Question About Ideal Gas and Average Free Movement of Molecules

[PeroK]

Using $v^2_{rel}$ the dot product term will vanish, and the computation is straightforward, but I don't think the $v_{rel}$ case (the correct way to do it) has been computed in these articles properly for any distribution. I don't think anyone has worked the $v_{rel}$ case with the Maxwell-Boltzmann distribution.

I'm just going by the result in Wikipedia. It says you can check the integral, which I admit I haven't done. I don't see that post #49 can compare with the calculation on Wikipedia.

 

[Charles Link]

I'm just going by the result in Wikipedia

Wiki =actually Hyperphysics does the general case, (they reference Maxwell-Boltzmann but work only the general case) and they do that somewhat incorrectly by writing $\bar{v_{rel}}=( \bar{v^2} +\bar{v^2}_2-2 \bar{\vec{v}_1 \cdot \vec{v}_2})^{1/2}$. This expression for the averages is only approximate and not exact, but Hyperphysics implies it is exact.

The bars for the average didn't come out properly in my Latex. Please go to the Hyperphysics article to see it more clearly: http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/menfre.html#c5

 

[Calstiel]

This expression for the averages is only approximate and not exact, but Wiki implies it is exact.

I'm currently working on a computer simulation of the topic here, so I'm a bit off the thread, but in general its often important to check the sources in Wiki, maybe someone here is doing that.

 

[PeroK]

Here are two more (longer) derivations that come to the same result. I suspect this result is well known:

https://chem346.class.uic.edu/wp-content/uploads/sites/521/2019/07/MB_ave_relative.pdf

https://chem.libretexts.org/Bookshe...ty_Density_Function_for_the_Relative_Velocity

 

[Charles Link]

@PeroK That makes me want to go back and check my result of post 49=I did get 4/3 for the $v_{rel}$ case where all speeds are the same. Thank you=this looks like two very good links that you gave us. :)

 

[PeroK]

@PeroK That makes me want to go back and check my result of post 49=I did get 4/3 for the $v_{rel}$ case where all speeds are the same. Thank you=this looks like two very good links that you gave us. :)

That may well be correct. All speeds the same is different from a M-B distribution.

 

[Charles Link]

That may well be correct. All speeds the same is different from a M-B distribution.

I'm almost a little surprised by the two different results, $\sqrt{2}$ and 4/3, especially when Hyperphysics does some incorrect handwaving to get the first result for what they seem to be showing as the case in general, regardless of the distribution. I have looked over my calculations of post 49 though, and I believe that I computed the case of all the same speed correctly. Thanks very much for your inputs. :)

 

[renormalize]

I have looked over my calculations of post 49 though, and I believe that I computed the case of all the same speed correctly.

So you've said repeatedly. But how is your calculation physically relevant? Can you cite any physical situation for which a gas is expected to have a mono-speed distribution (a Dirac delta function)?

 

[Charles Link]

So you've said repeatedly. But how is your calculation physically relevant? Can you cite any physical situation for which a gas is expected to have a mono-speed distribution (a Dirac delta function)?

This is more of an interest to me simply from a mathematical sense, because Hyperphysics seems to imply in the "link" of post 62 that the $\sqrt{2}$ is the result for the general case. The simple case of post 49 if my calculations are correct would show that this is not the case.

One reason I chose this case (of all speeds being the same) is that it is simple enough to solve. I still need to study the "links" that @PeroK provided in post 64. The mathematics for the M-B case looks somewhat complex from a first look at it.

 

[PeroK]

This is more of an interest to me simply from a mathematical sense, because Hyperphysics seems to imply in the "link" of post 62 that the $\sqrt{2}$ is the result for the general case.

The Hyperphysics page is clearly wrong in this respect. In general $E(X) \ne \sqrt{E(X^2)}$. But, if you do the full calculation for a Gaussian distribution of velocities, then $E(v_r) = \sqrt 2 E(v)$.