Problem Analysis

We are given an ideal gas expanding under a specific condition: $P T^{3} = \text{constant}$. We need to find the gas's coefficient of volume expansion, $\beta$. The ideal gas law is $PV = nRT$. The coefficient of volume expansion is defined as $\beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P$. However, the condition $P T^{3} = \text{constant}$ implies pressure ($P$) changes with temperature ($T$). Therefore, we calculate the effective coefficient along the given process path: $\beta_{eff} = \frac{1}{V} \frac{dV}{dT}$.

Calculating the Coefficient

1. Isolate Pressure (P): From the given condition $P T^{3} = K$ (where $K$ is a constant), we get: $ P = \frac{K}{T^3} $
2. Substitute P into the Ideal Gas Law: Using $PV = nRT$, substitute the expression for $P$: $ \left( \frac{K}{T^3} \right) V = nRT $
3. Solve for Volume (V): Rearrange the equation to express $V$ as a function of $T$: $ V = \frac{nRT^4}{K} $
4. Find the Rate of Change of Volume with Temperature: Differentiate $V$ with respect to $T$ along this specific process: $ \frac{dV}{dT} = \frac{d}{dT} \left( \frac{nRT^4}{K} \right) $ $ \frac{dV}{dT} = \frac{nR}{K} \frac{d}{dT}(T^4) = \frac{nR}{K} (4T^3) = \frac{4nRT^3}{K} $
5. Compute the Effective Volume Expansion Coefficient ($\beta_{eff}$): Use the formula $\beta_{eff} = \frac{1}{V} \frac{dV}{dT}$ and substitute the expressions for $V$ and $\frac{dV}{dT}$: $ \beta_{eff} = \frac{1}{\left( \frac{nRT^4}{K} \right)} \times \left( \frac{4nRT^3}{K} \right) $
6. Simplify the Result: Cancel terms to find the final coefficient: $ \beta_{eff} = \frac{K}{nRT^4} \times \frac{4nRT^3}{K} = \frac{4}{T} $

Final Answer

The coefficient of volume expansion for the gas under the given condition is $\frac{4}{T}$.