The pressure \(( P )\) and temperature \(( T )\) relationship of an ideal gas obeys the equation \(PT ^2=\) constant. The volume expansion coefficient of the gas will be:

\(P =\frac{ nRT }{ V } \)

\( PT ^2=\frac{ nRT }{ V } \times T ^2=\text { constant }\)

\(\Rightarrow T ^3= KV \)

\(\text { So, } \frac{ d }{ dT }( KV )=3 T ^2\)

\(\Rightarrow \frac{ KdV }{ dT }=3 T ^2\)

\(\Rightarrow dV =\frac{3 T ^2}{ K } dT\)

\(dV = V \gamma dT \quad \Rightarrow \gamma=\frac{3 T ^2}{ KV ^2}=\frac{3 T ^2}{ T ^3}=\frac{3}{ T }\)

\(\Rightarrow \gamma V =\frac{3 T ^2}{ K }\)