The reaction that cannot go all the way

Chemistry textbooks sort reactions into three kinds: those that go to completion, those that do not go at all, and those that reach an equilibrium somewhere in between. The first two kinds do not exist. Every reaction, left long enough in a closed vessel at a fixed temperature, stops with some of its reactants unconverted and some of its products made, and the only question is how much of each.

That is not a statement about slow kinetics or imperfect experiments. A reaction with a standard Gibbs energy of −818 kilojoules per mole — the burning of methane in oxygen — has an equilibrium constant of about $10^{143}$, and the methane left over is a number so small that no vessel in the observable universe could be expected to hold a single molecule of it. It is still not zero, and the reason it is not zero is the same reason a reaction with a positive standard Gibbs energy always makes some product. Both come from a logarithm, and the logarithm comes from counting.

The free energy of a half-finished reaction #

A system at fixed temperature and pressure settles where its Gibbs energy is least. For a reaction, the thing that can vary is how far it has gone — the extent, running from all reactant to all product — and so the question is what the Gibbs energy of a partly reacted mixture looks like as a function of that extent.

Take the simplest reaction there is: one molecule A turning into one molecule B, both ideal, at 298 K. If A and B did not mix — if the product accumulated in a separate compartment — the Gibbs energy would be a straight line from pure A to pure B, rising or falling by the standard reaction Gibbs energy $\Delta G^\circ$. A straight line has its minimum at an end, so the reaction would go all the way or not at all.

They do mix, and mixing adds the term that the entropy of mixing contributes, $RT[\xi\ln\xi + (1-\xi)\ln(1-\xi)]$, which is negative everywhere between the ends and zero at both. Added to the straight lines, it turns each into a curve with its lowest point strictly inside. At $\Delta G^\circ = -4$ kilojoules per mole the minimum is at 83.4 per cent B; at zero it is exactly half; at $+4$ it is at 16.6 per cent.

The right-hand panel is the case that matters most. A reaction with $\Delta G^\circ = +10$ kilojoules per mole has a straight line climbing steeply from pure A, and every intuition about energy says it should not start. Magnified near the pure end, its actual Gibbs energy falls first — to a minimum 43 joules per mole below pure A, at 1.74 per cent converted — and only then climbs. The dip is small in energy and it is not negotiable: the minimum sits where the ratio of B to A equals $e^{-\Delta G^\circ/RT}$, which is the equilibrium constant, and the search that located each minimum found that ratio to five figures.

The reason the curve must dip is the shape of $\xi\ln\xi$ near zero. Its value goes to zero there, but its slope, $\ln\xi + 1$, goes to minus infinity. Near a pure end the mixing term falls infinitely steeply, and no finite straight-line slope can overcome an infinite one. So no finite standard Gibbs energy can hold a reaction at nothing converted, and by the same argument at the other end, none can drive it to everything converted.

The slope is the chemical potential #

The slope of the Gibbs energy with respect to the amount of a substance is that substance’s chemical potential, $\mu$. For the reaction A ⇌ B, the slope of the curve above with respect to the extent is $\mu_B - \mu_A$: the Gibbs energy gained by turning one more mole of A into B at the present composition.

For an ideal mixture each chemical potential has the same form, $\mu = \mu^\circ + RT\ln x$, with $x$ the substance’s mole fraction. The difference is therefore $\Delta G^\circ + RT\ln[\xi/(1-\xi)]$, the dashed constant plus a logarithm of the ratio of product to reactant. Where that is negative, converting more A lowers the Gibbs energy and the reaction proceeds; where positive, it reverses; where zero, it stops. At a thousandth converted the logarithm is worth −17.1 kilojoules per mole, already more than the +10 it has to beat, and at none converted it is minus infinity.

This is the sense in which the chemical potential is to matter what temperature is to heat. Heat flows from a higher temperature to a lower one and stops when they are equal. Molecules move — across a membrane, between phases, from reactant to product — from a higher chemical potential to a lower one, and stop when the potentials balance. The condition for a reaction at equilibrium, $\sum \nu_i \mu_i = 0$ with the stoichiometric coefficients as weights, is exactly that statement, and the equilibrium constant is what it becomes once each $\mu$ is written out.

The analogy reaches further than the word suggests. The pressure that comes from counting across an osmotic membrane is water moving to where its chemical potential is lower, stopped by a pressure that raises it back. A gas that flows towards more of itself is doing what chemical potential requires rather than what concentration suggests. And the temperature of a boiling liquid depends on the pressure above it because liquid and vapour must have equal chemical potentials, and the vapour’s rises with the logarithm of its pressure.

