Why the triple point is a point

Water boils at a temperature that depends on the pressure, so “boiling point” names a curve rather than a number. Ice melts at a temperature that depends on the pressure too, along a different curve. Ice sublimes along a third.

The three curves cross at one place: $273.16$ K and $611.657$ Pa. That is a specific enough fact to have been used to define the kelvin, and the reason it is a point — rather than a short segment, or a small region, or three separate near-misses — is a piece of counting that mentions neither water nor heat.

The count #

The state of a system at equilibrium, as far as phase behaviour cares, is described by intensive quantities: temperature, pressure, and the composition of each phase present. Extensive quantities — how much of each phase there is — do not affect what phases can coexist, which is why a glass with one ice cube and a glass with fifty behave identically.

Count the intensive numbers. Temperature and pressure make two, shared by all phases in equilibrium. Each phase’s composition takes $C-1$ numbers for $C$ components, since the fractions sum to one. With $P$ phases that is

$$\text{variables} = 2 + P(C-1).$$

Now count the conditions. Equilibrium between two phases requires each component’s chemical potential to be equal in both. With $P$ phases there are $P-1$ independent equalities per component, so

$$\text{conditions} = C(P-1).$$

Subtract:

$$F = 2 + P(C-1) - C(P-1) = C - P + 2.$$

For a single substance, $C = 1$. One phase gives $F = 2$: temperature and pressure may both be chosen freely, which is an area of the diagram. Two phases give $F = 1$: choose the temperature and the pressure is determined, which is a curve. Three phases give $F = 0$: nothing may be chosen, and the state is a point.

Four phases give $F = -1$. That is not a system with less freedom than none; it is a system with more conditions than unknowns, which generically has no solution at all. Four phases of one substance do not coexist, anywhere, at any pressure, and the reason is that a set of four equations in three unknowns does not usually have a root.

The whole argument used one piece of thermodynamics — that chemical potentials are equal at equilibrium — and no property of any material. That is why the rule is quoted rather than derived in most places: there is not much to derive.

Why intensive variables and not extensive ones #

The step that is easiest to skip is the first one, and skipping it produces a rule that gives the wrong answer.

The variables counted are all intensive: they have the same value throughout a phase and do not depend on how much of it there is. Temperature, pressure and mole fraction qualify; volume, mass and energy do not. That restriction is what makes the count finite and what makes the answer independent of the size of the system.

The justification is that the equilibrium conditions are themselves conditions on intensive quantities. Thermal equilibrium equates temperatures; mechanical equilibrium equates pressures; chemical equilibrium equates chemical potentials. None of them says anything about amounts, so the amounts are free and are not counted — which is exactly the observation that a glass with one ice cube behaves like a glass with fifty.

It also explains why the rule is silent about something a chemist often wants to know. Given a temperature and a pressure inside the two-phase region, the rule says the state is fixed; it does not say what fraction is liquid. That fraction is an extensive question, answered by the lever rule from the overall composition, and it was excluded from the count at the first step.

Counting arrangements is the microscopic side of the same subject, and it is worth being clear that the two counts answer different questions. The phase rule counts equations: how many conditions the equality of chemical potentials imposes, against how many variables there are to satisfy them with. A count of arrangements gives the entropy, which is what fixes where the boundaries lie. One says how many degrees of freedom remain; the other says where in the plane they are.

What the count does not say #

The rule says how many points, curves and areas there are. It says nothing about where they are.

Where they are is Clapeyron’s, and it is a separate calculation. Along a coexistence curve the two phases have equal chemical potentials, and differentiating that equality along the curve gives

$$\frac{\mathrm{d}p}{\mathrm{d}T} = \frac{L}{T\,\Delta v},$$

with $L$ the latent heat and $\Delta v$ the change in specific volume. Both are properties of the substance and both have to be measured.

Heat going in with the temperature not rising is the plateau whose length is the latent heat, and it is one of the two numbers Clapeyron’s relation needs — the other being the volume change across the same boundary. Those two fix the slope of the boundary at every point, so the whole line can be integrated from calorimetry and densitometry without a single pressure measurement.

For water at the triple point the three come out as follows. Vaporisation: $L = 2500.9$ kJ/kg and the volume change is essentially the vapour’s, giving $44.4$ Pa/K. Sublimation: $L = 2834.4$ kJ/kg, giving $50.3$ Pa/K. Fusion: $L = 333.5$ kJ/kg and a volume change that is negative, because ice is less dense than water, giving $-1.35\times10^7$ Pa/K — $-138$ bar per kelvin, steep enough that on any diagram showing the other two curves it is a vertical line leaning imperceptibly to the left.

That negative slope is water’s oddity and it has consequences everywhere. It is why ice melts under pressure, why the deep base of a glacier can be liquid, and why the phase diagram’s solid region is to the left of the liquid rather than below it. It is not, however, relevant to the counting: the rule never asked which way a curve leaned.

