Question

The mean free path of moving gas molecules is expressed as $\lambda \propto n^x d^y$ where $n$ is the number of molecules per unit volume and $d$ is the size of the molecules. Then the values of $x$ and $y$ respectively, are

• A. 2,-2
• B. -1,-2
• C. 1,2
• D. 1,-2
• E. -2,-2

Hint:

Remember that as the number of molecules ($n$) or their size ($d$) increases, the chance of collision increases, thereby decreasing the mean free path. This implies inverse proportionality for both.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The mean free path ($\lambda$) is the average distance traveled by a moving particle between successive collisions. It depends on the concentration of particles and their collision cross-section.
Key Formula or Approach:
The standard formula for mean free path is:
\[ \lambda = \frac{1}{\sqrt{2} \pi n d^2} \]

Step 2: Detailed Explanation:
From the formula, we see that:
$\lambda \propto \frac{1}{n} \implies \lambda \propto n^{-1}$
$\lambda \propto \frac{1}{d^2} \implies \lambda \propto d^{-2}$
The expression given is $\lambda \propto n^x d^y$.
By comparison:
$x = -1$
$y = -2$

Step 3: Final Answer:
The values are $x = -1$ and $y = -2$.