Obtain the relation between the coefficient of performance and the efficiency of a heat engine.

(N/A) The coefficient of performance $(COP)$, denoted by $\alpha$, for a refrigerator is defined as the ratio of heat extracted from the cold reservoir $(Q_2)$ to the work done $(W)$ on the system.
$\alpha = \frac{Q_2}{W} = \frac{Q_2}{Q_1 - Q_2}$
Dividing the numerator and denominator by $Q_1$:
$\alpha = \frac{Q_2 / Q_1}{1 - Q_2 / Q_1} \dots (1)$
For a Carnot engine, the efficiency $\eta$ is defined as:
$\eta = 1 - \frac{Q_2}{Q_1}$
Rearranging this gives:
$\frac{Q_2}{Q_1} = 1 - \eta \dots (2)$
Substituting equation $(2)$ into equation $(1)$:
$\alpha = \frac{1 - \eta}{\eta}$
Thus, the relation is $\alpha = \frac{1}{\eta} - 1$.