The membrane that almost holds

The pressure that comes from counting derives van 't Hoff’s law and gets a striking result: the osmotic pressure of a solution depends on how many dissolved particles there are and on nothing else about them. A tenth-molar solution of anything develops the same pressure.

That result has a hidden condition. It describes the pressure developed across a membrane that passes water and stops the solute completely — and outside a thought experiment, no membrane does.

The number that measures the leak #

Staverman’s reflection coefficient $\sigma$ is defined as the fraction of the ideal pressure a membrane develops, and it runs from zero to one.

At $\sigma = 1$ the membrane reflects every solute molecule that arrives, and van 't Hoff’s law holds exactly. At $\sigma = 0$ the solute crosses as easily as the water does; there is no selectivity, and no pressure develops however concentrated the solution is. In between there is a real membrane.

The important structural point is that $\sigma$ is not a property of the membrane and is not a property of the solute. It is a property of the pair. The same membrane has $\sigma$ near one against sucrose and near zero against ethanol, and the same solute is reflected by one membrane and passed by another.

A single membrane therefore has as many osmotic behaviours as there are solutes, which is what makes the biological versions of this subject complicated and what makes the industrial versions tractable — a desalination membrane is engineered against one solute and characterised by one number.

How it is measured #

The measurement is a null one, which is why it is good.

Put the solution on one side, water on the other, and apply pressure until the flow stops. The stopping pressure is $\sigma \Pi$, and dividing it by the van 't Hoff pressure — which is known from the concentration alone — gives $\sigma$ directly. Neither the membrane’s area nor its water permeability enters, which are the two quantities hardest to know.

The same sweep gives the permeability for free, as the slope. So one experiment returns two independent numbers: what the membrane does to water, and what it does to the solute. That separation is the reason Kedem and Katchalsky’s formulation replaced the older one — it writes the flows in terms of coefficients each of which is separately measurable.

There is a third coefficient in their scheme, the solute permeability at zero volume flow, and the three together describe everything a membrane does to a two-component solution. They are not independent — a membrane that reflects perfectly must pass nothing — but they are not related by any general formula either, and all three are measured rather than derived.

The response that goes away #

The consequence with the most practical bite is that a leaky membrane’s response is transient, and the transient can be long enough to look like a steady state.

Water crosses a cell membrane far faster than any solute does — its permeability is larger by orders of magnitude, and in cells with aquaporins by more still. So the first thing that happens is always the same: the cell shrinks, at a rate set by the water permeability and by the full osmotic difference weighted by $\sigma$.

What happens next depends entirely on whether the solute follows. If it cannot, the shrinkage is permanent and the cell settles where what is trapped inside balances everything outside. If it can, the solute enters, the difference it was creating disappears, and the volume comes back.

A measurement taken in the first seconds cannot distinguish the two. That is not a hypothetical problem: it is why early measurements of “osmotic pressure” against biological membranes disagreed with each other and with the thermodynamics, and why the modern protocol is to watch the whole time course rather than to read a single number.

The same experiment, more gently #

Repeating the run at a gentler challenge is worth the space because it separates two things the first figure runs together.

The depth of the dip is set by the osmotic difference and by $\sigma$: halve the concentration and the cell shrinks half as far. The duration is set by the permeabilities and by the cell’s own size, through the ratio of its volume to its area — which is why a small cell equilibrates in milliseconds and a large one in seconds, and why measurements on giant algal cells were possible with nineteenth-century apparatus while measurements on red cells needed stopped-flow instruments.

The ordering of the curves is untouched. That is the useful invariance: $\sigma$ decides the shape of the response — whether it recovers and how completely — and the concentration decides its size. A single experiment at one concentration therefore returns $\sigma$ from the shape without needing the concentration to be known accurately, which is another reason the coefficient is a robust thing to measure.

A parameter that changes the shape of a curve is easier to extract than one that changes its scale, because a scale can be spoilt by a dozen calibration errors and a shape cannot.

Osmolarity is not tonicity #

The distinction the last figure draws has two words for it and they are constantly confused.

Osmolarity is a property of a solution: how many dissolved particles per litre. It is measured by freezing-point depression, it needs no membrane, and two solutions with the same osmolarity are thermodynamically equivalent in every respect a membrane-free measurement can see.

