Answer to Question #250022 in Electrical Engineering for Mangi

Question #250022

Derive the isentropic relation of state.


1
Expert's answer
2021-10-13T02:54:01-0400

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+dQdW=pdVdH=dU+pdV+VdpdU=dW+dQ=pdV+0    dU=nCvdT    dH=nCpdTdU=nCvdT=pdVdH=nCpdT=Vdp    γ=CpCv=dp/pdV/VHence p2p1=(V1V2)γdU=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}\\


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