A ratio in the exponent
At its defaults it draws 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.
boltzmann-factor 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.
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.
Hydrogen comes apart far below its binding temperature
The options are the ones The electron that leaves because there is room 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 fraction of hydrogen that is ionised in equilibrium, against temperature on a logarithmic axis, for densities of hydrogen nuclei of 10²³, 10¹⁸, 10¹², 10⁶ per cubic metre, from Saha's equation. Hydrogen's binding energy, 13.6 eV, corresponds to 157,803 K. It is half ionised at 13,980 K at 10²³ per cubic metre, 7,235 K at 10¹⁸ per cubic metre, 4,519 K at 10¹² per cubic metre, 3,270 K at 10⁶ per cubic metre — at a temperature where kT is between an eleventh and a fiftieth of the energy needed to free the electron. The thinner the gas, the cooler the temperature at which it comes apart, because a free electron in a thin gas has vastly more room to be in than a bound one.
The ratio of binding energy to kT at which hydrogen lets go
The options are the ones The electron that leaves because there is room 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 binding energy of hydrogen divided by kT at the temperature where it is half ionised, against the density of hydrogen nuclei, from Saha's equation. It is 11.3 at 10²³ per cubic metre, 34.9 at 10¹² and 48.3 at 10⁶. The dashed curve is ln(nQ/n) evaluated at the same temperatures, the logarithm of the number of states a free electron has room for per electron present: the two track each other within a unit or so, because at half ionisation the equation says exactly that the energy cost χ is paid for by an entropy k ln(nQ/n), less a small correction. A binding energy is not a temperature; what a bond can withstand depends on how much room there is to be free in.
Energy on one side, entropy on the other
The options are the ones The electron that leaves because there is room passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
For hydrogen at 10²³ nuclei per cubic metre, the energy it costs to free one electron, 13.6 eV, against the free energy its freedom is worth, kT ln(nQ/n) — the temperature times the entropy of choosing where to be among the states available to a free electron, per electron — against temperature. The two are equal at 14,923 K, where the entropy gained pays the energy spent, and the full Saha equation puts half ionisation at 13,980 K, close by. At lower temperatures the bound atom is cheaper; at higher ones the free pair has more freedom than its binding costs. The balance is the chemical potentials': the electron's plus the proton's equals the atom's, each chemical potential being an energy minus a temperature times an entropy.
Why the hydrogen lines are strongest in stars that are only warm
The options are the ones The electron that leaves because there is room 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 fraction of all hydrogen atoms in a stellar atmosphere that are neutral and sitting in their second energy level — the level that absorbs the visible Balmer lines — against temperature, at an electron pressure of 20 pascals, from the Boltzmann factor for the level multiplied by Saha's neutral fraction. It peaks at 9,900 K, where it is 8.7 × 10⁻⁶, and falls on both sides: below, too few atoms are excited to the second level; above, too few are left un-ionised to be excited at all. The stars with the strongest hydrogen lines, like Sirius, are therefore not the ones with the most hydrogen but the ones at the right temperature. Reading stellar spectra this way, Cecilia Payne showed in 1925 that the sequence of spectral types is a sequence of temperatures and that the stars are made mostly of hydrogen.
The universe's hydrogen holding on at a fortieth of its binding energy
The options are the ones The electron that leaves because there is room 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 ionised fraction of hydrogen in the early universe in Saha equilibrium, against the temperature of the radiation, with the density of hydrogen set by 1.6 billion photons for every nucleon. It is half ionised at 3,734 K, redshift 1,369, where the binding energy is 42.3 times kT: with so many photons for each atom, even the rare ones energetic enough to ionise are numerous enough to keep hydrogen apart until the radiation has cooled to a fortieth of the binding energy. Equilibrium is an idealisation here: the photons that recombination itself releases can ionise another atom, and the real universe fell behind Saha's curve; its light last scattered off free electrons near 2,970 K, at a redshift of about 1,090, when the microwave background was released.
What checks it
physicscheck asserts something about boltzmann-factor 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.
The electron that leaves because there is room
It takes 13.6 electronvolts to pull the electron off a hydrogen atom, the energy of a particle at a temperature of 158,000 kelvin. Hydrogen is half ionised at the surface of a star at 14,000 kelvin, in a thin nebula at 7,000, and in the early universe at under 4,000 — a fortieth of its binding energy. The electron leaves not because it has enough energy but because a free electron has so much more room to be in than a bound one, and at equilibrium room counts as much as energy. Saha's equation is that trade written down, and it turned out to be the key to what stars are made of.
ThermodynamicsThe 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.
ThermodynamicsThe pressure that ionises
Squeeze a hot gas of hydrogen and Saha's equation says it recombines: a free electron has less room to be in, and room was what paid for its freedom. Squeeze it far enough and it comes apart again, completely, at any temperature — not because the room has grown but because the atom has nowhere left to be. Crowded to within a few atomic radii of each other, hydrogen atoms have no bound state, and their electrons, freed by pressure alone, form a metal. Deep inside Jupiter most of the planet's hydrogen is in that state.
ThermodynamicsThe product doping cannot move
Add one phosphorus atom for every five million silicon atoms and the crystal's free electrons multiply a millionfold — and its holes fall a millionfold, although nothing was done to them. Add a drop of acid to water and the hydroxide ions vanish in exactly the proportion the hydrogen ions grow. Both are one law: when a reaction makes two species together, equilibrium fixes the sum of their chemical potentials, and so the product of their concentrations. Every semiconductor device is built on the minority carriers that law leaves behind.
ThermodynamicsThe protons a star cannot afford
A free neutron decays in fifteen minutes into a proton, an electron and an antineutrino. Inside a neutron star it cannot, and the star is made of neutrons because of it. Beta decay is a reaction like any other, and at equilibrium the neutron's chemical potential must equal the proton's plus the electron's. In a crowded star an electron can only be added at the top of a sea tens of megaelectronvolts deep. So the matter settles where protons are rare: half a per cent of the nucleons at nuclear density, if nucleons were free, and never more than one in nine at any density.
ThermodynamicsThe reaction that cannot go all the way
Chemistry speaks of reactions that go to completion and reactions that do not happen, and at equilibrium there are neither. The reason is a logarithm. The free energy of a half-finished reaction contains the entropy of mixing, whose slope is infinite at both pure ends, so every reaction's lowest point lies strictly inside — and the slope of that free energy, the chemical potential, is to particles what temperature is to heat.
ThermodynamicsThe site that fills like an electron level
A patch of catalyst holding a gas molecule, a haem in a muscle cell holding oxygen, an energy level in a metal holding an electron: each is a place that can hold one particle or none, in contact with a reservoir it trades particles with. Each fills according to the same function of one variable — how far the reservoir's chemical potential lies above the energy of the site. Langmuir's isotherm and the Fermi function are not analogous. They are the same law, and haemoglobin works because its sites break it.
ThermodynamicsThe temperature a molecule does not have
Temperature fixes a system's average energy and nothing more. The actual energy wanders, by an amount tied to the heat capacity, and the relative size of the wandering falls as one over the square root of the number of degrees of freedom — so a mole has a temperature and a molecule does not.
ThermodynamicsThe voltage that is a chemical potential
A battery, a nerve cell and a pH meter all turn a difference in chemical potential into a voltage, and the conversion is the same in each: fifty-nine millivolts for every factor of ten in concentration, at room temperature, whatever the ion. The voltage a battery holds as it discharges is the chemical potential of the atoms stored in its electrodes — and a battery whose voltage barely changes is one whose electrode is going through a phase transition.