Advanced Topic 392 of 1102 Thermodynamics & Energetics
Boltzmann Distribution
Populations of energy states fall off exponentially with energy at a given temperature.
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In-depth explanation of Boltzmann Distribution
The Boltzmann distribution gives the fraction of molecules occupying each energy level at thermal equilibrium. For two states with energies E1 and E2, the population ratio is N2 over N1 equal to (g2/g1) times exp(-(E2 - E1)/kT). Higher temperature or a smaller energy gap makes the excited state more populated, while at very low temperature almost all molecules remain in the ground state. This exponential dependence helps explain why chemical reaction rates rise sharply with temperature, because more molecules gain the energy needed to surmount activation barriers. It also explains the intensities of spectral lines, which depend on how many molecules occupy each level.
Going deeper: Boltzmann Distribution
Key ideas: Population ratio uses exp of minus delta E over kT times degeneracy ratio. At 300 K, kT is about 2.5 kJ per mole, so a gap of 25 kJ per mole means a ratio near 4.5 times 10 to the minus 5. Population inversion is impossible at equilibrium.
Common mistakes and connections: Do not forget degeneracy, which can make a higher level more populated than a lower one. The same exponential form appears in the Arrhenius equation. It explains the temperature dependence of rates, spectral intensities and vapor pressure.
Visualization of Boltzmann Distribution
Population of the upper level falls exponentially with energy gap and rises with temperature; drag T and gap to see occupancy change. The interactive lesson shows this as a live animation or chart.
Worked example and use case
At 300 K the value of kT is about 2.5 kJ per mole, so molecular vibrations with large energy spacings are mostly in the ground state.
Frequently asked questions about Boltzmann Distribution
What is Boltzmann Distribution?
Populations of energy states fall off exponentially with energy at a given temperature.
Is Boltzmann Distribution a beginner, intermediate or advanced topic?
Boltzmann Distribution is a advanced topic in Thermodynamics & Energetics. It builds on beginner and intermediate topics, so review those first if needed.
What should I learn before Boltzmann Distribution?
Study these earlier topics in Thermodynamics & Energetics first: Degrees of Freedom, Statistical Thermodynamics, Partition Function.
What comes after Boltzmann Distribution?
The next topics in Thermodynamics & Energetics are Chemical Potential and Fugacity and Activity.
Can I practise Boltzmann Distribution with quizzes?
Yes. The interactive lesson includes a self-check question on Boltzmann Distribution, and the Thermodynamics & Energetics quiz draws on every topic in this area.
Where Boltzmann Distribution fits in Thermodynamics & Energetics
Boltzmann Distribution is topic 64 of 91 in Thermodynamics & Energetics.