Solution

Statement

$5.9.14.$ Show that the efficiency of a heat engine in a cyclic process is maximal when the entropy of the system does not change.

Solution

In the problem statement, the "system" implies a complete closed thermodynamic system, which includes the working fluid of the heat engine as well as the heat reservoirs (the heater at temperature $T_1$ and the cooler at temperature $T_2$).

Over one complete cycle, the working fluid returns to its initial state. Since entropy $S$ is a state function, the change in entropy of the working fluid itself over a cycle is always zero:
$$\Delta S_{\text{wf}} = 0$$

Therefore, the total change in entropy of the closed system over a cycle consists solely of the change in entropy of the heat reservoirs:
$$\Delta S_{\text{system}} = \Delta S_{\text{heater}} + \Delta S_{\text{cooler}}$$

Suppose that during a cycle, the engine receives heat $Q_1$ from the heater and rejects heat $Q_2$ to the cooler (where $Q_1, Q_2 > 0$).
Then the heater loses entropy, and the cooler gains it:
$$\Delta S_{\text{system}} = -\frac{Q_1}{T_1} + \frac{Q_2}{T_2}$$

According to the second law of thermodynamics, the entropy of a closed system cannot decrease ($\Delta S_{\text{system}} \ge 0$). Therefore:
$$\frac{Q_2}{T_2} - \frac{Q_1}{T_1} \ge 0 \implies \frac{Q_2}{Q_1} \ge \frac{T_2}{T_1}$$

Now let us express the efficiency of the heat engine through the work $W = Q_1 - Q_2$:
$$\eta = \frac{W}{Q_1} = \frac{Q_1 - Q_2}{Q_1} = 1 - \frac{Q_2}{Q_1}$$

Let us substitute the obtained thermodynamic inequality $\frac{Q_2}{Q_1} \ge \frac{T_2}{T_1}$ into this expression:
$$\eta \le 1 - \frac{T_2}{T_1}$$

Conclusion:
The efficiency reaches its maximum possible value $\eta_{\text{max}} = 1 - \frac{T_2}{T_1}$ (Carnot cycle efficiency) if and only if the inequality turns into an equality:
$$\frac{Q_2}{T_2} - \frac{Q_1}{T_1} = 0 \implies \Delta S_{\text{system}} = 0$$
Thus, the efficiency of a heat engine is maximal if the process is completely reversible and the total entropy of the system does not change.