An ideal gas at pressure P and temperature T is expanding such that...

Given the process equation for the ideal gas:

$PT^3 = \text{constant}$

From the ideal gas equation, $PV = nRT$, which gives $P = \dfrac{nRT}{V}$.

Substituting $P$ into the process equation:

\left(\dfrac{nRT}{V}\right) T^3 = \text{constant}$<br><br>$\dfrac{T^4}{V} = \text{constant}$<br><br>$V = k T^4$<br><br>Differentiating both sides with respect to temperature$T$:<br><br>$\dfrac{dV}{dT} = 4k T^3$<br><br>The coefficient of volume expansion$\gamma$is defined as:<br><br>$\gamma = \dfrac{1}{V} \dfrac{dV}{dT}$<br><br>Substituting the values of$V and \dfrac{dV}{dT}$:<br><br>$\gamma = \dfrac{1}{k T^4} (4k T^3) = \dfrac{4}{T}$<br><br>Answer:$\dfrac{4}{T}