Question

Choose the correct option(s).

Here A is the Helmholtz free energy and G is the Gibbs free energy.

• A. A provides a criterion for equilibrium in a system with constant temperature and constant pressure
• B. G provides a criterion for equilibrium in a system with constant temperature and constant pressure
• C. A provides a criterion for equilibrium in a system with constant temperature and constant volume
• D. G provides a criterion for equilibrium in a system with constant temperature and constant volume

Hint:

Helmholtz free energy A is minimized at equilibrium under constant T and V; Gibbs free energy G is minimized at equilibrium under constant T and P.

Answer 1

The Correct Option is B, C

Step 1: Recall the definitions of A and G.
The Helmholtz free energy and the Gibbs free energy are both built from the internal energy \(U\), entropy \(S\) and enthalpy \(H\) of a system,
\[ A = U - TS \]
\[ G = H - TS = U + PV - TS \]
Both are defined so that they combine the first law (energy) and the second law (entropy) into a single quantity that tells us whether a process at fixed conditions can happen on its own.

Step 2: Derive the behaviour of A at constant T and V.
Taking the differential of \(A = U - TS\) and using the combined first and second law relation \(dU = TdS - PdV\) for a closed system,
\[ dA = dU - TdS - SdT = -PdV - SdT \]
At constant temperature and constant volume, both \(dT = 0\) and \(dV = 0\), so
\[ (dA)_{T,V} \leq 0 \]
for any spontaneous change, with equality holding only at equilibrium. So A falls until it reaches a minimum at constant T and V, and that minimum marks the equilibrium state. This is exactly what option (C) says, so option (C) is correct, and it also means option (A), which claims A is the criterion at constant T and P, is wrong.

Step 3: Derive the behaviour of G at constant T and P.
Taking the differential of \(G = U + PV - TS\),
\[ dG = dU + PdV + VdP - TdS - SdT \]
Substituting \(dU = TdS - PdV\),
\[ dG = VdP - SdT \]
At constant temperature and constant pressure, \(dT = 0\) and \(dP = 0\), so
\[ (dG)_{T,P} \leq 0 \]
for any spontaneous change, again with equality only at equilibrium. So G falls until it reaches a minimum at constant T and P, and that minimum marks equilibrium. This is exactly what option (B) says, so option (B) is correct, and option (D), which claims G is the criterion at constant T and V, is wrong.

Step 4: Match each free energy to its own fixed conditions.
The short way to remember this: the Helmholtz free energy pairs with volume being held fixed, since it is built to absorb the \(-PdV\) work term into itself and needs no \(PdV\) correction when V does not change. The Gibbs free energy pairs with pressure being held fixed, since it already carries the \(+PV\) term needed to account for boundary work at constant pressure. This is why A is the natural equilibrium criterion at constant T and V, while G is the natural equilibrium criterion at constant T and P, never the other way round.

Step 5: Final Answer.
A is the correct criterion at constant temperature and volume, and G is the correct criterion at constant temperature and pressure.
\[ \boxed{\text{(B) and (C) are correct}} \]