The entropy every liquid gains on boiling

A boiling point is a pressure, not a temperature found that a liquid boils when its vapour pressure reaches the pressure above it, so the “boiling point” printed in tables is the temperature at which that happens at one atmosphere, and nothing more fundamental. The heat that changes no temperature followed where the latent heat of boiling goes: into pulling molecules apart against their attraction and into pushing back the atmosphere to make room for the vapour.

Both essays treated each liquid separately. Put the latent heats of many liquids side by side, though, and a regularity appears that no study of one liquid would suggest. Frederick Trouton, a young Dublin physicist, noticed it in 1884: the latent heat per mole, divided by the boiling temperature in kelvin, is nearly the same for very different liquids. Benzene, chloroform, acetone, hexane, ether, carbon tetrachloride — molecules of different shapes, sizes and chemistry, boiling at temperatures from 35 °C to 80 °C — all come out within a few per cent of 87 joules per mole per kelvin. That quotient is an entropy, and the regularity is a statement about what boiling does to the number of ways the molecules can be arranged.

Trouton himself is better remembered for something else. A student of George FitzGerald at Trinity College Dublin, he went on in 1903, with H. R. Noble, to hang a charged capacitor from a fine fibre and look for the twist that the Earth’s motion through the supposed ether ought to give it. They found none, and the reason no twist appears became one of the small puzzles that special relativity resolved. The boiling rule was a note he published at twenty-one, two short paragraphs in a journal of the day, and it has outlived almost everything else from that period in the chemistry of liquids.

Twenty-one liquids on one line #

On logarithmic axes the latent heats of twenty-one substances span three decades and their boiling points two and a half, and the ordinary liquids fall along a single straight line of slope one. Liquid nitrogen needs 5.6 kilojoules per mole and boils at 77 K; benzene needs 31 and boils at 353 K; sodium needs 97 and boils at 1,156 K. Each is about 87 joules per mole for every kelvin of boiling point. The ordinary liquids boiling between 250 and 700 kelvin scatter about that value by only 3.3 per cent.

The practical use is immediate. A chemist who knows a new compound’s boiling point can estimate the heat needed to distil it without measuring anything else, and the estimate is good to a few per cent for most organic liquids. That is the form in which the rule appears in handbooks. Its physical content lies in the deviations, which come in three kinds, each with a reason.

Why an entropy should be universal #

The latent heat divided by the boiling temperature is the entropy gained on boiling: at the boiling point the liquid and vapour have equal Gibbs energy, so $\Delta H = T_b \Delta S$, and $\Delta S$ is what Trouton found to be constant.

Entropy is a count — the logarithm of the number of ways a system can be arranged — and the entropy gained on boiling is mostly the gain in the number of places each molecule can be. In the liquid a molecule is confined to a small free volume among its neighbours; in the vapour it roams the whole volume the vapour fills. The entropy gained per mole is roughly $R$ times the logarithm of the ratio of those volumes, together with smaller terms from rotation and from the liquid’s own structure.

The vapour’s volume is the part that is nearly universal. At one atmosphere and the boiling point, a mole of vapour fills $RT_b/P$ — about thirty litres for a liquid boiling near room temperature — and the free volume per mole in a liquid, the room each molecule has to rattle in, is a small and roughly similar fraction of its molar volume for all simple liquids. The ratio of the two is about the same for liquids of very different molecules, and so is its logarithm. That is all Trouton’s rule says: compared at the same vapour pressure, liquids gain about the same entropy on boiling because they gain about the same factor in room.

Liquid by liquid #

The sorted list makes the exceptions plain, and they divide into four groups.

The first is the hydrogen-bonded liquids: water at 108.9, ethanol at 109.7, methanol at 104.2, ammonia at 97.3. Their molecules link into networks through hydrogen bonds, which constrain not only where each molecule is but which way it points. The liquid is therefore more ordered than a simple liquid, and boiling, which destroys the network, gains extra entropy. Water’s high latent heat is not only a sign of strong attraction — strong attraction would raise its boiling point and leave its entropy of boiling ordinary — but of order. The same networks are why a spoonful of solute changes water’s freezing and boiling points by the amounts it does.

The second is acetic acid, at 60.6, well below the line. Its vapour at the boiling point is not single molecules but mostly pairs, held together by two hydrogen bonds facing each other, so boiling releases half as many independent particles as the formula suggests and gains correspondingly less entropy. Its latent heat per mole of monomer looks low for the same reason.

