Why use this premise behind the Maxwell-Boltzmann curve?

[snoopies622]

TL;DR Summary

Specifically, why are the molecular speeds in any given direction a normal curve? Why not the kinetic energies instead?

I'm trying to understand the Maxwell-Boltzmann gas molecule speed distribution. Suppose we have a container of gas such that all the molecules are identical.

At first I was under the mistaken impression that one starts with the premise that the distribution of their translational kinetic energies is a normal curve, and taking the square roots of these energies to get the speeds gives us the chi curve.

But after further reading, I guess instead the premise is that for each of the three dimensions of space, one assumes that the velocity components in that direction for all the molecules form a normal curve, then squaring those, adding them together and taking the square root to get the absolute speeds is what produces the chi distribution.

I realize that there may not be an answer to this question but, why is the normal curve in the speeds and not the energies? Why does nature prefer one over the other?

 

[Vanadium 50]

So, as I understand it, the question is why p and not E?

Most texts - including Wikipeda- derive it in terms of E starting from in terms of p.

 

[Ibix]

You know that the average velocity of gas molecules is zero in the bulk rest frame, so you have a symmetric probability distribution of velocity components in each dimension. So it's at least possible it's a normal distribution. But you can't say the same for energy - it is bounded at zero, so it cannot be normally distributed.

 

[snoopies622]

Thank you both. I've done a little more reading about this since i posted this question and realize it's not as simple a matter as i thought.

I'm always interested in the underlying premises, the axioms as it were.

Here's an interesting video where the function is apparently derived using the pressure/altitude formula - an approach which surprises me.

 

[DrClaude]

If you want to understand more fundamentally where the MB distribution comes from, starting from the barometric equation makes no sense, since it itself comes from a more fundamental basis. (The ideal gas law, which is the other thing used in the video, can also be derived from more basic principles, even though historically it is an empirical law.)

You should pick up a good book on statistical physics. Alternatively:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://www.sns.ias.edu/~tlusty/courses/statphys/statphys.pdf

 

[BvU]

And 'Physical Chemistry' by P. Atkins is also a good read ...

 

[snoopies622]

. . starting from the barometric equation makes no sense, since it itself comes from a more fundamental basis.

I did find it troubling that one could start with an approach that assumes a gravitational field to arrive at a relationship which doesn't involve gravity at all.

 

[DrClaude]

I did find it troubling that one could start with an approach that assumes a gravitational field to arrive at a relationship which doesn't involve gravity at all.

The thing is that probability of being in a given state goes as ##\exp(-E/kT)##.

 

[Vanadium 50]

the axioms as it were.

Really, physics doesn't work that way. It's more "lets see how far these guesses and assumptions will take us." Often with the guesses and assumptions being known to be wrong at some level.