# Multi-Species Ideal Gas Law and Mean Molecular Mass

1. Jun 13, 2013

### Allday

Hi forums. I have what I think is a simple question but I'm making myself confused. I'm trying to work out the relationship between energy density ( $u$ = energy per unit volume ) and temperature in a multi-species ideal gas (no molecules just different mass ions). The simplest example of something like this would be a hydrogen gas and the species would be neutral hydrogen, ionized hydrogen, and free electrons. I assume all species are at the same temperature. I know the monatomic answer is,

$$u = \frac{3}{2} n k T$$

where $n$ is the number density of all particles and $k$ is boltzmann constant. I'm trying to decide if I need to add a factor of the mean molecular mass ( $\mu$ = the mean mass of an ion divided by the proton mass) in this relation for the multi-species case. $\mu$ is dimensionless so I can't decide on the basis of dimensional analysis.

I suppose the question is equivalent to asking if the energy density of a box of hydrogen at a fixed temperature is equal to that of a box of helium at the same temperature .... the more I think about this, the more I'm thinking the factor of $\mu$ is not needed. Can anyone confirm this?
Any help appreciated.

Last edited: Jun 13, 2013
2. Jun 13, 2013

### gadong

Your expression is correct for the kinetic energy contribution to the energy density. Each particle species has mean kinetic energy 3/2kT, regardless of mass. The potential energy of a system of charged particles will be volume dependent though, so this expression may not get you very far.

3. Jun 14, 2013

### Allday

thanks gadong. I'm modeling the cooling curve for astrophysical plasmas (i.e. low density and globally charge neutral) so I'm just interested in that portion of energy which comes from kinetic motion of the particles.

4. Jun 14, 2013

### Staff: Mentor

Are you sure you can ignore rotational energy?

5. Jun 15, 2013

### gadong

To clarify, my comment referred to H/H+/e- particles, so I may have mislead you. For diatomic species like H2 the mean *classical* energy consists of:

translational kinetic energy (of entire molecule): 3/2kT
rotational: kT
vibrational: 0.5kT.

In total, 3kT, or the same as the two atoms considered in isolation (as it should be).

For a quantum system, the above values of the vibrational and rotational energies represent the high temperature limits.

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