Classical statistical mechanics
Helmholtz and Gibbs free energies
F=U−TS and G=H−TS are thermodynamic potentials naturally suited to fixed-temperature-volume and fixed-temperature-pressure conditions.
F=U−TS and G=H−TS are thermodynamic potentials naturally suited to fixed-temperature-volume and fixed-temperature-pressure conditions.
Definition: Quantities and meaning
For fixed composition, dF=−S dT−p dV and dG=−S dT+V dp. At fixed T,V stable equilibrium minimizes F; at fixed T,p it minimizes G. This assumes thermal contact with a bath and no unaccounted external work.
Quantitative relation
Example: Worked example
At fixed T,p a process has ΔG=−2.0 kJ·mol⁻¹. What does the sign imply about its spontaneous direction?
Solution
Under the stated constraints, the forward direction has ΔG<0 and lowers Gibbs free energy; at equilibrium ΔG=0.
Example: Comparing two states at fixed T,p
At the same T and p, suppose state A has a Gibbs energy 3.0 kJ·mol⁻¹ below state B. At fixed composition with only pV work, the change B→A has ΔG=G_A−G_B=−3.0 kJ·mol⁻¹ and is the spontaneous direction. At equilibrium, allowed infinitesimal changes have ΔG=0; this criterion alone says nothing about how fast conversion occurs.
A Legendre transform changes a potential’s natural variables: U(S,V) uses entropy and volume, F(T,V) replaces S with T, and G(T,p) also replaces V with p. Their differentials show that −S and V are the derivatives of G with respect to T and p. At low temperature, an entropy decrease can raise G; comparing the enthalpies of two states alone therefore does not determine spontaneity. If electrical, magnetic, or surface work is present, the appropriate potential must include the corresponding work terms for its minimization criterion to remain valid. For a chemical reaction at ordinary conditions, negative ΔG indicates thermodynamic favorability, not a guarantee that the reaction proceeds rapidly. Similarly, F falls during a spontaneous isothermal, isochoric change.
Quick check
Which thermodynamic potential is appropriate for equilibrium at fixed T,p?
Which equation defines the Helmholtz free energy?
References
- Charles Kittel and Herbert Kroemer (1980). Thermal Physics
- L. D. Landau and E. M. Lifshitz (1980). Statistical Physics