Active Reading Solution: Normalizing the Boltzmann Distribution

As you know, "normalizing" a probabiity distribution means multiplying every individual term by the same constant—in this case that constant is called 1/Z—so that the sum of all the probabilities is 1.

Because all 10 of the excited states in this exercise have the same probability, we just calculate that probability once and multiply by 10.

(1/Z)e0 + 10(1/Z)e-ε/(kT)=1

Z = e0 + 10e-ε/(kT) = 1 + 10(0.1445) = 2.445

Now you can plug that in to find that the probability of finding the system in its ground state is:

e0/Z = 1/2.445 = 0.41

So the probability is 41%. Just to check, we can calculate the combined probability of being in any of the excited states:

10 e-ε/(kT)/Z = 10 (0.1445)/2.445 = 0.59
So that's 59%, which checks out.