Tonicity is a property of a solution and a membrane: what the solution does to a cell bounded by that membrane. It is the sum of $\sigma_i c_i$ over the solutes, and it is what decides whether a cell shrinks, swells or does neither.

A 300 mol/m³ urea solution is isosmotic with blood and is not isotonic. Urea crosses a red-cell membrane readily, so the cell placed in it swells and lyses — the urea enters, the impermeant solutes inside cannot leave, and water follows them. Glycerol is worse and ethanol is worse still.

That is why an intravenous fluid is specified by what it does rather than by what it contains. Saline works because sodium and chloride are held out, and a solution of a permeant solute at the same osmolarity would kill.

What a cell does about it #

A membrane that leaks would be a problem for a cell if the cell had to rely on the membrane alone, and it does not — which is why the leakiness is tolerable rather than fatal.

The trick is to spend energy. Sodium crosses a cell membrane slowly but not never, and a cell that did nothing would gradually fill with it, swell, and burst. What prevents that is the sodium pump, which exports sodium continuously at the expense of a third or so of the cell’s entire energy budget. The pump does not make the membrane tighter; it makes the leak irrelevant by removing what leaks in as fast as it arrives.

That arrangement has a name — the pump-leak model — and it turns a steady-state volume into an actively maintained one. Its signature is that a cell poisoned with a pump inhibitor swells and lyses over hours, which is the same transient as in the figures above with the recovery removed. It is also why cold storage of tissue is a problem: the pumps slow far more than the leaks do, so a cooled cell gains sodium and swells, and the preservation solutions used for transplant organs are formulated with an impermeant solute to hold the volume mechanically while the pumps are stopped.

Selectivity and pumping are alternative solutions to the same problem, and living things use both — tight junctions and impermeant proteins where a barrier will do, and pumps where it will not. The pressure that comes from counting is the passive half of the story, and the reflection coefficient measures how much of the work the passive half can be asked to do.

Where the same number appears elsewhere #

The reflection coefficient turns up wherever selectivity is imperfect, which is nearly everywhere.

In desalination it is the rejection: a seawater membrane with $\sigma = 0.995$ passes half a per cent of the salt, which sets the product quality and, through the osmotic pressure it fails to develop, part of the energy cost that what it costs to take the salt out computes.

In capillary exchange it is why proteins matter and salts do not. The capillary wall has $\sigma$ near 0.9 for albumin and near zero for sodium, so the only solute developing an osmotic pressure across it is protein — the oncotic pressure, a few kilopascals against the several hundred the total osmolarity would give. Starling’s equation for fluid exchange is written in exactly the Kedem–Katchalsky form, with $\sigma$ in it, for that reason.

In soil and plants it explains why some solutes are osmotically active and others are taken up — and it sets the tension a root can generate, which is one of the terms in the column that is pulled not pushed. And in ion channels it is the beginning of the selectivity story, which is where a two-coefficient description stops being adequate.

Why the coefficient exists at all #

It is worth asking what physical fact a number between zero and one is standing for, because “the membrane is imperfect” is not a mechanism.

The cleanest picture is a pore. If the pore is much wider than the solute, the solute passes with the water and is not reflected: $\sigma$ is near zero. If it is narrower, the solute cannot enter at all: $\sigma$ is one. In between, the solute enters but is hindered — its centre cannot approach the wall closer than its own radius, so it samples only the faster-moving middle of the pore, and it is swept through at a different speed from the water around it.

That last statement is the origin of the coefficient, and it explains a feature that would otherwise be puzzling. $\sigma$ is not simply the fraction of solute molecules excluded; it involves the difference between the average velocity of the water and the average velocity of the solute, and it can be computed for a hard sphere in a cylindrical pore. The answer depends on the ratio of the two radii and nothing else, rising from zero to one across roughly a factor of two in that ratio.

Real membranes are not cylindrical pores, and the calculation is a caricature. What it establishes is that a single geometric ratio can produce the whole range of behaviour, and that the coefficient is a hindrance rather than an exclusion. A membrane can pass a solute and still reflect it, and that possibility is what makes σ a continuous quantity rather than a fraction of blocked pores.

