The net work done in the cycle is given as:
\(W = \left( {{P_2} - {P_1}} \right)\left( {{V_2} - {V_1}} \right)\)
Substitute all the values in the above equation.
\(\begin{aligned}W = \left( {2{P_0} - {P_0}} \right)\left( {2{V_0} - {V_0}} \right)\\W = {P_0}{V_0}\end{aligned}\)
The heat input of the engine is given as:
\(\begin{aligned}Q = \frac{{nR\left( {{T_2} - {T_1}} \right)}}{{\gamma - 1}} + \frac{{\gamma nR\left( {{T_3} - {T_2}} \right)}}{{\gamma - 1}}\\Q = \frac{{nR}}{{\gamma - 1}}\left( {{T_2} - {T_1} + \gamma \left( {{T_3} - {T_2}} \right)} \right)\\Q = \frac{{nR}}{{\gamma - 1}}\left( {\gamma \left( {{T_3} - 2{T_2}} \right) - {T_1}} \right)\end{aligned}\)
Here,\(R\)is the universal gas constant and\(\gamma \)is the specific heat ratio for diatomic gas.
The thermal efficiency of engine is given as:
\(\eta = \frac{W}{Q}\)
Substitute all the values in the above equation.
\(\begin{aligned}\eta = \frac{{{P_0}{V_0}}}{{\left( {\frac{{nR}}{{\gamma - 1}}\left( {\gamma \left( {{T_3} - 2{T_2}} \right) - {T_1}} \right)} \right)}}\\\eta = \frac{{\left( {\gamma - 1} \right){P_0}{V_0}}}{{nR\left( {\gamma \left( {{T_3} - 2{T_2}} \right) - {T_1}} \right)}}\end{aligned}\)
Therefore, the thermal efficiency of engine is \(\frac{{\left( {\gamma - 1} \right){P_0}{V_0}}}{{nR\left( {\gamma \left( {{T_3} - 2{T_2}} \right) - {T_1}} \right)}}\).
