Free Energy

Free Energy

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that we must determine 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 change (G) (or simply the free energy), and it is defined in terms of a system’s enthalpy and entropy as the following:

\(G=H-TS\)

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

\(\text{Δ}G=\text{Δ}H-T\text{Δ}S\)

(For simplicity’s sake, the subscript “sys” will be omitted henceforth.)

We can understand the relationship between this system property and the spontaneity of a process by recalling the previously derived second law expression:

\(\text{Δ}{S}_{\text{univ}}=\text{Δ}S+\phantom{\rule{0.2em}{0ex}}\cfrac{{q}_{\text{surr}}}{T}\)

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

\(\text{Δ}{S}_{\text{univ}}=\text{Δ}S-\phantom{\rule{0.1em}{0ex}}\cfrac{\text{Δ}H}{T}\)

ΔH is the enthalpy change of the system. Multiplying both sides of this equation by −T, and rearranging yields the following:

\(\text{−}T\text{Δ}{S}_{\text{univ}}=\text{Δ}H-T\text{Δ}S\)

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

\(\text{Δ}G=\text{−}T\text{Δ}{S}_{\text{univ}}\)

The free energy change is therefore a reliable indicator of the spontaneity of a process, being directly related to the previously identified spontaneity indicator, ΔS_univ. The table below summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators.