If n is the number density and d is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by :
- \( \frac{1}{\sqrt{2 n \pi d^2}} \)
- \( \sqrt{2 n \pi d^2} \)
- \( \frac{1}{\sqrt{2 \pi d^2}} \)
- \( \frac{1}{\sqrt{2 n^2 \pi^2 d^2}} \)
The Correct Option is C
Approach Solution - 1
The mean free path \( \lambda \) of a molecule is defined as the average distance that a molecule travels between two successive collisions. It is given by the formula:
\[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n}, \]
where:
- \( n \) is the number density of molecules (i.e., the number of molecules per unit volume),
- \( d \) is the diameter of the molecule,
- \( \pi \) is the mathematical constant.
Explanation: The formula for the mean free path is derived from kinetic theory, considering the probability of collisions between molecules in a given volume. The factor \( \sqrt{2} \) accounts for the random distribution of molecular velocities and the likelihood of collisions occurring.
Thus, the average distance covered by a molecule between two successive collisions is represented by:
\[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n}. \]
Therefore, the correct option is (3).
Approach Solution -2
The mean free path (λ) is the average distance a gas molecule travels between two successive collisions. It depends on the number density (n) of molecules and their diameter (d).
Step 2: Formula for mean free path.
The general formula for mean free path is given by:
λ = 1 / (√2 × π × d² × n)
Step 3: Understanding the terms.
Here,
- n = number of molecules per unit volume (number density)
- d = diameter of a molecule
- √2 accounts for the relative motion of molecules during collisions
Step 4: Simplify expression.
The expression for mean free path can be represented as:
λ = 1 / (√2 π d² n)
Final Answer: \( \frac{1}{\sqrt{2 \pi d^2 n}} \)
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