The exponential that decides everything

Almost every rate in the natural world — a reaction, a diffusion, a creep, a decay of a metastable state, an evaporation, the conduction of a semiconductor — is an attempt frequency multiplied by the same exponential. The attempt frequency varies by a few orders of magnitude between processes. The exponential varies by thirty, and it is therefore the only part of the expression that ever decides anything.

The important feature of that plot is not the slope but the axis. The horizontal quantity is a ratio, so an energy is never large or small in itself, only large or small compared with $kT$ — and $kT$ at room temperature is a fortieth of an electronvolt, which is very small compared with almost every energy in chemistry.

Where the factor comes from #

The derivation is a counting argument and does not mention forces, collisions or dynamics.

Consider a small system in contact with a large reservoir, with a fixed total energy between them. The probability that the small system is in a particular state of energy $E$ is proportional to the number of ways the reservoir can arrange the remaining energy — because every one of those arrangements is equally likely, and they are all compatible with the small system being in that one state.

The count the argument is about is the reservoir’s. Taking energy $E$ out of it reduces the number of arrangements available to it, and reduces it by a factor that is exponential in $E$ because entropy is a logarithm of that count and entropy is what changes linearly with energy. The whole derivation is that one sentence: a linear change in a logarithm is an exponential change in the thing.

Two things about that derivation are worth keeping. The temperature enters only through $\partial S/\partial E$, which is what temperature is in statistical mechanics — so the Boltzmann factor is not a separate assumption but the definition of temperature rearranged. And the expansion to first order requires the reservoir to be large compared with the system, which is the only approximation anywhere in it and the reason the factor fails for a system comparable in size with its surroundings.

Reading it as a population #

The most direct use is a ratio of populations between two states, in which the normalisation cancels and only the energy difference survives.

A ladder with a large first step shows what the factor does to a population. Hydrogen’s gap from ground to first excited state is 10.2 eV, which at 300 K is about 394 in units of $kT$ — so the excited fraction is $e^{-394}$, a number with no physical meaning whatever. Room-temperature hydrogen is entirely in its ground state, and “entirely” here is not an approximation.

That case is the general shape of the argument. Something that never happens at one temperature and dominates at another has not changed its physics; the ratio in the exponent has changed by a factor of twenty, and twenty in an exponent is nine orders of magnitude in the answer.

The same reading with a gravitational energy instead of an electronic one gives an atmosphere.

Density against height is the same factor with $E = mgh$. The scale height $kT/mg$ is the height at which the potential energy of a molecule equals $kT$, and above it the density falls by a factor of $e$ per scale height — which is why an atmosphere has no edge, and why the same arithmetic that governs excited states governs the barometer.

The slope that measures an invisible obstacle #

Because the factor is an exponential in $1/T$, a logarithm of a rate plotted against reciprocal temperature is a straight line whose slope is the barrier — and that is one of the most-used measurements in the whole of the physical sciences.

Arrhenius established the form empirically in 1889 for chemical reactions and had no derivation for it; the interpretation of the slope as an energy barrier came later, and the modern justification — transition-state theory — is essentially the argument above applied to the population of an unstable configuration at the top of the barrier.

The technique is not confined to chemistry. The same plot measures the activation energy of diffusion in a solid, of viscous flow in a glass, of dielectric relaxation in a polymer, of the conductivity of a semiconductor, and of the failure rate of a component in an accelerated-life test. In each case the line’s slope is an energy and its intercept is an attempt frequency, and in each case the slope is the part anyone trusts.

Diffusion in a solid is one of the processes such a slope belongs to: atoms hopping between lattice sites over a barrier, at a rate the factor fixes. Measuring the diffusion coefficient at several temperatures and plotting its logarithm against $1/T$ gives a straight line whose slope is the barrier — so an invisible activation energy is read off a graph, which is the practical use this whole subject is put to.

Why ten degrees is not a rule #

The most quoted consequence of the exponential is that a reaction rate doubles for every ten degrees, and it is worth doing the arithmetic to see what kind of statement that is.

So the rule of ten holds for a particular barrier at a particular temperature and nowhere else. It is not an accident that it is quoted so often — a great many biological and food-chemistry processes have effective barriers of 0.5 to 0.8 eV and are used between 273 and 310 K, which is where the rule works — but a process with a 0.2 eV barrier needs nearly thirty degrees to double at room temperature, and one with a 1.2 eV barrier needs five.

The practical form of the exponential is worth stating as its own sentence. Three kT is a factor of twenty; ten kT is a factor of twenty thousand; thirty kT is a factor of $10^{13}$. An energy of a few tenths of an electronvolt is nothing to a chemist and an absolute prohibition to a rate.

What the same factor forbids #

The exponential is also the reason a great many things are stable that have no business being stable.

A diamond at room temperature is thermodynamically unstable — graphite is the lower-energy form — and converts at a rate governed by a barrier of several electronvolts, which is some two hundred $kT$. The factor is $10^{-87}$, so the conversion takes longer than any timescale worth naming. The same argument protects every metastable material there is: window glass, which is a supercooled liquid; a supersaturated solution; the whole of organic chemistry, which sits in a deep local minimum a long way above the products of complete combustion.

