The pressure that comes from counting

A tube of water ten millimolar in anything at all — sugar, urea, potassium chloride counted as two — separated from pure water by a membrane that passes water and not solute, will stand two and a half metres higher than the pure water beside it. Ten millimolar is a thousandth of the strength of seawater. The concentration is barely detectable by taste and it lifts a column taller than a person.

The usual account of why is that the solute “draws water in”, and it is worth saying at the start that nothing in the apparatus pulls on anything. The solute never touches the left-hand arm. It exerts no force across the membrane, has no field, and would do exactly the same job if it were replaced by particles with no interactions of any kind.

The wrong picture, and why it survives #

Two things make the pulling picture attractive. It gets the direction right, and osmosis is usually met beside capillary rise, where something genuinely does pull.

In a capillary there really is a force. The liquid surface is under tension, the tube’s wall pulls on it round the whole perimeter of the meniscus, and the height is that force balanced against weight — which is why a narrower bore lifts water further. Osmosis produces heights of the same order by a mechanism with no force in it anywhere, and the two sit side by side in every introduction to the subject. That is most of why the pulling picture survives: it is borrowed from the demonstration on the next bench.

The second reason is that the alternative account requires giving up something that feels physical — a force — in favour of something that feels like bookkeeping: a count. It is the same trade that has to be made to understand why heat flows, and it is worth making once carefully.

What the membrane sees #

Consider the membrane from the point of view of the water molecules hitting it. Every molecule in the pure arm that arrives at the wall is a water molecule and may cross. In the solution, a fraction of the arrivals are solute and may not.

That is the entire mechanism, and it is the same argument that produces the pressure of a gas. A gas’s pressure is a rate of arrival at a wall: momentum delivered per unit area per unit time, with no attraction anywhere in the derivation.

Pressure is not a substance stored in a gas, it is a rate: so many molecules striking so much wall so often, each delivering its momentum, and the sum per unit area per unit time is the number a gauge reads. Nothing in that derivation attracts anything. The osmotic case is the same statement made about a wall that only one species is allowed through, and what is unbalanced there is simply the rate at which that species arrives from each side.

Put more carefully, in the language a thermodynamicist would use: the solvent’s chemical potential is lowered by mixing, so solvent flows toward the solution, and the flow stops when mechanical pressure on the solution has raised the solvent’s chemical potential back to what it is next door. The pressure required is the osmotic pressure.

Why the answer is the gas law #

Mixing lowers the solvent’s chemical potential because it raises the entropy, and the entropy of mixing depends on the proportions and on nothing else about the substances.

The entropy gained by mixing two species, per particle, depends on the proportion in which they are mixed and on nothing whatever about the species — the same curve for sugar in water as for argon in neon, peaking at equal parts and falling to zero at either pure end. That indifference is inherited whole by the osmotic pressure, and it is the reason the pressure counts how many particles the solute dissolved into rather than measuring anything about what they are.

Working the balance through for a dilute solution gives van 't Hoff’s result:

$$\Pi = c R T,$$

with $c$ the concentration of dissolved particles. Written out, that is $\Pi V = nRT$ — the ideal gas law, with the solute playing the gas and the solvent playing the vacuum it expands into. The resemblance is not an analogy. Both are the pressure of a dilute population of particles that do not interact with one another, and both are derived by counting.

The consequence is the one that surprises: a mole of glucose and half a mole of sodium chloride give the same osmotic pressure, because the salt dissociates into two. Nothing else about the two substances appears — not their size, their mass, their charge or their chemistry.

The indifference to mass deserves a sentence of its own, because it is the sharpest test of the counting picture and the one most people find hardest to believe. A sucrose molecule is nineteen times heavier than a water molecule and a sodium ion is a little heavier than one; on the pulling picture the heavy one should surely do more. It does not, and the reason is visible in the distribution of speeds.

A heavy particle at a given temperature moves more slowly, by exactly the factor that keeps its mean kinetic energy equal to everything else’s. So it arrives at the wall less often and hits harder when it does, and the two effects cancel exactly in the pressure. That cancellation is why the ideal gas law contains no mass at all, and the osmotic pressure inherits the same silence: nineteen times the mass, nineteen times less often, the same push.

A solute of any molecular weight whatever, at the same number per cubic metre, delivers the same pressure. That is a strong prediction, it is easy to falsify, and it survives: osmometry is used the other way round, to measure a molecular weight, precisely because the pressure counts particles and the mass then follows from weighing them.

Where the straight line stops #

Van 't Hoff’s law is the first term of an expansion, and the expansion is in the fraction of the solution that is solute.

This is worth stating in the form the site keeps returning to. The approximation is not good because somebody was lucky. It is good because the small quantity is the mole fraction of the solute, and even seawater is only two per cent solute by particle count. Naming what the small quantity is turns an assurance into a range, and the range here reaches much further than the derivation suggests it should.

The threshold, and what desalination costs #

Run the apparatus backwards. Push on the solution hard enough and water is forced out through the membrane against the concentration gradient, which is how most of the world’s desalinated water is made.

