The phase boundary of water, from one equation

At its defaults it draws the phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

phase-boundary 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.

Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

Temperature against heat added for 1 kilogram of water taken from -20 to 130 degrees Celsius. The two flat stretches are the melting and the boiling, where 334 and 2260 kilojoules go in and the temperature does not move. Melting costs as much as warming the water by 80 degrees; boiling costs as much as warming it by 541, which is 73 per cent of the whole journey.

Pressure against temperature for water on a logarithmic pressure axis. The vaporisation and sublimation curves are integrated from Clausius–Clapeyron with the latent heat held constant, and the fusion curve leans backwards with the slope Clapeyron gives, -13.5 megapascals per kelvin, because water expands when it freezes. Holding the latent heat constant puts the boiling point at one atmosphere at 382.3 kelvin against a measured 373.15, which is 2.5 per cent high.

The logarithm of vapour pressure against a thousand over the temperature, for water, carbon dioxide, nitrogen. On these axes Clausius–Clapeyron with a constant latent heat is exactly a straight line of slope −L/R, so the slope is not a summary of the curve — it is the latent heat, in different units. Each line here is drawn from its substance's triple point to its critical point, and the slope of the drawn polyline is then fitted by least squares and turned back into a latent heat: water went in at 40.65 kJ/mol and comes back at 40.65, 1 part per million out over 41 vertices; carbon dioxide went in at 15.33 kJ/mol and comes back at 15.33, 10 parts per million out over 41 vertices; nitrogen went in at 5.58 kJ/mol and comes back at 5.58, 4 parts per million out over 41 vertices. Each line covers only its own substance's liquid range — water from 273 to 647 kelvin, carbon dioxide from 217 to 304 kelvin, nitrogen from 63 to 126 kelvin — and the steeper the line, the more heat it costs to leave. Where each line ends, the constant-latent-heat model is visibly done — water's reaches 26.0 MPa at its critical temperature against a measured 22.1 MPa; carbon dioxide's reaches 6.00 MPa at its critical temperature against a measured 7.38 MPa; nitrogen's reaches 2.90 MPa at its critical temperature against a measured 3.40 MPa.

Pressure against temperature for carbon dioxide on a logarithmic pressure axis spanning 3.2 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 216.592 kelvin and 518 kPa and the critical point at 304.128 kelvin, with a single latent heat of 16.58 kilojoules per mole — the value the two published points on the curve imply. The measured latent heat at the triple point is 15.33, and the fitted value sits 8.2 per cent above it, which is the ideal-gas assumption showing: the vapour at these pressures is denser than an ideal gas, and an apparent latent heat fitted with the ideal law absorbs the difference. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 24.3 kilojoules per mole, which reaches 101 kPa at 193.3 kelvin against a measured 194.686. The melting curve is drawn at the slope Clapeyron gives it, 4.5 megapascals per kelvin, which is a volume ratio and nothing else: carbon dioxide's solid is 1562 against 1178 kilograms per cubic metre for its liquid, so melting expands it and the line leans forwards. Across the whole 3.2 decades of this axis that line moves 5.0 kelvin. At 1 atmosphere the boundary is crossed at 193.3 kelvin, where carbon dioxide sublimes without ever being a liquid. The one place the curve fails is its top end: a constant latent heat reaches 7.33 MPa at the critical temperature where the measured critical pressure is 7.38 MPa, 0.6 per cent low, because the latent heat falls to zero at the critical point and this curve does not know that.

physicscheck asserts something about phase-boundary 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.