Generator

A ratio in the exponent

One function in the thermal library, called 18 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 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.

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.

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.

A ratio in the exponent

The options are the ones The exponential that decides everything 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 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.

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.

A ratio in the exponent

The options are the ones The exponential that decides everything 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 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 4, 8, 12 times kT are factors of 10^-1.7, 10^-3.5, 10^-5.2. At 1000 K, kT is 86.2 meV, so a barrier of 0.7 eV is 8.1 kT and a factor of 3.0e-4. 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.

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 4, 8, 12 times kT are factors of 10^-1.7, 10^-3.5, 10^-5.2. At 1000 K, kT is 86.2 meV, so a barrier of 0.7 eV is 8.1 kT and a factor of 3.0e-4. 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.

The slope is the barrier

The options are the ones The exponential that decides everything 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 slope is the barrier. The logarithm of the Boltzmann factor against a thousand over the temperature, for barriers of 0.2, 0.35, 0.6 eV. Each is a straight line whose slope is the barrier divided by k, which is what makes the plot worth drawing: a rate measured at four temperatures gives the height of an obstacle nobody can see. The barriers read back from two points on each plotted line are 0.200 eV, 0.350 eV, 0.600 eV, against the values asked for. The lines fan out toward low temperature and converge at high, which is the same statement as before: heating does not lower a barrier, it makes the comparison with it less unfavourable.

The logarithm of the Boltzmann factor against a thousand over the temperature, for barriers of 0.2, 0.35, 0.6 eV. Each is a straight line whose slope is the barrier divided by k, which is what makes the plot worth drawing: a rate measured at four temperatures gives the height of an obstacle nobody can see. The barriers read back from two points on each plotted line are 0.200 eV, 0.350 eV, 0.600 eV, against the values asked for. The lines fan out toward low temperature and converge at high, which is the same statement as before: heating does not lower a barrier, it makes the comparison with it less unfavourable.

How many degrees double a 0.35 eV process

The options are the ones The exponential that decides everything passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

How many degrees double a 0.35 eV process. The temperature rise that doubles a process over a barrier of 0.35 eV, against the temperature it starts from. The familiar rule that a reaction doubles for every ten degrees is one point on this curve and not a law: it holds for a particular barrier at a particular temperature and nowhere else. From 250 K it takes 11.1 K; from 300 K it takes 16.2 K; from 350 K it takes 22.2 K; from 400 K it takes 29.3 K. The rise needed grows as the square of the temperature, because the exponent depends on 1/T and the change in 1/T for a given step shrinks — which is why a process that is sluggish at room temperature and brisk at 350 K is barely faster again at 700 K.

The temperature rise that doubles a process over a barrier of 0.35 eV, against the temperature it starts from. The familiar rule that a reaction doubles for every ten degrees is one point on this curve and not a law: it holds for a particular barrier at a particular temperature and nowhere else. From 250 K it takes 11.1 K; from 300 K it takes 16.2 K; from 350 K it takes 22.2 K; from 400 K it takes 29.3 K. The rise needed grows as the square of the temperature, because the exponent depends on 1/T and the change in 1/T for a given step shrinks — which is why a process that is sluggish at room temperature and brisk at 350 K is barely faster again at 700 K.

How many degrees double a 0.8 eV process

The options are the ones The exponential that decides everything passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

How many degrees double a 0.8 eV process. The temperature rise that doubles a process over a barrier of 0.8 eV, against the temperature it starts from. The familiar rule that a reaction doubles for every ten degrees is one point on this curve and not a law: it holds for a particular barrier at a particular temperature and nowhere else. From 280 K it takes 6.0 K; from 310 K it takes 7.3 K; from 350 K it takes 9.4 K. The rise needed grows as the square of the temperature, because the exponent depends on 1/T and the change in 1/T for a given step shrinks — which is why a process that is sluggish at room temperature and brisk at 350 K is barely faster again at 700 K.

The temperature rise that doubles a process over a barrier of 0.8 eV, against the temperature it starts from. The familiar rule that a reaction doubles for every ten degrees is one point on this curve and not a law: it holds for a particular barrier at a particular temperature and nowhere else. From 280 K it takes 6.0 K; from 310 K it takes 7.3 K; from 350 K it takes 9.4 K. The rise needed grows as the square of the temperature, because the exponent depends on 1/T and the change in 1/T for a given step shrinks — which is why a process that is sluggish at room temperature and brisk at 350 K is barely faster again at 700 K.

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 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.

The 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.

The 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.

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