The three slopes are not independent #

Here is the part that makes the crossing more than a coincidence.

Enthalpy is a state function, so the enthalpy change from solid to vapour is the same whether the route is direct or by way of the liquid. At the triple point, where all three phases are at the same temperature and pressure, that requires

$$L_{\text{sub}} = L_{\text{fus}} + L_{\text{vap}}.$$

Water’s measured values are $333.5$ and $2500.9$ against $2834.4$ kJ/kg. The sum of the first two is $2834.4$, agreeing to four parts in ten thousand, which is within the uncertainty of the measurements.

Because the vaporisation and sublimation curves have nearly the same $\Delta v$ — both dominated by the vapour’s volume — their slopes are in the ratio of their latent heats, and the difference of their slopes is

$$\left(\frac{\mathrm{d}p}{\mathrm{d}T}\right)_{\text{sub}} - \left(\frac{\mathrm{d}p}{\mathrm{d}T}\right)_{\text{vap}} = \frac{p\,L_{\text{fus}}}{R_v T^2} = 5.92\ \text{Pa/K},$$

which is what the drawn slopes differ by. Two of the three curves fix the third’s slope, and any measurement disagreeing with that would be a measurement of something inconsistent rather than a discovery.

The immediate consequence is that the sublimation curve is the steeper of the two, always, since the fusion latent heat is positive. So the two curves genuinely cross rather than touching, ice sublimes below the triple point and melts above it, and there is no substance for which the ordering is the other way round.

The slopes drawn honestly #

There is a difficulty in drawing this diagram that is worth admitting, because it is the reason most textbook versions are schematic.

The three slopes at water’s triple point differ by five orders of magnitude: $44$ and $50$ pascals per kelvin for the two vapour curves, and $1.35\times10^7$ for the melting curve. A diagram with a pressure axis wide enough to show the melting curve leaning would compress the other two into the horizontal axis; one with an axis fine enough to separate the two vapour curves shows the melting curve as exactly vertical.

The figure above takes the second option and says so. Over the $\pm0.9$ K drawn, the melting curve moves sideways by $6\times10^{-8}$ K per pascal — which over the $\pm46$ Pa of the vertical axis is $3\times10^{-6}$ K, or a fiftieth of the width of the drawn line. Nothing has been idealised: the line is vertical to well within the resolution of the picture, and a picture that showed it leaning would be exaggerating.

This is the ordinary hazard of a diagram whose quantities span decades, and the honest responses are to say what has been done or to use logarithmic axes. A schematic phase diagram with all three slopes comparable is neither, and it is what most readers have seen.

What a point with no freedom is good for #

A quantity that cannot be adjusted is a quantity that can be reproduced, and that is what a metrological fixed point needs.

To sit at the triple point of water, seal pure water in a glass cell, remove the air, and cool it until solid, liquid and vapour are all present. The temperature is then $273.16$ K and there is no setting to get wrong: not the pressure, which the system chooses; not the amount of each phase, which does not matter; not the container, provided it does not contaminate. Two cells made independently on different continents agree to a few tenths of a millikelvin, and the reproducibility is what made the triple point the definition of the kelvin from 1954 until the 2019 redefinition moved the base to a fixed value of Boltzmann’s constant.

The residual disagreement between good cells turns out to be almost entirely about isotopes. Ordinary water is a mixture of hydrogen and oxygen isotopes, and a mixture is not one component: heavy water’s triple point is $3.8$ K higher, so a cell filled from Antarctic melt differs measurably from one filled from ocean water. The definition therefore had to specify the isotopic composition, which is an admission that the substance was not as single as the phase rule assumed.

The other end of the liquid–vapour curve is also a point with no freedom left, and for a related reason. There the distinction between the two phases vanishes, which imposes an extra condition — the first and second derivatives of pressure with respect to volume both vanish — and two conditions on two variables leave nothing free. A triple point is fixed by three phases coexisting; a critical point by two phases becoming one. Different arithmetic, same conclusion: a point rather than a line.

The critical point is a second zero-freedom point on the same diagram, and it is worth noting that the phase rule as stated does not predict it. There, two phases become identical, which imposes an additional condition beyond the equality of chemical potentials — that the two are the same solution — and the extra condition removes the last degree of freedom. Counting phases alone would call it a two-phase state with one degree of freedom, and it has none.

Two components, and the rule earning its keep #

The counting is a curiosity for one substance and a working tool for two.

With $C = 2$ the rule gives $F = 4 - P$. One phase has three degrees of freedom — temperature, pressure and composition — which is a volume rather than an area, so a two-component phase diagram is usually drawn at fixed pressure and the diagrams everyone recognises are slices. Two phases leave two, which at fixed pressure is a region rather than a curve, and that is why an alloy melts over a range of temperature while a pure metal melts at a point. Three phases leave one, which at fixed pressure is a point: the eutectic, where liquid and two solids coexist at a single temperature and a single composition.

