Question

Prove that work done by an ideal gas in an adiabatic process is $W=\frac{P_i{V}_i-P_f{V}_f}{\gamma -1}$, using the intergration $\int PdV$

Answer 1

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

$⟹$ $P=V^{-\gamma }K$ ...............(1)

Work done in adiabatic process, $W=\int PdV=\int K{V}^{-\gamma }dV$

$⟹$ $W=K\times \frac{V^{1-\gamma }}{1-\gamma }{|}_{V_i}^{V_f}=\frac{\left(K{V}_f^{-\gamma }\right)V_f-\left(K{V}_i^{-\gamma }\right)V_i}{1-\gamma }$

Using (1) we get $W=\frac{P_f{V}_f-P_i{V}_i}{1-\gamma }$