Ex. 3

Heat engines revisited: From the law of non-decrease of entropy, show that the maximum efficiency of a heat engine operating between two reservoirs at $T_H$ and $T_C$ occurs when the engine is reversible.

This is a bit circular, as we used the properties of Carnot engines to derive this form of the second law! However we will see later in the course that it can be independently derived using statistical methods.

The change in entropy of the two reservoirs, which must be non-negative, is

$$\Delta S=\Delta S_C+\Delta S_H={Q_C\over T_C}-{Q_H\over T_H}\ge0 \qquad \Rightarrow\qquad {Q_C\over Q_H}\ge{T_C\over T_H}$$

with the equality being for $\Delta S=0$, ie for a reversible process. So the efficiency is

$$\eta={W\over Q_H}=1-{Q_C\over Q_H}\le 1-{T_C\over T_H}$$

This is maximum when the equality is satisfied, ie for a reversible engine.