Where the logarithm comes from #

The mixing term looks like a formula to be memorised. It is a count, and the cleanest way to see it is to leave the formula out.

Take N molecules. Any particular set of n of them can be the ones that have turned into B, and there are $\binom{N}{n}$ such sets. Each molecule that has converted carries a Boltzmann factor $e^{-\Delta G^\circ/RT}$ for doing so. The probability of finding exactly n converted is then the number of ways times the weight, normalised — and nothing else goes in.

Every one of the distributions is centred exactly on 23.0 per cent converted, the minimum of the smooth Gibbs curve for this reaction, and its width is $\sqrt{x(1-x)/N}$ — 9.4 per cent for twenty molecules, 3.0 per cent for two hundred, under one per cent for two thousand. The distributions were computed from the binomial count directly, and the smooth curve was never consulted.

The logarithm of the binomial count, divided by the number of molecules, is $-[\xi\ln\xi + (1-\xi)\ln(1-\xi)]$ in the limit of large numbers. That is the mixing term, arrived at as the logarithm of the number of ways to choose. It is entropy as a count applied to a single question — which molecules are the reacted ones — and its infinite slope at the ends is the statement that when almost none have reacted, converting one more increases the number of choices by a factor of order N, while the energy cost stays fixed.

Two things follow that the smooth curve hides. For a small system, equilibrium is a distribution rather than a composition, and a single enzyme or a nanoparticle of reacting material does fluctuate across the range drawn. And for a mole, where the width is a part in $10^{12}$, the distribution is so sharp that its centre is all that survives — which is why a smooth free energy with a minimum is the right description of a beaker and the wrong one for a cell compartment holding forty copies of a molecule.

How wide “partway” is #

The equilibrium fraction converted for A ⇌ B is $1/(1 + e^{\Delta G^\circ/RT})$, a smooth step from one to zero as $\Delta G^\circ$ runs from negative to positive. How sharp that step is decides which reactions look complete, which look absent, and which look like equilibria.

At room temperature the fraction falls from 99 to 1 per cent across a window 22.8 kilojoules per mole wide, and the window is $2RT\ln 99$, so it doubles at 600 K and doubles again at 1200 K. For comparison, a covalent bond is worth several hundred kilojoules per mole and a hydrogen bond about twenty. Reactions that make or break strong bonds almost always fall far outside the window, and are the ones called complete or impossible. Reactions whose products and reactants are nearly balanced in energy — isomerisations, the binding of a drug to a protein, the pairing of two strands of DNA near their melting temperature — fall inside it, and are the ones in which an equilibrium is visible.

Life is organised around that window. The hydrolysis of ATP has a standard Gibbs energy of about −30 kilojoules per mole, which on its own would leave a mixture overwhelmingly hydrolysed. A working cell holds the ratio of ATP to its products roughly ten orders of magnitude away from that equilibrium, so that the actual Gibbs energy released per mole is nearly twice the standard value. The reaction’s usefulness to the cell is precisely its distance from the minimum of the curve drawn above, maintained by constant work — and a cell whose ATP reached equilibrium would be dead.

The same reaction goes further with room #

Everything so far has been one molecule becoming one molecule, for which the composition alone decides the chemical potentials. When a reaction changes the number of molecules, the space available enters too, because the chemical potential of a gas depends on the logarithm of its own partial pressure, $\mu = \mu^\circ + RT\ln(p/p^\circ)$.

Dinitrogen tetroxide falls apart into two molecules of brown nitrogen dioxide with a standard Gibbs energy of +4.77 kilojoules per mole at room temperature. At a total pressure of one bar, 18.8 per cent of it is dissociated. Compressed to ten bar, 6.0 per cent; expanded to a hundredth of a bar, 88.6 per cent. The molecules and their energies have not changed at any point on the curve.

The reason is that compression raises every partial pressure by the same factor, and the product side has two molecules for every one on the reactant side, so its chemical potential rises by twice the logarithm while the reactant’s rises by once. The balance shifts back towards fewer molecules. That is Le Chatelier’s principle, which is usually stated as a tendency to oppose a disturbance and is here a statement about logarithms with coefficients — and the equilibrium constant, which is fixed, is satisfied at every point on the curve by a different composition.

