Carnot Refrigerator

Because every individual step of the Carnot cycle is, by construction, reversible (Section 4.11.2), the ENTIRE Carnot cycle — run as a whole — is also reversible. This means the whole cycle can simply be run in the opposite direction: absorbing heat $Q_C$ from the cold reservoir, having work $W$ done ON it, and rejecting heat $Q_H$ to the hot reservoir — exactly the behaviour of a refrigerator (Section 4.9). Run this way, the Carnot engine becomes a Carnot refrigerator.

Using the general refrigerator coefficient-of-performance formula, $\kappa = Q_C/(|Q_H| - Q_C)$ (Eq. 4.27), together with the Carnot relation $Q_C/Q_H = T_C/T_H$ (which follows from the same derivation used to reach Eq. 4.29), the coefficient of performance of a Carnot refrigerator simplifies to:

$\kappa = \frac{T_C}{T_H - T_C} \qquad \text{--- (4.31)}$ …