Illustrated Physics
physics, drawn
Concept

Osmotic pressure — where it appears

The pressure that must be applied to a solution to stop solvent flowing into it through a membrane that passes solvent and not solute. In the dilute limit it is the pressure the dissolved particles would exert as a gas of their own, 28.5 bar for seawater, and it fixes the least energy any desalination can cost.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

The column a dissolved thing holds up. Two arms of one vessel, joined below by a membrane that passes water and not solute. On the right is 10 mol/m³ of dissolved particles at 25 °C; on the left, pure water. Water crosses into the solution until the extra weight of the right-hand column has raised its pressure by the osmotic pressure — 24.8 kPa, which is 2.53 m of water, drawn here to scale. Nothing is pulling. The solvent is at a lower chemical potential where it is mixed, so it moves that way, and it stops when mechanical pressure has made up the difference. A solute a thousand times more dilute than seawater lifts a column taller than a person.

The pressure that comes from counting

Dissolve a teaspoon of anything in a litre of water, put a membrane between it and pure water, and the solution will hold up a column of water two and a half metres tall. Nothing is pulling. The pressure does not depend on what was dissolved, only on how many particles it made — which is the ideal gas law, with the solute in place of the gas.

fluids · Osmosis The least energy that fresh water can cost. The work needed to take fresh water out of a feed of 1150 mol/m³ of dissolved particles — seawater — against how much of the feed is taken, in kilowatt-hours per cubic metre of product. The lower curve is the reversible minimum, in which the pressure is raised continuously as the remaining feed gets saltier; the upper one is a single stage held throughout at the pressure the final, saltiest concentrate needs. At vanishing recovery both tend to the feed's own osmotic pressure, 28.5 bar or 0.79 kWh/m³, which is the floor for the first drop and is checked here against the limit of the formula. Taking more of the feed costs more per unit taken, because what is left behind is saltier: at 10% recovery 0.83 kWh/m³ reversibly and 0.88 in one stage, 30% recovery 0.94 kWh/m³ reversibly and 1.13 in one stage, 50% recovery 1.10 kWh/m³ reversibly and 1.58 in one stage, 60% recovery 1.21 kWh/m³ reversibly and 1.98 in one stage, 75% recovery 1.46 kWh/m³ reversibly and 3.17 in one stage. The horizontal line is what a good seawater plant actually uses, 3 kWh/m³, so the second-law efficiency of the industry is about 37 per cent. None of this is about membranes. It is the free energy of mixing salt into water, read backwards, and no technology of any kind can go below the lower curve.

What it costs to take the salt out

Salt dissolves in water because mixing is overwhelmingly the more probable arrangement, and separating the two again means paying back what the mixing gave away. The bill can be computed before any apparatus is chosen — 0.79 kilowatt-hours for the first cubic metre from seawater — and it rises with every further cubic metre taken, because what is left behind is saltier than what was started with.

fluids · Osmosis The fraction of the pressure a membrane can hold. The osmotic pressure actually developed across a membrane against a 300 mol/m³ solution at 298 K, as the reflection coefficient runs from a membrane the solute passes freely to one it cannot pass at all. The line is straight with the van 't Hoff pressure 743.69 kPa as its slope — checked against the drawn line — because the coefficient enters as a simple factor. At σ = 1 the pressure is 743.69 kPa; At σ = 0.6 the pressure is 446.21 kPa; At σ = 0.2 the pressure is 148.74 kPa. Van 't Hoff's law is the ceiling rather than the answer, and a membrane's coefficient against a given solute is as much a property of the pair as the concentration is of the solution.

The membrane that almost holds

Van 't Hoff's law gives the osmotic pressure a perfectly selective membrane would develop, and no membrane is. What a real one develops is a fraction of it — a number between zero and one that belongs to the membrane and the solute together, and that decides whether a solution is isotonic in effect or only on paper.

fluids · Osmosis The pressure a charged gel develops. The swelling pressure against the gel's fixed charge density, for several salt concentrations, on logarithmic axes. Each curve has two straight parts with different slopes, and both limits are checked against the solution rather than read off the plot. Where the fixed charge is small compared with the salt the pressure goes as the square of the charge divided by four times the salt: the bath's ions screen the fixed ones, and doubling the salt halves the pressure. Where the fixed charge dominates, every counterion it demands is an extra particle in the gel and the pressure goes as the charge itself. At 15 mM salt, a gel with 200 mM of fixed charge develops 427 kPa; At 50 mM salt, a gel with 200 mM of fixed charge develops 306 kPa; At 150 mM salt, a gel with 200 mM of fixed charge develops 150 kPa; At 500 mM salt, a gel with 200 mM of fixed charge develops 49 kPa. Cartilage carries a fixed charge of a couple of hundred millimolar from the sulphated sugars on its proteoglycans, sits in a bath of about 150 mM, and develops a swelling pressure of an atmosphere and a half — which is what holds a joint apart and carries the load across it.

