Illustrated Physics
physics, drawn
Concept

Intermolecular forces — where it appears

The residual attractions between neutral molecules, left over once the monopole term cancels, and weak enough to be paid for in surface energy. They fall off far faster than the inverse square — as the inverse seventh power for the dispersion force — which is why they act only across a molecular distance.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Surface at fixed volume. Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.

The skin that is not a skin

A drop of water behaves as though it were wrapped in a stretched membrane, and there is no membrane. What there is instead is an energy cost per unit of surface, and almost everything the apparent skin does follows from a liquid trying to have less of one.

fluids · Surface tension Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

The attraction that needs no charge

Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.

electromagnetism · The field concept A pore that lifts 100 m has to be 0.1 µm or finer. Capillary rise against pore radius, both logarithmic, for a liquid of surface tension 72.8 mN/m at a contact angle of 20°. The relation is a straight line of slope −1 — halve the pore and double the rise — and the two horizontal marks are the height in question, 100 m, and the 10.3 m that one atmosphere supports. 0.01 µm lifts 1394.7 m; 0.1 µm lifts 139.5 m; 1 µm lifts 13.9 m; 5 µm lifts 2.8 m; 20 µm lifts 69.7 cm; 50 µm lifts 27.9 cm. The conducting vessels of a tree are tens of microns across and lift under a metre; the pores in the membranes between them are tens of nanometres and would lift kilometres. Those are the same expression at two scales, and only one of them is a pipe.

The column that is pulled, not pushed

A capillary fine enough to lift a hundred metres is far too fine to carry any flow, and one wide enough to carry the flow lifts under a metre. Neither is how the water gets up a tree. The column is under tension — an absolute pressure of −0.88 MPa at the top, which a gas cannot have — held together by cohesion and prevented from tearing by pores a few tens of nanometres across.

fluids · Capillarity 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 Throttling a gas, and the curve that says which way it goes. Curves of constant enthalpy for a van der Waals gas, in temperature and pressure both measured against the critical values. A gas pushed slowly through a plug or a valve keeps its enthalpy, so it moves along one of these curves — from right to left, since the pressure falls. Where a curve slopes upward to the right the gas cools as it expands; where it slopes downward it warms. An ideal gas would give horizontal lines and no change at all, because its enthalpy depends on the temperature alone; every curve here is bent, and the bending is the attraction between molecules and the room they take up, fighting. The dashed line through the tops of the curves is the inversion curve, and the maxima were found on the drawn points rather than put there — they lie on the closed form to 0.65 per cent. Which side of it a gas starts on decides the sign of the effect, and that is the whole of why air can be liquefied by throttling at room temperature and hydrogen cannot: hydrogen has to be pre-cooled below its own inversion temperature first, which is why Dewar needed liquid air before he could get liquid hydrogen, and why Onnes needed liquid hydrogen before he could get helium.

The gas that cools by being let go

Push a gas through a plug and its temperature changes, although no work is done on anything and no heat goes anywhere. Which way it changes depends on where it starts: inside a dome in the pressure–temperature plane it cools, outside it warms, and hydrogen at room temperature is outside — which is why liquid hydrogen needed liquid air first.

thermodynamics · Phase change The pair potential, and the two things it does to a gas. The Lennard-Jones potential between two molecules, in units of its own depth and range, with the Mayer function it produces at 1, 3, 8 times the well depth in temperature. The virial coefficient is minus the integral of that function over volume, so the two parts of the potential contribute with opposite signs: the steep repulsive core makes the function minus one there, giving a positive contribution — molecules take up room — and the attractive well makes it positive, giving a negative one. At low temperature the attraction dominates and a gas is easier to compress than an ideal one; at high temperature the core dominates and it is harder. Between them is one temperature at which they cancel.

The first correction to the gas law

An ideal gas has no forces between its molecules. The first correction to what it does is computable from those forces alone — one integral over the pair potential — and its sign flips at a temperature where a real gas obeys the ideal law without being ideal at all.

thermodynamics · Kinetic theory

Named alongside it

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

The Boltzmann factorCapillary lengthEquation of stateKinetic theoryPressureSurface tensionVan der waalsActivation energyArrheniusCapillarityCavitationCohesion

All concepts