The degree a spoonful of solute buys

The reaction that cannot go all the way found the logarithm at the heart of chemical equilibrium: a substance’s chemical potential rises as the logarithm of its concentration, so diluting anything costs nothing and gains entropy, and no reaction can use up its last traces. Later essays found that same logarithm making light-emitting diodes glow with a voltage, filling binding sites like electron levels, fixing the product of a semiconductor’s carriers, setting the voltage of a battery, ionising hydrogen where there is room and converting the protons of a neutron star. Every one of them used the logarithm to decide which way a particle would go.

This essay uses it for the most familiar effect of dissolving anything: a solution freezes at a lower temperature and boils at a higher one than its pure solvent. Road salt melts ice; sea water freezes at −1.9 °C; antifreeze keeps a car’s coolant liquid in a hard winter. The cause is one line in the chemical potential of the solvent. What is less familiar is that the two shifts are very different in size — the freezing point always moves three and a half times as far as the boiling point in water — and that the reason is not the solute at all but the solvent’s own latent heats.

One line moved down #

A phase transition happens where two phases have the same chemical potential. For pure water near 0 °C, the chemical potentials of ice and of liquid water are two lines against temperature, each sloping down with a slope equal to minus that phase’s molar entropy, and they cross at the melting point. Below it ice has the lower potential and wins; above it liquid does. Near 100 °C the same is true of liquid and vapour, as a boiling point is a pressure found from the other side.

Dissolve a solute that does not enter the ice and does not evaporate, and the liquid water is diluted. Its mole fraction falls from one to $1 - x$, and the logarithm lowers its chemical potential by $RT\ln(1-x)$, which for a dilute solution is $-RTx$. The ice is still pure ice and the vapour still pure vapour; their lines do not move. Only the liquid’s line drops.

The figure draws the lines for a solute at five per cent mole fraction, about three moles per kilogram, a strong solution chosen so the shifts are visible. Both crossings move outward. At the cold end the lowered liquid line now meets the ice line at a lower temperature: the solution freezes 5.2 degrees below zero. At the hot end it meets the vapour line at a higher temperature: the solution boils 1.4 degrees above a hundred. The liquid, made more stable by the solute’s entropy of mixing, wins over a wider range of temperature at both ends.

Why the cold end moves further #

The same vertical drop, $RTx$, moves the two crossings by different amounts, and the figure shows why. A crossing moves by the vertical shift divided by the difference in the slopes of the two lines that cross, and the slopes are entropies. Ice and liquid water differ in entropy by the entropy of melting, 22 joules per mole per kelvin; their lines are nearly parallel, and pushing one down slides their crossing a long way. Liquid and vapour differ by the entropy of boiling, 109 joules per mole per kelvin; their lines cross steeply, and the same push hardly moves the crossing.

So the freezing point falls by $RT_f x/\Delta S_\text{fus}$ and the boiling point rises by $RT_b x/\Delta S_\text{vap}$, and their ratio for water is 3.63, independent of what is dissolved. The asymmetry is a statement about how alike ice and water are compared with how different water and steam are. A solid and its liquid are both condensed, with molecules touching their neighbours, and melting changes their arrangement only modestly; boiling throws the molecules into a gas a thousand times less dense. The heat that changes no temperature found latent heat as the cost of that change of arrangement, and here it reappears in a second role: as the stiffness of a phase boundary against being pushed.

The same shift, read as a vapour pressure #

The boiling end has a second, older description that says the same thing in terms anyone can measure. A liquid boils when its vapour pressure reaches the pressure of the air above it. Raoult found in the 1880s that dissolving a non-volatile solute lowers a solvent’s vapour pressure in proportion to the solvent’s mole fraction: with five per cent of the molecules solute, the vapour pressure is ninety-five per cent of the pure solvent’s. Fewer of the molecules at the surface are water, so fewer escape per second, while the rate at which vapour molecules return is set by the vapour alone. To reach atmospheric pressure the solution must then be heated further, by the amount the steep rise of vapour pressure with temperature requires — and that amount, worked through with the Clausius–Clapeyron relation, is exactly $RT_b^2x/\Delta H_\text{vap}$. The chemical-potential picture and the vapour-pressure picture are the same statement: the solvent’s tendency to leave the liquid has been reduced by the logarithm of its dilution.

