An ideal gas undergoes through following cyclic process.

1-2: Reversible adiabatic compression from ${\mathrm{P}}_1{\mathrm{V}}_1{\mathrm{T}}_1$ to ${\mathrm{P}}_2{\mathrm{V}}_2{\mathrm{T}}_2$

2-3 : Reversible isochoric heating from ${\mathrm{P}}_2{\mathrm{V}}_2{\mathrm{T}}_2$ to ${\mathrm{P}}_3{\mathrm{V}}_3{\mathrm{T}}_3$

3-4 : Reversible adiabatic expansion from ${\mathrm{P}}_3{\mathrm{V}}_3{\mathrm{T}}_3$ to ${\mathrm{P}}_4{\mathrm{V}}_4{\mathrm{T}}_4$

4-1 : Reversible isochoric cooling from ${\mathrm{P}}_4{\mathrm{V}}_4{\mathrm{T}}_4$ to ${\mathrm{P}}_1{\mathrm{V}}_1{\mathrm{T}}_1$

Efficiency of the cycle is :

$\begin{array}{l}{\mathrm{V}}_2={\mathrm{V}}_3\\ {\mathrm{w}}_{1-2}=\frac{\mathrm{nR}\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)}{\mathrm{\gamma }-1}\\ {\mathrm{w}}_{3-4}=\frac{\mathrm{nR}\left({\mathrm{T}}_4-{\mathrm{T}}_3\right)}{\mathrm{\gamma }-1}\\ {\mathrm{T}}_1{\mathrm{V}}_1^{\mathrm{\gamma }-1}={\mathrm{T}}_2{\mathrm{V}}_2^{\mathrm{\gamma }-1};{\mathrm{P}}_1{\mathrm{V}}_1^{\mathrm{\gamma }-1}={\mathrm{P}}_2{\mathrm{V}}_2^{\mathrm{\gamma }-1}\\ \frac{{\mathrm{P}}_3}{{\mathrm{T}}_3}=\frac{{\mathrm{P}}_2}{{\mathrm{T}}_2};{\mathrm{T}}_3{\mathrm{V}}_2^{\mathrm{\gamma }-1}={\mathrm{T}}_4{\mathrm{V}}_1^{\mathrm{\gamma }-1};\end{array}$

${\mathrm{P}}_3{\mathrm{V}}_2^{\mathrm{\gamma }-1}={\mathrm{P}}_4{\mathrm{V}}_1^{\mathrm{\gamma }-1}$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{{\mathrm{P}}_1}{{\mathrm{P}}_2}=\frac{{\mathrm{P}}_4}{{\mathrm{P}}_3};\frac{{\mathrm{P}}_4}{{\mathrm{P}}_1}=\frac{{\mathrm{T}}_4}{{\mathrm{T}}_1};\frac{{\mathrm{T}}_3}{{\mathrm{T}}_2}=\frac{{\mathrm{T}}_4}{{\mathrm{T}}_1}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{{\mathrm{T}}_3-{\mathrm{T}}_2}{{\mathrm{T}}_2}=\frac{{\mathrm{T}}_4-{\mathrm{T}}_1}{{\mathrm{T}}_1}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{{\mathrm{T}}_4-{\mathrm{T}}_1}{{\mathrm{T}}_3-{\mathrm{T}}_2}=\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}\end{array}$

$q_{\mathit{\text{supplied }}}\mathit{=}n{C}_{\mathit{v}}\left(T_3-T_2\right)\mathit{=}\frac{\mathit{n}\mathit{R}}{\mathit{\gamma }\mathit{-}\mathit{1}}\left(T_3-T_2\right)$

Work obtained $={\mathrm{w}}_{1-2}+{\mathrm{w}}_{3-4}$

$\begin{array}{r}=\frac{\mathrm{nR}}{\mathrm{\gamma }-1}\left[{\mathrm{T}}_2-{\mathrm{T}}_1+{\mathrm{T}}_4-{\mathrm{T}}_3\right]\\ \mathrm{\eta }=\frac{\left|\mathrm{w}\right|}{\mathrm{q}}=1-\left(\frac{{\mathrm{T}}_4-{\mathrm{T}}_1}{{\mathrm{T}}_3-{\mathrm{T}}_2}\right)\\ =1-\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}=1-{\left(\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1}\right)}^{\mathrm{\gamma }-1}\\ {\mathrm{q}}_{4-1}=\frac{\mathrm{nR}}{\mathrm{\gamma }-1}\left({\mathrm{T}}_1-{\mathrm{T}}_4\right)\end{array}$

$\Rightarrow \mathrm{\eta }=1-\left|\frac{{\mathrm{q}}_{4-1}}{{\mathrm{q}}_{2-3}}\right|$.