Illustrated Physics
physics, drawn
Concept

Compressibility — where it appears

The fractional change in a substance's volume per unit of applied pressure, which for a gas is enormous and for a liquid nearly nothing. It is also the mean square density fluctuation in disguise, so a substance that compresses easily is one that varies in density on its own.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

The part of the curve no fluid follows. Van der Waals' isotherms in reduced units, at 5 temperatures either side of the critical one, so that nothing about any particular substance appears. Above the critical temperature the pressure falls monotonically as the volume grows, which is what a fluid does. Below it the curve develops a loop with a rising section in the middle, and that section says the pressure increases as the substance expands — a material with negative compressibility, which cannot exist, because any fluctuation would run away. What happens instead is drawn as the horizontal line: the substance separates into two phases at one pressure, and the volume moves along the line as the proportions change. Its height, 0.6470 of the critical pressure, is fixed by requiring the two areas the line cuts off to be equal, which is the condition that the two phases have the same Gibbs energy. It meets the curve at volumes 0.603 and 2.349, a ratio of 3.9, and those are the densities of the liquid and its vapour. The two turning points of the loop, at 0.72 and 1.53, bound the part that is not merely unobserved but impossible; between them and the construction the substance can be made to sit, superheated or supercooled, until something nucleates.

The part of the curve no fluid follows

One equation for a real gas produces isotherms with a rising middle section, which says a substance would expand as the pressure on it grows. Nothing does that. What replaces it is a horizontal line whose height is fixed by making two areas equal, and the condition is not a convenience.

thermodynamics · Phase change Buoyancy that falls away as the body sinks. The net upward force on a body containing a little gas, against how deep it has been taken, for 3 gas fractions. The weight does not change with depth. The buoyancy does, because the gas obeys Boyle's law and the pressure rises by an atmosphere every ten metres, so a body that displaced its own weight at the surface displaces less at depth. Every curve therefore slopes downward, and that slope is the whole point: where a curve crosses zero the body is in equilibrium, and the crossing is always from above, which makes every one of these equilibria unstable. Push the body a little deeper and the force does not push back — it turns downward and grows. The crossings drawn are at 8.2 m, 10.0 m, 13.6 m, and a body sitting at one of them is balanced in the sense that a pencil is balanced on its point. This is why a diver at neutral buoyancy has to keep adjusting, why a submarine's depth is held by hydroplanes and not by ballast alone, and why a fish that loses the use of its swim bladder sinks rather than drifting.

The depth past which it must sink

A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.

fluids · Buoyancy Two straight lines that are not the same line. Half the difference between the liquid and vapour densities, in units of the critical density, against the distance from the critical temperature — both logarithmic, over 5 decades. The van der Waals curve is solved for the coexisting pair at each temperature and its slope in the last decade is 0.500, which is the mean-field ½. Real fluids give 0.326. The two differ by 11 per cent at t = 0.1 and by a factor of 7.9 at the bottom of the axis, which is why an equation with the wrong exponent in it looked right for eighty years.

The point at which the two become one

Heat a sealed tube of carbon dioxide and the meniscus inside it does not boil away — it fades, the two densities converging until there is nothing to separate. Twenty millikelvin before that happens the fluid turns milky, and the exponent describing the last approach is a number van der Waals got wrong and could not have got right.

thermodynamics · Phase change N when the phases are random, N² when they are not. Scattered intensity against the number of scatterers, both logarithmic, for two ways of adding the same amplitudes. The lower curve averages 400 draws of N unit amplitudes with independent random phases and grows as N^1.000; the upper one adds them in phase and grows as N². At 3000 scatterers the two differ by a factor of 2921. Nothing about the scatterers is different between the two — same number, same strength, same wavelength. Only the arrangement is, and it is worth three decades here.

Why a litre of water is not blue for the reason the sky is

The same molecules that make the sky blue also make the refractive index of air, and the two numbers agree because the sideways sum has random phases and the forward one does not. Condense those molecules into a liquid and the sideways sum collapses by a factor of sixteen — and what is left is thirty-four times smaller than the absorption that actually colours the water.

optics · Scattering 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 A body at an interface, and the difference that holds it. The net upward force on a ten-centimetre cube straddling the boundary between two fluids, against how far its underside sits below that boundary — each curve scaled by its own largest value so that three very different cases fit on one axis. The displaced weight has to be counted twice, once with each density, so the force is linear in the displacement with a stiffness set by gravity, the cube's cross-section and the difference of the two densities — the difference, and neither of them alone. air over water holds the cube with its underside 60 per cent of the way through, at a stiffness of 97.8 newtons per metre; oil over water holds the cube with its underside 47 per cent of the way through, at a stiffness of 14.5 newtons per metre; water over mercury holds the cube with its underside 24 per cent of the way through, at a stiffness of 1229.4 newtons per metre. Both numbers are read off the drawn curve rather than substituted. The consequence is the one worth the figure: a body at an oil–water interface is held six times less stiffly than the same body at an air–water one, because the density difference is six times smaller, and the ordinary intuition that a denser fluid holds a body more firmly is exactly wrong — what matters is the contrast across the surface the body is sitting in.

