The voltage that is a chemical potential

The reaction that cannot go all the way introduces the chemical potential as the slope of free energy with respect to the number of particles — what temperature is to heat, the chemical potential is to particles, and they flow from high to low until it is equal everywhere. The product doping cannot move ends by pointing at the case where the particles are charged and the flow is between two different materials: electrons in a metal and ions in a solution. There, a difference of chemical potential becomes a voltage, and that conversion is what every battery, every nerve cell and every pH meter is built on.

The conversion is simple enough to state in one line and far-reaching enough to organise a whole field. A charged particle feels two things: the chemical potential $\mu$, which depends on how crowded it is, and the electrical potential energy $ze\phi$, which depends on where it is in a field. Its equilibrium is set by their sum, the electrochemical potential. If the sum is the same everywhere, nothing flows — however different the concentrations are, and however large the voltage.

Fifty-nine millivolts for every factor of ten #

For ions dilute enough to ignore their interactions, the chemical potential depends on concentration as $kT\ln c$ — the entropy of choosing where to put each ion, which mixing what is already mixed computes for gases. Put a boundary between two solutions of the same ion at different concentrations, let the ion cross it, and it will cross from the concentrated side to the dilute one until a voltage builds up to stop it. Equilibrium requires $kT\ln c_1 + ze\phi_1 = kT\ln c_2 + ze\phi_2$, so

$$\phi_2 - \phi_1 = \frac{kT}{ze}\ln\frac{c_1}{c_2}.$$

At room temperature, $kT/e$ is 25.7 millivolts, and times the natural logarithm of ten it is 59.16 millivolts: a factor of ten in concentration is worth 59 millivolts, whatever the ion. The number rises with temperature, to 61.5 millivolts at body temperature, and nothing else about the system appears in it — not the size of the ion, not the solvent, not the nature of the boundary. It is the Boltzmann factor of the exponential that decides everything read backwards: a potential step of 59 millivolts reduces the probability of finding an ion on its high side by exactly a factor of ten.

That is the Nernst equation, stated by Walther Nernst in 1889, and it is a statement about entropy dressed as electricity. The concentration difference is a store of free energy only because a dilute ion has more ways to arrange itself than a concentrated one; the voltage is that entropy, per unit charge, multiplied by the temperature.

Two slopes that cancel #

The drawing separates the two parts. On the concentrated side the chemical part is high and the electrical part low; across the boundary they swap; their sum, drawn thick, does not move. An ion on the left is pushed rightwards by its concentration and pushed back exactly as hard by the field, and the voltage of 89 millivolts is the number that makes the two pushes equal. Nothing flows.

This is the equilibrium that the swelling a membrane cannot stop finds across a charged gel, where a Donnan potential of tens of millivolts arises with no pump and no current. It is also, with electrons instead of ions, the built-in voltage of a semiconductor junction: one level and the field that bends the bands shows two pieces of silicon with different electron concentrations developing a voltage across their junction until the electrons’ electrochemical potential — the Fermi level — is flat through both. Same equation, different particles.

Four ions, four voltages, one cell #

A nerve cell is a bag of potassium in a bath of sodium, and the Nernst equation says what voltage each would hold on its own.

Potassium is twenty-eight times more concentrated inside, and on its own would hold the inside at −89 millivolts. Sodium is twelve times more concentrated outside, and would hold it at +67. Calcium, ten thousand times more concentrated outside and doubly charged, would hold it at +132. The membrane cannot satisfy all of them at once, so it is not in equilibrium; it sits at a voltage that balances the currents of all the ions it lets through, weighted by how easily each one passes. At rest a neuron’s membrane is far more permeable to potassium than to sodium, and the Goldman–Hodgkin–Katz equation — the Nernst equation generalised to several ions leaking at different rates — puts it at −73 millivolts, close to potassium’s value and far from sodium’s.

That arrangement makes a prediction a voltmeter can test. Raise the potassium in the bath and the resting potential should follow potassium’s Nernst voltage, climbing by about sixty millivolts for each factor of ten. Experiments on squid axons in the 1940s found exactly that slope once the bath held more than a few tens of millimolar potassium, and a flatter one below it, where sodium’s small leak is no longer negligible beside potassium’s. The bend in that curve is how the permeability ratio was first measured, and it is the Nernst slope, not any property of the axon, that sets the straight part.

