Boltzmann Factor Calculator — Relative Probability of Two States (P₁/P₂ = e^(−ΔE/k_BT))

What is the Boltzmann Factor?

The Boltzmann factor tells you the relative probability with which two states of a system appear when the system is in thermal equilibrium at temperature T. Use this Boltzmann factor calculator to compute the ratio of probabilities of two energy states, or the Boltzmann factor of a single state. The relative probability of two states with energies E₁ and E₂ is:

where:

• P₁, P₂ — the probabilities that the system occupies state 1 and state 2;
• E₁, E₂ — the energies of the two states;
• k_B = 1.380649 × 10⁻²³ J/K — the Boltzmann constant;
• T — the absolute temperature in kelvins (K).

The Boltzmann (Gibbs) Distribution

The Boltzmann distribution (also called the Gibbs distribution) specifies the probability with which an individual state of a system appears at thermal equilibrium characterized by the temperature T:

where Z is the normalization constant (the partition function), E is the energy of the state (in joules), and P is the probability that this state occurs. An essential feature of the Boltzmann distribution is that the probability P depends only on the energy E of the state. The Boltzmann distribution is central to our understanding of condensed matter and statistical physics — it is the close relative of the Maxwell–Boltzmann distribution of particle velocities.

How the Boltzmann Factor Works

Dividing the Boltzmann distribution for two states cancels the normalization constant Z, so the relative probability depends only on the difference in energies, ΔE = E₁ − E₂:

• Two states of the same energy (ΔE = 0) are equally probable — the ratio P₁/P₂ = 1.
• The lower-energy state is always the more probable one.
• The other factor that plays a role is temperature: the lower the temperature, the more strongly the system favors the lower-energy state. As T → ∞, both states become equally likely.
• The quantity k_B·T is the characteristic thermal energy. At room temperature (≈ 300 K), k_B·T ≈ 0.0259 eV ≈ 4.14 × 10⁻²¹ J.

Boltzmann Factor Example

Consider two states separated by ΔE = 0.1 eV at room temperature (T = 25 °C = 298.15 K):

• k_B·T = 1.380649 × 10⁻²³ × 298.15 ≈ 4.116 × 10⁻²¹ J ≈ 0.0257 eV
• ΔE/(k_B·T) = 0.1 / 0.0257 ≈ 3.89
• P₁/P₂ = e^−3.89 ≈ 0.0204

So the higher-energy state (state 1) is only about 2% as likely as the lower-energy state (state 2) — the system spends most of its time in the lower state.

Boltzmann Constant vs. Gas Constant

The Boltzmann constant k_B measures energy per particle. Its molar counterpart is the universal gas constant R = N_A · k_B = 8.3145 J/(K·mol), where N_A = 6.022 × 10²³ mol⁻¹ is Avogadro's number. This calculator accepts energies both per particle (eV, meV, J, zJ) and per mole (kJ/mol, kcal/mol) — molar energies are automatically divided by N_A to obtain the per-particle value used with k_B.

Unit Systems Supported

This Boltzmann factor calculator supports both metric (SI) and American/imperial conventions:

• Energy: eV and meV (atomic/solid-state physics), J and zJ (SI), kJ/mol and kcal/mol (chemistry; kcal/mol is common in American biochemistry).
• Temperature: K (Kelvin, SI), °C (Celsius, metric), °F (Fahrenheit, American).

Enter your values in the most convenient units and the calculator returns the probability ratio, the normalized share of each state, the thermal energy k_B·T, and the energy difference in several units.