The Boltzmann Distribution over Molecular Levels
A sample holds an enormous number of molecules spread across the levels the quantum models produced. The Boltzmann expression says what fraction sits in each level at a given temperature, and it is where temperature enters molecular chemistry.
What a learner can do afterwards
- Writes the ratio of populations of two levels in terms of their energy gap and the temperature
- Calculates the fraction of molecules in the first excited vibrational state at room temperature and comments on the size
- Explains why rotational levels are heavily populated while vibrational levels are barely populated at the same temperature
- Uses degeneracy correctly when two levels of different degeneracy are compared
1 · Read
A sample holds a huge number of molecules spread across energy levels. For two levels, N2 / N1 = (g2 / g1) x exp(-delta E / kT), where g counts the states in each level and k is the Boltzmann constant. Each single state holds less than any state below it, and the share falls exponentially with energy.
Vibrational gaps dwarf thermal energy. For a 2000 cm^-1 vibration, delta E = h c times the wavenumber = 3.97e-20 J, while kT at 298 K is 4.12e-21 J. The ratio is exp(-9.65), about 6e-5, so the first excited level is nearly empty. Infrared absorption starts from the ground state, and hot bands barely register.
Rotations behave the opposite way. Their gaps are small next to kT, so many rotational levels hold sizable populations. When degeneracies differ, multiply by their ratio: a level built from more states holds more molecules at equal energy, because there are more ways to occupy it.
Heating flattens the distribution and feeds population into higher levels. Count the ways molecules can share a fixed total energy and one distribution wins by sheer numbers. That winner is the Boltzmann distribution, which is also why entropy grows as energy spreads.
Lower levels take the biggest share with an exponential falloff, so vibrations sit empty while rotations fill up at room temperature.
The rewritten readings were written for this dataset. The retained library readings derive from OpenStax College Physics and University Physics.
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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.