Question

Find the molar heat capacity of an ideal gas with adiabatic exponent ${}^{\prime }{\gamma }^{\prime }$ for the polytropic process $P{V}^n=constant$.

Answer 1

We have, using the first law of thermodynamics
$C=C_v+\frac{PdV}{n^{\prime }dT}(n=$ number of moles)
We have, $P{V}^n=constant$
From the ideal gas equation, $PV=nRT$
Taking ratio, $T{V}^{n-1}=Constant$
Differentiating we get $\frac{dV}{dT}=-\frac{V}{T(n-1)}$
Putting it in the equation for ${}^{\prime }{C}^{\prime }$.
$C=C_v-\frac{PV}{n^{\prime }T(n-1)}=C_v-\frac{n^{\prime }RT}{n^{\prime }T(n-1)}=C_v-\frac{R}{n-1}$
$C=\frac{R}{\gamma -1}-\frac{R}{n-1}$.