If one mole of an ideal gas at ( P _ 1 , V _ 1 ) is allowed to...

$A - B =$ isothermal process

${W}_{AB} = {P}_{1}{V}_{1}\ln \left[\dfrac{2{V}_{1}}{{V}_{1}}\right] = {P}_{1}{V}_{1}\ln \left(2\right)$

$B - C \to$ Isochoric process

${W}_{BC} = 0$

$C - A \to$Adiabatic process

${W}_{CA} = \dfrac{{P}_{1}{V}_{1} - \dfrac{{P}_{1}}{4} \times 2{V}_{1}}{1 - \gamma} = \dfrac{{P}_{1}{V}_{1}\left[1 - \dfrac{1}{2}\right]}{1 - \gamma} = \dfrac{{P}_{1}{V}_{1}}{2\left(1 - \gamma\right)}$

${W}_{net} = {W}_{AB} + {W}_{BC} + {W}_{CA}\left\{{P}_{1}{V}_{1} = RT\right\}$

$= {P}_{1}{V}_{1}\ln \left(2\right) + 0 + \dfrac{{P}_{1}{V}_{1}}{2\left(1 - \gamma\right)}$

${W}_{net} = RT\left[\ln \left(2\right) - \dfrac{1}{2\left(\gamma - 1\right)}\right]$