Entropy counts the microstates behind what you can see. When a liquid boils, the same molecules gain about sixteen hundred times the room, and the count of arrangements explodes. Worked here for water, with a result the counting alone does not fully reach: the volume term explains roughly half of the measured number, and the page says which half is missing.
Boltzmann’s statistical definition (1877) is $S = k_B \ln W$, where $W$ is the number of microscopic arrangements consistent with the macroscopic state. It is a count, not a measure of “mess”. The energy and entropy chapter tells the history, including the H-theorem (1872) and the replies to the reversibility objection; this page does one calculation with the formula and checks how far it goes.
Give each of $N$ molecules a choice among $M$ cells of space. Then $W = M^{N}$. Double the cells and every molecule has twice the choices:
Per mole, $N_A k_B = R$, so one mole whose volume doubles gains $R\ln 2 = 5.76$ J/(mol·K). This is the same counting move as in ch-bose, where the choice of how to count (distinguishable or not) changes the answer; here it is only the number of cells that changes.
At 100 °C and 1 atm an ideal gas occupies $RT/P = 30.6$ L per mole. Liquid water at that temperature occupies roughly 18.8 cm³ per mole (an approximate value, from a density near 0.96 g/cm³, not taken from a steam table for this page). The ratio is about 1,600. At the boiling point, heat taken in reversibly gives $\Delta S = \Delta H_{\text{vap}}/T_b$. With $\Delta H_{\text{vap}} = 40.65$ kJ/mol:
| quantity | value, J/(mol·K) |
|---|---|
| Trouton’s rule, ordinary liquids | about 85–88 |
| toluene / chloroform / benzene | 87.30 / 87.92 / 89.45 |
| water, from 40.65 kJ/mol at 373.15 K | 108.9 |
| position-only estimate $R\ln(V_{gas}/V_{liq})$, water | about 61 |
Two things are visible. Water sits well above Trouton’s range, which is usually attributed to hydrogen bonding making the liquid unusually ordered; this page does not derive that. And the volume-ratio estimate gives about 61, which is roughly 56% of 108.9 and lies below even the ordinary-liquid range. Free volume alone does not reproduce the measured entropy of vaporization.
The Clapeyron relation $dP/dT = \Delta S/\Delta V$ reads the entropy jump as a slope of the coexistence line. With the numbers above, and the ideal-gas volume for the vapour, it gives about 3.6 kPa/K. This is computed from the inputs on this page; it is not compared with a measured saturation curve here. The geometric reading (tangent planes, common tangent, Gibbs’ surface) is in ch-clapeyron-gibbs.
Adsorbing a molecule into a pore takes it from a large volume to a small one, so the same $N k_B \ln(V_2/V_1)$ term appears with the opposite sign, and is one reason adsorption needs an energy gain to win. That is the standard textbook account. It is the same functional form, not shared invariant as the self-trapping radius in the zeolite paper: nothing here derives one from the other, and no pore-entropy number is computed on this page.
The counting above is the position part only. It does not include the change in rotational and vibrational freedom, the interactions in the liquid, or hydrogen bonding, and together those make up the missing part of 108.9. The liquid molar volume is approximate. The Trouton values and $\Delta H_{\text{vap}}$ are typed in from a secondary source, not measured or derived here. “Entropy is disorder” is not the claim this page makes; it is a count.
Boltzmann 1877 L. Boltzmann, on the relation between the second law and probability calculus; the formula $S = k\ln W$ is attributed to Boltzmann and was written in this form by Planck. (Cited as history; the paper was not re-read for this page.)
data Wikipedia, “Enthalpy of vaporization” (water, 373.15 K: 40.65 kJ/mol in the text, 40.66 in its table) and “Trouton’s rule” (85–88; toluene, benzene, chloroform values above). Secondary sources.
in-corpus ch-energy-entropy · ch-bose · ch-clapeyron-gibbs
verification book7/ch-boltzmann-verify.py —
standard library only; its last block holds controls that must fail. It checks the
arithmetic and the page structure, not that the typed-in reference values are correct.
No priority is claimed for anything on this page. Every relation used is classical thermodynamics or statistical mechanics.