Library / Thermodynamics / THE-22
Thermodynamics
Boltzmann entropy from the number of microstates
Statistical entropy of a system that can be arranged in Ω equally likely microscopic ways while looking the same on the large scale.
- Ref
- THE-22
- Rev
- v1
- Field
- Thermodynamics
- Result
- S [J/K]
Ω is the number of microscopic arrangements (microstates) that all look the same on the large scale. Boltzmann’s constant k_B is 1.380649 × 10⁻²³ J/K. Everyday systems have an astronomically large Ω, so real calculations work with ln Ω directly; here Ω is limited to 10³⁰⁰ for the arithmetic.
Statistical entropy of a system that can be arranged in Ω equally likely microscopic ways while looking the same on the large scale. More ways means more entropy. Because Ω is astronomically large, entropy depends on its logarithm.
Calculator
- S/k_B = ln Ω52.96
- Entropy if the microstates doubled7.408×10−22 J/K
- Entropy added by doubling, k_B ln 29.57×10−24 J/K
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- 1×1023 equally likely arrangements correspond to 7.312×10−22 J/K. Entropy grows only with the logarithm, so doubling Ω adds a fixed k_B ln 2 = 9.57×10−24 J/K.
- One bit of lost information, a doubling of the possibilities, is k_B ln 2 of entropy: the link between thermodynamics and information.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J/K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ω Number of microstates | 1 to 1e300 | 0 | 9.537e-21 | 9.537e-21 | +10% | +0.18 % |
Sweep of Number of microstates
| Number of microstates | Entropy (J/K) |
|---|---|
| 1 | 0 |
| 3.162e37 | 1.192e-21 |
| 1e75 | 2.384e-21 |
| 3.162e112 | 3.576e-21 |
| 1e150 | 4.769e-21 |
| 3.162e187 | 5.961e-21 |
| 1e225 | 7.153e-21 |
| 3.162e262 | 8.345e-21 |
| 1e300 | 9.537e-21 |
Sweep one variable
Variables and how each one contributes
Entropy
What the equation gives you. Everything else feeds it.
Number of microstates
The number of equally likely microstates. Raises the entropy only logarithmically: a 10 % rise in Ω adds a tiny amount. Raising it by 10 % (to 1.1×1023) moves the result by +0.18 %, from 7.312×10−22 to 7.325×10−22 J/K.
Worked example with the values above
- Natural log: ln Ω = ln(1×1023) = 52.9595.
- S = k_B ln Ω = 1.381 × 10⁻²³ × 52.9595.
- S = 7.31184×10−22 J/K.
Assumptions and limits
- All Ω microstates are equally likely (an isolated system in equilibrium).
- Ω counts the arrangements of the macroscopic state, not the labelling of identical particles.
Common mistakes
- Using log₁₀ instead of the natural logarithm.
- Counting microstates for a state that is not at equilibrium.
- Entering Ω below 1, which is not a valid count.
Statistics and uncertainty
Because S depends on ln Ω, a large relative error in Ω changes S very little: even a factor of 10 uncertainty in Ω changes S by only k_B ln 10, which is negligible for a system with 10²³ particles.
Where it is used
- Explaining why a gas spreads into a larger volume: more ways to arrange the molecules.
- Estimating the entropy of a crystal with random defects.
Maths you need
- Natural logarithm: ln
- Probability: equally likely outcomes