Derive the isentropic relation of state.
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}\\dU=dW+dQdW=−pdVdH=dU+pdV+VdpdU=dW+dQ=−pdV+0⟹dU=nCvdT⟹dH=nCpdTdU=nCvdT=−pdVdH=nCpdT=−Vdp⟹γ=CvCp=dV/Vdp/pHence p1p2=(V2V1)γ
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