Boltzmann equation and energy level occupancy at infinitely high temp

[Haynes Kwon]

Let's look at the Boltzmann equation
$$ \frac {p_{i}} {p_{j}} = e^{\frac{E_{j}-E_{i}} {kT}},$$

and take infinitely high temperature, the RHS becomes 1. I interpreted that this means every energy level is occupied by equal number of electrons. But if T is high enough, wouldn't the hydrogen atom be ionized and not a single energy level is occupied by an electron?

 

[BvU]

Yes. You're in the domain of plasma physics ...

every energy level is occupied by equal number of electrons

Namely zero

 

[hutchphd]

Why zero?

 

[PeterDonis]

if T is high enough, wouldn't the hydrogen atom be ionized and not a single energy level is occupied by an electron?

If T is high enough, the energy levels of the system change (for example, from those of a hydrogen atom to those of a free electron in a plasma), and hence the Fermi energy in the Boltzmann equation changes. So there is no single "Boltzmann equation" that applies at all temperatures.

 

[hutchphd]

So there is no single "Boltzmann equation" that applies at all temperatures.

As a practical matter we are certainly incapable of writing down a Boltzmann equation valid for all temperatures. This seems to me to be different from your statement. Are you saying that no such equation exists in principal?? Please elucidate.

 

[HomogenousCow]

You can't apply that formalism to the hydrogen atom without modification of the problem since the probability distribution isn't normalizable.

 

[PeterDonis]

re you saying that no such equation exists in principal?

I'm saying that the Boltzmann equation includes the Fermi energy, and the Fermi energy is not the same over the entire range of temperature from zero to infinity, because of phase changes like the change from atoms to plasma, which change the energy levels of the system. So you can't take, for example, the Boltzmann equation for the hydrogen atom, which includes the Fermi energy for that system, and apply it at temperatures which are high enough that the hydrogen will be ionized, because the Fermi energy is different in that temperature range.

 

[hutchphd]

Ah yes the phase change...thanks. Do treatments of real plasma need to include bound states (I profess total ignorance here) . It is called a fourth phase of matter for a reason... so little time ... so much to know.

 

[PeterDonis]

Do treatments of real plasma need to include bound states

In general, probably not, since the relevant degrees of freedom are the free motions of the electrons and ions.

 

[BvU]

Why zero?

Just from looking at the formula. There is a factor $(kT)^{-{3\over 2}}$ in front of the exponent.

Slighly sarcastic remark from my part, I must admit. Of course the formula itself goes invalid for extremely high $T$ as @PeterDonis pointed out.

 

[Demystifier]

But if T is high enough, wouldn't the hydrogen atom be ionized and not a single energy level is occupied by an electron?

No. When temperature is high (but not infinite), the number of non-ionized atoms is small (but not zero).

 

[hutchphd]

The important physics in my opinion is that at high temperatures all states are approximately equally likely to be occupied.