Problems 1 through 5 will use the following system. Suppose that we have a two state system, in which the first state has energy and the second, energy 3. Give the ratio of the probability of occupancy of the first to the probability of occupancy of the second, and simplify.
We can take the ratio of the Boltzmann factors to get the ratio of the probabilities:
Calculate the partition function of the system.
Calculate the absolute probability of finding the system in the state with energy .
What happens to the occupation of the state with energy as τ→ 0 and as τ→∞?
As τ→ 0, the term of Z that is e-3/τ becomes insignificant in comparison to the term e-/τ. Therefore the absolute probability simplifies to:
As τ→∞, all terms go to 1, and therefore we find that:
These results make sense. If the temperature is very low in comparison to , often stated τ, there will be little thermal excitation that can promote the system from the first state to the second. In that case, we can be almost certain to find the system in the state of lower energy. If the temperature is very high, or τ, then the gap between the states becomes insignificant, and the system becomes about equally likely to be in either state.
This kind of analysis, looking at the limits of your answers, is an excellent way to check if you are on the right track. If your answers don't make sense at the limits then you have probably made a mistake somewhere.
Calculate the free energy of the system.