Explain the working of refrigerators/heat pumps and their coefficient of performance.

(N/A) If a cyclic process performed in a heat engine is reversed,it acts as a refrigerator or a heat pump.
The working substance in a refrigerator/heat pump draws heat $Q_{2}$ from a cold reservoir at a lower temperature $T_{2}$,external work $W$ is performed on the working substance,and heat $Q_{1}$ is released into the hot reservoir at a higher temperature $T_{1}$.
In a refrigerator,the working substance (in gaseous form) goes through the following steps:
$(a)$ Sudden expansion of the gas from high to low pressure,which cools it and converts it into a vapour-liquid mixture.
$(b)$ Absorption of heat by the cold fluid from the region to be cooled,converting it into vapour.
$(c)$ Heating up of the vapour due to external work done on the system.
$(d)$ Release of heat by the vapour to the surroundings,bringing it to the initial state and completing the cycle.
If it is used to cool a space inside a chamber when its surroundings are at a higher temperature,it is called a refrigerator.
If it is used to heat a space or a room when its surroundings are at a lower temperature,it is called a heat pump.
The ratio of the quantity of heat $Q_{2}$ extracted from the cold reservoir to the work $W$ done on the system (the refrigerant) is known as the coefficient of performance $(\alpha)$ of a refrigerator:
$\alpha = \frac{Q_{2}}{W} \dots(1)$
For a heat pump,the coefficient of performance is:
$\alpha = \frac{Q_{1}}{W}$
In a heat engine,efficiency $\eta$ can never exceed $1$,while for a heat pump,$\alpha$ can be more than $1$. From the law of conservation of energy:
$Q_{1} = W + Q_{2}$
$\therefore W = Q_{1} - Q_{2}$
From equation $(1)$:
$\alpha = \frac{Q_{2}}{Q_{1} - Q_{2}}$