Obtain the formula for the coefficient of performance of a Carnot refrigerator.

The efficiency of a cyclic heat engine is given by $\eta = 1 - \frac{Q_2}{Q_1}$.
For a Carnot engine,the efficiency is $\eta = 1 - \frac{T_2}{T_1}$.
Equating these,we get $\frac{Q_2}{Q_1} = \frac{T_2}{T_1}$,which implies $\frac{Q_1}{Q_2} = \frac{T_1}{T_2}$.
Subtracting $1$ from both sides: $\frac{Q_1}{Q_2} - 1 = \frac{T_1}{T_2} - 1$,which simplifies to $\frac{Q_1 - Q_2}{Q_2} = \frac{T_1 - T_2}{T_2}$.
Taking the reciprocal,we get $\frac{Q_2}{Q_1 - Q_2} = \frac{T_2}{T_1 - T_2}$.
The coefficient of performance $\alpha$ for a refrigerator is defined as $\alpha = \frac{Q_2}{W} = \frac{Q_2}{Q_1 - Q_2}$.
Therefore,the formula is $\alpha = \frac{T_2}{T_1 - T_2}$.