(a) The fraction of gas molecules which traverses distances...

(a) The fraction of gas molecules which traverses distances exceeding the mean free path without collision is just the probability to traverse the distance $s=\lambda$ without collision.
Thus
$$
P=e^{-1}=\frac{1}{e}=0.37
$$
(b) This probability is
$$
P=e^{-1}-e^{-2}=0 \cdot 23
$$

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Key Concepts

Survival Probability Calculation

The survival probability is the probability that a molecule traverses a given distance without colliding. This is computed using the exponential decay function by evaluating the exponential at that distance relative to the mean free path. Additionally, determining the probability of collisions occurring within a specific range involves calculating the difference between cumulative probabilities evaluated at the endpoints of the range.

Exponential Distribution

In the context of molecular collisions, the distances traveled between collisions follow an exponential distribution. This distribution describes the probability that a molecule will travel a certain distance without a collision, with the probability decaying exponentially with distance. It reflects the random, memoryless nature of collisions.

Mean Free Path

The mean free path is the average distance a molecule in a gas travels before colliding with another molecule. It is a fundamental concept in kinetic theory and helps in quantifying the molecular motion and the likelihood of collisions under given conditions such as pressure and temperature.

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