The column a dissolved thing holds up

At its defaults it draws 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.

osmotic-cell is one function in lib/figures/fluids.js — matter that will not hold a shape, and the forces in it. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

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 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 probability that a single counterion, alone beside a charged rod, is found within a distance r0 of it rather than anywhere in a cell a million times wider, against how much thinner than r0 the rod is, on a logarithmic axis. The counterion's Boltzmann weight near a line charge goes as the distance to the power −2ξ, and the probability is a ratio of two integrals of that weight, each evaluated numerically and held against its closed form. For ξ = 0.5 it reaches 0.000; for ξ = 0.9 it reaches 0.063; for ξ = 1 it reaches 0.667; for ξ = 1.1 it reaches 0.996; for ξ = 2 it reaches 1.000 when the rod is a trillionth of r0. Below ξ = 1 the integral converges as the rod thins and the counterion wanders off into the cell, because the entropy it gains by going far away grows faster than the energy it pays. At ξ = 1 the two grow at the same logarithmic rate, and above it the integral diverges at the rod and the counterion is captured. Condensation is that divergence, and the other counterions, by screening the chain, are what stop it at ξ = 1.

The osmotic coefficient of a solution of charged rods with no added salt — the osmotic pressure divided by what the counterions would exert as an ideal gas — against the charge parameter ξ, from the Poisson–Boltzmann cell model, where the pressure is the counterion density at the cell's edge times kT. Cells of 10², 10⁴, 10⁸ rod radii are successively more dilute. The solid line is Manning's limiting law: 1 − ξ/2 below the threshold and 1/2ξ above it. At ξ = 4.2 the computed coefficient is 0.154, 0.130, 0.122 for the three cells, falling towards Manning's 0.119. A solution of DNA with a mole of counterions therefore exerts roughly a tenth of the osmotic pressure a mole of free salt ions would, and the reason is not that the counterions are attached to anything but that the chain's field holds them within a few nanometres.

The swelling pressure of a charged gel carrying 0.5 M of fixed charge on chains with ξ = 2.85 — sodium polyacrylate, a charge every 0.25 nm — against the salt in its bath, on logarithmic axes. The dashed curve counts every fixed charge in the Donnan balance; the solid curve counts only the charge left after condensation, the fixed charge divided by ξ. In a low-salt bath the pressure is the fixed charge counted as a gas, and condensation cuts it by ξ, from 1235 to 430 kPa at 1 mM. In a bath at physiological salt, 150 mM, the two give 702 and 118 kPa, a factor of 6.0, because the screening regime goes as the square of the charge. A nappy's absorbency in distilled water and in urine differs by both effects at once, and the charge the polymer was synthesised with is not the charge the water sees.

The fraction of a chain's charge neutralised by condensed counterions against the charge parameter ξ, for counterions carrying one, two and three charges. A counterion of charge z feels z times the chain's field, so it condenses until the effective charge parameter is 1/z and neutralises 1 − 1/zξ of the chain. For DNA, ξ = 4.2, the Poisson–Boltzmann cell at the inflection of each profile gives 76.2 per cent for monovalent ions, 88.1 per cent for divalent ions, 92.1 per cent for trivalent ions, matching Manning's form. The dashed line is 89 per cent, near which DNA in solution is observed to collapse from an extended chain into tightly packed toroids: sodium cannot take it there, magnesium falls just short, and the trivalent polyamine spermidine does, which is how DNA is packed into sperm heads and viral capsids.

physicscheck asserts something about osmotic-cell that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.