0; thus, there is no enthalpy change associated with a pressure change for an ideal gas. This is in accord with assumptions about an ideal gas: namely, that there are no forces between molecules, hence no energy is stored as potential energy of attraction between molecules.
The isothermal pressure dependence of entropy is given by eqn. 2.106. We substitute 1/T for α and RT/P for V and integrate from P1 to P2:
The whole enthalpy and entropy changes are the sum of the changes in these three steps:
2.11 FREE ENERGY
We can now introduce two free energy functions, the Helmholtz free energy and Gibbs free energy. Gibbs free energy is one of the most useful functions in thermodynamics.
2.11.1 Helmholtz free energy
We can rearrange eqn. 2.58 to read dU − TdS = − PdV. The −PdV term is the work term and the TdS term is the heat function. TdS is the energy unavailable for work. Therefore, dU − TdS is the amount of internal energy available for work, or the free energy. We define it as A, the Helmholtz free energy:
(2.119)
As usual, we are interested in the differential form (since we are more interested in changes than in absolutes):
(2.120)
or substituting eqn. 2.58 into 2.120:
(2.121)
2.11.2 Gibbs free energy
2.11.2.1 Derivation
The Gibbs free energy is perhaps misnamed. By analogy to the Helmholtz free energy, it should be called the free enthalpy (but enthalpy is an energy), because it is derived as follows:
(2.122)
and
(2.123)
or
which reduces to:
(2.124)
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