The rate law is not the equation

How fast a reaction goes does not follow its balanced equation — and that mismatch is how mechanisms are discovered. The rate law reports on the slowest step alone, which means a flask and a clock can describe a molecular event nobody can see.

1850 – 193516 min

By the end you should be able to

Explain why a balanced equation cannot tell you the rate law, and what can
Identify a reaction order from data, using the plot that comes out straight
Say what a first-order and a second-order rate law each imply about the slow step
Describe how stereochemistry confirms a mechanism that kinetics only suggested

Here is a fact that ought to be startling and usually is not. You can balance an equation perfectly, account for every atom on both sides, know the products exactly — and still have no idea how fast the reaction goes, or what its speed depends on. 2 N₂O₅ → 4 NO₂ + O₂ is a complete and correct statement about matter. It is silent about time. It does not tell you whether the rate doubles when you double the N₂O₅, or quadruples, or does not move at all. And there is no way to get from one to the other. The equation is an accounting identity — the thing Lavoisier’s balance established. The rate law is a separate experimental fact about the same reaction, and it has to be measured. That sounds like a limitation. It is the opposite. It is the single most useful gift chemistry has ever been handed, and this lesson is about why.

The first person to measure a rate law properly was Ludwig Wilhelmy, in Heidelberg in 1850, and he chose his reaction for the instrument rather than for the chemistry. Cane sugar in dilute acid slowly turns into an equal mixture of glucose and fructose. Sugar solution rotates the plane of polarised light to the right; the mixture it becomes rotates it to the left. So the composition of the flask can be read off a polarimeter, continuously, without taking a sample, without quenching anything, without disturbing the reaction in the slightest. Wilhelmy watched the rotation swing from right to left and found that the rate at any moment was proportional to how much sugar was left. He wrote that down as a differential equation, integrated it, and got an exponential. It is the first rate law in the history of chemistry, and it was published in a physics journal, and it was almost entirely ignored for thirty years. Note what the instrument bought him: a continuous reading. Almost every advance in kinetics has been an advance in watching a reaction without stopping it.

The rate law reports the slowest step, and nothing else

Almost no reaction happens in one go. Most proceed through a sequence of elementary steps — a bond breaks, a fragment turns over, something collides with something else — and a sequence runs no faster than its slowest member. So the measured rate law is a statement about that one step. It names the species that have to be present, and in what numbers, to get over the highest hill. Everything that happens after that hill is fast, and therefore invisible to the stopwatch. Read that again, because it is the whole lesson. A measurement of a concentration in a flask, against a clock on the wall, tells you the composition of a single molecular arrangement that exists for perhaps a hundred femtoseconds and that nobody can see, isolate or weigh. That is a preposterous amount of information to get out of a stopwatch, and the reason it works is precisely the mismatch we started with. If the rate law simply followed the equation, it would carry no information at all — you would already know it. It is because the exponents are free to disagree with the coefficients that measuring them tells you something.

Take the readings and watch which of the three plots on the right comes out straight. Then wind "follow it for" down to half a half-life and try to decide again.

Loading the rate curves…

The diagnostic, as it is actually done. Each integrated rate law is a straight line in a different quantity — [A] for zero order, ln[A] for first, 1/[A] for second — so plotting all three and asking which is straight identifies the order without assuming it. The second half of the exercise matters more than the first: stop the run early, or add a per cent of scatter, and all three fit beautifully. Kinetics is not hard because the mathematics is hard. It is hard because you have to follow the reaction a long way.

Once you can measure an order without assuming one, the mismatches start arriving, and they are not rare curiosities. They are the normal case. 2 N₂O₅ → 4 NO₂ + O₂. Two molecules of N₂O₅ on the left. The rate is k[N₂O₅] — first order. One molecule falls apart on its own in the slow step, and the second one is consumed later, in a fast step the clock cannot see. 2 NO₂ + F₂ → 2 NO₂F. Two molecules of NO₂ on the left. The rate is k[NO₂][F₂] — first order in NO₂. The slow step is one NO₂ meeting the F₂ and taking one fluorine atom off it; the loose fluorine atom then finds a second NO₂ almost instantly. H₂ + Br₂ → 2 HBr. A one-to-one equation, and one of the ugliest rate laws in chemistry. Bodenstein and Lind measured it in 1907 and found the rate goes as [H₂] times the square root of [Br₂], all divided by a term containing [HBr] over [Br₂]. A square root. And the product in the denominator, so the reaction slows itself down as it proceeds. No equation could suggest a square root. That fractional order is the fingerprint of a molecule splitting into two halves that then do the work — a chain reaction — and it took until 1919 for anyone to say so. The rate law arrived first, and the mechanism was reverse-engineered from it.

You might think

The exponents in the rate law are the coefficients in the balanced equation, so 2 A + B → P must have rate = k[A]²[B].

