Illustrated Physics
physics, drawn
Generator

Two straight lines that are not the same line

One function in the thermal library, called 12 times across 3 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

At its defaults it draws 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.

critical-point is one function in lib/figures/thermal.js — cycles, distributions and the statistics underneath them. 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.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

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.

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.

Throttling a gas, and the curve that says which way it goes

The options are the ones The gas that cools by being let go passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

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.

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 flattest curve in thermodynamics

The options are the ones The gas that cools by being let go passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The flattest curve in thermodynamics. How much the pressure changes when the density is changed, at exactly the critical temperature. The van der Waals isotherm has a slope of 3.000 on these axes — the first and second derivatives both vanish at the critical point, so the leading term is a cube. Real fluids are flatter still, at 4.8. A fluid at its critical point is so soft that its own weight compresses it measurably over the height of the vessel, which is one of the reasons the exponent was hard to measure.

How much the pressure changes when the density is changed, at exactly the critical temperature. The van der Waals isotherm has a slope of 3.000 on these axes — the first and second derivatives both vanish at the critical point, so the leading term is a cube. Real fluids are flatter still, at 4.8. A fluid at its critical point is so soft that its own weight compresses it measurably over the height of the vessel, which is one of the reasons the exponent was hard to measure.

The dome outside which a gas warms as it expands

The options are the ones The gas that cools by being let go passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The dome outside which a gas warms as it expands. The inversion curve of a van der Waals gas in reduced pressure and temperature. The dots are computed: for each of 90 enthalpies, the temperature's turning point along that curve of constant enthalpy, found by scanning the curve. The line is the closed form 24√(3T) − 12T − 27, and the two agree to 0.030 in reduced pressure. Inside the dome a throttled gas cools; outside it, above or below or to the right, the same gas warms. The curve meets zero pressure at 0.750 and 6.750 times the critical temperature and peaks at (3.00, 9.00), all three of which come out of the equation of state with nothing put in by hand. The upper number is the one that matters industrially, because it says a gas can only be liquefied by throttling if it starts below 6.75 times its critical temperature. Real gases sit between about five and nine times theirs — helium at 8.3, hydrogen at 6.1, nitrogen at 4.9, carbon dioxide at 4.9 — so the model gets the ratio right to a factor well under two while getting the mechanism exactly right, which is the usual bargain with van der Waals. Nitrogen's inversion temperature of 621 K is why air liquefies in a Linde machine starting from room temperature, and hydrogen's 202 K is why hydrogen does not.

The inversion curve of a van der Waals gas in reduced pressure and temperature. The dots are computed: for each of 90 enthalpies, the temperature's turning point along that curve of constant enthalpy, found by scanning the curve. The line is the closed form 24√(3T) − 12T − 27, and the two agree to 0.030 in reduced pressure. Inside the dome a throttled gas cools; outside it, above or below or to the right, the same gas warms. The curve meets zero pressure at 0.750 and 6.750 times the critical temperature and peaks at (3.00, 9.00), all three of which come out of the equation of state with nothing put in by hand. The upper number is the one that matters industrially, because it says a gas can only be liquefied by throttling if it starts below 6.75 times its critical temperature. Real gases sit between about five and nine times theirs — helium at 8.3, hydrogen at 6.1, nitrogen at 4.9, carbon dioxide at 4.9 — so the model gets the ratio right to a factor well under two while getting the mechanism exactly right, which is the usual bargain with van der Waals. Nitrogen's inversion temperature of 621 K is why air liquefies in a Linde machine starting from room temperature, and hydrogen's 202 K is why hydrogen does not.

A fluid that stops resisting

The options are the ones The gas that cools by being let go passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

A fluid that stops resisting. The isothermal compressibility along the critical isochore, approaching the critical temperature from above. The van der Waals slope measured off the drawn curve is -1.000, so the compressibility goes as 1/t exactly; real fluids diverge faster, at 1.24. Across the 5 decades drawn it rises by 5.4 decades, and the compressibility is also the mean square density fluctuation — so the same axis says how large a density difference the fluid will produce unprompted.

The isothermal compressibility along the critical isochore, approaching the critical temperature from above. The van der Waals slope measured off the drawn curve is -1.000, so the compressibility goes as 1/t exactly; real fluids diverge faster, at 1.24. Across the 5 decades drawn it rises by 5.4 decades, and the compressibility is also the mean square density fluctuation — so the same axis says how large a density difference the fluid will produce unprompted.

The dome outside which a gas warms as it expands

The options are the ones The gas that cools by being let go passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The dome outside which a gas warms as it expands. The inversion curve of a van der Waals gas in reduced pressure and temperature. The dots are computed: for each of 90 enthalpies, the temperature's turning point along that curve of constant enthalpy, found by scanning the curve. The line is the closed form 24√(3T) − 12T − 27, and the two agree to 0.031 in reduced pressure. Inside the dome a throttled gas cools; outside it, above or below or to the right, the same gas warms. The curve meets zero pressure at 0.750 and 6.750 times the critical temperature and peaks at (3.00, 9.00), all three of which come out of the equation of state with nothing put in by hand. The upper number is the one that matters industrially, because it says a gas can only be liquefied by throttling if it starts below 6.75 times its critical temperature. Real gases sit between about five and nine times theirs — helium at 8.3, hydrogen at 6.1, nitrogen at 4.9, carbon dioxide at 4.9 — so the model gets the ratio right to a factor well under two while getting the mechanism exactly right, which is the usual bargain with van der Waals. Nitrogen's inversion temperature of 621 K is why air liquefies in a Linde machine starting from room temperature, and hydrogen's 202 K is why hydrogen does not.

The inversion curve of a van der Waals gas in reduced pressure and temperature. The dots are computed: for each of 90 enthalpies, the temperature's turning point along that curve of constant enthalpy, found by scanning the curve. The line is the closed form 24√(3T) − 12T − 27, and the two agree to 0.031 in reduced pressure. Inside the dome a throttled gas cools; outside it, above or below or to the right, the same gas warms. The curve meets zero pressure at 0.750 and 6.750 times the critical temperature and peaks at (3.00, 9.00), all three of which come out of the equation of state with nothing put in by hand. The upper number is the one that matters industrially, because it says a gas can only be liquefied by throttling if it starts below 6.75 times its critical temperature. Real gases sit between about five and nine times theirs — helium at 8.3, hydrogen at 6.1, nitrogen at 4.9, carbon dioxide at 4.9 — so the model gets the ratio right to a factor well under two while getting the mechanism exactly right, which is the usual bargain with van der Waals. Nitrogen's inversion temperature of 621 K is why air liquefies in a Linde machine starting from room temperature, and hydrogen's 202 K is why hydrogen does not.

What checks it

physicscheck asserts something about critical-point 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.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Thermodynamics

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

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

The transition with nothing to order

Every transition in this collection so far has an order parameter — a quantity that is zero on one side and not on the other. In two dimensions a continuous symmetry cannot break at any temperature above zero, so there is nothing for such a quantity to be, and by the usual reckoning there can be no transition. There is one anyway, and what changes at it is whether vortices are bound in pairs.

The whole library · All essays