As ideal gas is expanding such that PT³ = constant. The coefficient of volume expansion of the gas is:

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

$4\times \ln \mathrm{T}-1\times \ln \mathrm{v}=\ln \left(\text{ constant}\right)$
$\begin{array}{r}\Rightarrow 4\frac{\mathrm{dT}}{\mathrm{T}}=\frac{\mathrm{dV}}{\mathrm{V}} \to (\text{i) }\\ \mathrm{dV}=\mathrm{\gamma V}\left(\mathrm{dT}\right) \to \left(\text{ ii }\right)\end{array}$
From (i) & (ii),

$\mathrm{\gamma }=\frac{4}{\mathrm{T}}$