Statistical mechanics · two-level equilibrium
Boltzmann Distribution Calculator
Translate an energy gap and a thermodynamic temperature into an equilibrium state population. This independent physics workspace keeps energy conversion, degeneracy, the partition function, entropy, heat capacity, and the temperature response on one auditable page.
Equilibrium solution
Thermal population ledger
Energy landscape
State occupancy and thermal response
The left panel connects the entered energy separation to the normalized populations. The right curve uses the universal thermal ratio kBT/ΔE, so the current marker moves without hiding the role of degeneracy.
Lower state Upper state / current point High-temperature limit
Temperature ladder
How the same energy gap responds when temperature changes
Every row preserves the entered gap and degeneracies. Only absolute temperature changes, making the exponential response, state population, and heat-capacity peak directly comparable.
| Scenario | Temperature (K) | kBT (eV) | ΔE/kBT | Boltzmann factor | Upper population | C (J/mol·K) |
|---|
Detailed calculation process
From energy units to normalized populations
- Convert the entered temperature to kelvins and the entered gap to joules per particle.
- Calculate thermal energy kBT and the dimensionless separation x = ΔE/kBT.
- Evaluate the raw Boltzmann factor B = exp(−x), which describes energy suppression before degeneracy.
- Multiply B by the degeneracy ratio g₁/g₀ to obtain N₁/N₀.
- Normalize both statistical weights through Z = g₀ + g₁B to obtain p₀ and p₁.
- Use those normalized populations to calculate average excitation energy, entropy, and heat capacity.
kB = 1.380 649 × 10⁻²³ J·K⁻¹ (exact)
B = exp[−(E₁ − E₀)/(kBT)] = exp(−ΔE/kBT)
N₁/N₀ = (g₁/g₀)B
Z = g₀ + g₁B; p₀ = g₀/Z; p₁ = g₁B/Z
U − E₀ = p₁ΔE
S/R = ln Z + (ΔE/kBT)p₁
C/R = (ΔE/kBT)²p₁(1 − p₁)
Live substitution
Current values inserted step by step
—
—
—
—
—
Energy-gap cross-check
—
—
—
—
Symbol ledger
Meaning and unit of every term
| Symbol | Meaning | Unit |
|---|---|---|
| kB | Boltzmann constant | J/K |
| T | Thermodynamic temperature | K |
| E₀, E₁ | Lower and upper state energies | J per particle |
| ΔE | Positive energy gap E₁ − E₀ | J, eV, kJ/mol, cm⁻¹, or K equivalent |
| g₀, g₁ | Lower and upper state degeneracies | dimensionless |
| B | Raw Boltzmann factor exp(−ΔE/kBT) | dimensionless |
| Z | Partition function relative to E₀ = 0 | dimensionless |
| p₀, p₁ | Normalized state populations | fraction or percent |
| R | Molar gas constant NAkB | J/(mol·K) |
Interpretation
Degeneracy competes with the energy penalty
The exponential term always penalizes the higher energy at positive temperature. Degeneracy pushes in the opposite direction by providing more upper-level microstates. That is why the raw Boltzmann factor and the normalized upper-state population are not interchangeable.
- When ΔE ≫ kBT, the upper level is strongly suppressed.
- When ΔE ≈ kBT, both energy and degeneracy materially affect population.
- When kBT ≫ ΔE, energy becomes unimportant and degeneracy sets the limiting shares.
- A population ratio can exceed one when upper-level degeneracy is sufficiently large.
Model boundary
What this page deliberately does not assume
This is an equilibrium two-level canonical model. It does not calculate a continuous molecular speed distribution, non-equilibrium relaxation, transition rates, inverted populations, chemical reactions, quantum coherence, level broadening, interactions, or a multi-level molecular partition function.
Enter degeneracies and energy differences that belong to the same physical model. Spectroscopic term symbols, vibrational ladders, rotational levels, and electronic states may require many more levels and additional selection or symmetry rules.
Boltzmann calculator FAQ
Questions about factors, populations, and temperature
What does the Boltzmann factor mean?
The Boltzmann factor exp(-ΔE/kBT) measures the thermal suppression associated with an energy gap ΔE at absolute temperature T. A smaller gap or a higher temperature moves the factor closer to one.
Why does degeneracy change the state population?
Degeneracy counts how many microstates share the same energy. The population weight of a level is its degeneracy multiplied by its Boltzmann factor, so a highly degenerate upper level can hold more population than the energy gap alone suggests.
Can the upper-state population exceed 50%?
Yes. If the upper level has greater degeneracy, its high-temperature limit is g1/(g0+g1), which can exceed 50%. For equal degeneracies, the positive-temperature upper-state population approaches 50% from below.
Does this calculate a Maxwell-Boltzmann speed distribution?
No. This page models equilibrium occupation of two discrete energy levels. A Maxwell-Boltzmann speed distribution is a continuous distribution that also depends on particle mass.
Why must temperature be expressed on an absolute scale?
The exponential uses thermodynamic temperature. Celsius and Fahrenheit entries are converted to kelvins before any calculation, and temperatures at or below absolute zero are rejected.
What is the high-temperature population limit?
As temperature grows, the energy penalty becomes negligible and population is divided according to degeneracy. The upper-state limit is therefore g1/(g0+g1), not automatically 50%.
Reference basis
Constants and physical interpretation
The calculation uses the exact SI defining values kB = 1.380 649 × 10⁻²³ J/K, h = 6.626 070 15 × 10⁻³⁴ J·s, c = 299 792 458 m/s, e = 1.602 176 634 × 10⁻¹⁹ C, and NA = 6.022 140 76 × 10²³ mol⁻¹.