The Equilibrium Populations of Molecular States

So far in this chapter we have studied the quantum-mechanical states of isolated atoms or molecules. In a dilute gas, the molecules do not significantly interfere with each other, and we can apply these states to the individual molecules. However, all molecular states will not be occupied by the same numbers of molecules in a dilute gas at equilibrium. The Boltzmann probability distribution gives the probability of a molecular state of energy e in a system at thermal equilibrium  

Each state in an energy level has the same energy so it will have the same population. If g is the degeneracy of the level. 

In Chapter 9 there is a derivation of the Boltzmann probability distribution for classical dilute gases. There is a derivation of this probability distribution for a quantum dilute gas in Part 4. For now, we introduce it without derivation. The important fact about the Boltzmann probability distribution is that states of energy much larger than k T are 

To a good approximation, the energy of a molecule is a sum of four different 

The probabilities of each of these energies is independent of the others, so that the probability of an energy level is the product of four Boltzmann factors  

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