An ideal gas at pressure \(P\) and temperature \(T\) is expanding such that \(PT^3 = \) constant. The coefficient of volume expansion of the gas is _______.

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:

\(\left(\dfrac{nRT}{V}\right) T^3 = \text{constant}\)

\(\dfrac{T^4}{V} = \text{constant}\)

\(V = k T^4\)

Differentiating both sides with respect to temperature \(T\):

\(\dfrac{dV}{dT} = 4k T^3\)

The coefficient of volume expansion \(\gamma\) is defined as:

\(\gamma = \dfrac{1}{V} \dfrac{dV}{dT}\)

Substituting the values of \(V\) and \(\dfrac{dV}{dT}\):

\(\gamma = \dfrac{1}{k T^4} (4k T^3) = \dfrac{4}{T}\)

Answer: \(\dfrac{4}{T}\)