Formula for molar specific heat capacity in polytropic process

I found this formula for a polytropic process, defined by $PV^n = {\rm constant}$, in a book:

$$C = \frac R{\gamma-1} + \frac R{1-n}$$ where $C$ is molar specific heat and $\gamma$ is adiabatic exponent. I do not know how it was derived, can someone guide me?

• What is the book that shows this result? – Kyle Kanos Apr 18 '14 at 13:05
• Actually, it's my coaching institute's textbook – user34304 Apr 18 '14 at 13:06
• And which specific heat, $C_V$ or $C_p$, is that in the equation? – Kyle Kanos Apr 18 '14 at 13:12
• Neither, the book just says C – user34304 Apr 18 '14 at 13:44
• @KyleKanos It is specific heat of the given process – evil999man Apr 18 '14 at 14:40

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 $$P_iV_i^z=P_fV_f^z$$

$i = \text{initial}$

$f=\text{final}$

Focusing on integral only,

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

$$PV^z\left[\frac{V^{-z+1}}{-z+1}\right]^{V_f}_{V_i}$$

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

$$-\frac{P_iV_i^zV_i^{-z+1}}{-z+1}+\frac{P_fV_f^zV_f^{-z+1}}{-z+1}$$

$$-\frac{P_iV_i}{-z+1}+\frac{P_fV_f}{-z+1}$$

Note that $PV=nRT$

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

where $\Delta T=T_f-T_i$

Final equation :

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

$$C=C_v+\frac{R}{1-z}$$

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

$$C_p/C_v=\gamma$$

$$C_p-C_v=R$$

Using $C_p=\gamma C_v$,

$$C_v\left(\gamma-1\right)=R$$

$$C_v=\frac{R}{\gamma-1}$$

Substituting in original equation,

$$C=\frac{R}{\gamma-1}+\frac{R}{1-z}$$

• I know it's bad to spoon feed, but I am not able to get my equation in the end. Can u pls help – user34304 Apr 18 '14 at 14:55
• @user34304 were you able to convert that $C_v$ expression? – evil999man Apr 18 '14 at 15:45
• No, sorry. I tried, pls believe me. – user34304 Apr 18 '14 at 15:46
• @user34304 both the integrals and $C_v$? will edit – evil999man Apr 18 '14 at 15:46
• @user34304 That was rather obvious! :P – evil999man Apr 18 '14 at 16:56

We can also derive the result without integration:

$$PV^n=constant$$ can be written as $$TV^{n-1}=constant$$ $$nCdT=dU+PdV$$ Dividing this equation throughout by $$dT$$, differentiating $$TV^{n-1} = constant$$ with respect to temperature, and plugging $${dV/dT}$$ into the equation will give the desired result.