What a learner can do afterwards
- Writes the partition function as a sum over levels and evaluates it for a two-level system
- Sums the vibrational series as a geometric series and states the closed form
- Explains what the limiting values at low and high temperature mean physically
- Separates the total into translational, rotational, vibrational and electronic factors and says why the separation is allowed
1 · Read
Summing the Boltzmann factors over every level gives one number, q, that counts how many levels are effectively in use. Each level adds its degeneracy times e to minus its energy over kT, so accessible states add a lot and remote ones almost nothing. A large q means many states are in reach, not that any one state is favoured.
With the ground level set to zero energy, a two-level system gives q equal to 1 plus e to minus gap over kT. If the excited factor is 0.3, q is 1.3. A vibrational ladder of equally spaced levels forms a geometric series, giving q equal to 1 over 1 minus the factor: with a factor of 0.5, q is 2.
Near zero temperature only the ground level counts, so q sinks toward 1, while high temperature lets countless levels contribute. The total separates into translational, rotational, vibrational and electronic factors multiplied together, because those motions are nearly independent. Dividing one level's weighted factor by q gives its population fraction, so q is the normalising sum behind every average.
To reach bulk properties, build the sample's partition function Q from q: for N identical, independent molecules, Q = q^N / N!. The Helmholtz energy is then A = -kT ln Q, and other thermodynamic quantities follow by differentiation.
The partition function sums weighted levels into one count of states in use, and its logarithm opens the door to bulk thermodynamics.
Rewritten readings were written for this dataset.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.