Question

What is the relation between the efficiency $\mathrm{\eta }$ of a heat engine and the co-efficient of performance $\mathrm{\alpha }$ of a refrigerator?

Answer 1

Step 1: Definition

A machine that turns heat into work is called a heat engine. The efficiency of such a heat engine is greatest for reversing engines such as the Carnot heat engine.

Step 2: The relation between the efficiency $\mathrm{\eta }$ of a heat engine and the co-efficient of performance $\mathrm{\alpha }$ of a refrigerator

We know,

The efficiency of a heat engine is given by

$$\begin{gathered} \mathrm{\eta }=1-\frac{{\mathrm{T}}_2}{{\mathrm{T}}_1} \\ =\frac{{\mathrm{T}}_1-{\mathrm{T}}_2}{{\mathrm{T}}_1} \\ =\frac{1}{1+{\displaystyle \frac{{\mathrm{T}}_2}{{\mathrm{T}}_1+{\mathrm{T}}_2}}} \\ =\frac{1}{1+\mathrm{\alpha }} \end{gathered}$$
Here $T_{1}and{T}_2$are temperatures

Therefore, The relation between the efficiency $\mathrm{\eta }$ of a heat engine and the co-efficient of performance $\mathrm{\alpha }$ of a refrigerator is $\mathrm{\eta }=\frac{1}{1+\mathrm{\alpha }}$.