Boltzmann Factor Calculator
Calculate relative probabilities of energy states in thermal equilibrium using statistical mechanics
Calculate Boltzmann Factor
Energy of the first quantum state
Energy of the second quantum state
Absolute temperature of the system
Boltzmann Factor Results
Formula: P₁/P₂ = exp((E₂ - E₁) / (k_B × T))
Energy difference: ΔE = 0.000 eV
k_B × T: 0.000 meV
Example: Electronic States in Atoms
Energy State 1: 0.1 eV
Energy State 2: 0.2 eV
Temperature: 273.15 K (0°C)
Energy difference: 0.1 eV
Boltzmann factor: ≈ 70
Result: Lower energy state is 70× more probable
Temperature Examples
Absolute Zero
0 K
Theoretical minimum temperature
Liquid Helium
4.2 K
Superfluid helium
Liquid Nitrogen
77 K
Cryogenic applications
Room Temperature
298 K
Standard conditions
Human Body
310 K
Physiological temperature
Water Boiling
373 K
Phase transition
Physical Constants
Boltzmann Constant
k_B = 1.381 × 10⁻²³ J/K
Fundamental constant relating energy and temperature
Electron Volt
1 eV = 1.602 × 10⁻¹⁹ J
Common energy unit in atomic physics
Thermal Energy (300K)
k_B T ≈ 25.9 meV
Characteristic energy scale at room temperature
Understanding the Boltzmann Factor
What is the Boltzmann Factor?
The Boltzmann factor describes the relative probability of finding a system in different energy states at thermal equilibrium. It's fundamental to statistical mechanics and thermodynamics, determining how particles distribute among available energy levels.
Physical Significance
- •Lower energy states are more probable at low temperatures
- •Higher temperatures lead to more uniform state distribution
- •Exponential dependence on energy difference
- •Foundation for partition functions and thermodynamic properties
Mathematical Foundation
P₁/P₂ = exp((E₂ - E₁) / (k_B × T))
P ∝ exp(-E / (k_B × T))
- P₁, P₂: Probabilities of states 1 and 2
- E₁, E₂: Energies of the states
- k_B: Boltzmann constant (1.381 × 10⁻²³ J/K)
- T: Absolute temperature (Kelvin)
Key Insight: The factor depends only on the energy difference ΔE = E₂ - E₁
Applications in Physics
Atomic Physics
Electronic state populations, emission and absorption spectra, laser operation
Condensed Matter
Fermi-Dirac and Bose-Einstein distributions, phase transitions, magnetic systems
Chemical Physics
Reaction rates, equilibrium constants, molecular vibrations and rotations