74 Free Energy
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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:
(For simplicity’s sake, the subscript “sys” will be omitted henceforth.)
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 qsurr = −qsys, and at constant pressure qsys = ΔH, so this expression may be rewritten as:
Multiplying both sides of this equation by −T, and rearranging yields the following:
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, being directly related to the previously identified spontaneity indicator, ΔSuniv. (Figure) summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators.
| Relation between Process Spontaneity and Signs of Thermodynamic Properties | ||
|---|---|---|
| ΔSuniv > 0 | ΔG < 0 | spontaneous |
| ΔSuniv < 0 | ΔG > 0 | nonspontaneous |
| ΔSuniv = 0 | ΔG = 0 | at equilibrium |
What’s “Free” about ΔG?
In addition to indicating spontaneity, the free energy change also provides information regarding the amount of useful work (w) that may be accomplished by a spontaneous process. Although a rigorous treatment of this subject is beyond the scope of an introductory chemistry text, a brief discussion is helpful for gaining a better perspective on this important thermodynamic property.
For this purpose, consider a spontaneous, exothermic process that involves a decrease in entropy. The free energy, as defined by
may be interpreted as representing the difference between the energy produced by the process, ΔH, and the energy lost to the surroundings, TΔS. The difference between the energy produced and the energy lost is the energy available (or “free”) to do useful work by the process, ΔG. If the process somehow could be made to take place under conditions of thermodynamic reversibility, the amount of work that could be done would be maximal:
However, as noted previously in this chapter, such conditions are not realistic. In addition, the technologies used to extract work from a spontaneous process (e.g., automobile engine, steam turbine) are never 100% efficient, and so the work