A gas is held in a rigid container (constant volume) in thermal contact with a heat bath (constant temperature). Which condition correctly describes thermodynamic equilibrium?
AInternal energy U is minimized
BEntropy S is maximized without constraint
CHelmholtz free energy F is minimized
DTemperature T is equalized between system and bath
At constant T and V (the canonical ensemble conditions), the equilibrium condition is that Helmholtz free energy F = U − TS is minimized. This follows from the fundamental relation dF = −S dT − P dV: at constant T and V, both differentials vanish and the system evolves to lower F. Minimizing U alone ignores entropy; maximizing S alone ignores energy constraints. Temperature equalization is a precondition (the bath enforces it), not the equilibrium criterion. The power of F is precisely that it folds both energy and entropy considerations into one minimization principle.
Question 2 Multiple Choice
What does the 'free' in Helmholtz free energy physically represent?
AEnergy that is freely conserved regardless of the second law of thermodynamics
BThe total internal energy U available at constant volume
CThe portion of internal energy available to perform useful work, after paying the entropy cost TS
DEnergy stored in the thermal fluctuations of the system's molecules
The Helmholtz free energy F = U − TS separates internal energy into two parts: TS, which is 'locked up' in thermal disorder and cannot be extracted as ordered work (the entropy tax imposed by the second law), and F, which is the remainder — the energy 'free' to do work. In a reversible process at constant T, the maximum work the system can perform equals −ΔF. This is why TS is sometimes called the 'unavailable energy': even a perfect engine cannot convert it to work without violating the second law.
Question 3 True / False
Helmholtz free energy F = −k_BT ln Z provides a direct bridge between the partition function of the canonical ensemble and measurable thermodynamic quantities like entropy, pressure, and average energy.
TTrue
FFalse
Answer: True
This is the central result that makes F so powerful. Once Z is computed from the energy spectrum of the system, F = −k_BT ln Z yields all thermodynamic properties by differentiation: S = −(∂F/∂T)_V gives entropy, P = −(∂F/∂V)_T gives pressure, and U = F + TS = −T²(∂(F/T)/∂T)_V gives internal energy. This single function consolidates what would otherwise require separate calculations for each observable — making it the standard starting point for statistical mechanics calculations.
Question 4 True / False
A spontaneous process at constant temperature and volume usually decreases the system's internal energy U.
TTrue
FFalse
Answer: False
At constant T and V, the criterion for spontaneity is ΔF ≤ 0, not ΔU ≤ 0. Since F = U − TS, a process can increase U and still be spontaneous if the entropy gain ΔS is large enough that TΔS > ΔU, making ΔF = ΔU − TΔS < 0. Mixing of ideal gases is a classic example: the internal energy barely changes, but entropy increases substantially, driving the spontaneous mixing. Conflating ΔU < 0 with spontaneity is a common error that ignores the entropic contribution.
Question 5 Short Answer
Explain why the maximum work a system can perform at constant temperature equals −ΔF, and what the TS term represents physically.
Think about your answer, then reveal below.
Model answer: The first law gives ΔU = Q − W, and the second law requires Q ≤ TΔS (with equality for reversible processes). Combining: W ≤ TΔS − ΔU = −ΔF. The maximum work occurs in a reversible process where W_max = −ΔF. The TS term represents the energy locked into thermal disorder — microscopic random motion that is incoherent and cannot be organized into useful work without violating the second law. Even a perfect engine cannot extract this energy; it is the irreducible entropy tax on any thermodynamic process.
The name 'free energy' carries this meaning: F is the portion of internal energy that is 'free' to become work. When a system releases ΔF of free energy, some goes to useful work and the rest (if the process is irreversible) is dissipated as heat. Only in the idealized reversible limit does all ΔF convert to work. This framing also clarifies why ΔF ≤ 0 signals spontaneity: the system is releasing free energy, and nature proceeds in the direction that extracts the most available work.