follows. We know that transformations tend to go in the direction of the lowest energy state (e.g., a ball rolls down hill). We have also learned that transformations go in the direction of increased entropy (if you drop a glass it breaks into pieces; if you drop the pieces they don't re-assemble into a glass). We must consider both the tendency for energy to decrease and the tendency for entropy to increase in order to predict the direction of a chemical reaction. This is what the Gibbs free energy does. Example 2.7 illustrates how Gibbs free energy of reaction is used to predict equilibrium.
2.11.4 Temperature and pressure dependence of the Gibbs free energy
One reason why the Gibbs free energy is useful is that its characteristic variables are temperature and pressure, which are the “external” variables of greatest interest in geochemistry. Since it is a state variable, we can deduce its temperature and pressure dependencies from eqn. 2.124, which are:
(2.128)
(2.129)
Example 2.7 Using Gibbs free energy to predict equilibrium
Using the thermodynamic data given in Table 2.2, calculate ΔGr for the reaction:
at 298 K and 0.1 MPa. Which mineral assemblage is more stable under these conditions (i.e., which side of the reaction is favored)? Which assemblage will be favored by increasing pressure? Why? Which side will be favored by increasing temperature? Why?
Answer: We can calculate ΔGr from ΔHf and ΔSf, values listed in Table 2.2:
ΔH is calculated as:
To find out which side will be favored by increasing pressure and temperature, we use equations 2.128 and 2.129 to see how ΔG will change. For temperature,
Equations 2.128 and 2.129 allow us to predict how the Gibbs free energy of reaction will change with changing temperature and pressure. Thus, we can predict how the direction of a reaction will change if we change temperature and pressure. To obtain the ΔGr at some temperature T' and pressure P' we integrate:
(2.130)
