Answer to Question #234954 in Mechanical Engineering for raz

Question

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)

Answer 1

We know that

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

Therefore:

$$C_V=C_p-R,\\ C_p=kC_V.\\ C_V=kC_V-R,\\ C_V(1-k)=-R,\\\space\\ C_V=\frac{R}{k-1}.$$

For $C_p$, we have

$$C_p=R+C_V,\\\space\\ C_V=\frac{C_p}{k}.\\\space\\ C_p=R+\frac{C_p}{k},\\\space\\ C_p\bigg(1-\frac 1k\bigg)=R,\\\space\\ C_p\bigg(\frac {k-1}{k}\bigg)=R,\\\space\\ C_p=\frac{kR}{k-1}.$$