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 is the average distance a gas molecule travels between successive collisions. It depends on the density of the gas molecules and their cross-sectional area.
Step 2: Key Formula or Approach:
The theoretical formula for the mean free path (\(\lambda\) or \(\alpha\)) derived from kinetic theory is:
\[ \alpha = \frac{1}{\sqrt{2} \pi n d^2} \] Where \(n\) is the number density (molecules/volume) and \(d\) is the collision diameter.
Step 3: Detailed Explanation:
From the formula, we can isolate the dependencies on \(n\) and \(d\):
\[ \alpha \propto \frac{1}{n \cdot d^2} \] Rewrite this relationship using negative exponents:
\[ \alpha \propto n^{-1} d^{-2} \] Comparing this to the given expression \(\alpha \propto n^x d^y\), we can map the variables:
\(x = -1\)
\(y = -2\)
Step 4: Final Answer:
The values of \(x\) and \(y\) are -1 and -2 respectively.