Overview

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs. This new property is called the Gibbs free energy (G) (or simply the free energy), and it is defined in terms of a system’s enthalpy and entropy as the following:

Free energy is a state function, and at constant temperature and pressure, the free energy change (ΔG) may be expressed as the following:

The relationship between this system property and the spontaneity of a process may be understood by recalling the previously derived second law expression:

The first law requires that q_surr = −q_sys, and at constant pressure q_sys = ΔH, so this expression may be rewritten as:

Multiplying both sides of this equation by −T and rearranging yields the following:

For simplicity’s sake, the subscript “sys” can be omitted, and the expression becomes

Comparing this equation to the previous one for free energy change shows the following relation:

The free energy change is, therefore, a reliable indicator of the spontaneity of a process, as it is directly related to the previously identified spontaneity indicator, ΔS_univ.

If ΔS_univ > 0, ΔG < 0 and the reaction is spontaneous.

If ΔS_univ < 0, ΔG > 0 and the reaction is nonspontaneous.

If ΔS_univ = 0, ΔG = 0 and the reaction is at equilibrium.

This text is adapted from Openstax, Chemistry 2e, Chapter 16.4: Free Energy.

Procedure

The second law of thermodynamics states that for all spontaneous processes, the entropy of the universe—the sum of the entropy of the system and the entropy of the surroundings—increases.

While the entropy of the system can be calculated from standard molar entropies, the entropy of the surroundings is more difficult to calculate or measure.

Therefore, J. Willard Gibbs defined a new thermodynamic function, which allows spontaneity to be determined solely through the entropy and enthalpy of the system and not the surroundings.

Recall that under constant pressure and temperature conditions, the ΔS of the surroundings is equal to the negative ΔH of the system divided by the temperature, T. This term can be substituted into the equation representing the second law.

When both sides are multiplied by negative T, the equation now becomes: −TΔS_univ = ΔH_sys − TΔS_sys. The thermodynamic functions on the right side of the equation—enthalpy and entropy—are both solely dependent on the system.

Because both enthalpy and entropy are state functions, a new state function can be defined as −TΔS_univ. This new term is called Gibbs free energy and is denoted by the letter G.

The equation for ΔG leads to a new criterion for spontaneous reactions. The difference between the enthalpy change and the temperature or entropy change must be less than zero.

ΔG is also known as chemical potential because it is similar to the mechanical potential energy of a system.

Just as a ball will always roll downhill to lower its potential energy, a chemical reaction proceeds to lower its chemical potential.

Thus, at constant temperature and pressure, if the free energy of the system decreases (that is, ΔG < 0), the reaction is spontaneous.

Conversely, if the free energy of the system increases, ΔG > 0, and the reaction is not spontaneous.

If ΔG = 0, then the reactants and products are in equilibrium.