Question

Two boxes A and B contain an equal number of molecules of the same gas. If the volumes are $V_A$ and $V_B$, and $\lambda_A$ and $\lambda_B$ denote respective mean free paths, then

• A. $\lambda_A = \lambda_B$
• B. $\dfrac{\lambda_A}{V_A} = \dfrac{\lambda_B}{V_B}$
• C. $\dfrac{\lambda_A}{V_A^{1/3}} = \dfrac{\lambda_B}{V_B^{1/3}}$
• D. $\lambda_A V_A = \lambda_B V_B$

Hint:

When the number of molecules is fixed, number density scales as $1/V$, and mean free path scales directly with volume.

Answer 1

The Correct Option is B

Step 1: Use the formula for mean free path.
Mean free path for a gas: $\lambda \propto \dfrac{1}{n}$, where $n$ is number density.
Since both boxes contain equal number of molecules, $n_A = \dfrac{N}{V_A},\; n_B = \dfrac{N}{V_B}$.

Step 2: Relate the mean free paths.
$\lambda_A \propto \dfrac{V_A}{N}, \lambda_B \propto \dfrac{V_B}{N}$.
Thus $\lambda_A \propto V_A,\; \lambda_B \propto V_B$.

Step 3: Compare with molecular spacing.
Mean molecular separation $\propto V^{1/3}$. Hence the ratio $\lambda / V^{1/3}$ is constant for equal number of molecules.

Step 4: Conclusion.
Therefore, $\dfrac{\lambda_A}{V_A^{1/3}} = \dfrac{\lambda_B}{V_B^{1/3}}$.