Question

\[-w = \frac{P_1 V_1 - P_2V_2}{\gamma - 1}\]

Answer 1

Alright, so the P-V curve is given by \(PV^{\gamma} = \rm constant \). Fair enough.

Answer 2

this looks bit like thermodynamics, reverse adiabatic expansion?

Answer 3

Yes.

Answer 4

\[w = - \int P dV = - {\rm constant }\int \limits _{V_1}^{V_2} \frac{dV}{V^{\gamma }} = -{\rm constant } \left[ \frac{V^{1 - \gamma }}{1- \gamma } \right]_{V_1 }^{V_2}\]\[= {\rm constant }\frac{V_2^{1-\gamma}-V_1^{1-\gamma }}{\gamma - 1}\]\[= \frac{PV^{\gamma} (V_2^{1-\gamma} - V_1^{1-\gamma})}{\gamma-1 }\]\[= \frac{P_2 V_2 - P_1 V_1}{\gamma - 1}\]Incredible, I did it!