The distribution whose tail does the work matters because everything above is about a fraction of a population rather than about any molecule. The factor does not say that a molecule cannot have the energy; it says how few do. And since “how few” is exponential, the answer swings between “essentially all” and “essentially none” over a range of energies that is narrow in absolute terms and wide in units of $kT$.

And there is a class of process the factor does not govern at all, which is the sharpest way to see what it is claiming. Radioactive decay has a rate that is entirely independent of temperature, because the barrier is crossed by tunnelling rather than by thermal excitation and the nucleus is not in thermal contact with anything in the relevant sense. A process whose rate does not respond to heating is announcing that its mechanism is not thermal, and that is a diagnostic used routinely.

A rate with no temperature in it is worth setting alongside, because the two exponentials look identical and are not. Radioactive decay falls exponentially in time from a constant probability per unit time; the Boltzmann factor falls exponentially in energy from a constant entropy per unit energy. Same shape, different variable, and no temperature anywhere in the first.

The same exponent, three ways of being useful #

It is worth separating the three quite different jobs the factor does, because they are constantly run together and only one of them is about rates.

As a population. Two states, one energy difference, one ratio. Nothing dynamic is involved and the answer is exact in equilibrium. This is how the intensity ratio of two spectral lines gives the temperature of a flame or a star, how the polarisation of a nuclear spin sample is worked out, and how the fraction of a semiconductor’s carriers sitting in the conduction band is found.

As a rate. The population at the top of a barrier, multiplied by an attempt frequency. This is the chemical and materials use, and it is an approximation of a different order: it assumes the barrier is high compared with $kT$ so that the population there is small, and it assumes anything reaching the top goes over rather than coming back.

As a fluctuation. The probability that a system in equilibrium departs from its average state by an energy $E$ — which is the same exponential read as a statement about excursions rather than about states. That is where the size of thermal noise in a resistor comes from, why a colloidal particle jiggles, and why the fluctuations in any macroscopic quantity are so small: a departure large enough to see in a mole of gas has an energy of order $10^{23}kT$ and a probability nobody has a notation for.

The third reading is the one that connects to the direction of time. A cup of tea does not spontaneously reheat, and the reason is not that it is forbidden but that the exponent is $10^{22}$ or so — which is a probability that differs from zero in a way that has no operational meaning at all. Irreversibility, in this language, is the statement that the exponent is enormous rather than that anything is prohibited.

Why a smooth exponential looks like a threshold #

Nothing in the Boltzmann factor is discontinuous. It is a smooth function of temperature with no kink, no step and no critical value anywhere in it. Yet almost every process it governs is described in the language of thresholds: an ignition temperature, a point at which a glue cures, a temperature above which a component fails. That gap between the smooth formula and the switch-like behaviour is worth closing, because the closing is one line and it explains a great deal.

Ask how strongly the rate responds to a fractional change in temperature — the logarithmic derivative, which is the local power-law index. Differentiating the exponential gives

$$\frac{\mathrm{d}\ln k}{\mathrm{d}\ln T} = \frac{E_a}{kT},$$

which is the barrier measured in units of $kT$ and nothing else. So a process with a 30 $kT$ barrier behaves locally like the thirtieth power of the temperature.

That is what makes it look like a threshold. A ten per cent rise in absolute temperature multiplies a thirtieth-power quantity by seventeen; a thirty per cent rise multiplies it by two thousand. Nothing switched on — the rate was always there and was simply too small to see, and then within a small span of temperature it became too large to ignore. The apparent sharpness is a statement about the size of the exponent rather than about any feature of the curve.

The same arithmetic run in reverse explains the other half of the impression: below the apparent threshold the rate falls just as steeply, so a process that is comfortable at one temperature is not slow at ten per cent lower but absent. There is very little middle ground when the local index is thirty, and the intuition that something either happens or does not is a reasonable summary of a power of thirty.

This is the same phenomenon as a combustion process running away. The reaction generates heat, the heat raises the temperature, and the rate responds as a thirtieth power — so a small excess of generation over loss becomes a large one immediately. The ignition temperature of a fuel is not a property of the fuel’s chemistry so much as the point where that feedback loop closes, which is why it depends on the geometry of the container and on how fast heat can leave.

Two products, and which one arrives #

A second consequence follows from the factor appearing in two different roles at once, and it is one of the few places where the same exponent gives contradictory advice.

Suppose a system can go to either of two products. One is lower in final energy — more stable — and the other has a lower barrier on the way. Which one is obtained?

The rates are governed by the barriers, so at short times the product with the lower barrier accumulates faster, in a ratio $e^{-\Delta\Delta E^{\ddagger}/kT}$ where the difference is between the two barrier heights. The equilibrium populations are governed by the final energies, in a ratio $e^{-\Delta\Delta E/kT}$ using the difference between the products. The two exponents are different differences, and they can point in opposite directions.