The intercept is the same osmotic pressure the solution would generate on its own, and it sets a floor on the energy: 0.75 kilowatt-hours per cubic metre if a negligible fraction of the feed is taken, rising to 1.04 at fifty per cent recovery because the brine left behind gets saltier as fresh water leaves it. Working plants use about three, and the gap between three and one is the pumps, the membrane’s own resistance and the pressure that has to be thrown away with the brine.

That structure — an unavoidable floor set by a state function, and a working machine some multiple above it — is exactly the structure of the ceiling on a heat engine, and the two are worth holding side by side. In both cases the bound is derived from equilibrium arguments and says nothing about rate, and in both cases a machine that approached it would deliver nothing per hour.

What the number is worth, in ordinary places #

Three readings, because the magnitude is the part that is hard to believe from the derivation.

Blood plasma carries about 300 moles of dissolved particles per cubic metre, most of them small ions, and its osmotic pressure is therefore about 7.4 bar — a hundred pounds per square inch, held across a membrane a few nanometres thick. An intravenous drip has to match that number to within a few per cent or the red cells either swell and burst or crumple, which is the whole reason saline is made up at nine grams per litre and not at some rounder figure.

Seawater at 1,100 mol/m³ of ions gives 27 bar, which is the pressure at 270 metres depth. A membrane holding fresh water on one side of it and the sea on the other is under the load of a small dam.

And a plant cell at 500 mol/m³ has 12 bar inside it, contained by a cellulose wall a fraction of a micrometre thick. That pressure is not a nuisance the plant tolerates; it is the plant’s structural system. A wilted leaf is a leaf whose cells have lost their osmotic pressure, and it recovers not by growing anything but by taking up water.

None of those numbers required knowing what was dissolved. Each is a particle count multiplied by RT, and RT at room temperature is 2,478 joules per mole — the one constant the whole subject turns on, and the same one that sets the spread of molecular speeds in a still room.

The height it cannot reach #

Osmosis is quantitative, which means it can be checked against a job it is regularly credited with.

Root pressure is real — cut a vine’s stem near the ground in spring and sap runs out under pressure — and it is enough for a shrub. It is not remotely enough for a redwood, and the mechanism that is enough is the opposite of a push: water in the xylem is under tension, pulled from above by evaporation at the leaves, in a continuous thread held together by its own cohesion.

That thread is under negative pressure, several megapascals of it, which is a state water should not survive: a liquid column under enough tension cavitates, and the ceiling on a siphon is exactly where the hydrostatic pressure at the crown meets the vapour pressure. Xylem sap passes that ceiling routinely because the water is degassed, the vessels are narrow, and the metastable state persists — until it does not, and a vessel embolises with an audible click.

The same law under other names #

The pressure of a dissolved population turns up wherever a boundary lets one species past and not another, and it is usually met under a different name.

A cell in pure water bursts because its contents are a solution and its membrane is semipermeable: at 300 mol/m³ of internal solutes the pressure is 7.4 bar, which is about a hundred pounds per square inch across a structure a few nanometres thick, and it is why animal cells need a pump and plant cells need a wall.

A boiling point rises and a freezing point falls in a solution for the same reason and by the same counting — a boiling point is the temperature at which the vapour pressure reaches the surroundings, and dissolving anything lowers the solvent’s escaping tendency by exactly the factor that produces the osmotic pressure. All three effects are proportional to the particle count and to nothing else, and the family name for them is a good one: colligative, from colligere, to collect together.

And a battery’s separator, a fuel cell’s membrane and a dialyser are all the same apparatus with different species allowed through.

The table of numbers that turned out to be the gas law #

The relation $\Pi = cRT$ was not derived and then confirmed. It was noticed, in somebody else’s published measurements, ten years after they were made.

The obstacle for most of the nineteenth century was mechanical. A semipermeable membrane strong enough to hold several atmospheres did not exist; the natural ones — a pig’s bladder, a piece of gut — leaked and burst. Traube found in 1867 that a film of copper ferrocyanide, precipitated where two solutions meet, passes water and stops sugar, but the film was gossamer and could hold nothing.

Wilhelm Pfeffer, a botanist, solved it in 1877 by precipitating that film inside the pore walls of an unglazed clay pot, so that the clay carried the pressure and the film did the selecting. With that he built a manometer that worked, and he measured the osmotic pressure of sugar solutions at a range of concentrations and temperatures and published the table.

Van 't Hoff read it a decade later and saw that the pressures were proportional to the concentration, that they were proportional to the absolute temperature, and — the part nobody had checked — that the constant of proportionality was $R$, the same constant that appears in the gas law, to the accuracy of Pfeffer’s manometer. He published the observation in 1887, and in 1901 it earned the first Nobel Prize in Chemistry ever awarded.

The episode is worth telling for two reasons. The first is that the discovery was a recognition rather than an experiment: the numbers had been in print for ten years, correct, and what was missing was somebody computing $\Pi/cT$ and recognising the answer.

