Formula for molar specific heat capacity in polytropic process

That $C$ is the specific heat for the given cycle, i.e. $$dQ=nCdT$$ This is for $n$ moles of gas.(not the $n$ you stated in question)

I will assume $$PV^z=\text{constant}$$

$$nCdT=dU+PdV$$ $$\int nCdT=\int nC_vdT+\int PdV$$

$$nC\Delta T=nC_v \Delta T+\int \frac{PV^z}{V^z}dV$$

As numerator is a constant, take it out!

Also note that $$P_iV_i^z=P_fV_f^z$$

$i = \text{initial}$


Focusing on integral only,

$$PV^z\int V^{-z}dV$$


Note that the $PV^z$ is same for initial and final step. So, we multiply it inside and do this ingenious work:



Note that $PV=nRT$

$$\frac{nR\Delta T}{-z+1}$$

where $\Delta T=T_f-T_i$

Final equation :

$$nC\Delta T=nC_v \Delta T+\frac{nR\Delta T}{-z+1}$$


This will bring you the original equation, you can find $C_v$ by



Using $C_p=\gamma C_v$,



Substituting in original equation,