At low temperature both exponents are large, so the lower-barrier product forms overwhelmingly faster and nothing has enough energy to go back over its barrier and re-sort. The mixture is whatever the rates delivered, frozen. This is kinetic control.

At high temperature both exponents are small, so both products form quickly and both revert quickly, and the mixture settles into the ratio the final energies dictate. This is thermodynamic control.

So the same reaction gives different products at different temperatures, and the changeover is not a change of mechanism. Both paths are open throughout; what changes is whether the system has time and energy to undo its first answer. The textbook demonstration is the addition of hydrogen bromide to butadiene, which gives predominantly one isomer at eighty degrees below zero and predominantly the other at forty above, with the same reagents in the same flask.

The general principle is broader than chemistry and is worth having in that form. A system relaxing toward equilibrium reaches the nearest accessible state first and the most stable state eventually, and which of those it is found in depends on how its observation time compares with the slowest barrier crossing. Glass is the standing example: it is the kinetically controlled outcome of cooling a liquid too fast for the crystal to form, and it remains one for as long as anybody has been watching.

What it costs, and where the model stops #

It says nothing about how long. The factor gives an equilibrium probability. Turning that into a rate needs the attempt frequency as well, and the attempt frequency is the hard part — it depends on the shape of the barrier, on how many ways there are over it, and on how strongly the system is coupled to its surroundings. The exponential decides orders of magnitude and the prefactor decides the answer.

It requires a reservoir. The derivation expanded the reservoir’s entropy to first order, which needs the reservoir to be much larger than the system. For a nanoparticle in a small droplet, or for a system whose energy is comparable with its surroundings’, the expansion fails and the distribution is not exponential.

It requires equilibrium. A system driven by an external source — a laser, a current, a metabolism — has populations that are not Boltzmann-distributed at any temperature, and inverting a population is the whole operating principle of a laser. Quoting an effective temperature for such a system is a convenience and sometimes a negative number.

A degeneracy is not a detail. The probability of an energy is the factor times the number of states at that energy, and in a large system that number grows fast enough to compete with the exponential. The competition between the two is what free energy is, and it is why a system at high temperature moves to a state of higher energy and higher entropy rather than staying in its ground state.

Where the numbers come from in practice #

Three values of $kT$ are worth carrying, because almost every estimate in this subject is made by comparing something to one of them.

At room temperature, 293 K, $kT$ is 25.3 meV. At the boiling point of nitrogen, 77 K, it is 6.6 meV. At the surface of the sun, 5,800 K, it is 0.50 eV. The first is why chemistry happens at all at room temperature and why semiconductor devices work; the second is why cooling a detector to liquid-nitrogen temperature reduces its thermal noise by a factor of four and its thermally generated carriers by a factor of $10^{13}$; the third is why a stellar photosphere ionises hydrogen only in its tail and why stellar spectra are as informative as they are.

A fourth comparison is the one that makes biology possible. A hydrogen bond is worth about 0.2 eV, which is 8 $kT$ at body temperature — strong enough to hold a structure together against thermal agitation, weak enough to be broken by an enzyme without a catastrophe. A covalent bond is 3 to 4 eV, or 150 $kT$, and is never broken thermally at all. The gap between those two numbers is why a cell can have parts that stay assembled and parts that come apart on demand, and both of those facts are the same exponential evaluated at two energies.

The other way past a barrier does not consult the temperature at all. Tunnelling transmits a wave through a wall it has not the energy to cross, at a rate exponential in the barrier’s width rather than in $kT$ — so a process with both channels open is thermally activated at high temperature and temperature-independent at low, and the crossover is visible as a kink in exactly the plot described above.

What the picture cannot show #

The plots on this page are of the factor, not of a rate. A measured rate is the factor times a prefactor that also depends on temperature — as a square root for a collision rate, or with its own weak dependence for a solid-state process — and an Arrhenius plot’s slope is therefore an effective activation energy that includes any temperature dependence hidden in the prefactor. Over a narrow range the two are indistinguishable, which is why Arrhenius plots are usually drawn over a narrow range.

Nor can a plot of a single barrier show what happens when there are two. A process with two paths of different barriers switches from following one to following the other as the temperature changes, and the Arrhenius plot then has a kink rather than a slope. That kink is one of the more informative things a rate measurement can produce, and it is invisible in any figure drawn for a single mechanism.

The domain of validity is: a system in equilibrium with a much larger reservoir, at a well-defined temperature, with the energies of its states known. That is a strong set of conditions, and the factor’s astonishing reach comes from the fact that a very great many things in the world satisfy them to an excellent approximation.

The ladder from here #

Later rungs on this anchor: the partition function as the normalising sum, and the extraction of every thermodynamic quantity from its logarithm; free energy as the competition between the factor and the degeneracy; transition-state theory, which supplies the prefactor; the Kramers treatment, in which the escape rate depends on how strongly the system is damped; and negative temperature, which is what a population inversion is when the same formula is applied to it backwards.

The neighbouring ladders are entropy as a count, which is what the derivation maximises, the barometric distribution, which is this factor with a gravitational energy, and tunnelling, which is the way over a barrier that does not use the factor at all.