Coefficient of Volume Expansion for Ideal Gas Expanding via PT Power Three Constant

The coefficient of volume expansion $\gamma$ for a gas undergoing a thermodynamic process is defined as: $\gamma = \frac{1}{V} \frac{dV}{dT}$

For an ideal gas, the equation of state is given by: $PV = nRT \implies P = \frac{nRT}{V}$

The process equation given in the question is: $P T^3 = C \quad (\text{where } C \text{ is a constant})$

Substituting the expression for pressure $P$ into the given relation: $\left(\frac{nRT}{V}\right) T^3 = C$ $\frac{nRT^4}{V} = C$ $V = \frac{nR}{C} T^4$

Now, differentiating volume $V$ with respect to temperature $T$: $\frac{dV}{dT} = \frac{nR}{C} \cdot \frac{d}{dT}(T^4) = \frac{4 n R T^3}{C}$

Substituting $\frac{dV}{dT}$ and $V$ into the expression for $\gamma$: $\gamma = \frac{1}{V} \frac{dV}{dT} = \frac{1}{\left(\frac{nRT^4}{C}\right)} \left(\frac{4nRT^3}{C}\right) = \frac{4}{T}$

Hence, the coefficient of volume expansion of the gas is $\frac{4}{T}$.

Correct Answer: Option C ($\frac{4}{T}$)