The site that fills like an electron level

Three questions, from three subjects that rarely meet. What fraction of the active sites on a platinum catalyst are carrying a hydrogen molecule at a given pressure? What fraction of the myoglobin in a muscle cell is carrying oxygen? What is the chance that an energy level in a metal is occupied by an electron? The chemist, the physiologist and the solid-state physicist each have a formula, a name for it and a textbook chapter. The formulas are the same formula. Each describes a place that can hold one particle or none, trading particles with a large reservoir, and each gives the fraction of time the place is full as one function of one number: how far the reservoir’s chemical potential lies above the energy of the site.

Two states and their weights #

A site that holds one particle or none has two states. In contact with a reservoir at temperature $T$ and chemical potential $\mu$, the probability of each state is proportional to a Boltzmann factor, and the relevant energy is not just the energy of the state but the energy the reservoir gives up to create it. Moving a particle from the reservoir onto the site costs the reservoir $\mu$ and gains the site $\varepsilon$, so the filled state carries the weight $e^{(\mu - \varepsilon)/kT}$ against a weight of one for the empty state. The fraction of time the site is full is the filled weight divided by the sum:

$$\theta = \frac{e^{(\mu-\varepsilon)/kT}}{1 + e^{(\mu-\varepsilon)/kT}} = \frac{1}{1 + e^{(\varepsilon-\mu)/kT}}.$$

That is the whole of the argument. It does not ask what the particle is, what holds it to the site, or whether quantum mechanics is involved. It needs only that the site holds at most one and that the reservoir is large enough for its chemical potential not to change when one particle leaves.

For an electron level in a metal, the site is the level, the reservoir is the rest of the metal’s electrons, $\mu$ is the Fermi level, and the formula is the Fermi–Dirac distribution. The “at most one” is the exclusion principle, which forbids two electrons of the same spin in one state — the counting that fills a metal’s electron sea to a temperature of tens of thousands of kelvin. For a molecule on a surface, the site is an adsorption site, the reservoir is the gas above it, and “at most one” is simply that a site is the size of a molecule. There is no exclusion principle there, only crowding, and the formula comes out the same.

Langmuir’s isotherm is the Fermi function #

What makes the surface version look different is the variable it is written in. A chemist measures the gas pressure, not its chemical potential. For an ideal gas the two are related by a logarithm,

$$\mu = kT \ln\frac{p}{p_Q},$$

where $p_Q$ is a pressure set by the mass of the molecule and the temperature — the pressure at which the gas’s molecules would be packed as closely as their own quantum wavelengths. Substituting turns the Boltzmann weight into a ratio of pressures, $e^{(\mu-\varepsilon)/kT} = p/p_{1/2}$ with $p_{1/2} = p_Q\,e^{\varepsilon/kT}$, and the occupancy becomes

$$\theta = \frac{p}{p + p_{1/2}}.$$

Here $\varepsilon$ is the energy of a molecule on the site measured from the gas, so it is negative for a site that binds, and a binding energy of 0.35 eV means $\varepsilon = -0.35$ eV. That is Langmuir’s isotherm, derived by Irving Langmuir in 1918 from the kinetics of molecules striking a surface and leaving it, and it is the Fermi function wearing a logarithm.

On a logarithmic pressure axis, Langmuir’s isotherm has the Fermi function’s S-shape exactly, with a half-full point where the pressure equals $p_{1/2}$ — where the gas’s chemical potential equals the site’s energy — and a width of a few $kT$ either side. On a linear pressure axis it is a hyperbola, rising steeply at low pressure and flattening towards full coverage, and that is how it is usually drawn and why nobody recognises it. The shape of the Fermi function is hidden in the most common way of plotting the most common isotherm.

The isotherms in the first figure are the same curve at three temperatures, and the way they move is itself a Boltzmann factor. The half-full pressure $p_{1/2} = p_Q e^{\varepsilon/kT}$ rises with temperature, because a warmer surface lets molecules escape more easily and needs a denser gas to keep half its sites filled. Between 250 and 350 kelvin the half-full pressure of a 0.35 eV site rises by a factor of two hundred and forty. That is why heating a catalyst clears its surface, why a gas mask’s charcoal releases what it has trapped when warmed, and why a chemical engineer controlling a surface reaction thinks about temperature and pressure as two handles on the same one quantity.

Reading a binding energy off a slope #

The dependence on temperature is also the way the binding energy is measured, because at a single temperature the coverage depends on the energy and the pressure together and cannot separate them.

