Answer
$COP=\frac{1-e}{e}$
Work Step by Step
We know that the efficiency of the Carnot engine is given as
$e=1-\frac{T_c}{T_h}$
Coefficient of the refrigerator is given as
$COP=\frac{Q_c}{W}$
$\implies COP=\frac{Q_c}{Q_h-Q_c}$
$COP=\frac{1}{\frac{Q_h}{Q_c}-1}$
$COP=\frac{1}{\frac{T_h}{T_c}-1}$
$COP=\frac{1}{\frac{1}{1-e}-1}$
$COP=\frac{1-e}{e}$
