The work input of a compressor is minimum when the law followed by compression is

Explanation:

Consider a single-stage reciprocating air compressor:

We know that the area under the P-V graph gives us the work done.

In the graph, we can clearly see that adiabatic compression requires more work than the isothermal process and the remaining two processes given in the options will have their curve above isothermal because of the higher polytropic index(n). n isothermal compression, the temperature of the gas remains constant throughout the compression process. This means that all of the work input is used to increase the pressure of the gas, and none of it is wasted on increasing the temperature of the gas. In contrast, in adiabatic compression, the temperature of the gas increases during the compression process. This means that some of the work input is wasted on increasing the temperature of the gas, and less of it is available to increase the pressure of the gas.

Hence, we can conclude that isothermal compression requires the minimum work to be done.

Work done by a single-stage air compressor (PVn = C):

$W=\frac{n}{n-1}{P}_1{V}_1\left[{\left(\frac{P_2}{P_1}\right)}^{\frac{n-1}{n}}-1\right]$