As ideal gas is expanding such that ${\mathrm{PT}}^3=$constant. The coefficient of volume expansion of the gas is:

$\begin{array}{l}{\mathrm{PT}}^3=\text{ constant }\\ \left(\frac{\mathrm{nRT}}{\mathrm{V}}\right){\mathrm{T}}^3=\text{ constant }\\ {\mathrm{T}}^4{\mathrm{V}}^{-1}=\text{ constant }\\ [\text{ using }\mathrm{PV}=\mathrm{nRT}]\end{array}$
Taking log on both sides,

 $4\ln \left(\mathrm{T}\right)-1\ln \left(\mathrm{V}\right)=\ln \left(\mathrm{constant}\right)$ 
Deafferenting both sides,
$\begin{array}{l}4\frac{\mathrm{dT}}{\mathrm{T}}=\frac{dV}{\mathrm{V}}\to \left(\mathrm{i}\right)\\ \mathrm{dV}=\mathrm{V\gamma \Delta T}\to \left(\mathrm{ii}\right)\end{array}$
From (i) & (ii)
$\mathrm{\gamma }=\frac{4}{\mathrm{T}}$