For adiabatic process, Poisson's equation is given by
$P V^\gamma=\text { constant }$(1)
Ideal gas relation is
$\Rightarrow V=\frac{R T}{P}$(2)
From Eqs. (1) and (2), we get
$ P\left(\frac{R T}{P}\right)^\gamma=\text { constant }$
$\Rightarrow \frac{T^\gamma}{P^{\gamma-1}}=\text { constant }$(3)
where $\gamma$ is ratio of specific heats of the gas.
Given, $P \propto T^C$ (4)
On comparing with Eq. (3), we have, $C=\frac{\gamma}{\gamma-1}$
For a monoatomic gas $\gamma=\frac{5}{3}$
$\therefore $ We have, $C=\frac{\frac{5}{3}}{\frac{5}{3}-1}=\frac{5}{2}$