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?

  • $\begingroup$ What is the book that shows this result? $\endgroup$ – Kyle Kanos Apr 18 '14 at 13:05
  • $\begingroup$ Actually, it's my coaching institute's textbook $\endgroup$ – user34304 Apr 18 '14 at 13:06
  • $\begingroup$ And which specific heat, $C_V$ or $C_p$, is that in the equation? $\endgroup$ – Kyle Kanos Apr 18 '14 at 13:12
  • $\begingroup$ Neither, the book just says C $\endgroup$ – user34304 Apr 18 '14 at 13:44
  • 3
    $\begingroup$ @KyleKanos It is specific heat of the given process $\endgroup$ – 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 $$P_iV_i^z=P_fV_f^z$$

$i = \text{initial}$


Focusing on integral only,

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


Note that the $PV^z$ is same for initial and final step. So, we write 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,


| cite | improve this answer | |
  • $\begingroup$ I know it's bad to spoon feed, but I am not able to get my equation in the end. Can u pls help $\endgroup$ – user34304 Apr 18 '14 at 14:55
  • $\begingroup$ @user34304 were you able to convert that $C_v$ expression? $\endgroup$ – evil999man Apr 18 '14 at 15:45
  • $\begingroup$ No, sorry. I tried, pls believe me. $\endgroup$ – user34304 Apr 18 '14 at 15:46
  • $\begingroup$ @user34304 both the integrals and $C_v$? will edit $\endgroup$ – evil999man Apr 18 '14 at 15:46
  • 1
    $\begingroup$ @user34304 That was rather obvious! :P $\endgroup$ – 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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.