Find the molar heat capacity of an ideal gas in a polytropic process pVn = const if the adiabatic exponent of the gas is equal to γ.
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That C is specific heat for given cycle, i.e. dQ=nCdT. This is for n moles of gas. Let assume, PVz=constant or, nCdT=dU+PdV or, ∫nCdT=∫nCvdT+∫PdV or, nCΔT=nCvΔT+∫PVzVzdV. As the numerator is constant, we must take it out. PiVzi=PfVzf, i=initial, f=final. We focus on the integral only, PVz∫V−zdV or, PVz[V−z+1−z+1]VfVi. PVz is same for the initial and final step. −PiVziV−z+1i−z+1+PfVzfV−z+1f−z+1or, −PiVi−z+1+PfVf−z+1. We know, PV=nRT or, nRΔT−z+1 where ΔT=Tf−Ti. nCΔT=nCvΔT+nRΔT−z+1 or, C=Cv+R1−z. It would bring you the original equation, you can find Cv by Cp/Cv=γ or, Cp−Cv=R. Using Cp=γCv or, Cv(γ−1)=R. Cv=Rγ−1. After Substituting the original equation, we get C=Rγ−1+R1−.
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