The last trace of anything costs the most to remove #

The infinite slope at the ends has a second consequence, and it runs in the direction chemists and engineers feel most. If no reaction can go all the way, no separation can either: the chemical potential of a substance present as a trace, $\mu^\circ + RT\ln x$, falls without limit as its fraction $x$ goes to zero. A molecule of impurity dissolved in a pure solvent is in a state of extremely low chemical potential, and pulling it out means raising it back, at a cost per molecule that grows as the logarithm of how dilute it has become.

Written as work per mole of impurity removed, the cost is $RT\ln(1/x)$. At room temperature that is 11 kilojoules per mole to take a substance from a part in a hundred, 34 kilojoules per mole from a part in a million, and 57 kilojoules per mole from a part in ten thousand million. Each factor of a thousand in purity costs another 17 kilojoules for every mole of impurity still to be extracted, and the total never converges to a finite figure for perfect purity. The cost is set by counting alone, before any membrane, column or furnace is chosen, which is the same argument that fixes the minimum work of taking the salt out of seawater.

That is why purity is always quoted as a number of nines. Semiconductor silicon at eleven nines — one foreign atom in a hundred billion — is not a triumph over a technical obstacle that a better process would remove; it is a point on a curve whose next point costs more for every atom, for a reason that has nothing to do with silicon. Zone refining, which melts a narrow band and sweeps it along a bar so that impurities preferring the liquid are carried to one end, works because it exploits a small difference in chemical potential between two phases many times over rather than trying to overcome the logarithm in a single step.

The same slope explains a familiar observation from the other side. Drop a crystal of anything into a solvent in which it is almost insoluble, and some of it still dissolves, because the first few dissolved molecules are at a chemical potential that heads to minus infinity and more than repays the energy of leaving the crystal. The solubility is where the dissolved chemical potential rises to meet the solid’s, and it is never exactly zero. Water that has stood in a gold cup contains gold.

What the ideal picture assumes #

The mixtures are ideal. The logarithm of the mole fraction is the whole of the composition dependence only when A and B interact with each other exactly as each does with itself. In real solutions the chemical potential carries an activity coefficient, which can be far from one in concentrated electrolytes or mixtures of dissimilar liquids, and it can reverse the conclusion of this essay in one important case: if unlike molecules repel strongly enough, the mixture separates into two phases, and the logarithm then decides the composition of each phase rather than forcing a single one.

Equilibrium is assumed to be reached. A thermodynamic minimum says where a reaction would stop, not whether it gets there. Diamond is not graphite at room temperature because a barrier has to be climbed and the rate of climbing is set by an exponential in the barrier’s height. Most of practical chemistry is about reactions held away from their minima by barriers, and catalysts change how fast a minimum is approached without moving it.

The standard Gibbs energy is a function of temperature. Every curve is drawn at one temperature. How the minimum moves as the temperature changes is governed by the reaction’s enthalpy, through the van 't Hoff relation, and a reaction that sits at one end of the window when cold can cross it when hot.

The open vessel that never reaches its minimum #

Each curve is a free energy at one fixed composition of the surroundings — a closed vessel, one temperature, one pressure. What they cannot show is a reaction in an open system, fed with reactants and drained of products, where the composition never reaches the minimum at all and the rate is set by how fast the system is driven. That is the condition of every living cell, every industrial reactor and most of the atmosphere, and it is described by the distance from the minimum rather than by the minimum.

Nor do the pictures show time. The count of molecules converted is drawn as a probability, and a single sample of twenty molecules wanders through that distribution one reaction at a time. How fast it wanders, and how long a fluctuation to all-B would take to appear, are questions about rates.

What has a chemical potential of nothing #

Every chemical potential so far has been a number set by what a substance is and how much of it is present. One kind of particle ordinarily has none: a photon in thermal radiation. The gas that nobody counted takes as many photons as the temperature says, because the walls create and destroy them freely, and a particle whose number is free has its chemical potential forced to zero by exactly the argument of this essay — the Gibbs energy is least with respect to a number that nothing holds fixed.

What happens when something does hold the number of photons fixed, or pushes it away from the thermal value, is the question the glow that carries a voltage takes up. A light-emitting diode is such a device, and its light has a chemical potential set by the voltage across it.

The habit worth carrying away is to ask what a minimum is a minimum of. An energy alone has its minimum at an end; an energy with an entropy of mixing has it strictly inside, and the logarithm that moves it there is a count of arrangements, not a property of any particular substance. The same logarithm that keeps a combustion from ever being complete keeps a reaction uphill in energy from ever being absent.