Very basic form of the diffusion law for gasses

In summary, the statement ##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 can be proven at the undergraduate level, but it requires a lot of work in order to get the correct factor of 1/3. Many treatments make approximations that simplify the argument but result in the wrong numerical factor, and it is stated that a more refined treatment will yield the correct factor. Feynman's lectures, particularly section 43-5, offer a more detailed explanation.
  • #1
Coffee_
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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.
 
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  • #2
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.
 
  • #3
TSny said:
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.
 

1. What is the basic form of the diffusion law for gasses?

The basic form of the diffusion law for gasses is given by Fick's first law, which states that the flux of a gas is directly proportional to the concentration gradient of the gas and the diffusion coefficient.

2. How does diffusion occur in gasses?

Diffusion in gasses occurs due to random molecular motion, where molecules move from an area of high concentration to an area of low concentration until equilibrium is reached.

3. What factors affect the rate of diffusion in gasses?

The rate of diffusion in gasses is affected by the temperature, pressure, and molecular weight of the gas. Higher temperatures, lower pressures, and lighter molecules all lead to faster diffusion rates.

4. Can the diffusion law be applied to all gasses?

Yes, the diffusion law can be applied to all gasses as it is a fundamental law of physics that governs the movement of molecules in gasses.

5. How is the diffusion coefficient determined for a specific gas?

The diffusion coefficient for a specific gas can be determined experimentally by measuring the diffusion rate of the gas in a controlled environment and using the diffusion law equation to calculate the coefficient.

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