The swelling a membrane cannot stop

Give the thing a membrane holds back an electric charge and the small ions that can cross are no longer free to distribute themselves. Two conditions — neutrality on each side, and equal chemical potential for the salt — fix where every ion goes, and they leave the charged side with more particles than the other. That excess is what holds a joint apart, and it is why a cell has to spend a third of its energy pumping.

fluids · Osmosis Most of a chain's counterions never leave it. The fraction of a charged rod's counterions lying within a distance r of it, against r in rod radii on a logarithmic axis, for a charge parameter ξ = 4.2 — the Bjerrum length of water, 0.7135 nm, over a charge spacing of 0.17 nm, which is DNA's. Each curve is the Poisson–Boltzmann solution for the rod at the centre of a cell of radius 10², 10⁴, 10⁶ rod radii, with its counterions checked to neutralise it exactly. Diluting the solution widens the cell by four decades at a time, and a counterion free to go anywhere in the cell ought to spread with it; instead each curve keeps a plateau near the rod whose height does not change. At the inflection of every curve the enclosed fraction is 0.762, which is Manning's 1 − 1/ξ, and the plateau sits there: 76 per cent of the counterions stay bound to the chain however dilute the solution, and only 24 per cent spread through it.

The counterions that never leave the chain

Dilute a solution of DNA a million times and its counterions ought to scatter through the whole volume. Three quarters of them do not. A line of charges closer together than the Bjerrum length — 0.71 nm in water — holds on to its counterions however much room they are given, until the chain's charge is cut back to one per Bjerrum length, and every osmotic pressure, swelling gel and packed virus built from such chains is set by that length rather than by the chemistry.

fluids · Osmosis Mean field always pushes; correlations pull. The pressure between two planes of equal charge with only their own counterions between them, against their separation, in units of the Gouy–Chapman length μ for the separation and 2πℓ_Bσ²kT for the pressure. The upper curve is the Poisson–Boltzmann result, solved from k·tan(kd/2) = 1: it is the density of counterions at the midplane and is positive at every separation, falling from the ideal-gas 2/d at contact to π²/d² far apart — 1.71 at 1μ, 0.290 at 4μ. The lower curve is the strong-coupling limit, 2/d − 1, which the same ions reach when their valence and the surface charge are high. It crosses zero at d = 2μ and is negative beyond, tending to −1: the two like-charged planes attract, and the separation 2μ is where they come to rest.

The like charges that pull together

Two surfaces carrying the same charge, with nothing between them but the ions that neutralise them, ought to repel, and the standard mean-field theory proves that they always do. With calcium or spermine as the counterions they attract, and come to rest a fraction of a nanometre apart. The mean field misses it because it averages the ions into a smooth cloud, and multivalent ions are too strongly repelled by each other to form one. Each keeps a patch of surface to itself, and the pressure between the plates becomes a single ion's business.

fluids · Osmosis Chains counted, and then blobs. The osmotic pressure of a polymer solution in a good solvent against concentration, on logarithmic axes, for chains of 1,000, 10,000, 100,000 segments. Dilute, each chain is one particle and the pressure is van 't Hoff's, proportional to the number of chains: a line of slope one, lower for longer chains by the ratio of their lengths. Past the overlap concentration of each, marked, the coils interpenetrate and the curves bend up onto a single line of slope 2.31 that does not depend on chain length at all. The overlap concentration falls as the chains lengthen, by a factor of 5.8 for each tenfold increase in length, which is why the longest chains leave the counting of molecules first.

The pressure that stops counting molecules

Van 't Hoff's law counts dissolved particles, and for a dilute polymer each chain is one particle, however many thousand segments it has. That is how osmometers weigh polymers. Concentrate the solution until the coils overlap and the count changes. The pressure stops depending on how long the chains are and grows as the 2.3 power of the concentration, the same for chains a thousand segments long and a hundred thousand. The unit being counted is no longer a molecule. It is a stretch of chain as long as the mesh of its neighbours allows, and it has no chemical identity at all.

fluids · Osmosis A cell swells until its membrane runs out. The volume of a red cell of 90 fL against the salt concentration of the solution round it, in per cent sodium chloride by weight, with 0.9 per cent isotonic. By the Boyle–van 't Hoff law the water inside follows the outside's tonicity: the cell's volume is an inactive part, 40 per cent of the normal volume in haemoglobin and other solids, plus a part inversely proportional to the concentration. Its membrane's 135 μm² can enclose at most 147.5 fL, as a sphere (dashed). The swelling curve reaches that at 0.44 per cent salt: below it the cell has no more shape to give up and bursts. In a 1.5 per cent solution the same cell shrinks to 68 fL and crenates, with membrane to spare.

