Illustrated Physics
physics, drawn
Concept

Semipermeable membrane — where it appears

A barrier that one species may cross and another may not, which is all that osmosis requires. The pressure it can hold up depends on how many particles are dissolved and on nothing about their chemistry, reaching twenty-seven bar for seawater against fresh.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The entropy of mixing, and the entropy of not mixing. The entropy gained when two ideal gases at the same temperature and pressure are allowed to mix, per particle and in units of Boltzmann's constant, against the proportion of the mixture that is the first gas. The curve has no property of either gas in it — not their masses, not their sizes, not how strongly they interact, since ideal gases do not — and it is largest at 0.500, where it reaches ln 2 = 0.6931. Below it is the same quantity for two samples of the SAME gas, which is zero at every proportion: removing the partition between two halves of a box of nitrogen changes nothing that can be measured, and putting it back recovers the original state. The two results are correct and they do not join up. Make the two gases more and more alike — two isotopes, then two nuclear spin states, then nothing at all — and the upper curve does not descend to meet the lower one; it stays exactly where it is until the two species become identical, and then jumps. What the figure is really about is that the jump is in the counting and not in the gas.

Mixing what is already mixed

Let two different gases into each other's halves of a box and the entropy rises by a fixed amount that contains nothing about either gas. Do it with the same gas on both sides and it rises by nothing. Make the gases more and more alike and the answer does not converge — it jumps.

thermodynamics · Entropy 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 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

Named alongside it

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

Entropy of mixingOsmotic pressureFree energyOsmosisChemical potentialColligativeConcentration polarisationDiffusionEfficiencyEntropyEquilibriumExtensivity

All concepts