Plotting the logarithm of the half-full pressure against inverse temperature turns the Boltzmann factor into a straight line whose slope is the binding energy. The same plot, with a rate on the vertical axis instead of a pressure, reads an activation energy off the rates of a reaction, and a vapour pressure plotted the same way reads off the latent heat — which is why a boiling point is really a pressure. In each case the quantity of interest sits in an exponent, and the only way to get it out is to vary the temperature and look at the slope.

A site that holds two, and a site that holds any number #

The restriction to one particle per site is what produces the S-shape, and relaxing it shows what the restriction was doing.

Allow a site to hold up to two particles and the weights become $1$, $x$ and $x^2$, with $x = e^{(\mu - \varepsilon)/kT}$; the occupancy is now $(x + 2x^2)/(1 + x + x^2)$, which rises through one as $\mu$ passes $\varepsilon$ and saturates at two. Allow any number and the weights form a geometric series, $1 + x + x^2 + \cdots = 1/(1 - x)$, whose mean is

$$\bar n = \frac{1}{e^{(\varepsilon - \mu)/kT} - 1},$$

the Bose–Einstein distribution. It diverges as $\mu$ approaches $\varepsilon$, because the series stops converging: a site with room for any number of particles, offered them by a reservoir at a chemical potential equal to its own energy, takes them all. That is why the chemical potential of a gas of bosons can never rise to the energy of its lowest state, and why, when the gas is cooled and compressed until $\mu$ would have to, particles pour into the lowest state instead — Bose–Einstein condensation, which is what makes liquid helium flow without resistance. It is also why light in a cavity has no chemical potential at all: photons can be made and destroyed freely, their number adjusts to minimise the free energy, and $\mu$ sits at zero.

So the three distributions of quantum statistics are three answers to one question — how many particles may a site hold — asked of the same two-line derivation, and the classical Boltzmann distribution is what all three reduce to when the site is nearly always empty, far below its half-full point, where $\theta \approx e^{(\mu - \varepsilon)/kT}$ and the difference between holding one, two or any number never comes up.

The same reservoir, seen from other sites #

Because the formula cares only about the chemical potential of the reservoir, two quite different kinds of site in contact with the same reservoir must agree about it, and that agreement is often the useful fact.

In a semiconductor under illumination or bias, electrons and holes are driven out of equilibrium with each other and each population settles to its own chemical potential — a quasi-Fermi level — and the occupancy of every electron state is the Fermi function in the electrons’ $\mu$. The gap between the two quasi-Fermi levels is a voltage, and the light a diode emits carries exactly that voltage as its own chemical potential. A dopant atom in the same crystal is a site that holds one electron or none, so its occupancy is a Fermi function in the same $\mu$, and measuring how many dopants are ionised is a way of reading where the Fermi level sits.

On a surface exposed to a vapour, the same chemical potential that fills the adsorption sites also governs condensation in any narrow pore nearby. At low pressure the sites fill one molecule at a time, as the figures show; as the pressure approaches the vapour’s saturation, the curved surface of liquid in a pore lowers the pressure at which liquid can form, and a narrow pore fills with liquid from air well below saturation. Both are one reservoir setting one $\mu$, read by two kinds of receptacle: a site that holds one molecule, and a space that holds liquid once $\mu$ passes the value its curvature allows. The measured uptake of a porous material is the first at low pressure and the second near saturation, and the handover between them is how the size of its pores is measured.

And for electrons in a metal, where the site is a quantum state and the restriction to one per state is the exclusion principle, the same formula at room temperature says that only states within a few $kT$ of the Fermi level are partly filled — which is why only a sliver of a metal’s electrons can take part in carrying heat or current. The formula’s width, a few $kT$ in chemical potential, is the same width that forces haemoglobin to cooperate.

Where a site responds most #

The S-shape has a second meaning that is easy to miss: its slope is a fluctuation.

A site is either full or empty at any moment, so its occupancy fluctuates, and the variance of a quantity that is 1 with probability $\theta$ and 0 otherwise is $\theta(1 - \theta)$. That same expression is $kT$ times the slope of the S-curve. The identity is a special case of a general rule — the response of a system to a change in the variable conjugate to some quantity is proportional to the size of that quantity’s spontaneous fluctuations — and here it says that a site responds most to a change in the reservoir exactly where its own occupancy fluctuates most: half full. Far below that point the site is almost always empty and indifferent to small changes; far above, almost always full and equally indifferent.

The same fact is the principle of a buffer. A collection of sites whose energies are spread over a range holds the chemical potential of its reservoir steady within that range, because any change in $\mu$ is met by sites near their half-full points taking up or releasing particles in quantity. A pH buffer is a population of proton-binding sites half full at the buffer’s pH; a semiconductor’s dopant levels pin its Fermi level in the same way; and the oxygen-storing myoglobin in a muscle holds the oxygen supply near its own half-full pressure while demand fluctuates. Each one works best at its half-full point for the reason the figure shows.