An action potential is that balance shifting. When sodium channels open, sodium’s permeability rises a hundredfold for a millisecond, and the membrane swings towards sodium’s +67 millivolts before potassium channels open and pull it back. The swing and its recovery are a change in which chemical potential the membrane is listening to; the voltages themselves were set all along by the concentrations, and the pumps that maintain those concentrations — spending, by most estimates, a large fraction of the brain’s energy — are what keep the battery charged.

A battery’s voltage is a free energy per charge #

A battery works the same way with a chemical reaction in place of a concentration ratio. In a zinc–copper cell, zinc metal dissolves as zinc ions at one electrode and copper ions plate out as copper metal at the other; the electrons the zinc gives up travel through the external circuit to the copper. The reaction has a free energy, and at equilibrium — with no current flowing — the voltage between the electrodes is that free energy divided by the charge that has to move for one unit of reaction.

The standard voltage, 1.10 volts, is the difference between how strongly the two metals hold on to their outer electrons, measured against each other; the concentrations enter only through the logarithm, with a slope of 29.6 millivolts per decade because two electrons move for every ion. A factor of a hundred million in the ratio of zinc to copper ions moves the cell’s voltage by less than a quarter of a volt. That is why a battery’s voltage is a signature of its chemistry — 1.1 volts for a Daniell cell, 1.5 for an alkaline cell, 2.0 per cell for lead–acid, 3.2 to 4.2 for lithium cells — and only secondarily of its state of charge.

The temperature slope is the reaction’s entropy #

Because the voltage is a free energy per charge, it inherits everything thermodynamics says about free energy, and one consequence is easy to measure with nothing more than a voltmeter and a water bath. The free energy of a reaction is $\Delta G = \Delta H - T\Delta S$, so its slope with temperature is minus the reaction’s entropy change. Divide by the charge moved, and the slope of the cell’s open-circuit voltage with temperature is $\Delta S/nF$, where $F$ is the charge of a mole of electrons. A cell whose voltage rises when it is warmed is running a reaction that increases entropy; one whose voltage falls is running a reaction that decreases it.

For the zinc–copper cell the slope is small — a fraction of a millivolt per degree — because dissolving one metal and plating another leaves the disorder of the system nearly unchanged, and almost all of the 213 kilojoules per mole is enthalpy. For other reactions it is large enough to matter: it is the reason a lithium cell warms or cools slightly while it is charged slowly, over and above its resistive heating, since at a fixed voltage the reversible heat $T\Delta S$ has to go somewhere. Electrochemists use the slope the other way round, reading the entropy of a reaction off a voltage measured at two temperatures with a precision calorimetry struggles to match. A voltmeter, read carefully, is an instrument for measuring entropy.

A voltage is always measured against something #

The Nernst equation fixes a difference, and every voltage in electrochemistry is one. There is no instrument that reads the electrical potential of a single electrode, because connecting a voltmeter to a solution means putting a second metal in it, and the second metal has its own chemical potential step. Chemists solve this by agreement: the standard hydrogen electrode, a platinum surface in acid at unit activity under hydrogen gas at one bar, is assigned zero volts, and every other electrode’s standard potential is quoted against it. Zinc sits at −0.76 volts and copper at +0.34, and their difference is the 1.10 volts the figure starts from.

The same is true of the nerve cell. An electrophysiologist’s −73 millivolts is the potential inside the cell relative to the bath, measured with two matched electrodes whose own chemical steps cancel. What is physical is the difference in the electrochemical potential of an ion between two places, and that is what decides whether the ion moves.

The equation applies only when no current flows. A battery delivering current loses some of its voltage to the resistance of its electrolyte and to the extra push needed to drive the electrode reactions at a finite rate, and the energy it delivers is less than the free energy by exactly those losses, which appear as heat. The Nernst voltage is the ceiling; everything a real battery does is below it.

Where the voltage of a lithium cell comes from #

Modern rechargeable cells do not dissolve their electrodes. They store lithium ions in the gaps of a crystal lattice — graphite at one end, a metal oxide or phosphate at the other — and the voltage of each electrode is set by the chemical potential of the lithium in its lattice. How that chemical potential depends on how full the lattice is decides the shape of the battery’s discharge curve.

If the lithium ions in the lattice ignored each other, each site would behave like the single site of the site that fills like an electron level, filling according to the same function of the chemical potential that describes gas on a catalyst and electrons in a metal. Inverted, that function gives the voltage as $(kT/e)\ln((1-x)/x)$: a curve that falls steeply at the start and end and slowly in the middle, the discharge curve of a lithium cell with a sloping voltage.

