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## Homework Statement

Show that the number of molecules hitting unit area of a surface per second with speeds between v and v+dv and angle between θ and θ+dθ to the normal is

dΦ=0.5vnf(v)dvsinθcosθdθ

where f(v) is the distribution of molecular speeds and n is the number density.

Hence calculate the effusion rate from a small hole, and obtain the speed and angle distributions of the emerging molecules.

## The Attempt at a Solution

So this is just bookwork, but I'm not too sure about my understanding in terms of the speed and angle distributions of the emerging molecules and I can't find anywhere that really goes into any detail on this.

So stating the answers:

the velocity distribution is proportional to v

^{3}exp(-mv

^{2}/2kT),

the angle distribution is proportional to sinθcosθ

I'm not too sure about how I would logically explain how this derives from

dΦ=0.5vnf(v)dvsinθcosθdθ

The best I can do at explaining is

dΦ is the number of molecules hitting unit area in unit time with the molecules in the interval [v,v+dv] and [θ,θ+dθ]. We can think of splitting this into it's speed and angular parts, so

dΦ=0.5vnf(v)dvsinθcosθdθ=n(0.5sinθcosθdθ)[vf(v)dv]

(I've kept the 0.5 in with the angles as this is where it derives from). Then we can think of the θ part as being the probability of a molecule hitting the wall in the [θ,θ+dθ] interval, and the v part as being the probability of a molecule hitting the wall in the [v,v+dv] interval, hence the above results.

However this far from satisfies me - if the bracketed expressions were each probabilities as I suggest above, they would be multiplying the number per unit volume which doesn't make any sense in terms of getting the number hitting the wall per unit area per unit time. Can anyone offer a logical link between dΦ and the speed and angle distributions please? Thankyou :)

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