Illustrated Physics
physics, drawn
Concept

Transport coefficient — where it appears

The number relating a flow to the gradient that drives it — a conductivity, a viscosity, a diffusivity. The coefficients relating each flow to the other flows' driving forces form a matrix, and Onsager's relations say that matrix is symmetric, so two apparently separate experiments measure one number.

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

One word, two mechanisms, opposite signs. Viscosity on a logarithmic axis against temperature over the range 280 to 360 kelvin, where gases and liquids can both be measured. The gases rise and the liquids fall, and the two families are separated by three decades of magnitude as well as by sign. The logarithmic slopes at the middle of the range are 0.76 for air, 0.69 for helium, -6.13 for water, -5.41 for ethanol, -23.00 for glycerol, so the steepest liquid responds 30 times more strongly than the gas and in the other direction. Nothing about the word viscosity requires this: what is being measured in both cases is the ratio of a shear stress to a shear rate, and that definition says nothing about what carries the momentum. In a gas it is molecules in free flight, so heating speeds up the carriers; in a liquid the molecules are permanently in contact and what has to happen is one of them getting past its neighbours, so heating removes an obstacle rather than adding a carrier. The obvious question this raises is what a dense gas near its critical point does, where neither picture holds, and the honest answer is that neither formula on this chart applies there at all.

The thickness that goes both ways

Heat a liquid and it thins; heat a gas and it thickens. The two are not a strong effect and a weak one but opposite signs, differing by a factor of thirty in size as well — and the word viscosity names one measurement made on two mechanisms that have almost nothing in common.

fluids · Viscosity Two experiments, two computations, one coefficient. The Seebeck coefficient of a resonant conductor against where its resonance sits relative to the chemical potential, together with the Peltier coefficient divided by the temperature. The first is obtained by applying a temperature difference and finding the voltage that stops the current; the second by applying a voltage at uniform temperature and taking the ratio of the heat flow to the current. Different driving, different measurement, different integral — and the two curves agree to 3.8e-7 of the sweep's own scale across the whole of where they pass through zero and change sign. That equality is Kelvin's relation Π = ST, guessed in 1854 from an argument its author knew was not sound and proved by Onsager in 1931 from microscopic reversibility. It is not a property of this conductor; it holds for every one.

The second experiment that cannot disagree

Heat one end of a wire and a voltage appears across it. Pass a current through the same wire at uniform temperature and it carries heat. Those are two different experiments with two different apparatus, and the coefficient in front is the same number in both — not approximately, and not for some materials. The reason is that the equations of motion underneath look the same run backwards.

thermodynamics · Diffusion Effusion rate against molecular mass. The rate at which a gas escapes through a small hole, against its molar mass, normalised to hydrogen. The rate is a quarter of the number density times the mean speed times the area, and the mean speed goes as the inverse square root of the mass — so the rate does too, which is Graham's law of 1848. Hydrogen escapes four times faster than oxygen and 13.3 times faster than uranium hexafluoride. The practical consequence is isotope separation, and its difficulty is on this chart. The two uranium hexafluorides differ by 3 out of 352 in mass, so a single stage enriches by a factor of only 1.00429 — four parts in a thousand. Reaching 90 per cent from natural uranium's 0.72 per cent therefore takes about 1665 ideal stages, and a real cascade needs more because each stage is imperfect. That number is why gaseous-diffusion plants were among the largest industrial structures ever built, and why centrifuges — which separate by mass directly rather than by the square root of it — replaced them.

The gradient that drives the other thing

A concentration gradient drives a flow of matter and a temperature gradient drives a flow of heat. Each also drives the other, by coefficients that are equal — a relation nobody could have guessed and which follows from the fact that the underlying motion runs the same forwards and backwards in time.

thermodynamics · Diffusion Nitrogen that runs the wrong way. The nitrogen mole fraction in each of two bulbs joined by a capillary, as in Duncan and Toor's experiment: one bulb starts with 0.50086 nitrogen and the rest carbon dioxide, the other with 0.49879 nitrogen and the rest hydrogen, at 35 °C. Solid curves: the capillary solved at each instant from the Maxwell–Stefan equations with the three pairs' diffusivities, 83.8, 68.0 and 16.8 mm²/s. Dashed: Fick's law for nitrogen alone, which can only let the two start values relax together. At the start the nitrogen gradient is 0.00207, yet nitrogen flows at 309 times the rate that gradient would drive. From 0.1 to 6.5 hours it flows from the bulb with less nitrogen into the bulb with more, opening a difference of 0.1468 at 6.5 hours; at 6.6 hours its flux passes through zero with a difference of 0.1468 still in place. Each gas is conserved to a part in 10⁹. The carbon dioxide moving out of the first bulb drags nitrogen with it, because the nitrogen–carbon dioxide pair has by far the smallest diffusivity and so the strongest friction.

The gas that flows towards more of itself

Fick's law says a substance diffuses from where there is more of it to where there is less. In a mixture of three gases, nitrogen can do the opposite for hours — flowing into the bulb that already holds more nitrogen, and then stopping while a difference remains — and in a welded bar of steel, carbon crosses into the side that is already richer. Nothing is wrong with the second law. Diffusion flattens chemical potential, and with more than two components, or a second element changing it, that is not the same as flattening concentration.

thermodynamics · Diffusion

Named alongside it

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

IrreversibilityCross effectDetailed balanceDiffusionEntropy productionEquilibriumOnsager relationsReciprocityActivation energyArrheniusEntropyGlass transition

All concepts