Question

An ideal gas is expanding such that \[ P^2T=\text{constant}. \] The coefficient of volume expansion of the gas is \[ (P=\text{Pressure},\; T=\text{Temperature}) \]

• A. \(\dfrac{1}{T}\)
• B. \(\dfrac{2}{T}\)
• C. \(\dfrac{3}{T}\)
• D. \(\dfrac{3}{2T}\) \bigskip

Hint:

If \[ V\propto T^n, \] then \[ \alpha = \frac{1}{V}\frac{dV}{dT} = \frac{n}{T}. \] Here, \[ V\propto T^{3/2}, \] hence \[ \alpha=\frac{3}{2T}. \]

Answer 1

The Correct Option is D

Concept: \(P^2T = \text{constant} \Rightarrow P \propto T^{-1/2}\). Ideal gas \(V = nRT/P \propto T/T^{-1/2} = T^{3/2}\). \(\alpha = \frac{1}{V}\frac{dV}{dT} = \frac{1}{CT^{3/2}} \cdot \frac32 CT^{1/2} = \frac{3}{2T}\).

Step 1: Write the final answer. \(\boxed{\alpha=\frac{3}{2T}}\)