A Carnot heat engine with thermal efficiency $\frac{1}{3}$is run backward as a Carnot refrigerator. What is the refrigerator’s coefficient of performance?

The efficiency of Carnot engine$=\frac{1}{3}$

Formula used :

We know that the thermal efficiency of heat engine is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.

Here, $T_H$is the temperature of the hot reservoir that is source and $T_c$is the temperature of cold reservoir i.e. sink. The efficiency is given by

$\eta =1-\frac{T_C}{T_H}\left(I\right)$

$\eta =\frac{T_H-T_C}{T_H}$

Therefore, role="math" localid="1648467374481" $\frac{T_C}{T_H}=1-\eta$

$\frac{T_H}{T_C}=\frac{1}{1-\eta }\left(II\right)$

Coefficient of performance of refrigerator is defined as the reciprocal of the ideal efficiency of the Carnot engine, where k is the coefficient of performance of a refrigerator.

$$\begin{gathered} k=\frac{T_c}{T_H-T_C} \\ k=\frac{1}{\frac{T_H}{T_C}-1} \end{gathered}$$

Substituting the value of $\frac{T_H}{T_C}$in above equation, the coefficient of performance of a refrigerator is calculated as

$$\begin{gathered} k=\frac{1}{\frac{1}{1-\eta }-1} \\ k=\frac{1-\eta }{\eta }\left(III\right) \end{gathered}$$

Substituting the value of $\eta$in the above equation

$$\begin{gathered} k=\frac{1-\frac{1}{3}}{\frac{1}{3}} \\ k=2 \end{gathered}$$

Conclusion :

Refrigerator coefficient of performance is$2$.