The body that displaces two things

Every earlier argument has a body wetted by one fluid, so the displaced weight is a volume times a density. A body at an interface displaces two, and what holds it is the *difference* between them — so the same block is held six times less stiffly at an oil–water boundary than at an air–water one. In a continuously stratified column the neutral depth becomes stable, which is the exact opposite of the compressible case.

fluids · Buoyancy Water gets denser the deeper it is. The density of seawater at 0 °C and a salinity of 35 against depth, from the surface to the 10.99 km of the Challenger Deep, with the pressure integrated down the column from the equation of state rather than assumed. At the surface it is 1028.1 kg/m³. At the ocean's mean depth of 3.68 km it is 1045.3, 1.67 per cent more; at the bottom of the deepest trench it is 1076.5, 4.71 per cent more. An incompressible ocean would be the vertical line. The curve is almost straight; what curvature it has, too little to see here, is the water stiffening as it is squeezed: its secant bulk modulus rises by 3.24 pascals for every pascal of pressure.

The sea that stands lower because water gives

Water is the textbook incompressible fluid, and at the bottom of the ocean it is 4.7 per cent denser than at the top. That squeezing lowers the whole sea by about thirty metres, adds the weight of 266 metres of water to the pressure in the deepest trench, and speeds sound up so steadily with depth that it digs a channel in which a low note can cross an ocean.

fluids · Hydrostatics The deepest water reads warmer than the water above it. A column of seawater 10.9 km deep whose potential temperature — the temperature each parcel would have if brought to the surface without gaining or losing heat — falls from 20 °C at the top to 1 °C in the deep, and the temperature a thermometer lowered into it would read. The two agree near the surface and part company with depth: the thermometer reads a minimum of 1.38 °C near 3.9 km and then rises to 2.35 °C at the bottom, although the water there is the same water, with the same potential temperature, 1.00 °C, as the water two kilometres higher. The warming is compression alone; no heat reaches the deep trench from anywhere.

The water that warms on the way down

Lower a thermometer into the Mariana Trench and it reads colder and colder for four kilometres and then starts to warm, reaching about two and a half degrees at the bottom, eleven kilometres down. No heat is coming up from the sea floor to do it. The water at the bottom is the same water as the water two kilometres higher, carried down and squeezed, and squeezing warms it. In fresh water near four degrees the same squeezing does nothing at all, and below four degrees it cools — so a parcel of cold lake water pushed deep enough keeps sinking on its own.

fluids · Hydrostatics The lift of a zero-pressure balloon as it climbs. The gross lift, in tonnes, of a zero-pressure balloon whose envelope holds at most 1.1 million cubic metres, against height, for launches with 3,500 and 5,000 cubic metres of helium; the dashed line is its 3 tonnes of envelope, payload and ballast. At launch the envelope is mostly empty. As the balloon climbs the gas expands exactly as fast as the air thins, so the mass of air it displaces stays the same and the lift stays at 3.69 or 5.28 tonnes all the way up. The envelope fills at 37.5 km with the larger fill and 39.9 km with the smaller; above that, gas vents through the open base, the volume is fixed and the lift falls with the air's density. Both reach the same float, 41.3 km, where the lift has fallen to the weight. More helium only makes the climb faster; the ceiling is set by the envelope's size and the mass it carries.

The balloon that floats at a density

A helium balloon let go rises until something stops it, and what stops it depends entirely on how it is built. A party balloon or a weather balloon stretches until it bursts. An open-bottomed science balloon climbs at constant lift — its gas expanding exactly as fast as the air thins — until its envelope is full, then floats where the air is thin enough. A sealed, unstretchable balloon floats where its whole density equals the air's and bobs back if pushed away. None of them floats at a height; each floats at a density. The difference between the three is how compressible the balloon is compared with the air around it, the same comparison that decides whether a Cartesian diver sinks.

fluids · Buoyancy The rise a watt-year buys at each depth. The rise in sea level, in millimetres, from one watt-year per square metre of heat (3.16 × 10⁷ joules) stored in a thin layer at each depth, for two water columns: a subtropical one, 24 °C at the surface cooling to 1.5 °C in the deep, and a polar one, 1.5 °C at the surface and 0.5 °C below. In the warm column the yield is 2.36 mm at the surface, falls to 1.02 mm near 2.0 km where the water is cold but the pressure still modest, and recovers to 1.49 mm at 6 km as pressure takes over. In the polar column the same heat buys 0.63 mm at the surface and 1.13 mm at 4 km. A joule stored in cold, deep water raises the sea about twice as much as one stored in the cold water above it.

The rise a joule of heat buys

Warm a column of seawater and it stands taller without weighing any more, so the sea rises while the pressure on the sea floor stays exactly the same. How much it rises depends on where the heat goes. Seawater's expansion coefficient grows with temperature and with pressure, so a joule stored in warm tropical water lifts the sea more than six times as far as the same joule in Arctic surface water. Cold water in the deep, squeezed by four kilometres above it, expands about twice as much per joule as the same cold water at the surface. That is why the budget of sea-level rise turns on thousandths of a degree in the least-measured part of the ocean.

fluids · Hydrostatics

Named alongside it

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

Critical pointEquation of stateBuoyancyHydrostatic pressureStabilityArchimedes principleDensityEquilibriumFluctuationsHydrostaticsThe ideal gas lawNeutral buoyancy

All concepts