Let the ions attract their neighbours, and the middle flattens. At an attraction of $2kT$ the slope at half-filling vanishes; beyond it the lattice would rather not be half full at all. It separates into a lithium-poor phase and a lithium-rich phase, coexisting side by side, and as the battery discharges, lithium moves from one phase to the other at a fixed chemical potential — a fixed voltage — until the poor phase is gone. The flat plateau between 2 per cent and 98 per cent filled in the drawing is the coexistence region of a first-order phase transition, exactly as a pot of water boiling at a fixed temperature is, in the heat that changes no temperature: energy is going in, and the intensive quantity that would normally measure it stays put.

A battery whose voltage barely changes as it discharges is one whose electrode is undergoing a phase transition. Lithium iron phosphate cells, with their famously flat 3.4 volt discharge, are the best-known example; the flatness that makes them easy to use is what makes their state of charge hard to read from their voltage.

A pH meter and a thermometer #

The same fifty-nine millivolts measure acidity. A glass electrode has a thin glass membrane that exchanges hydrogen ions with the solution on each side, and its voltage is the Nernst voltage of the hydrogen-ion ratio across the membrane: 59 millivolts per unit of pH at 25 degrees. A pH meter is a very high-impedance voltmeter reading a chemical potential. Because the slope is proportional to the absolute temperature, every pH meter has a temperature compensation built in, and a meter calibrated at room temperature and used on a sample at 37 degrees is wrong by four per cent unless it corrects for the change in $kT/e$.

That temperature dependence can be turned round. A concentration cell with a known ratio is a thermometer, reading the absolute temperature from a voltage with no reference to any property of any material; in practice other thermometers are more convenient, but the principle is the one the glow that carries a voltage uses in the other direction, where a voltage applied to a light-emitting diode sets the chemical potential of the photons it emits.

Where the Nernst voltage stops #

Dilute solutions. The chemical potential $kT\ln c$ is exact only when the ions do not interact. In real electrolytes, at concentrations of a tenth of a mole per litre and above, each ion is surrounded by a cloud of opposite charge that lowers its chemical potential; chemists write the concentration as an “activity” that absorbs the correction, and the Nernst equation is exact in activities and approximate in concentrations.

Equilibrium. Every voltage here is an open-circuit voltage, measured with no current flowing. The membrane potential of a living cell is a steady state rather than an equilibrium, maintained by pumps, and the Goldman–Hodgkin–Katz equation describes it only under assumptions about how ions cross the membrane.

A lattice gas with one interaction. The lithium electrode is modelled as sites with a single nearest-neighbour attraction treated in the simplest average way. Real electrodes have long-range elastic interactions, several kinds of site and ordered arrangements of their ions, and their voltage curves show steps and plateaux the model does not; the model captures why plateaux exist, not where any particular one falls.

One temperature. The electrochemical potential is flat only in thermal equilibrium. A temperature difference across a junction drives currents of its own — the thermoelectric effects of the second experiment that cannot disagree — and adds voltages this essay does not count.

The thin layer where the charge sits #

Every figure here is a voltage, and none shows where the charge that makes it sits. The 89 millivolts across a nerve cell’s membrane is held by an imbalance of about one ion in a hundred thousand, crowded into a layer a nanometre thick on either side of the membrane; the cell as a whole is neutral to that precision. The battery’s voltage sits across a double layer of charge at each electrode’s surface, a few ångströms thick, where the electric field reaches hundreds of millions of volts per metre. The figures show the potential difference and not the enormous, thin field that carries it, which is where the electrochemistry actually happens.

Still open: what limits a battery’s approach to its own voltage #

The open-circuit voltage is set by thermodynamics and is well understood. What a battery can deliver at a useful rate is set by how fast ions move through the electrodes and across their interfaces, and there the physics is much less settled. In phase-separating electrodes such as lithium iron phosphate, whether the new phase grows particle by particle, as a front through each particle, or by a more complicated path depends on the rate and on the particles’ size, and the path changes how much of the free energy is lost as heat. How to design electrode materials that keep most of their thermodynamic voltage at high current, and whether a solid electrolyte can carry ions fast enough to replace liquid ones, are among the most actively worked questions in materials physics.

The habit worth carrying away is to add the chemical potential before asking why something does not flow. Charged particles move down the slope of their electrochemical potential, not of their concentration or of the voltage alone, and a boundary can hold any voltage at all with nothing flowing, provided a concentration difference pays for it. Fifty-nine millivolts is the price of a factor of ten, and batteries, nerves and pH meters are all ways of charging it.