Thermodynamics

Trouton's Rule: Why the Entropy of Vaporization Is Always ~85 J/mol·K

Divide the heat it takes to boil benzene (30.7 kJ/mol) by its boiling point (353 K) and you get 87 J/mol·K. Do the same for chloroform, hexane, carbon tetrachloride, or diethyl ether and you land on almost exactly the same number: about 85–88 J/mol·K. This uncanny constancy across chemically unrelated liquids is Trouton's Rule, discovered by the Irish physicist Frederick Thomas Trouton in 1884.

Trouton's Rule states that the molar entropy of vaporization at the normal boiling point — the enthalpy of vaporization ΔH_vap divided by the boiling temperature T_b — is approximately constant at ~85 J/mol·K (roughly 21 cal/mol·K, or ~10.5R) for most non-associated liquids. It is one of the oldest and most useful empirical shortcuts in physical chemistry, letting you estimate a latent heat from nothing more than a boiling point.

TypeEmpirical thermodynamic rule
IntroducedFrederick T. Trouton, 1884
Key equationΔS_vap = ΔH_vap / T_b ≈ 85 J/mol·K
Typical value85–88 J/(mol·K) (~10.5R, ~21 cal/mol·K)
Applies toNon-associated (non-H-bonding) liquids
Measured byCalorimetry / vapor-pressure (Clausius–Clapeyron)

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What Trouton's Rule Is and Where It Applies

Trouton's Rule is an empirical generalization about the phase change from liquid to gas. When a liquid boils at its normal boiling point (the temperature at which its vapor pressure equals 1 atm), the molar entropy change is:

ΔS_vap = ΔH_vap / T_b ≈ 85 J/mol·K

where ΔH_vap is the molar enthalpy (latent heat) of vaporization and T_b is the absolute boiling temperature. The remarkable observation is that this ratio is nearly the same constant for a huge range of substances, from noble-gas liquids to hydrocarbons to halogenated solvents.

Applies well to: non-polar or weakly polar "non-associated" liquids — benzene, toluene, CCl₄, chloroform, hexane, diethyl ether, argon, nitrogen.
Fails for: hydrogen-bonded liquids (water, alcohols, carboxylic acids, HF) and very light liquids.

Its practical power is inversion: knowing only T_b, you can estimate ΔH_vap ≈ 85 × T_b J/mol without any calorimetry.

The Derivation: Why the Constant Emerges

The constancy is not a coincidence of measurement — it follows from statistical thermodynamics. Entropy of vaporization measures the disorder gained when molecules escape the condensed phase into the gas phase.

By the Sackur–Tetrode relation, the translational entropy of an ideal gas depends only weakly (logarithmically) on molecular mass and on temperature. The gas-phase molar volume at 1 atm is fixed by the ideal-gas law, V_gas = RT_b/P, so the dominant term — the volume expansion from a compact liquid (~tens of cm³/mol) to a dilute gas (~30,000 cm³/mol at typical T_b) — is roughly the same for everything.

Step by step:

Liquid molar volumes are all small and similar (~50–150 cm³/mol).
Gas molar volume scales as RT_b/P, and P = 1 atm by definition of T_b.
The entropy gain is dominated by ΔS ≈ R·ln(V_gas/V_liq), which is nearly constant because the expansion ratio (~10³) is similar for all.

Adding a small, roughly universal contribution from freed translational and rotational motion gives ~10.5R ≈ 87 J/mol·K.

Key Numbers and a Worked Example

The Trouton constant is commonly quoted three ways — pick the units carefully:

85–88 J/(mol·K) (SI, the usual textbook figure)
~21 cal/(mol·K) (older literature)
~10.5R (in units of the gas constant R = 8.314 J/mol·K)

Worked example — estimating a latent heat. Bromine boils at 59 °C = 332 K. Trouton's Rule predicts:

ΔH_vap ≈ 85 J/mol·K × 332 K ≈ 28,200 J/mol ≈ 28.2 kJ/mol.

The experimental value is 29.96 kJ/mol — an error of only ~6% from a single boiling point.

Reverse check — benzene. With ΔH_vap = 30.7 kJ/mol and T_b = 353 K, ΔS_vap = 30,700/353 = 87.0 J/mol·K, squarely on target. Chloroform gives 87.9, toluene 87.3, CCl₄ 85.7 — all within ±3 of the nominal constant.

How ΔH_vap and ΔS_vap Are Measured

Two independent routes supply the numbers that Trouton's Rule correlates.

1. Direct calorimetry. A known amount of liquid is boiled and the electrical energy needed to vaporize it (at constant pressure, 1 atm) is measured. Dividing by moles gives ΔH_vap directly; dividing by T_b gives ΔS_vap.

2. Vapor-pressure / Clausius–Clapeyron. Measuring vapor pressure P at several temperatures and plotting ln(P) versus 1/T yields a straight line whose slope is −ΔH_vap/R:

ln(P) = −(ΔH_vap/R)(1/T) + C

This is often preferred because it needs no calorimeter — just a manometer and a thermostat.

In practice, Trouton's Rule is used in the opposite direction as a sanity check or a first estimate: chemical engineers screen unknown or newly synthesized compounds, checking that a reported ΔH_vap is consistent with 85·T_b, and flagging associating liquids when it is not.

