Answer to Question #234954 in Mechanical Engineering for raz

Question #234954

The ratio of specific heats is k=Cp/Cv=R, a constant. Combine these two expressions and show that Cv=R/(k-1) and Cp=kR/(k-1)


1
Expert's answer
2021-09-10T00:02:45-0400

We know that


CpCV=R, k=CpCV.C_p-C_V=R,\\\space\\ k=\frac{C_p}{C_V}.

Therefore:


CV=CpR,Cp=kCV.CV=kCVR,CV(1k)=R, CV=Rk1.C_V=C_p-R,\\ C_p=kC_V.\\ C_V=kC_V-R,\\ C_V(1-k)=-R,\\\space\\ C_V=\frac{R}{k-1}.

For CpC_p, we have


Cp=R+CV, CV=Cpk. Cp=R+Cpk, Cp(11k)=R, Cp(k1k)=R, Cp=kRk1.C_p=R+C_V,\\\space\\ C_V=\frac{C_p}{k}.\\\space\\ C_p=R+\frac{C_p}{k},\\\space\\ C_p\bigg(1-\frac 1k\bigg)=R,\\\space\\ C_p\bigg(\frac {k-1}{k}\bigg)=R,\\\space\\ C_p=\frac{kR}{k-1}.


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