Degrees per mole #

In the units chemists use — moles of dissolved particles per kilogram of solvent — the dilute laws read $\Delta T = K m$, with the cryoscopic constant $K_f = RT_f^2 M/\Delta H_\text{fus}$ for freezing and the ebullioscopic constant $K_b = RT_b^2 M/\Delta H_\text{vap}$ for boiling, both computed from nothing but the solvent’s own properties: its transition temperature, molar mass and latent heat. For water they are 1.86 and 0.513 kelvin per mole per kilogram.

That settles a kitchen argument at once. Ten grams of salt in a litre of pasta water — a generous pinch — is 0.17 moles of sodium chloride, 0.34 moles of ions per kilogram, which raises the boiling point by 0.18 of a degree. The salt is for flavour; the water boils no hotter in any way a cook could detect. At the other extreme, seawater carries about 1.1 moles of dissolved particles per kilogram, and the ideal law predicts it freezes 2.1 degrees below zero; it is measured to freeze at −1.9 °C, the small difference coming from the ions’ attraction to one another, which makes them behave as slightly fewer independent particles than their number. That freezing point matters to the whole ocean: the mixture heavier than either water found that sea water, unlike fresh, has no density maximum above its freezing point, which is why the polar oceans convect all the way down to freezing before ice forms.

Counting particles, not kinds #

The laws contain the number of dissolved particles and nothing about what they are. A gram of sugar in a hundred grams of water lowers the freezing point by 0.054 of a degree; a gram of urea, a much lighter molecule, by 0.31; a gram of salt, lighter still per formula unit and splitting into two ions, by 0.64. Calcium chloride splits into three ions but each formula unit is heavier, and a gram of it does a little less than a gram of salt. This is why such effects are called colligative, from the Latin for “bound together”: they depend on the collection of particles, not on their nature. The pressure that comes from counting found the osmotic pressure doing the same, and it is the same lowering of the solvent’s chemical potential, balanced there by a pressure instead of by a change of temperature.

That property made the freezing point a balance for molecules. Raoult showed in the 1880s that equal molalities of very different organic substances lowered the freezing point of a solvent by equal amounts, and chemists then used the depression to find unknown molar masses: dissolve a weighed amount, measure the fall in freezing point with a very sensitive thermometer, and the number of moles follows from the constant. The depressions of salt solutions were the anomaly. They came out nearly twice what the formula predicted for sodium chloride and nearly three times for calcium chloride, as if each formula unit were several particles. In 1887 Arrhenius drew the conclusion that they were: in water, salts exist as separate ions. The freezing point of salt water was among the first evidence for the ionic theory of solutions, which earned him a Nobel Prize.

Solvents that make better thermometers #

The constants depend on the solvent, and water’s are small. Water has light molecules and an enormous latent heat of melting for its size, both of which make its freezing point hard to move. Benzene’s freezing point moves nearly three times as far per mole; cyclohexane’s eleven times, because it melts with very little latent heat — its molecules already rotate freely in the solid, so melting changes little — and camphor’s twenty times, for a similar reason and because its molecules are heavy. Camphor’s constant, about 38 kelvin per mole per kilogram, made it the standard solvent for measuring molar masses with an ordinary thermometer: a few milligrams of an unknown compound dissolved in molten camphor lowers its freezing point by several degrees, easily read. The Rast method, as it was called, was a laboratory staple until mass spectrometry replaced it.

The boiling constants vary less, because latent heats of boiling are much more alike from liquid to liquid — the entropy of boiling of most ordinary liquids is close to 85–90 joules per mole per kelvin, a regularity known as Trouton’s rule, with water above it because its hydrogen bonds must be broken. That is another way of seeing why boiling points are harder to move than freezing points: boiling is a big, standard change, and melting can be a small, idiosyncratic one.

Where the simple law gives out #

The linear laws assume a dilute solution. The exact relation for an ideal solution keeps the logarithm, $\ln(1 - x) = -(\Delta H/R)(1/T - 1/T_f)$, and the figure shows it parting only slowly from the straight line: at a fifth of the molecules being solute the two differ by about half a degree. The bigger departure comes from the solution not being ideal. Ethylene glycol mixed half and half with water by volume, the usual antifreeze, has about a quarter of its molecules glycol, and the ideal relation predicts it freezes at −26 °C. It actually freezes near −37 °C. Glycol and water attract each other through hydrogen bonds more strongly than either attracts its own kind, so mixing them lowers the liquid’s potential further than the ideal entropy of mixing alone, and the liquid survives to lower temperatures. Pushed further, by adding more glycol, the freezing point does not keep falling; the mixture reaches a lowest freezing temperature and then rises towards the freezing point of pure glycol, the behaviour why the triple point is a point and its relatives treat as a phase diagram of two components.

