4.1 Why free energy: system and reservoir

The thermodynamics chapter gave the rules of bookkeeping; this one gives the rule of prediction. The second law says an isolated system moves toward maximum entropy — but the systems we usually care about are not isolated. A reaction in a beaker, a crystal in contact with its melt, a magnet in a thermostat all exchange heat with their surroundings. For these the quantity that is extremised is not the entropy but a free energy.

The total-entropy argument

Put the system in thermal contact with a much larger reservoir at fixed temperature $T$. The combination “system + reservoir” is isolated, so its total entropy can only increase. The reservoir is so large that it stays at $T$; the entropy it gains is the heat it absorbs divided by $T$, and at fixed system volume that heat is $-\Delta U_\text{sys}$. So

where $F \equiv U - TS$ is the Helmholtz free energy. The second law $\Delta S_\text{tot}\ge 0$ is therefore exactly equivalent to $\Delta F \le 0$. The free energy is the system-level bookkeeping that has already absorbed the reservoir’s entropy change, so that a statement about the universe becomes a statement about the system alone.

Slide $\Delta U$ and $\Delta S_\text{sys}$. The total-entropy verdict — “allowed” or “forbidden” — flips exactly at the $\Delta F = 0$ line: a change with $\Delta F < 0$ is allowed, one with $\Delta F > 0$ is forbidden. That is why a system at fixed $T$ and $V$ relaxes toward the minimum of $F$.

For a system held at fixed temperature and pressure — the usual case under the atmosphere — the same argument, now allowing the reservoir to do pressure work, gives the Gibbs free energy

with the criterion $\Delta G \le 0$. Which free energy to minimise is set entirely by which variables the surroundings hold fixed.

Free energies as Legendre transforms

The definitions $F = U - TS$ and $G = U + pV - TS$ are not arbitrary: each is the Legendre transform of the internal energy, trading a natural variable for its conjugate. $U(S,V)$ has natural variables $S$ and $V$, with $T = (\partial U/\partial S)_V$ and $-p = (\partial U/\partial V)_S$. The transform $F(T,V) = U - TS$ swaps the entropy $S$ for the temperature $T$ — geometrically, $F$ is the intercept of the tangent to $U(S)$ whose slope is $T$.

Slide $T$ — the slope of the tangent line. The point of tangency on $U(S)$ moves, and the tangent’s intercept is $F(T) = U(S^*) - T S^*$. The transform carries exactly the same information as $U(S)$, repackaged so that the control variable is the temperature the experimenter actually fixes. Differentiating the definitions gives the working forms

from which the entropy, pressure, and volume are read off as first derivatives — the machinery the rest of the chapter uses to locate equilibrium.