Illustrated Physics
physics, drawn
Concept

Activation energy — where it appears

The height of the barrier a process has to get over, read off the slope of the logarithm of a rate against one over the temperature. For a liquid's viscosity it is the energy a molecule needs to squeeze past its neighbours: about 1,960 kelvin for water and 7,360 for glycerol, which is why glycerol thins so much faster on heating.

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

A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 10, 20, 30 times kT are factors of 10^-4.3, 10^-8.7, 10^-13.0. At 300 K, kT is 25.9 meV, so a barrier of 0.35 eV is 13.5 kT and a factor of 1.3e-6. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens.

The exponential that decides everything

Maximising the number of ways a reservoir can arrange what is left after taking E out of it gives one factor, e to the minus E over kT. Its exponent is a ratio, which is why a barrier of a third of an electronvolt — nothing at all by chemical standards — is the difference between instantly and never.

thermodynamics · Entropy 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

Named alongside it

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

ArrheniusArrhenius lawBarrierThe Boltzmann factorEntropyGlass transitionHard-sphereIntermolecular forcesKinetic theoryMean free pathMomentum diffusionPartition function

All concepts