Where the picture fails is where the solute interacts with the membrane chemically — a charged solute against a charged membrane, a lipid-soluble one against a lipid bilayer — and there $\sigma$ is still measurable and no longer geometric.

Where the model stops #

The description is linear. Kedem–Katchalsky is a first-order irreversible-thermodynamic scheme, valid when the driving forces are small. At the pressures a desalination membrane runs at — sixty bar or more — the coefficients themselves depend on the operating point, and the linear form is fitted locally rather than derived.

The solution is dilute. Van 't Hoff’s law is itself the leading term of an expansion, and seawater is not dilute: its actual osmotic pressure is about ten per cent above the ideal value. The reflection coefficient multiplies whichever pressure is used, and quoting one without saying which is a common source of confusion.

The two coefficients are tied together here. The figures make the solute permeability proportional to $1 - \sigma$, so that one number selects the curve. In reality they are separately measurable and only loosely related, and a membrane can be found with intermediate values of both.

Temperature is held fixed. Both permeabilities rise with temperature, and they rise at different rates, so σ measured warm is not σ measured cold — a variation of the same kind as the thickness that goes both ways, and one reason membrane data are quoted with a temperature attached.

And concentration polarisation is ignored. Solute rejected at a membrane accumulates against it, so the concentration the membrane actually sees exceeds the bulk value, and the apparent $\sigma$ measured without stirring is lower than the true one. Most of the art in a real measurement is in defeating that.

Where the pressure is, and where it is not #

One more confusion is worth clearing, because it is the reason the coefficient is sometimes described as a fudge factor.

The osmotic pressure is not a pressure the solute exerts. Nothing in the solution is pushing on the membrane in the way a gas pushes on a piston, and the striking part of van 't Hoff’s law — that the pressure equals what the solute would exert as an ideal gas in the same volume — is a coincidence of the free energy rather than a statement about collisions. The driving quantity is the chemical potential of the water, which the solute lowers, and the pressure is what has to be applied to raise it back.

Read that way the reflection coefficient stops looking like a correction and starts looking like a statement about which chemical potentials are coupled. A membrane that passes the solute freely has the same solute chemical potential on both sides, so there is nothing to equilibrate and no pressure to develop; a membrane that stops it holds a difference. What σ measures is the fraction of that difference the membrane converts into a difference in the water’s own chemical potential.

The same reframing explains why the pressure is developed at the membrane and not in the bulk solution. A solution in a beaker has no osmotic pressure in any measurable sense — a gauge lowered into it reads the hydrostatic pressure and nothing else, exactly as the pressure that only knows depth requires. The osmotic pressure exists only across a boundary that distinguishes the components, and a boundary that half-distinguishes them develops half of it.

A quantity that requires a membrane to exist is a quantity about a pair of systems, and the coefficient is the honest bookkeeping for how completely the pair is separated.

What the pictures cannot show #

The transient figure draws volume against time and cannot show what a cell is doing mechanically. A red cell shrinking is crenating rather than staying spherical, its membrane area is fixed while its volume is not, and past about a forty per cent swelling it lyses rather than continuing along the curve. The drawing continues where the cell does not.

The species figure treats each solute as though it acted alone. In a mixture the flows are coupled — solvent drag carries one solute while another is being reflected — and the tonicity of a mixture is not always the sum of its parts. The additivity used here is the leading approximation and is what the linear scheme predicts.

Where the ladder goes next #

The osmosis ladder began with the pressure that comes from counting, where the pressure turns out to depend on the number of particles and nothing else, and continued with what it costs to take the salt out, which is the thermodynamic floor under desalination. This rung asks what a real membrane does and finds a coefficient between the ideal answer and none. The rungs after it: the coupled transport of two solutes, where a third coefficient is needed; active transport, which pumps against the gradient and is not described by any of this; and selectivity in an ion channel, where the mechanism is a structure rather than a number.

The habit worth carrying away is that an idealisation usually has a measurable distance from reality, and the distance is often a better quantity than the ideal. The reflection coefficient is the departure made into an instrument, and a number that is one for a perfect membrane and zero for no membrane at all is a more useful thing to measure than the perfection it is a departure from.