Question

An ideal gas is expanding such that \( PT^2 = \text{constant} \). The coefficient of volume expansion of the gas is

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

Hint:

If \( V \propto T^n \), then coefficient of volume expansion is \( \frac{n}{T} \).

Answer 1

The Correct Option is D

Step 1: Given relation.
\[ PT^2 = \text{constant} \]

Step 2: Express pressure in terms of temperature.
\[ P \propto \frac{1}{T^2} \]

Step 3: Use ideal gas equation.
\[ PV = nRT \]

Step 4: Substitute \( P \).
\[ \frac{1}{T^2} \cdot V \propto T \]

Step 5: Simplify relation.
\[ V \propto T^3 \]

Step 6: Define coefficient of volume expansion.
\[ \beta = \frac{1}{V}\frac{dV}{dT} \]

Step 7: Differentiate.
Since \( V \propto T^3 \):
\[ \beta = \frac{3}{T} \]