IGNOU PHE 6 2018 - the average height at which a molecule makes its last collision before

Question

the average height at which a molecule makes its last collision before crossing
the plane is

2λ.

3

27 Jan 2018

Answer :

Word Count : 354

You’re asking about the average height at which a molecule makes its last collision before crossing a plane, and the statement suggests the answer is $\mathbf{2\lambda/3}$, where $\lambda$ is the mean free path. Let’s solve it manually and justify it numerically. --- ### Step 1: Setup Let $z$ be the distance from the plane (the plane is at $z = 0$), and consider molecules moving towards it. The probability that a molecule travels a distance $z$ without collision is exponential: $$ P(z) = \frac{1}{\lambda} e^{-z/\lambda} $$ Here $\lambda$ is the mean free path. We want the average last collision height $\langle z \rangle$ before crossing the plane, which is given by the expected value: $$ \langle z \rangle = \int_0^\infty

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