The third is helium and hydrogen, far below the line at 19.6 and 44.6. In these liquids quantum mechanics dominates: the atoms are so light, and the liquid so cold, that their zero-point motion already spreads each atom over a large fraction of the space between its neighbours. The liquid is far less ordered than a classical liquid at that density, and boiling has less to gain. Helium is the extreme case, a liquid that does not freeze at all at ordinary pressures, whose entropy of boiling is a fifth of Trouton’s value.

The fourth group is more interesting than it first looks: the ordinary liquids that simply boil cold. Neon at 63, nitrogen at 72, argon at 74, oxygen at 76 and methane at 73 are as simple as liquids get, with no hydrogen bonds, no association and no strong quantum effects, and they still lie well below the line.

Comparing at the same concentration #

The reason cold-boiling liquids lie low is not in the liquid at all. At one atmosphere and 77 K, a mole of nitrogen vapour occupies only about six litres, a fifth of the volume a mole of vapour occupies at 350 K. The vapour is denser because it is colder, so the gain in room on boiling is smaller, and so is the entropy. Comparing all liquids at one atmosphere compares their vapours at different concentrations, which builds a drift with boiling point into the numbers.

Joel Hildebrand proposed in 1915 to compare liquids instead at the temperature where their vapours have the same concentration — the same number of molecules per litre — which removes that drift. Adjusting each entropy by $R\ln$ of the ratio of concentrations does exactly that, and it pulls the cold-boiling liquids up and the hot ones down. For the ordinary liquids in the figure the spread falls by more than half, and the trend with boiling point disappears: neon and sodium, separated by a factor of forty in boiling temperature, gain nearly the same entropy on evaporating into vapour of the same density. The hydrogen-bonded liquids stay above even after the adjustment, which is the sign that their excess is real — a property of their liquid — rather than an artefact of where they were compared.

Melting has a rule of its own, and a smaller one #

Freezing and melting have a corresponding regularity, found by Richards in 1897: for many metals the entropy of melting is close to the gas constant itself, about 8 to 10 joules per mole per kelvin, roughly a ninth of the entropy of boiling. The contrast is the point. Melting changes the volume by a few per cent, so it adds almost no room; what it adds is the freedom of each atom to leave its lattice site and wander among its neighbours, a disorder of arrangement rather than of space, and that is worth about $R$ per mole for simple atoms. Boiling adds a factor of a few thousand in volume, and $R$ times the logarithm of a few thousand is about $8R$, the bulk of Trouton’s $10.5R$.

The two rules are much less alike than their form suggests. Richards’s rule fails badly for molecules that rotate freely in the solid, or that must find one orientation in the crystal and lose it on melting, and its exceptions run from a fraction of $R$ to many times it. Trouton’s rule holds better because the room a vapour adds dwarfs everything else, and the details of the liquid’s structure enter only through a small correction to a large logarithm. Both are measured from the same kind of experiment — the heat that changes no temperature at a transition, divided by the temperature of the transition — and why the triple point is a point is where the two transitions, and their two entropies, meet.

What the rule predicts about pressure #

Combined with the Clausius–Clapeyron relation — the slope of the vapour-pressure curve is the entropy of boiling divided by the change of volume, which decides how a coexistence line leans — Trouton’s constant predicts a liquid’s whole vapour-pressure curve near its boiling point from the boiling point alone. With the vapour treated as an ideal gas and the latent heat held constant, $\ln(P/P_0) = (\Delta S/R)(1 - T_b/T)$, and with $\Delta S/R = 10.5$ everything else is fixed.

Written the other way round, the same relation says something odd. A vapour pressure is, to a first approximation, a Boltzmann factor: the chance that a molecule has the energy to leave the liquid, times a prefactor. $P = P_0\,e^{\Delta S/R}\,e^{-\Delta H/RT}$, and at the boiling point it equals one atmosphere. Trouton’s rule fixes the prefactor: $e^{10.5}$ atmospheres, about thirty-six thousand, the same for every ordinary liquid. Every such liquid’s vapour-pressure curve, extended naively to infinite temperature, heads for the same thirty-six thousand atmospheres, and what distinguishes one liquid from another is only how far down the exponential its latent heat puts it at room temperature. The extension is not physical — every real curve ends at its critical point, at a few tens of atmospheres — but the shared prefactor is, and it is the rule restated.