Actually

That rule is true for exactly one kind of equation: an elementary step, one that describes a single molecular event as written. There, the equation really is the collision, and the number of each species that must arrive really is the power. For an overall equation — which is a sum of steps, most of them invisible — the exponents are experimental quantities with no obligation to the coefficients whatsoever. They can be smaller than the coefficients (2 N₂O₅ decomposes first order). They can be fractional (the square root in H₂ + Br₂). They can be negative, so that adding a substance slows the reaction down — 2 O₃ → 3 O₂ has [O₂] in the denominator, meaning oxygen inhibits its own formation. And the rate law can contain a species that does not appear in the equation at all, which is exactly what a catalyst is. If you could read the exponents off the coefficients, kinetics would be a branch of arithmetic and would have told us nothing about mechanism. The disagreement is the data.

Now the cleanest case in the subject, and the reason this lesson has a date on it. Take a carbon atom carrying a halogen — a leaving group — and offer it something that wants to bond to carbon: a nucleophile. The halogen goes, the nucleophile takes its place. One product, one balanced equation. There are two ways that can happen. Either the halogen leaves first, on its own. The carbon is briefly left with only three bonds and a positive charge, flat as a starfish, and the nucleophile arrives afterwards. Two steps, and the first one is the slow one, and only one species is involved in it. Or the nucleophile pushes in from behind at the same moment the halogen departs. One step. Both partners must meet, so both must be in the rate law. The equation is identical. The products are identical. Everything you can put on a piece of paper about what went in and what came out is identical. So you cannot argue your way to the answer. You have to measure something.

One of the two routes is running in this flask and you are not told which. Double the substrate; then double the nucleophile. Decide before you press reveal.

Loading the rate curves…

Doubling the substrate doubles the rate — which settles nothing, because both mechanisms predict it. Everything hangs on the second measurement. If doubling the nucleophile leaves the rate exactly where it was, the nucleophile is absent from the slow step, and the only way that happens is if the C–X bond breaks first, by itself. The decisive reading, once again, is the one where nothing moves.

The experiment

Separating two mechanisms by kinetics alone

Edward D. Hughes and Christopher Kelk Ingold, with a large group of research students · 1933 – 1935, in a series of papers in the Journal of the Chemical Society · University College London

The question: A nucleophile replaces a halogen on a saturated carbon. Is that one molecular event or two? And can a stopwatch tell the difference?
The apparatus: Alkyl halides dissolved in mixed water–alcohol solvents, with the halide ion released titrated against time to follow the reaction. The nucleophile’s concentration was then varied independently by adding a salt of it. Crucially, the total ionic strength was held constant with a spectator salt, because ionic strength changes reaction rates all by itself — the salt effect would otherwise have been mistaken for a dependence on the nucleophile.

Theory predicted

If there is one mechanism, every substrate should give the same kind of rate law. Under the one-step picture, everything is second order: rate = k[RX][Nu]. Under the two-step picture, everything is first order: rate = k[RX], with the nucleophile irrelevant.

They measured

Neither. Tertiary halides gave clean first-order kinetics, and adding nucleophile did not change the rate at all. Primary halides gave clean second-order kinetics, first order in each. The same overall transformation was going by two different routes, and which route depended on the structure of the substrate.

How sure could they be? The discriminating measurement is a null one: the rate of the tertiary substrate does not move when the nucleophile concentration is raised several-fold. That is why the ionic-strength control matters so much — a spurious salt effect of a few per cent would have looked like a weak dependence on the nucleophile, and turned a clean answer into a muddy one.

Why it mattered

Ingold named them: S_N1 for substitution, nucleophilic, unimolecular — one species in the rate-determining step — and S_N2 for the bimolecular one. The names are about the slow step, not about the number of steps in the reaction, which is a distinction almost every textbook blurs. And note what has just been achieved: two molecular routes have been told apart by a titration, without anyone seeing anything.

And now the part that makes the case airtight, because it arrives from a completely different instrument and has nothing to do with speed. Think about the shape each route demands. If the halogen leaves first, the carbon it leaves behind has three bonds and a vacancy, and it flattens: the three remaining groups spread into a plane. A flat centre has two identical faces. The nucleophile can arrive at either with equal ease. So if you start with a single mirror image of the molecule, you must finish with both, in equal amounts — a mixture that rotates polarised light not at all. Optically dead. If instead the nucleophile pushes in as the halogen departs, it can only come in on the side the halogen is not. It arrives at the back, the other three bonds sweep past the carbon like an umbrella turning inside out, and the product is the opposite mirror image of the starting material. Every time. No exceptions. Paul Walden had found exactly that inversion in 1896, in Riga, and could not explain it. He converted one enantiomer of chlorosuccinic acid into malic acid, and then back again by a different route, and ended up with the mirror image of what he started with. A closed cycle that returns you to the wrong hand. It was a famous puzzle for nearly forty years before it turned out to be a photograph of a mechanism.

The experiment

Racemisation at exactly twice the rate of exchange

E. D. Hughes, F. Juliusburger, S. Masterman, B. Topley and J. Weiss · 1935 · University College London

The question: Does every bimolecular substitution invert the centre, or only some of them? A rate law cannot answer that. A ratio can.
The apparatus: Optically active 2-iodooctane, dissolved in dry acetone, with radioactive iodide ions added as a salt. The iodide swapping onto the molecule is chemically identical to the one leaving, so the product is the same compound — nothing changes except which atoms are where. Two instruments then watch the same flask: a counter following radioactivity appearing in the organic material, which counts substitutions, and a polarimeter following the rotation, which counts loss of handedness.

