Question

An ideal gas is expanding such that \( P T^3 = {constant} \). The coefficient of volume expansion of the gas is:

• A. \( \frac{1}{T} \)
• B. \( \frac{2}{T} \)
• C. \( \frac{4}{T} \)
• D. \( \frac{3}{T} \)

Hint:

The coefficient of volume expansion for an ideal gas can be derived using the relationship between pressure, temperature, and volume. For a gas expanding with \( P T^3 = {constant} \), \( \beta = \frac{4}{T} \).

Answer 1

The Correct Option is C

Step 1: The equation of state is given by the relation: \[ P T^3 = {constant}. \] This indicates that the product of the pressure \( P \) and the cube of the temperature \( T \) remains constant during the expansion. 
Step 2: The coefficient of volume expansion \( \beta \) is defined as: \[ \beta = \frac{1}{V} \left( \frac{\Delta V}{\Delta T} \right)_P, \] which can be related to the change in temperature under constant pressure. 
Step 3: To relate the volume expansion to the temperature, we differentiate the given equation \( P T^3 = {constant} \). Differentiating with respect to \( T \) gives: \[ P \cdot 3T^2 \cdot \frac{dT}{dT} + T^3 \cdot \frac{dP}{dT} = 0 \quad \Rightarrow \quad \frac{dP}{dT} = -\frac{3P}{T}. \] 
Step 4: The relationship for the volume change is now obtained by considering that: \[ \frac{dV}{V} = \beta \, dT \quad \Rightarrow \quad \beta = \frac{4}{T}. \] 
Step 5: Hence, the coefficient of volume expansion is \( \beta = \frac{4}{T} \).