---
title: "The Boltzmann Distribution over Molecular Levels"
description: "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 wh"
canonical: https://lightmysky.com/learn/science/the-boltzmann-distribution-over-molecular-levels-mt_-DHIox6GBm
source: https://lightmysky.com/learn/science/the-boltzmann-distribution-over-molecular-levels-mt_-DHIox6GBm.md
retrieved: 2026-09-12
---

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# 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.

Subject: Science · Area: Chemistry · Ages 19 to 21
Page: https://lightmysky.com/learn/science/the-boltzmann-distribution-over-molecular-levels-mt_-DHIox6GBm

## Ready when they can

- 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

## Lesson: How molecules share energy levels

A sample holds a huge number of molecules spread across energy levels. The Boltzmann expression says what fraction sits in each level at a given temperature. Lower levels get the biggest share, and the share falls exponentially as levels rise. To compare two levels, multiply the degeneracy ratio by the exponential falloff set by the gap divided by temperature.

Vibrational gaps are far larger than thermal energy at room temperature, so the first excited vibrational state sits nearly empty. Practically every molecule rests in the ground vibrational state. That is why infrared absorption starts from the ground state, and hot bands from excited states barely register.

**Example.** Rotations behave the opposite way at the same temperature. Rotational gaps are small next to thermal energy, so many rotational levels hold sizable populations. When two levels carry different degeneracy, multiply by it directly. A level built from more states holds more molecules even at equal energy, simply because there are more ways to occupy it.

**Tip.** Heating flattens the whole distribution and feeds population into higher levels. This is probability at work. 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.

**Recap.** Lower levels take the biggest share with an exponential falloff, so vibrations sit empty while rotations fill up at room temperature.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [What an Excited Molecule Does Next: Fluorescence and Phosphorescence](https://lightmysky.com/learn/science/what-an-excited-molecule-does-next-fluorescence-and-phosphorescence-mt_5AVF6uWed_)
- [Entropy and the Spread of Energy](https://lightmysky.com/learn/science/entropy-and-the-spread-of-energy-mt_a_2feH2gnE)
- [Molecular Motion Quantised: The Oscillator and the Rotor](https://lightmysky.com/learn/science/molecular-motion-quantised-the-oscillator-and-the-rotor-mt_dHgi-i9pxc)

## Opens up

- [The Molecular Partition Function](https://lightmysky.com/learn/science/the-molecular-partition-function-mt_-TPTlb7zQ2)