Theory predicted

Three possibilities, and they give three different numbers. If every substitution inverts, each event converts one molecule of one hand into one of the other, which destroys two units of excess — so racemisation runs at twice the exchange rate. If substitution goes half by inversion and half by retention, the ratio is one. If it always retains, the exchange is invisible to the polarimeter and the ratio is zero.

They measured

The rate of racemisation was twice the rate of exchange, within experimental error.

How sure could they be? A ratio is a far more robust thing to measure than either rate on its own. Errors in the temperature, the concentration, the purity of the sample and the calibration of the instruments largely cancel, because both clocks are watching the same flask at the same moment. The experiment does not need either rate constant to be accurate. It needs their ratio to be two rather than one or zero, and those are not close together.

Why it mattered

Every single bimolecular substitution event inverts the centre. Not most of them; all of them. That is a statement about the geometry of an individual molecular collision, established by a Geiger counter and a polarimeter pointed at a beaker. And it agrees with the kinetics, which was arrived at by a completely unrelated measurement. Two independent methods that could easily have disagreed, and did not — which is the only kind of evidence that ever really settles anything.

The part the story usually rounds off

Textbooks say S_N1 gives a racemic product. Measured, it usually does not — it gives a slight excess of inversion, commonly ten to twenty per cent, and sometimes much more. The reason is worth knowing, because it is the mechanism refusing to be as tidy as the diagram. The halide that has just left does not instantly vanish into the solvent; for a moment it sits beside the flat carbon as an ion pair, screening the face it departed from. During that moment the nucleophile finds the other face easier to reach. Wait longer and the halide diffuses away and the excess disappears — which is why the effect is bigger in solvents that separate ions slowly. The larger caveat is that S_N1 and S_N2 are not two boxes. They are the ends of a continuous range, and secondary carbons sit in the middle, where the leaving group is partly gone and the nucleophile partly arrived in the same transition state. Arguments about the borderline ran for decades; in the 1960s and 70s Richard Sneen proposed abandoning the distinction altogether in favour of a single ion-pair scheme, and the proposal was resisted, and something like a truce was reached in which both descriptions are limiting cases of one picture. The clean cases really are clean. A tertiary halide in water is S_N1 and a methyl halide with a strong nucleophile is S_N2, and no honest data disputes either. The middle is genuinely difficult, and a lesson that pretends otherwise is teaching a filing system rather than chemistry.

Problem

How fast does the rotation die?

A run like Hughes’s: optically active 2-iodooctane in acetone, with enough iodide present that the exchange follows a simple first-order decay. The exchange rate constant is measured at 3.0 × 10⁻⁴ s⁻¹, and every substitution inverts the centre. How long does the sample’s optical rotation take to fall to half its starting value?

Exchange rate constant: k_ex = 3.0 × 10⁻⁴ s⁻¹
Stereochemical outcome: inversion at every event
Order of the exchange: first

1850Wilhelmy follows sugar inversion with a polarimeter and writes the first rate law. It is ignored for thirty years.
1884Van ’t Hoff’s Études de dynamique chimique makes "order of reaction" an empirical quantity in its own right, separate from stoichiometry.
1896Walden closes a cycle of substitutions and ends up with the mirror image of what he started with. Nobody can explain it.
1907Bodenstein and Lind measure the H₂ + Br₂ rate law and find a square root, and the product inhibiting the reaction.
1919Christiansen, Herzfeld and Polanyi independently explain that square root: a chain reaction started by bromine molecules splitting.
1933Hughes and Ingold begin the series of papers that will separate the two substitution mechanisms by kinetics.
1935The radio-iodide experiment: racemisation at twice the rate of exchange, so every bimolecular substitution inverts.
1937Ingold’s nomenclature — S_N1, S_N2, E1, E2 — is in general use. It is still in use.

Where this leaves us

The balanced equation tells you what is conserved. The rate law tells you what has to be present at the top of the hill — and the two are free to disagree, which is precisely why measuring the second one is worth doing. Out of that gap came something nobody could have expected: a description of a single molecular event, arrived at by titrating a flask and watching a needle. The kinetics said the leaving group goes first. The polarimeter, from a completely unrelated direction, said the same thing. Two measurements with nothing in common agreeing is the only sort of evidence that ever really closes an argument. And all of it now hangs on one quantity: the height of one hill. Everything about how fast a reaction goes is the height of the barrier in its slowest step. Which raises a question with rather a lot of money attached to it. If you could lower that hill — offer the reaction a different route over the same ground — it would go faster. Perhaps enormously faster, because the height sits inside an exponential. Something that does exactly that, and walks away from the reaction unchanged, had been on sale since 1823. A German chemist sold a lamp that lit itself on a sponge of platinum, and nobody could say why it worked.