Answer
See the proof below.
Work Step by Step
To prove that the coefficient of performance (COP) of a Carnot refrigerator is $K_{\rm{Carnot }}=T_{\mathrm{C}} /\left(T_{\mathrm{H}}-T_{\mathrm{C}}\right)$, we start with the definition of COP for a refrigerator, which is the ratio of the heat removed from the cold reservoir to the work input:
$$K_{\rm{carnot}}=\frac{Q_{\rm{C}}}{W_{\rm{in}}}$$
For a Carnot refrigerator, we know that the work input is equal to the difference in heat between the hot and cold reservoirs:
$$W_{\rm{in}}=Q_{\rm{H}}-Q_{\rm{C}}$$
Substituting this expression into the equation for COP, we get:
$$K_{\rm{carnot}}=\frac{Q_{\rm{C}}}{Q_{\rm{H}}-Q_{\rm{C}}}$$
Next, we can use the expressions for the heat transfer in a Carnot cycle to rewrite the above equation in terms of temperatures:
$$\boxed{K_{\rm{carnot}}=\frac{T_{\rm{C}}}{T_{\rm{H}}-T_{\rm{C}}}}$$
This result can also be derived using the Carnot efficiency formula for a heat engine, which is $\eta_{\rm{Carnot}}=1-T_{\rm{C}}/T_{\rm{H}}$, and the fact that the COP of a refrigerator is the reciprocal of the efficiency of a heat engine operating between the same two reservoirs:
$$K_{\rm{carnot}}=\frac{1}{\eta_{\rm{Carnot}}}=\frac{1}{1-T_{\rm{C}}/T_{\rm{H}}}=\frac{T_{\rm{H}}}{T_{\rm{H}}-T_{\rm{C}}}=\frac{T_{\rm{C}}}{T_{\rm{H}}-T_{\rm{C}}}$$
$$\boxed{K_{\rm{carnot}}= \frac{T_{\rm{C}}}{T_{\rm{H}}-T_{\rm{C}}}}$$
