Question

Derive the isentropic relation of state.

Answer 1

For a closed system, the total change in energy of a system is the sum of the work done and the heat added:

$dU=dW+dQ\\ dW=-pdV\\ dH=dU+pdV+Vdp\\ dU=dW+dQ= -pdV+0\\ \implies dU=nC_vdT\\ \implies dH=nC_pdT\\ dU=nC_vdT=-pdV\\ dH=nC_pdT=-Vdp\\ \implies \gamma= \frac{C_p}{C_v}= \frac{dp/p}{dV/V}\\ Hence \space \frac{p_2}{p_1}= (\frac{V_1}{V_2})^{\gamma}\\$