Trouton's Rule vs. Its Close Cousins

Several related rules sharpen or extend the same idea:

Hildebrand's Rule (1915, Joel Hildebrand): compares entropies of vaporization at temperatures where the vapors have equal molar concentration rather than at 1 atm. This removes the T_b dependence and gives an even tighter constant (~92 J/mol·K at a common vapor concentration), explaining why light liquids like methane (73 J/mol·K by raw Trouton) fall in line once their low boiling points are accounted for.
Trouton–Hildebrand–Everett rule: adds a temperature-dependent correction term, ΔS_vap ≈ 4.5R + R·ln(T_b), improving accuracy over a wider T_b range.
Watson correlation: estimates how ΔH_vap changes with temperature away from T_b, complementing Trouton's single-point estimate.

Distinguish Trouton's Rule from the Clausius–Clapeyron equation (an exact thermodynamic relation between P, T, and ΔH_vap) — Trouton's Rule is only an empirical approximation for the value of ΔS_vap, not a fundamental law.

Exceptions, Hydrogen Bonding, and Why It Matters

The famous failures of Trouton's Rule are as instructive as its successes.

Hydrogen-bonded liquids (positive deviation): water gives ΔS_vap = 40.7 kJ/mol ÷ 373 K = 109 J/mol·K, and ethanol ~110 J/mol·K. The liquid is abnormally ordered by an extended hydrogen-bond network, so boiling releases extra disorder — pushing ΔS_vap well above 85. Formic acid and HF behave similarly.
Carboxylic acids (negative deviation): acetic acid dimerizes in the vapor via double H-bonds, so fewer independent gas particles form and ΔS_vap can drop below 85 J/mol·K.
Light/low-boiling liquids: methane, neon, and hydrogen boil below ~150 K and give 65–75 J/mol·K because the raw 1-atm reference under-counts their translational entropy.

Why it matters: the very deviations that break the rule are a quantitative fingerprint of intermolecular association. Trouton's Rule thus doubles as a diagnostic — a large positive deviation is a red flag for strong hydrogen bonding, valuable in solvent selection, distillation design, and estimating properties of newly made compounds.

Entropy of vaporization at the normal boiling point for representative liquids (ΔS_vap = ΔH_vap/T_b). Non-associated liquids cluster near 85 J/mol·K; hydrogen-bonded and light liquids deviate.
Liquid	T_b (K)	ΔH_vap (kJ/mol)	ΔS_vap (J/mol·K)
Benzene	353	30.7	87.0
Toluene	384	33.5	87.3
Chloroform	334	29.4	87.9
Carbon tetrachloride	350	30.0	85.7
Water (H-bonded)	373	40.7	109.1
Ethanol (H-bonded)	351	38.6	110.0
Methane (light)	112	8.2	73.2

Frequently asked questions

Why is the entropy of vaporization approximately constant at 85 J/mol·K?

Because vaporization is dominated by the volume expansion from a compact liquid to a dilute gas, and at the normal boiling point that expansion ratio is about the same (~1000-fold) for nearly all liquids since the vapor pressure is fixed at 1 atm. The entropy gain, roughly R·ln(V_gas/V_liq) plus a near-universal contribution from freed motion, therefore works out to ~10.5R ≈ 87 J/mol·K regardless of the substance.

Who discovered Trouton's Rule and when?

Frederick Thomas Trouton, an Irish physicist, published the correlation in 1884 while at Trinity College Dublin. He noticed that the latent heat of vaporization divided by absolute boiling point was nearly constant across many liquids. Joel Hildebrand later refined it in 1915 with a more consistent reference state.

Why does water violate Trouton's Rule?

Water's entropy of vaporization is about 109 J/mol·K, far above the 85 J/mol·K norm. Its extensive hydrogen-bond network makes the liquid unusually ordered (low entropy), so vaporizing it releases more disorder than for a non-associated liquid. Ethanol, formic acid, and HF show the same positive deviation for the same reason.

How do I use Trouton's Rule to estimate a heat of vaporization?

Multiply the constant by the normal boiling point in kelvin: ΔH_vap ≈ 85 J/mol·K × T_b. For bromine (T_b = 332 K), that gives ~28.2 kJ/mol versus the experimental 29.96 kJ/mol — about 6% error. It is a quick first estimate when you know only the boiling point.

What is the difference between Trouton's Rule and Hildebrand's Rule?

Trouton's Rule evaluates the entropy of vaporization at the 1-atm normal boiling point. Hildebrand's Rule instead compares liquids at temperatures where their vapors have the same molar concentration, which cancels the boiling-point bias and yields a tighter constant. Hildebrand's version correctly brings low-boiling liquids like methane into agreement.

Is Trouton's Rule the same as the Clausius–Clapeyron equation?

No. Clausius–Clapeyron is an exact thermodynamic relation linking vapor pressure, temperature, and ΔH_vap, and is used to measure ΔH_vap from vapor-pressure data. Trouton's Rule is only an empirical approximation that predicts the numerical value of the entropy of vaporization (~85 J/mol·K). They are complementary, not equivalent.