# Non homogeneous density in a gas

1. May 23, 2014

### JamesWolf

Hi, have searched around but can't find what im looking for (probably because im not entirely certain what its called)

An elemental gas in a contained volume, say microcanonical for simplicity, will have atoms that bounce against the walls. Is there an equation for the average density of the gas with respect to position?

Everywhere I see (such as Van der Walls equation) presumes a constant density across the volume. Is this simplification justified? Or do atoms on average hang by the walls, or in the middle?
Presumably in a more complicated system, gravity included, they hang closer to the bottom. If we allow energy exchange between the walls and the gas, will this average position change? Speed is related to the temperature kT, is this position also?

Pointing me in the right direction on this will be greatly appreciated. Thanks.

2. May 23, 2014

### WannabeNewton

See section 3.5: http://www.pma.caltech.edu/Courses/ph136/yr2011/1103.1.K.pdf

Clearly from eq. (3.37a) if the gas (which may consist of multiple components) is at equilibrium, meaning constant temperature and constant chemical potential, then the number density is also constant.

This is reasonable. Eq. (3.37a) is derived assuming the particles are non-relativistic, classical, free particles so that their occupation number obeys the Boltzmann distribution $\mathcal{N} = \frac{g_s}{h^3}e^{(\mu - p^2/2m)/k_B T}$ and as such, given the isotropy and lack of interactions of significant time scales (meaning collisions don't matter) there is no reason there would be overdense regions and underdense regions of the gas in the volume.

Last edited: May 23, 2014
3. May 23, 2014

### JamesWolf

Thanks for answering me, ill have a read of the pdf :)

4. May 23, 2014

### Jorriss

In general, it depends on what you want to do, but no the assumption is often not justified. Infact, to get the van der waals equation one has to enforce homogeneous density. As the radial distribution function corresponding to a van der waals gas is not the same as for an ideal gas, the assumption is not exactly right and one can easily show the true density of a van der waals gas is not constant across the gas.

Last edited: May 23, 2014
5. May 24, 2014

### JamesWolf

great thanks, this is the kind of thing I was looking for. The earlier pdf is thick reading for me, but ill get there.

6. May 25, 2014

### Jorriss

You certainly should. WBN introduced those notes to me too and they are very informative.