Deriving the boltzmann factors

[aaaa202]

My classmate has already asked about this I think, but since I can't find the post I'll ask again..

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?

 

[Valerie Park]

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.