# Deriving the boltzmann factors

• aaaa202
In summary: This is why the probability of finding the atom in the ground state is overwhelmingly more probable, even if the multiplicities may seem similar. In summary, the conversation discusses the concept of finding the probability of an atom being in different energy states when placed in a reservoir. The Boltzmann factor, which is a measure of the probability of finding an atom in a given energy state, is affected by the temperature of the reservoir. This explains why the probability of finding the atom in the ground state is much higher compared to the first excited state, even if the multiplicities may seem similar.
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Take a single atom with degenerate energy levels E1, E2 etc. We place the atom in a resevoir and want to find the probability of probability for finding the atom in the different energy states. So we note that the ratio of probabilities must be the ratio of the multiplicities for the heat resevoir corresponding to the energy levels. Thus we have:

p2/p1 = Ω2/Ω1

which using S = kln(Ω) can be written as:

p2/p1 = exp((S2-S1)k)

Invoking the thermodynamic identity using dS = -(U2-U1)/T we get:

p2/p1 = exp(-(U2-U1)kT)

and we have arrived at the Boltzmann factors, which only depend on temperature. For me this is a fantastic result, but I just wish, that I understood it. In my view, it should somehow depend on the characteristica of the resevoir such as how many particles and energy it stores. - these do after all determine the multiplicity! Let's take an example:

Let's say you have an resevoir with an energy of 1010J. If our atom is excited to a state with higher energy it "steals" some energy from the resevoir. But this energy portion is tiny compared to the total energy, and thus the multiplicity of the resevoir shouldn't reduce significantly. But if you calculate the ratio of the Boltzmann factors at 300K for the ground state in hydrogen and the first state, you get that the probability of finding the atom in the ground state is overwhelmingly more probable - another way of saying that the multiplicity for the ground state system is far bigger. How is this possible when the multiplicities are almost the same from my logic?

The answer lies in the fact that the Boltzmann factor is a measure of the probability of finding an atom in a given energy state, not the multiplicity. The energy levels of the atom still have the same multiplicities regardless of the amount of energy in the reservoir - they are just the number of ways that the atom can be in a given state. The Boltzmann factor is a measure of how likely it is for the atom to be in a particular state, and is affected by the temperature of the reservoir. In this case, since the first excited state of hydrogen has a much higher energy than the ground state, the Boltzmann factor will be much lower, meaning that it is much less likely to find the atom in the first excited state at 300K.

## 1. What is the Boltzmann factor and why is it important in scientific research?

The Boltzmann factor is a mathematical term used to describe the probability of a particle occupying a certain energy state in a system. It is important in scientific research, particularly in statistical mechanics and thermodynamics, as it helps us understand the behavior and distribution of particles in a system.

## 2. How is the Boltzmann factor derived?

The Boltzmann factor is derived from the Boltzmann distribution, which describes the probability of particles occupying different energy states in a system at a given temperature. It is derived using the principles of statistical mechanics and the laws of thermodynamics.

## 3. What are the key assumptions made when deriving the Boltzmann factor?

There are two key assumptions made when deriving the Boltzmann factor: 1) the system is in thermal equilibrium, meaning the temperature is constant throughout the system, and 2) the particles in the system do not interact with each other, also known as the non-interacting particle approximation.

## 4. How is the Boltzmann factor used in practical applications?

The Boltzmann factor is used in many practical applications, such as calculating the equilibrium distribution of particles in a gas, predicting the behavior of molecules in a chemical reaction, and determining the stability of a protein structure. It is also used in statistical mechanics to understand the properties of materials at the atomic scale.

## 5. Can the Boltzmann factor be applied to all systems, regardless of size or complexity?

The Boltzmann factor can be applied to most systems, as long as the key assumptions are met. However, it becomes increasingly difficult to use in extremely large or complex systems, as the calculations become more computationally intensive. In these cases, approximations and simplifications may need to be made to apply the Boltzmann factor.

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