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## Homework Statement

Derive an expression for the mean collision time in a gas where the collision cross-section is σ and the number density is n.

## The Attempt at a Solution

I'm just using my book to do this, and I can't get past the first bit...

It says

Consider a particular molecule moving at speed v with all other molecules in the gas stationary. In a time dt, the molecule sweeps out a volume σvdt, and if another molecule lies inside this volume, there will be a collision. With n molecules per unit volume, the probability of a collision in time dt is therefore nσvdt.

I'm sure the quantity nσvdt is just the number of molecules in the volume it sweeps out in time dt, so how does this give a probability? And it can't be normalized can it?

The remainder of the derivation is

define P(t) as the probability of a molecule not colliding up to time t.

Then P(t+dt)=P(t)+(dP/dt)dt

However P(t+dt) is the probability of a molecule not colliding up to time t multiplied by not colliding in time dt, i.e

P(t+dt)=P(t)(1-nσvdt)

Then

(1/P)dP/dt=-nσv

P(t)=exp(-nσvt)

The probability of not colliding up to time t, then colliding in the next dt is

P(t)nσvdt=exp(-nσvt)nσvdt

which is a normalized probability distribution with mean time 1/nσv which is the required result.

This all makes sense assuming the first bit, but I just can't see why it is right, can anyone help please?

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