Step-by-step Solution:

1. Understand the Goal: We need to find the coefficient of volume expansion ($\\gamma$) for an ideal gas undergoing a specific process described by the equation $PT^2 = \\text{constant}$.

2. Definition of Coefficient of Volume Expansion: The coefficient of volume expansion for a specific process is defined as the fractional change in volume per unit change in temperature. Mathematically, it is expressed as: $\gamma = \frac{1}{V} \frac{dV}{dT}$ While the standard definition is for an isobaric (constant pressure) process, when a specific process is given, we calculate this value for that process.

3. Use the Ideal Gas Law: For an ideal gas, the equation of state is: $PV = nRT$ where $P$ is pressure, $V$ is volume, $T$ is temperature, $n$ is the number of moles, and $R$ is the universal gas constant. We can express the pressure $P$ from this equation as: $P = \frac{nRT}{V}$

4. Combine the Process Equation and the Ideal Gas Law: The given process is $PT^2 = K$, where $K$ is a constant. Substitute the expression for $P$ from the ideal gas law into this process equation to establish a relationship between volume $V$ and temperature $T$: $\left( \frac{nRT}{V} \right) T^2 = K$ $\frac{nRT^3}{V} = K$

5. Express Volume as a Function of Temperature: Rearrange the equation from the previous step to solve for $V$: $V = \left( \frac{nR}{K} \right) T^3$ Since $n$, $R$, and $K$ are all constants, we can let $C = \frac{nR}{K}$. The relationship simplifies to: $V = CT^3$ This shows that for this particular process, the volume of the gas is proportional to the cube of its absolute temperature.

6. Calculate the Derivative $\frac{dV}{dT}$: Differentiate the expression for $V$ with respect to $T$: $\frac{dV}{dT} = \frac{d}{dT}(CT^3) = C \cdot (3T^2) = 3CT^2$

7. Calculate the Coefficient of Volume Expansion: Now, substitute the expressions for $V$ and $\frac{dV}{dT}$ into the formula for $\\gamma$: $\gamma = \frac{1}{V} \frac{dV}{dT} = \frac{1}{CT^3} (3CT^2)$ The constant $C$ and the term $T^2$ cancel out: $\gamma = \frac{3}{T}$

8. Conclusion: The coefficient of volume expansion for the gas undergoing the process $PT^2 = \\text{constant}$ is $\frac{3}{T}$. This matches option C.