Derive an expression for co-efficient of performance of refrigerator.

Explanation

Coefficient of performance (COP)`(beta)`: COP is a measure of the efficiency of a refrigerator. It is defined as the ratio of heat extracted from the cold body (sink) to the external work done by the compressor W.
COP `= beta = (Q_(L))/W` ...(1)
From the equation `Q_(L)+W=Q_(H)`
`beta = (Q_(L))/(Q_(H) - Q_(L))`
`beta = (1)/((Q_(H))/(Q_(L)) - 1)` ...(2)
But we know that `(Q_(H))/(Q_(L)) = (T_(H))/(T_(L))`
Substituting this equation into equation (1) we get `beta = (1)/((T_(H))/(T_(L)) - 1) = (T_(L))/(T_(H) - T_(L))`