Understand Effusion of a Gas: Speed & Angle Distribution

[physiks]

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 v³exp(-mv²/2kT),
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 :)

 

[Bystander]

http://en.wikipedia.org/wiki/Knudsen_diffusion

 

[physiks]

http://en.wikipedia.org/wiki/Knudsen_diffusion

Sorry but that doesn't seem to answer the question...

 

[Bystander]

That wasn't the best article in the world. Didn't critique it before linking it, because that's all been "shot, skinned, cut, dried, stuffed, mounted, and gathering dust on library shelves" for a century.

Daniels, Williams, Bender, & Alberty, p-chem lab book had a good discussion of Knudsen vapor pressure measurements --- good luck finding it on library shelves. Can't think where else to point you for further discussion on the topic.

 

[physiks]

That wasn't the best article in the world. Didn't critique it before linking it, because that's all been "shot, skinned, cut, dried, stuffed, mounted, and gathering dust on library shelves" for a century.

Daniels, Williams, Bender, & Alberty, p-chem lab book had a good discussion of Knudsen vapor pressure measurements --- good luck finding it on library shelves. Can't think where else to point you for further discussion on the topic.

Hmm, well this just seems to be a more of a mathematical concept that the books and websites I've looked at tend to brush over by just saying things like 'you can see that the new speed distribution goes like v³ by looking at the differential flux expression' (that is in explaining how would you logically explain the jump from the expression for the differential flux to giving the effused speed and angle distributions) - I don't think any sources on Knudsen vapor pressure measurements, which from what I know just seems to apply the results for the effusion rate, would help that much...

I was expecting it to be quite an obvious thing that somebody could explain given that the books and notes I've read talk about it as though it's obvious.

 

[Bystander]

This is me going back fifty years, so I guarantee nothing, but DWBA did discuss diameter and length of the hole as far as the angular distribution went. It was a standard text for labs, so should be around somewhere.

 

[physiks]

This is me going back fifty years, so I guarantee nothing, but DWBA did discuss diameter and length of the hole as far as the angular distribution went. It was a standard text for labs, so should be around somewhere.

I would have a look, but I don't have access to books (apart from the ones I already have) anytime in the near future, so hopefully somebody else will know about this...

May be able to get it online though.

Edit: Found it, unfortunately of no help. It's just a mathsy conceptual thing I would like somebody to help me see really (rather than just accept it).