Question

Derive an expression for work done in adiabatic expansion

Answer 1

Adiabatic process : $P{V}^{\gamma }=K$

So, $P=K{V}^{-\gamma }$

Work done $W=\int PdV$

Or $W=\int K{V}^{-\gamma }dV$

Or $W=K\times \frac{V^{-\gamma +1}}{1-\gamma }{|}_{V_1}^{V_2}$

Or $W=\frac{K}{1-\gamma }\times \left[V_2^{-\gamma +1}-V_1^{-\gamma +1}\right]$

Or $W=\frac{1}{1-\gamma }\times \left[K{V}_2^{-\gamma +1}-K{V}_1^{-\gamma +1}\right]$

Or $W=\frac{1}{1-\gamma }\times \left[P_2{V}_2^{\gamma }{V}_2^{-\gamma +1}-P_1{V}_1^{\gamma }{V}_1^{-\gamma +1}\right]$

Or $W=\frac{P_2{V}_2-P_1{V}_1}{1-\gamma }$