That is the rule’s most-used consequence. A eutectic exists, is unique for a given pair at a given pressure, and melts sharply like a pure substance despite being a mixture — which is why solder is a eutectic alloy and why an ice-and-salt bath sits at $-21.1$ °C and not at whatever temperature is wanted.

Gibbs, and why nobody read him #

The rule is in On the Equilibrium of Heterogeneous Substances, published in the Transactions of the Connecticut Academy between 1875 and 1878 — a journal with almost no circulation, in a paper of some three hundred pages, written in a style that states results and leaves the reader to supply the enthusiasm.

It was not ignored so much as unread. Maxwell recognised its value immediately and died in 1879; Ostwald translated it into German in 1892 and van der Waals into Dutch; the phase rule reached working chemistry mainly through Roozeboom, who spent the 1880s and 1890s showing that it organised the confusing experimental literature on salt hydrates and alloys into something with a shape.

What Roozeboom’s campaign demonstrated is worth the paragraph. Before the rule, a phase diagram was a collection of measured curves with no reason to have any particular arrangement, and a report of four phases coexisting in a one-component system was an experimental claim to be argued about. After it, such a report is arithmetically impossible and the argument becomes about what the experiment actually contained — an impurity, a second component, a metastable phase. The rule’s main practical use is negative: it says which observations cannot be right, and that is a more useful thing to know than most positive predictions.

Where it stops #

A “phase” has to be uniform and a “component” independently variable. Both are less obvious than they look. A solid solution is one phase; two solids in contact are two. A chemical reaction between the components reduces the number of independent ones by the number of independent reactions, which is why the rule for a reacting system is $F = C - R - P + 2$ and why applying the plain version to something that reacts gives a wrong answer.

Surfaces are not counted. The derivation assumes bulk phases, so it ignores surface energy. For a very small system that is not negligible: a droplet’s chemical potential depends on its radius, so a small drop does not coexist with its vapour at the bulk pressure, and a nanoparticle melts below the bulk melting point by tens of kelvin. The phase rule for finite systems has to add the surface area as a variable, and then a triple point can be smeared into a region.

Two more variables can always be added. The $2$ in the rule is the number of intensive variables other than composition, and it is $2$ because temperature and pressure were the only ones considered. In a magnetic field it becomes $3$, in an electric field $3$, and under uniaxial stress rather than hydrostatic pressure it becomes larger still. Every such addition adds a degree of freedom, so a triple point in a magnetic field becomes a curve.

Other substances have more solids than one. Water has at least eighteen crystalline ices, and each pair of them that coexists has its own curve, and each place where three of them meet is another triple point. Counting still gives $F = 0$ at every one, so a substance may have many triple points — water has more than a dozen on its full diagram — and what the rule forbids is not several triple points but a fourth phase at any one of them. The everyday triple point is simply the one involving the vapour, which is the only one at a pressure anybody encounters.

And metastability makes a mess of the picture without touching the rule. Water is routinely liquid at $-20$ °C, in a state the phase diagram says is solid, because nothing has nucleated. The rule describes equilibrium, and a supercooled liquid is not in it — it is in a state that will not last and may last a long time, because something has to nucleate first.

What the diagram cannot show #

Two absences are worth naming, because both are things a reader reasonably expects a phase diagram to contain.

It shows no rates. Nothing on it says how long a transition takes, and the answer is sometimes geological: diamond is the metastable phase at room conditions and graphite is the stable one, so the diagram says a diamond should turn into graphite and gives no hint that the conversion takes longer than the age of the universe. Everything on the diagram is a statement about where a system would end up given unlimited time and something to nucleate on.

And it shows no amounts, for the reason the counting excluded them. A point inside the two-phase region is a state in which the proportions of the two phases can be anything between all of one and all of the other, and the diagram represents that entire family by one dot. A reader who takes the dot to mean a definite mixture is reading in something the axes cannot carry — which is the recurring hazard of a diagram whose coordinates are intensive and whose subject is often extensive.

The ladder from here #

Later rungs on this anchor: the phase rule for reacting systems, and the bookkeeping of independent reactions that reduces the component count; the Gibbs–Konovalov rules for azeotropes, where a two-component mixture behaves like a pure substance and the phase rule explains why; the triple point of helium, which does not exist because helium has no solid at low pressure and whose absence is as informative as water’s presence; and phase diagrams under stress rather than pressure, where the number of intensive variables rises and the topology changes.

The neighbouring ladders are a boiling point is a pressure, which is one coexistence curve taken seriously; the point at which the two become one, which is the diagram’s other zero-freedom point; and the heat that changes no temperature, which is the latent heat these slopes are computed from.