# Boltzmann equation and energy level occupancy at infinitely high temp

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• Haynes Kwon
In summary, at infinitely high temperatures, every energy level is occupied by an equal number of electrons, but at high temperatures (but not infinitely high), the number of non-ionized atoms is small and all states are approximately equally likely to be occupied. However, the Boltzmann equation includes the Fermi energy, which changes at different temperatures due to phase changes in the system. Therefore, there is no single Boltzmann equation that applies at all temperatures.
Haynes Kwon
Gold Member
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?

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

Haynes Kwon said:
every energy level is occupied by equal number of electrons
Namely zero

Why zero?

Haynes Kwon said:
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.

PeterDonis said:
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.

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

hutchphd said:
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.

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.

hutchphd said:
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.

hutchphd said:
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.

Last edited:
Haynes Kwon said:
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).

vanhees71 and BvU
The important physics in my opinion is that at high temperatures all states are approximately equally likely to be occupied.

Demystifier

## 1. What is the Boltzmann equation?

The Boltzmann equation is a fundamental equation in statistical mechanics that describes the relationship between the energy level occupancy of a system and its temperature. It was developed by Austrian physicist Ludwig Boltzmann in the late 19th century.

## 2. How does the Boltzmann equation relate to energy level occupancy at infinitely high temperatures?

At infinitely high temperatures, the energy level occupancy of a system approaches a state of maximum disorder, known as thermal equilibrium. The Boltzmann equation can be used to calculate the probability of a particle occupying a particular energy level at this state.

## 3. What is the significance of infinitely high temperatures in the Boltzmann equation?

Infinitely high temperatures serve as a theoretical limit in the Boltzmann equation, allowing for the calculation of the maximum possible energy level occupancy in a system. This is useful in understanding the behavior of systems at extreme temperatures.

## 4. How does the Boltzmann equation relate to the concept of entropy?

The Boltzmann equation is directly related to the concept of entropy, which is a measure of the disorder or randomness in a system. The equation shows that as temperature increases, the entropy of a system also increases, leading to a more disordered state.

## 5. What are some real-world applications of the Boltzmann equation?

The Boltzmann equation has many applications in various fields, including physics, chemistry, and engineering. It is used to understand the behavior of gases, the properties of materials, and the kinetics of chemical reactions. It also plays a crucial role in the development of technologies such as refrigeration and thermoelectric devices.

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