The cell that bursts by geometry

Put a red blood cell in salt water weaker than blood and water flows in. The cell does not stretch to take it, because its membrane will not stretch: it can change shape almost for nothing and change its area by about two per cent before it tears. So the cell swells by changing shape, from a dimpled disc towards a sphere, and when it reaches the sphere its membrane can enclose, it has no more shape to spend. That happens at about 0.44 per cent salt, half the strength of blood, and the clinical test that measures it — called a test of fragility — turns out to measure a ratio of area to volume, not the strength of anything.

fluids · Osmosis The waterfall hidden in a river mouth. The height of waterfall that would release as much energy per cubic metre of fresh water as mixing it into a large body of salt solution, against the solution's salt content by mass, for sodium chloride at 25 °C: the osmotic pressure divided by the weight of a cubic metre of water. Seawater is worth a 290 m fall; the brine rejected by a desalination plant, about twice as salty, 617 m; a saturated salt solution 3864 m. The curve bends upward because concentrated brine is less ideal than dilute and its osmotic coefficient rises above one. Every river mouth on Earth is a waterfall of this height that nobody sees, since all of its energy goes into the stirring of the estuary.

The waterfall at every river mouth

Where a river meets the sea, fresh water mixes into salt and the free energy of mixing is released as nothing more useful than a little warmth and stirring. It is not a little energy: a cubic metre of river water entering the sea gives up as much as if it fell 290 metres. A desalination membrane run backwards can catch some of it, and the arithmetic of doing so is the arithmetic of every source with an internal resistance — the most power comes at half the pressure, and half the work is the price of getting it quickly.

fluids · Osmosis An attraction found by counting. How often two large discs are found at each gap between their surfaces, relative to how often they are found far apart, in a direct simulation of two discs among 270 small particles that pass through each other freely and do not enter the discs (dots), and the prediction from counting the small particles' available area (line); small particles a quarter of the large radius. Nothing in the simulation attracts anything: every move that does not put a small particle inside a large one is accepted. Yet the discs are found touching 2.7 times as often as apart (2.6 predicted), because touching returns the overlap of their exclusion zones to the small particles, and the probability of a separation is proportional to the number of arrangements the small particles have at it. Beyond a gap of twice the small radius the zones no longer overlap and the effect vanishes. Written as a potential, −kT times the logarithm of that ratio, it is an attraction about 1.0 kT deep.

The attraction that is the small ones' room

Put large particles into a crowd of small ones that cannot get inside them, and the large ones begin to stick together — though nothing in the mixture attracts anything. Each large particle keeps the small ones' centres out of a thin shell around it. When two large particles touch, their shells overlap, and the overlap is handed back to the small particles as room to move in. More room is more arrangements, more arrangements is more entropy, and the large particles are pushed together by the pressure of the small ones on their outer sides. Fumio Oosawa and Sho Asakura worked it out in 1954. The force is now used to purify DNA, sort colloids onto patterned surfaces and explain why the inside of a cell behaves as though its proteins were stickier than they are.

thermodynamics · Entropy The tube that is stiff because it is full. The bending moment a thin-walled tube 1 mm in radius can carry when inflated, in newton-millimetres, against its internal pressure in megapascals: the moment at which the side in compression goes slack and wrinkles, πPr³/2, and the moment at which the whole section folds, πPr³. Both are proportional to the pressure and contain nothing about the wall's own stiffness: a membrane that cannot resist compression at all carries bending only by being pre-stretched, and the pressure is the pre-stretch. At 0.74 MPa the tube wrinkles at 1.17 N·mm; with no pressure, at nothing. That is the whole of how a lettuce leaf, a petal or a young shoot keeps its shape. It is also how an inflatable boat, a bouncy castle or the boom of an inflatable antenna holds up a load, and how they fail: by wrinkling on the compressed side first.

The stem that stands by pressure

A lettuce leaf has no bones, and a young sunflower stem has hardly any wood, yet both hold their shape against gravity and wind. They do it with pressure. Each cell holds more dissolved solute than the water around it, water flows in until the cell wall pushes back, and the wall ends up stretched taut by a pressure of seven or eight atmospheres — three times a car tyre's. A tube of membrane under tension resists bending in exact proportion to its pressure, whatever the membrane is made of, so a soft plant is stiff exactly as long as its cells are full. Lose a few per cent of their water and the pressure is gone, and the plant wilts.

fluids · Osmosis

Named alongside it

The objects these essays reach for when they reach for this one.

OsmosisElectrostaticsEquilibriumFree energySemipermeable membraneBjerrum lengthChemical potentialCounterion condensationDiffusionEntropyEntropy of mixingMean-field theory

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