That is the arithmetic behind vacuum distillation. A liquid that boils at 200 °C at one atmosphere boils at about 62 °C at ten torr, a pressure a simple water-jet pump can reach, and a compound that would decompose at its normal boiling point can be distilled a hundred and forty degrees cooler. Chemists used a printed nomograph built on exactly this rule for most of the twentieth century; it works well for ordinary liquids and poorly for water and alcohols, which is the same exception again. Water itself boils at 11 °C at ten torr, twenty degrees warmer than the rule predicts for a liquid of its boiling point, because its larger entropy of boiling makes its vapour pressure fall faster as it cools.

The same constant says how steeply vapour pressure rises with temperature near the boiling point. The rise that doubles it is $\ln 2 \cdot T_b/(\Delta S/R)$, about one fifteenth of the absolute boiling temperature for any ordinary liquid: six kelvin for liquid nitrogen, twenty-three for benzene, seventy-nine for sodium. In proportion to their boiling points, all of them are equally sensitive. It is why a pressure cooker at two atmospheres raises water’s boiling point by twenty degrees and why liquid nitrogen in an open flask holds its temperature to within a few degrees as the weather changes.

Why one number spans so much #

Trouton’s rule is a corresponding-states law in disguise. Liquids whose molecules interact through forces of the same shape — a repulsive core and an attraction, differing only in size and strength — behave alike when temperatures are measured in units of the attraction’s strength and volumes in units of the core’s size. Their boiling points at one atmosphere are then at nearly the same fraction of their critical temperatures, about two thirds, and their entropies of boiling at that point are nearly the same. The point at which the two become one and the tension that vanishes where the phases meet found universality near the critical point, in the exponents; Trouton’s rule is a looser, older universality far below it, in the size of the jump.

The looseness has a source. Molecules are not all spheres with one kind of attraction. Long chains like hexane have more ways to arrange themselves in the vapour than in the liquid, which raises their entropy of boiling slightly; rigid, compact molecules have fewer. That is why hexane sits at 84 and acetone at 88, and why chemists have refined Trouton’s constant with corrections for molecular shape. The corrections are small because, for liquids that boil near room temperature, the dominant entropy is the gain in room, and it is nearly the same for everyone.

What the pictures cannot show #

The latent heats and boiling points are standard tabulated values, each measured once and checked many times, and they are drawn as points with no error bars because their uncertainties are a fraction of a per cent, far smaller than the deviations that matter. The Hildebrand adjustment uses the ideal-gas concentration of the vapour, which is a good approximation at one atmosphere for most of these substances and a poorer one for the highest-boiling liquids, whose vapours at their boiling points are dense enough to depart from ideality by a few per cent.

The vapour-pressure predictions assume a latent heat that does not change with temperature, which is reasonable over a range of a hundred degrees and fails approaching the critical point, where the latent heat falls to zero along with the difference between liquid and vapour. They are drawn as curves to four decades below one atmosphere, and below about a hundredth of an atmosphere they should be read as estimates.

The domain of Trouton’s rule is ordinary liquids — molecules without strong association, at temperatures where quantum effects are small — boiling at around one atmosphere. Inside it the entropy of boiling is about 10.5R. Outside it, the deviations name what makes a liquid unusual.

Still open: how much order is in a liquid #

Trouton’s rule says that most liquids gain about the same entropy on boiling, and the exceptions say that some liquids are more ordered than others. How much more, and of what kind, is harder to say. The entropy of a liquid has no simple formula: it is not a gas, whose entropy follows from counting free particles, nor a crystal, whose entropy follows from counting its vibrations. Simulations can compute liquid entropies for model molecules, and for water the excess entropy of boiling can be attributed in part to the hydrogen-bond network and in part to orientational order, but how to divide it, and whether “the order in a liquid” is a single quantity at all, are argued about. The answer matters for anything that depends on how liquids hold together, from the solubility of drugs to the behaviour of water near surfaces.

The rule itself is a single observation that has survived for a hundred and forty years. The heat to boil an ordinary liquid divided by its boiling temperature is about 87 J/(mol K), 10.5 times the gas constant, because boiling at one atmosphere gives every molecule about the same gain in room; compared at equal vapour concentration the ordinary liquids agree still better, and the liquids that miss — water and alcohols above, acetic acid and helium below — miss because their liquids are more ordered, their vapours are pairs, or their atoms are already quantum-spread. A latent heat is mostly a boiling point in disguise, and the exceptions are where the chemistry is.