JEE Main & Advanced Physics Thermodynamical Processes Refrigerator or Heat Pump

A refrigerator or heat pump is basically a heat engine run in reverse direction.

It essentially consists of three parts

(1) Source : At higher temperature \[{{T}_{1}}\].

(2) Working substance : It is called refrigerant liquid ammonia and freon works as a working substance.

(3) Sink : At lower temperature \[{{T}_{2}}\].

The working substance takes heat \[{{Q}_{2}}\] from a sink (contents of refrigerator) at lower temperature, has a net amount of work done W on it by an external agent (usually compressor of refrigerator) and gives out a larger amount of heat \[{{Q}_{1}}\] to a hot body at temperature \[{{T}_{1}}\] (usually atmosphere). Thus, it transfers heat from a cold to a hot body at the expense of mechanical energy supplied to it by an external agent. The cold body is thus cooled more and more.

The performance of a refrigerator is expressed by means of "coefficient of performance" \[\beta \] which is defined as the ratio of the heat extracted from the cold body to the work needed to transfer it to the hot body.

i.e.  \[\beta =\frac{\text{Heat extracted}}{\text{Work done}}=\frac{{{Q}_{2}}}{W}=\frac{{{Q}_{2}}}{{{Q}_{1}}-{{Q}_{2}}}\]

A perfect refrigerator is one which transfers heat from cold to hot body without doing work

i.e. \[W=0\] so that \[{{Q}_{1}}={{Q}_{2}}\] and hence \[\beta =\infty \]

(1) Carnot refrigerator : For Carnot refrigerator \[\frac{{{Q}_{1}}}{{{Q}_{2}}}=\frac{{{T}_{1}}}{{{T}_{2}}}\]

\[\Rightarrow \] \[\frac{{{Q}_{1}}-{{Q}_{2}}}{{{Q}_{2}}}=\frac{{{T}_{1}}-{{T}_{2}}}{{{T}_{2}}}\] or \[\frac{{{Q}_{2}}}{{{Q}_{1}}-{{Q}_{2}}}=\frac{{{T}_{2}}}{{{T}_{1}}-{{T}_{2}}}\]

So coefficient of performance \[\beta =\frac{{{T}_{2}}}{{{T}_{1}}-{{T}_{2}}}\]

where \[{{T}_{1}}=\] temperature of surrounding, \[{{T}_{2}}=\] temperature of cold body. It is clear that \[\beta =0\] when \[{{T}_{2}}=0\]

i.e. the coefficient of performance will be zero if the cold body is at the temperature equal to absolute zero.

(2) Relation between coefficient of performance and efficiency of refrigerator 

We know \[\beta =\frac{{{Q}_{2}}}{{{Q}_{1}}-{{Q}_{2}}}\]\[=\frac{{{Q}_{2}}/{{Q}_{1}}}{1-{{Q}_{2}}/{{Q}_{1}}}\]                                 ... (i)

But the efficiency \[\eta =1-\frac{{{Q}_{2}}}{{{Q}_{1}}}\] or \[\frac{{{Q}_{2}}}{{{Q}_{1}}}=1-\eta \]            ...(ii)

From (i) and (ii) we get, \[\beta =\frac{1-\eta }{\eta }\]