In a large refrigeration plant it is necessary to compress a fluid, which we will assume to be an ideal gas with constant heat capacity, from a low pressure P₁ to a much higher pressure P₂.

a. If the compression is done in a single compressor that operates reversibly and adiabatically, obtain an expression for the work needed for the compression in terms of the mass flow rate, P₁, P₂, and the initial temperature, T₁. 

b. If the compression is to be done in two stages, first compressing the gas from P₁ to P^∗, then cooling the gas at constant pressure down to the compressor inlet temperature T₁, and then compressing the gas to P₂, develop an expression for the work needed for the compression. What should the value of the intermediate pressure be to accomplish the compression with minimum work?