Key Idea Poisson's equation for adiabatic process is given by
$p V^{\gamma}$ =constant
For adiabatic process, Poisson’s equation is given by
$PV^{\gamma}$ = constant $\dots (i)$
Ideal gas relation is
$pV=RT$
$\Rightarrow V=\frac{RT}{p}\, \dots(ii)$
From Eqs. (i) and (ii), we get
$p\left(\frac{RT}{p}\right)^{\gamma}=$ constant
$\Rightarrow \frac{T^{\gamma}}{p^{\gamma-1}}=$ constant $\dots (iii)$
where $\gamma$ is ratio of specific heats of the gas
Given, $p \propto T^{C}$ $\dots (iv)$
On comparing with Eq. (iii), we have
$C=\frac{\gamma}{\gamma-1}$
For a monoatomic gas $\gamma=\frac{5}{3}$
We have
$C=\frac{\frac{5}{3}}{\frac{5}{3}-1}=\frac{5}{2}$