Very basic form of the diffusion law for gasses

[Coffee_]

I'm looking for a proof of the following statement at a level an early undergrad would understand:

$J=D \vec{\nabla} \vec{n}$ where $D=\frac{v_{th}l}{3}$ with $l$ being the mean free path and $v_{th}$ the thermal agitation velocity, $J$ is the particle current density.

I really did try google a lot but no luck. Either the proof is way too complicated for me to understand or it just tells the results without work. I'm looking for a very raw/approximating derivation that doesn't really have to be formal at all, something that can be understood in like 10 minutes, hard assumptions are allowed. I would really appreciate it if someone who knows where I can find one, could inform me.

 

[TSny]

To get the correct factor of 1/3 apparently requires a lot of work. The treatments that I have seen at the undergraduate level make approximations that simplify the argument but end up getting the wrong numerical factor. They then just state that a more refined treatment will yield the 1/3. See for example Feynman's lectures http://www.feynmanlectures.caltech.edu/I_43.html especially section 43-5.

 

[Coffee_]

To get the correct factor of 1/3 apparently requires a lot of work. The treatments that I have seen at the undergraduate level make approximations that simplify the argument but end up getting the wrong numerical factor. They then just state that a more refined treatment will yield the 1/3. See for example Feynman's lectures http://www.feynmanlectures.caltech.edu/I_43.html especially section 43-5.

I see, this is exactly the answer I was looking for. The reason why I'm asking is that in class I remember some very very handwaving derivation that got the wrong factor. When I then looked up the law it had a different factor and I thought we were just blatantly wrong.