Derive the following expression (P 2 /P 1 ) s=const =(V 1 /V 2 ) k (idealgas) De

Question

Derive the following expression (P2/P1)s=const=(V1/V2)k            (idealgas) Derive the following expression (P2/P1)s=const=(V1/V2)k            (idealgas)

Explanation / Answer

dU = n*Cv*dT = q + W q = 0 for adiabatic processs n*Cv*dT = W = -PdV W =-PdV (PV work relationship to work) PV = nRT =>        PdV+ VdP = nRdT ndT = (P/R)dV + (V/R)dP n*Cv*dT = -PdV = (P)(Cv/R)dV + (V)*(Cv/R)dP R*-PdV = P*Cv*dV + V*Cv*dP -(P)*(Cv + R)*dV =V*Cv*dP                  recall that Cv + R = Cp   [ dH/dT = dU/dT +d(PV)/dT   and PV = nRT] -P*Cp*dV = V*Cv*dP -(Cp/Cv)*dV/V = dP/P Cp/Cv = k -k*ln(V2/V1) = ln(P2/P1) = -ln[V2/V1]k =ln[V1/V2]k P2/P1 = (V1/V2)k

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