Question

The coefficient of performance of a refrigerator is 5. If the temperature inside freezer is $-20{}^{o}C$, the temperature of the surroundings to which it rejects heat is

Answer 1

A refrigerator works in a reverse Carnot cycle. It has three main components; the source, the sink and the working substance. Its performance is analyzed by the coefficient of performance which is the ratio of heat released to the sink to the work done to release the heat. Substituting corresponding values in the formula, we can calculate the temperature of the surroundings.
Formulas used:
$C=\dfrac{Q}{W}$

Complete answer:
The refrigerator is a reverse system of the Carnot engine. It has three parts; The source, the working substance and the sink. The heat is absorbed from the sink by the working substance and given back to the source.
The coefficient of performance of a refrigerator is defined as the heat removed from the working substance divided by the work done to remove the heat.
$C=\dfrac{Q}{W}$
Here, $C$ is the coefficient of performance
$Q$ is the heat released
$W$ is the work done
Coefficient of performance is also given as-
$C=\dfrac{{{T}_{fr}}}{{{T}_{sur}}-{{T}_{fr}}}$ - (1)
Here, ${{T}_{fr}}$ is the temperature of freezer
${{T}_{sur}}$ is the temperature of the surroundings
Given, $C=5$
$\begin{aligned} & {{T}_{fr}}=-20{}^{o}C \\\ & \Rightarrow {{T}_{fr}}=(273-20)K \\\ & \therefore {{T}_{fr}}=253K \\\ \end{aligned}$
We substitute in eq (1), to get,
$\begin{aligned} & C=\dfrac{{{T}_{fr}}}{{{T}_{sur}}-{{T}_{fr}}} \\\ & \Rightarrow 5=\dfrac{253}{T-253} \\\ & \Rightarrow T-253=50.6 \\\ & \Rightarrow T=(253+50.6) \\\ & \Rightarrow T=303.6K \\\ & \Rightarrow T=303.6-273 \\\ & \therefore T=30.6{}^{o}C \\\ \end{aligned}$
Therefore, the temperature of the surroundings is $30.6{}^{o}C$.

Note:
The sink is at lower temperature, while the source is at a higher temperature. The working substance is at a constant temperature. A Carnot engine is a thermodynamic engine which converts heat into work with the greatest efficiency. No engine can have a greater efficiency than a Carnot cycle. The coefficient of performance of a refrigerator does not have a unit as it is a ratio of quantities with the same units.