Free-energy criteria — In spontaneous processes, how do the Helmholtz free energy (A) and the Gibbs free energy (F here, commonly written G) change when the system interacts with reservoirs at fixed temperature and/or pressure?

Introduction:
Spontaneity criteria depend on the constraints imposed by the surroundings. Two free energies are commonly used: Helmholtz free energy A = U − T S (natural variables T, V) and Gibbs free energy G (denoted F in some texts) = H − T S (natural variables T, P). Correctly identifying when each decreases clarifies which potential predicts equilibrium.

Given Data / Assumptions:

• System can exchange heat with a thermal reservoir at temperature T₀.
• Volume and pressure constraints come from rigid walls (V fixed) or pressure reservoirs (P fixed).
• No non-PV work unless noted; classical criteria applied.

Concept / Approach:
At constant T and V, the second law implies that the Helmholtz free energy A of the system decreases for a spontaneous change, reaching a minimum at equilibrium. At constant T and P, the Gibbs free energy G decreases similarly. Thus each potential decreases under its own natural constraints. In many practical processes, the system exchanges heat with a thermostat at T₀ and PV work against an ambient pressure P₀; then the appropriate function (G at constant T, P) decreases.

Step-by-Step Solution:
For T, V fixed: use A = U − T S → spontaneous direction has ΔA < 0.For T, P fixed: use G = H − T S → spontaneous direction has ΔG < 0.Interpretation: both free energies provide minimum principles, each in its own constraint set.

Verification / Alternative check:
From the combined first and second laws and Legendre transforms, dA ≤ 0 (T,V const) and dG ≤ 0 (T,P const). Equality holds at equilibrium.

Why Other Options Are Wrong:

• “Only” one decreases ignores the other’s role under its proper constraints.
• Increase of free energies contradicts the second law criteria.
• No directional tendency is inconsistent with observed relaxation to equilibrium.

Common Pitfalls:
Mixing symbols where F is used for G in some curricula; assuming a single criterion applies regardless of constraints.

Final Answer:
Both F and A decrease (each under its natural constraints)