Isn't the change in Gibbs free energy for a reversible process zero?

Question

In problem 6, part 6.5 of the 2020 IChO (PDF), a thermodynamic cycle is given for “one mole of monoatomic perfect gas”:

• $\mathrm{A \to B}$; isothermal reversible expansion receiving $\pu{250 J}$ by heat transfer $(q_H)$ at a temperature of $\pu{1000 K}$ $(T_H)$ from a hot source.
• $\mathrm{B \to D}$; reversible adiabatic expansion.
• $\mathrm{D \to C}$; isothermal reversible compression at a temperature of $\pu{300 K}$ $(T_C)$ releasing some amount of heat $(q_C)$ to a cold sink.
• $\mathrm{C \to A}$; reversible adiabatic compression.

6.5 Calculate the entropy change $(\Delta S)$ for $\mathrm{A \to B}$ and $\mathrm{D \to C}$ processes in the heat engine in terms of $\pu{J K^-1}.$

To my understanding, the answer should be $\pu{0 J}$ for both steps since $\Delta G = 0$ for any reversible process, which $\mathrm{A \to B}$ and $\mathrm{D \to C}$ are. But the solution below gives non-zero values of $\Delta G.$ $ΔS$ for both steps was calculated in previous parts, $\pu{+0.25 J}$ for $\mathrm{A \to B},$ $\pu{-0.25 J}$ for $\mathrm{D \to C}.$ What is wrong with my reasoning here?

$$\Delta G = \Delta H - T\,\Delta S,\tag{1}$$

but $\Delta H = 0$ for an isothermal process, so

$$\Delta G = - T\,\Delta S.\tag{2}$$

$$ \begin{align} \Delta G_\mathrm{A \to B} &= \pu{-0.25 J K^-1}\times\pu{1000 K} = \pu{-250 J}\tag{3}\\ \Delta G_\mathrm{D \to C} &= \pu{0.25 J K^-1}\times\pu{300 K} = \pu{75 J}\tag{4} \end{align} $$

Answer 1

The cycle shown is a classic reversible Carnot cycle. An intuitive reason for the change in Gibbs free energy not being equal to zero for such a step is that the pressure is not constant: the internal and external pressure are throughout in equilibrium, but since $p_\mathrm{ext} = p = nRT/V = \mathrm{constant}/V$, $p$ is not constant. A condition for applying $\Delta G$ as a criterion of spontaneity (reversibility) is that the pressure is constant.

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In fact you can show (by extension of the calculations show below) that the free energy change is equal to the expansion or compression work done in that step: $$\Delta G_\mathrm{isothermal} = w$$ as follows.

Computing the entropy changes for the isothermal steps from the heats is straightforward using the first law of thermodynamics, since

• for an ideal gas undergoing an isothermal expansion or compression $\Delta U=0$ so that $w=-q$
• for a reversible isothermal process $q=T \Delta S$

such that we can compute

• for the first step, $\Delta S = \frac{\pu{250 J}}{\pu{1000 K}} = \pu{0.250 J K-1}$

For the second isothermal step ($D\rightarrow C$) we use the fact that the total entropy change for the system, given that it is a cyclic process, is $\Delta S = 0$, therefore

• for the second isothermal step, $\Delta S_\mathrm{DC} = - \Delta S_\mathrm{AB} = \pu{-0.250 J K-1}$

Regarding now the free energy computation, you would use (sure enough) $$\Delta G_\mathrm{isothermal} = \Delta H_\mathrm{isothermal} - T \Delta S_\mathrm{isothermal} = - T \Delta S_\mathrm{isothermal}=-q=w$$

since $\Delta H_\mathrm{isothermal}=0$ for such a process.