An ideal gas at pressure $$P$$ and temperature $$T$$ is expanding such that $$PT^3 = \text{constant}$$. The coefficient of volume expansion of the gas is :

$$PT^3 = \text{constant}$$

For an ideal gas, the ideal gas equation of state is:

$$PV = nRT \implies P = \frac{nRT}{V}$$

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

$$\frac{T^4}{V} = \frac{\text{constant}}{nR}$$

Since $$n$$ (number of moles) and $$R$$ (universal gas constant) are constants, we can replace the entire right side of the equation with a new constant, let's call it $$K$$:

$$V = KT^4$$

volume expansion ($$\gamma$$)

$$\gamma = \frac{1}{V} \frac{dV}{dT}$$

$$\frac{dV}{dT} = \frac{d}{dT}(KT^4)$$

$$\frac{dV}{dT} = 4KT^3$$

$$\gamma = \frac{1}{KT^4} (4KT^3)$$

$$\gamma = \frac{4KT^3}{KT^4}$$

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