Prove that the probability, $P(L)$, that a molecule in a gas travels a distance \newline L without suffering a collision is $P(L) = l^{-1} \exp(-L/l)$, where $l$ is the \newline mean free path of a molecule in the gas.
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The mean free path, l, is defined as the average distance a molecule travels before it collides with another molecule. Show more…
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Key Concepts
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It is shown in the kinetic theory of gases that the probability for the distance a molecule travels between collisions to be between $x$ and $x+d x$, is proportional to $e^{-x / \lambda} d x,$ where $\lambda$ is a constant. Show that the average distance between collisions (called the "mean free path") is $\lambda$. Find the probability of a free path of length $\geq 2 \lambda$.
Probability and Statistics
Continuous Distributions
(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 $$ P=e^{-1}-e^{-2}=0 \cdot 23 $$
Thermodynamics And Molecular Physics
Transport Phenomena
In a sample of ideal gas the probability that a gas molecule will travel a distance between r1 and r2 before colliding with another molecule is given by P(r1;r2) = ∫_{r1}^{r2} Ae^{-cr} dr where A and c are constant parameters to be determined. Note, ∫_{0}^{∞} r^{n}e^{-cr} dr = n!/c^{n+1}. (a) The probability that a gas molecule will travel between r = 0 and r → ∞ is one hundred percent, P(0; ∞) = 1. Use this to determine A in terms of c. (b) Compute the average distance traveled before colliding with another molecule, ⟨r⟩. (c) Equate the average distance to the mean free path, ⟨r⟩ = λ, to determine the constant c in terms of λ. (d) Rewrite P(r1;r2) using the mean free path λ as the only parameter.
Frank D.
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