Helmholtz vs Gibbs Free Energy

Helmholtz Free Energy (A) vs Gibbs Free Energy (G)

Property	Helmholtz Free Energy A (or F)	Gibbs Free Energy G (or F)
Definition	A = U − T S	G = H − T S = U + P V − T S
Natural Variables	T, V, N (or T, V, {Nᵢ})	T, P, N (or T, P, {Nᵢ})
Most Useful When	Constant temperature and constant volume	Constant temperature and constant pressure (most real processes)
Differential Form	dA = –S dT – P dV + μ dN	dG = –S dT + V dP + μ dN
Criterion of Spontaneity	(ΔA)T,V ≤ 0  process is spontaneous	(ΔG)T,P ≤ 0  process is spontaneous
Equilibrium Condition	(dA)T,V = 0	(dG)T,P = 0 (most chemical reactions)
Physical Meaning	Maximum work that can be extracted at constant T and V (= useful work when only if volume isochoric)	Maximum non-expansion work (e.g. electrical work in batteries) at constant T and P
Chemical Potential	μᵢ = (∂A/∂Nᵢ)T,V,Nⱼ	μᵢ = (∂G/∂Nᵢ)T,P,Nⱼ ← most used definition
Typical Applications	– Statistical mechanics – Explosions, combustion bombs – Theoretical physics – Ideal gas expansions in fixed volume	– Chemical reactions in open flasks – Phase equilibria – Electrochemistry – Biochemistry – Almost all industrial processes
Standard Tables Use	Very rare	ΔG°f, ΔG°rxn tables everywhere
Relation Between A and G	G = A + P V

Rule of Thumb (memorize this):

• If the process happens in a closed container with fixed volume (bomb calorimeter, diamond synthesis, many theoretical calculations) → use Helmholtz A.

• If the process happens at constant pressure (open beaker, reactor open to atmosphere, living cells, almost every real chemical reaction) → use Gibbs G.

Quick Summary Table

Situation	Use This Free Energy
Chemical reaction in open flask	Gibbs (G)
Battery or fuel cell (constant P)	Gibbs (G) → ΔG = –nFE
Explosion in a rigid bomb	Helmholtz (A)
Phase diagram calculation	Gibbs (G)
Statistical mechanics partition function	Helmholtz (A)