Salt on a road behaves the same way, more dramatically. Brine of increasing strength freezes lower and lower until, at 23 per cent salt by mass, it reaches −21 °C, the eutectic, below which ice and solid salt crystallise together. Below about −10 °C on a real road, the salt dissolves too slowly into the little liquid there is to be much use, and roads in very cold climates are sanded instead.

Ice that leaves the salt behind #

Because the solute does not enter the ice, freezing a solution separates them. As sea water freezes, the growing ice crystals exclude the salt, which collects in pockets and channels of concentrated brine between them and slowly drains downward, so that sea ice a year old is far less salty than the water it grew from, and multi-year ice is fresh enough to drink when melted. The ice that grows more slowly the thicker it gets treated the growth of that ice; its saltiness is set by the same exclusion that lowers its freezing point. The rejected brine, denser than the water around it, sinks and helps drive the deep circulation of the oceans from the polar seas.

The same separation has been tried, many times, as a way to make fresh water from the sea: freeze it, wash the brine from the crystals, melt them. The minimum work any method needs is the same in every case, because it is the work of undoing the solute’s lowering of the water’s chemical potential — the quantity what it costs to take the salt out computed, 0.79 kilowatt-hours per cubic metre for the first drop taken from sea water, rising as the brine left behind concentrates. Freezing, boiling and pushing water through a membrane are three ways of paying it, and none can pay less. Real plants pay several times more, for the speed they need: a process run at a finite rate has to drive the water across a finite difference of chemical potential, and the difference is lost as heat.

Melting ice by adding salt #

Road salt raises an obvious question: where does the heat come from? Sprinkle salt on ice at −5 °C and the ice melts, though nothing has warmed it. The answer is that the ice provides the heat itself. Salt dissolves into the thin film of liquid on the ice’s surface, making brine whose freezing point is below −5 °C; ice in contact with that brine is above the brine’s freezing point and melts, and melting draws latent heat from its surroundings — from the ice, the road and the brine — which cools them. A mixture of ice and salt cools itself well below zero, which is how ice cream was frozen in a bucket of salted ice for two centuries before refrigerators. The chemical potential argument says the melting happens; the latent heat says it costs heat, and the heat is taken from what is already there.

What the pictures cannot show #

The figures use ideal-solution and dilute-solution laws with constants computed from tabulated latent heats, and treat the solute as entering neither the ice nor the vapour. Real solutions depart from ideality at concentrations of a few per cent for salts — whose ions interact strongly — and at higher concentrations for most molecular solutes; the seawater and antifreeze points are measured values marked to show the departure, not results of the model. The chemical-potential lines are drawn with an assumed entropy for liquid water, so their absolute heights mean nothing, and only their slopes and shifts carry information. The solvent constants are computed from rounded literature values of the latent heats, and camphor’s latent heat in particular is uncertain by several per cent.

Still open: how life survives below the freezing point #

Some fish in polar seas, whose blood would freeze at about −0.7 °C by its salt content alone, swim in water at −1.9 °C without freezing. They do not rely on colligative depression, which would need their blood to be as salty as the sea. They carry antifreeze proteins, which bind to the surfaces of tiny ice crystals and stop them growing, lowering the temperature at which ice grows without lowering the melting point — a gap between freezing and melting that no colligative effect can open. Insects, plants and bacteria that survive freezing use related proteins, sugars and controlled ice formation. How the proteins recognise ice, why they stop growth at one face and not another, and whether synthetic versions can protect frozen cells and organs for transplant, are being worked out now.

The habit worth carrying away is to ask which phases a change touches and how steeply their lines cross. A solute lowers only the liquid’s chemical potential, by RTx, and a transition moves by that amount divided by its entropy change: water’s freezing point falls 1.86 K per mole of particles per kilogram and its boiling point rises 0.513, 3.63 times less, because ice and water differ in entropy five times less than water and steam. The effect counts particles, which is how it weighed molecules and found ions.