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Boltzmann Entropy Formula

Relates the entropy of a system to the number of microscopic configurations (microstates) that correspond to its macroscopic state.

Written by the Vorzep Team

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Boltzmann Entropy CalculatorS = k · ln(W)

S = k · ln( W )
S = entropy (J/K)  ·  k = Boltzmann constant (J/K)  ·  W = number of microstates
⟹ Solve⟷S, k, W
J/K
J/K
—
k = 1.380649 × 10⁻²³ J/K (exact)
Please fix the errors above.
Solve for:
Presets:
Entropy (S)—
—
S: —k: —W: —
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Entropy Magnitude—
Low (< 1e-23) Medium (1e-23–1e-21) High (1e-21–1e-19) Very High (> 1e-19)
S = k · ln(W)  ·  k = 1.380649 × 10⁻²³ J/K
S = k * ln(W)
Boltzmann Entropy Formula

Variables

SymbolQuantityUnit
SEntropyJ/K
kBoltzmann constant1.381e-23 J/K
WNumber of accessible microstates (dimensionless)

What it means

The Boltzmann entropy formula is the statistical definition of entropy, relating the entropy S of a system to the number of microscopic configurations W (microstates) consistent with its macroscopic state. The constant k is Boltzmann’s constant (1.38×10⁻²³ J/K). This equation is engraved on Boltzmann’s tombstone. It provides a probabilistic interpretation of entropy and explains why systems tend towards equilibrium (maximum W). This formula is the foundation of statistical mechanics and underpins the second law of thermodynamics. It is used to calculate entropy in gases, crystals, and other systems. Understanding the Boltzmann entropy is essential for connecting thermodynamics to atomic theory.

Worked example

Boltzmann Entropy – Two Examples

Real‑World
Scenario: A system has W = 10 microstates. Find its entropy (k = 1.38×10⁻²³ J/K).
ParameterValue
W10
1S = k·ln(W) = 1.38e-23 × ln(10) = 1.38e-23 × 2.303 = 3.18×10⁻²³ J/K
Result 3.18×10⁻²³ J/K ✓ Small
Scenario: W = 100. Find S.
ParameterValue
W100
1S = 1.38e-23 × ln(100) = 1.38e-23 × 4.605 = 6.36×10⁻²³ J/K
Result 6.36×10⁻²³ J/K ✓ Larger
Key insight: S = k·ln(W) – entropy measures the number of microscopic configurations.

Common mistakes

  • Boltzmann constant k: k = 1.381×10⁻²³ J/K – use the correct value.
  • Number of microstates W: The number of ways the system’s microscopic configuration can be arranged.
  • Natural logarithm ln: Use natural log, not log₁₀.
  • Entropy S: In J/K – a measure of disorder.
  • Statistical mechanics: This formula connects microscopic states to macroscopic entropy.

Applications

Boltzmann's entropy formula, S = k·ln(W), connects the macroscopic property of entropy to the number of microscopic configurations (W). It is fundamental to statistical mechanics and is used to understand phase transitions, information theory, and the behaviour of gases. Engineers apply it in polymer science to model chain configurations, in nanotechnology to study molecular assemblies, and in computational chemistry. The formula also underpins the concept of information entropy in communications. By using this relation, professionals can bridge the microscopic and macroscopic worlds and design systems that exploit entropic effects.

  • Statistical mechanics modelling of gases and liquids
  • Polymer science – conformational entropy calculations
  • Nanotechnology – self‑assembly and molecular motors
  • Information theory and data compression
  • Computational simulation of molecular systems