Why haemoglobin is not a Langmuir site #

The oxygen carriers of the body make the case for the formula and then the case against it.

Myoglobin, the oxygen store in muscle, has one haem group and one binding site, and its binding curve is exactly Langmuir’s: a hyperbola in pressure, a Fermi function in chemical potential, half full at a partial pressure of about 2.8 millimetres of mercury. That makes it an excellent store and a useless transporter. A single independent site goes from 10 per cent to 90 per cent full over a factor of eighty-one in pressure — the Fermi function’s width of $2\ln 9 \approx 4.4\,kT$, converted through the logarithm — and the difference between the oxygen pressure in the lungs and in working tissue is a factor of five. Myoglobin is nearly full in both, and a carrier that is full in both places carries nothing.

Haemoglobin has four haems, and the binding of oxygen at one changes the shape of the whole molecule in a way that makes the others bind more readily. Its sites are not independent, and the derivation above, which assumed one site trading with a reservoir, does not apply. The binding curve is steeper than any single site can give — sigmoidal on a linear axis, a sharpened S on a logarithmic one — and it goes from mostly full in the lungs to much less full in the tissues, handing over a large fraction of its load across a pressure range that would move a Langmuir site hardly at all. The steepness is usually summarised by Hill’s coefficient, 2.8 for haemoglobin against 1 for an independent site; the coefficient is an empirical description rather than a mechanism, and the models that explain it treat the molecule as switching between two shapes with different affinities. What the figure shows without any model is the necessity: a transporter has to be a site that breaks the rule of independence, because the rule forces a width of several $kT$ in chemical potential and transport needs a narrower one.

Where the two-state site stops being the whole story #

Sites interact. Langmuir’s derivation assumes that a molecule on one site does not change the energy of its neighbours. On many real surfaces adsorbed molecules attract or repel each other, and the binding energy then depends on the coverage; the isotherm is distorted, and at strong attraction the adsorbed layer can condense abruptly, like a gas turning into a liquid in two dimensions.

Sites are not all alike. A real surface has steps, defects and different crystal faces with different binding energies. The measured isotherm is a sum of Fermi functions with a spread of $\varepsilon$, broader than any one of them, and fitting it with a single site energy gives an average that describes no actual site.

Only one layer is allowed. The derivation stops at one molecule per site. At pressures approaching the gas’s condensation, molecules pile onto molecules, and the multilayer isotherms used to measure the area of porous materials are built by stacking the two-state argument on itself.

The reservoir is ideal. The step $\mu = kT\ln(p/p_Q)$ is exact for an ideal gas and becomes approximate for a dense or strongly interacting one, where the pressure has to be replaced by a fugacity. For electrons in a metal the reservoir is the rest of the electron sea and the formula is exact; for oxygen dissolved in blood it is the dissolved concentration, and partial pressure stands in for it.

A site that is only ever full or empty #

Every curve here is an average: the fraction of time a site is full, or the fraction of a large number of sites that are full at one instant. A single site is never partly full. It flips between empty and full at random, spending on average a fraction $\theta$ of its time full, and the rate at which it flips — how long a molecule sits on a catalyst site, how long oxygen stays on a haem — is not in these pictures at all. Two sites with the same isotherm can differ in residence time by orders of magnitude, and for a catalyst the residence time is often what matters. Equilibrium says how full; it never says how fast.

Still open: what the curve says when the sites are single molecules #

The two-state average is exact for one site observed for a long time and for many sites observed at once. Experiments now watch single molecules — one haem, one catalytic site, one ion channel — switching between states, and they find that the switching rates of nominally identical molecules differ, and that a single molecule’s rate can itself change over time, as though the molecule wandered slowly among many slightly different versions of itself. Whether the equilibrium curve of a population is enough to describe what each member does, or whether the spread and memory seen in single molecules carry information that bulk measurements average away, is an active question in single-molecule biophysics, and it bears on how enzymes and binding proteins actually achieve the selectivity that their average curves describe.

The next question about chemical potential turns from a site to a reaction whose products are made in pairs — an electron and the vacancy it leaves, a proton and a hydroxide ion — where each species has its own chemical potential and only their sum is fixed. The habit worth carrying from here is to ask of any filling curve what variable it is being plotted against. A Fermi function drawn against pressure looks like a hyperbola, and a hyperbola drawn against the logarithm of pressure is a Fermi function — and the law that fills a catalyst, a muscle and a metal is visible only when the axis is the chemical potential.