The second is that the mechanism van 't Hoff offered along with the law was the wrong one, and it is precisely the wrong one this essay opened by rejecting. He described the solute particles as bombarding the membrane, exerting a pressure on it in the way a gas exerts pressure on a wall, and pushing the solvent through. That picture explains the formula and is not what happens: the solute does not push the water, and a membrane that reflected solute would feel a force whether or not any water crossed. What is actually unbalanced is the rate at which water arrives from each side. The law survived; the picture that came with it had to be replaced, and a good deal of the confusion about osmosis in textbooks descends directly from it.

Weighing a molecule by counting #

The law’s indifference to what was dissolved turns it into a measuring instrument, and the instrument does something nothing else does as cleanly.

Since the pressure counts particles, a solution of known mass concentration and unknown molecular weight gives the weight directly: with $\rho$ kilograms of solute per cubic metre and $M$ kilograms per mole, the particle concentration is $\rho/M$, so

$$M = \frac{\rho RT}{\Pi}.$$

Weigh the solute, measure the pressure, divide.

What makes this worth having rather than merely available is which average it returns. Osmometry counts molecules, so it gives the number-average molecular weight — the total mass divided by the number of molecules. Light scattering, by contrast, responds more strongly to large molecules and gives the weight-average. For a polymer, where the molecules are not all the same length, those two averages differ, and their ratio is the standard measure of how broad the distribution is. Two instruments returning different averages is not a discrepancy; it is two pieces of information, and one of them exists because osmotic pressure is a count.

The practical range follows from the arithmetic. A one per cent solution of a polymer of molar mass a hundred thousand grams gives a tenth of a mole of molecules per cubic metre, hence about 250 pascals, which is two and a half centimetres of water — comfortably measurable. Ten times the molar mass gives two and a half millimetres, which is at the edge, and osmometry runs out somewhere around a million.

There is one further quantity available for free. Plotting $\Pi/\rho$ against $\rho$ and extrapolating to zero concentration gives $RT/M$ as the intercept — the honest molecular weight, uncontaminated by the solute–solute interactions the ideal expression ignores. And the slope of that line measures those interactions: it is positive when the polymer prefers the solvent to itself and negative when it does not, which is how a solvent is graded as good or poor. The departure from the ideal law, which the previous section treated as a small nuisance, is a second measurement hiding inside the first.

Where the model stops #

The membrane is idealised out of existence. Everything above treats it as a wall with a rule, and a real membrane rejects imperfectly, passes some solute, fouls, and has a pore structure whose interactions with the solute change the answer. The correction is a reflection coefficient multiplying $\Pi$, and for a biological membrane it is often well below one.

Nothing here computes a rate. The osmotic pressure is an equilibrium quantity: it says where the flow stops, not how fast it gets there. The flux is a separate measurement, governed by the membrane’s permeability, and the two are related by nothing in this essay.

How long it takes is a wholly separate calculation, and it is the one that decides whether osmosis is useful. A concentration profile spreads on a timescale set by the square of the distance, which is why the same physics is fast across a cell membrane a few nanometres thick and hopeless across a room. The equilibrium the pressure describes is reached in microseconds in one case and never in the other.

Charged solutes bring their own complication. An ion cannot cross without its counter-ion or a compensating current, so a membrane permeable to one ion and not another sets up a potential difference as well as a pressure difference, and the equilibrium is a balance of both. That is a real and common case and it is a different calculation.

And the solution is assumed ideal beyond the dilute expansion. Solute–solute interactions, which are ignored here entirely, matter for polymers and for concentrated electrolytes, and are handled by replacing the concentration with an activity — which is to say, by measuring the departure rather than predicting it.

What the pictures cannot show #

The counting figure exaggerates the solute fraction by a factor of nine hundred, and says so, because a real ten-millimolar solution is one solute particle in five and a half thousand water molecules. Any honest drawing of the real ratio would be a picture of pure water. The figure is therefore correct about the mechanism and mute about the magnitude, and the magnitude is what makes the effect surprising: one particle in five thousand lifts two and a half metres.

Nor can the column figure show that nothing is happening at equilibrium. It is drawn as a static picture, and the static picture is right — but the underlying state is water crossing the membrane in both directions at an enormous and exactly equal rate, which is a different thing from water not crossing. Every equilibrium in this collection has that property and no drawing of one conveys it.

Where this ladder goes next #

What has been established is that a pressure can arise from counting alone: no force between the parties, no field, no attraction, and a number that depends on how many particles there are and on nothing else about them. The nearest neighbour to that idea in this collection is the pressure a gas exerts, and the nearest cousin is what a system actually minimises when it is held at a fixed temperature, which is where the chemical-potential language above comes from.

The habit worth carrying away is a test. When something is described as being pulled, ask what would happen if the attraction were switched off. Here the answer is: exactly the same thing, because there was no attraction. The solute’s only role is to occupy places at the membrane that water would otherwise occupy, and a picture in which it reaches across and grabs is not merely imprecise — it predicts a dependence on the solute’s chemistry that measurement flatly denies.

What is left on this ladder is the charged case, where the same counting produces a voltage as well as a pressure, and where a membrane that stops one ion and passes another turns a concentration difference into a battery.