The explanation with the wrong sign

Put two methyl groups on one carbon of a chain and the chain closes to a ring faster. Very much faster: the standard measurements have gem-dimethyl accelerating the formation of a five-membered lactone by a factor of 250, and tetramethyl by 11,000.

The explanation in every textbook is an angle. Two methyls on one carbon push each other apart, so the angle between them opens; the four angles at a tetrahedral carbon are not independent, so the one opposite — the ring’s own internal angle — closes; and a smaller internal angle is what a small ring needs. It is a tidy argument, it has a picture, and it is called the Thorpe–Ingold effect. Hybridisation does not explain is the general form of the complaint made here: an account that reproduces a fact is not thereby the cause of it.

A bending term is available. Applying it to that argument gives the wrong sign for every ring the effect has been measured on.

The angle account, computed #

The model is the bending term and nothing else, the same one the angle a ring cannot have uses. Every carbon pays $k(\theta - \theta_0)^2$ for sitting at the ring’s geometric interior angle, and a substituted carbon’s own preference $\theta_0$ is reduced by a stated amount. The force constant is quoted; the shrink is quoted; and what is computed is a difference of two strains using the same two numbers, so most of the arbitrariness cancels.

The relief a substituted carbon provides is the difference between the unsubstituted ring’s angle strain and the substituted one’s. For a three-ring, whose interior angle is fifty degrees below the preference, shrinking one carbon’s preference moves it towards the ring: relief of 21.30 kJ/mol. For a four-ring, 7.55.

For a five-ring the interior angle is 108 degrees, which is already 1.47 below the tetrahedral preference. Shrinking a carbon’s preference moves it further from what the ring offers, and the relief is −0.70. For a six-ring, at 120 degrees, it is −6.20. For a seven-ring, −10.13.

Converted to a room-temperature factor, the account predicts 0.754 for the five-ring against a measured 250, and 0.082 for the six-ring against a measured 10. Not too small. The wrong way — and by a factor of three hundred and thirty for the five-ring, which is the ratio of the measured acceleration to the predicted retardation.

Why the sign is not negotiable #

There is no parameter that rescues it, and it is worth seeing why.

The whole effect is $k(\theta_{\text{ring}} - \theta_0 + \delta)^2 - k(\theta_{\text{ring}} - \theta_0)^2$, which expands to $k\delta(2(\theta_{\text{ring}} - \theta_0) + \delta)$. The sign is the sign of $\theta_{\text{ring}} - \theta_0$ — whether the ring’s interior angle is below or above the preference — and nothing else. A larger force constant scales it, a larger shrink scales it, and neither can flip it.

So the argument is not “the effect is small in this model”. It is: the mechanism, whatever its magnitude, helps the rings where the effect is not measured and hinders the rings where it is.

The torsional profile a rotation about a carbon–carbon bond runs over is a threefold barrier with three minima, and everything in the account that works is a statement about which of those minima the closure can use. That is the coordinate the argument lives in, and it is not an angle at any atom.

The account with the right sign #

The mechanism that does work is not about angles at all, and it is the one Bruice and Pandit proposed on kinetic grounds. It is a population argument rather than an energy one, which is the same move the conformer family makes for a medium ring.

A chain closes to a ring only from a conformation in which its two ends are near one another. A chain with three internal rotations has twenty-seven staggered arrangements to be in, and only the two in which every rotation is gauche and all of one sign curl it round. So the closable population is small, and anything that makes the extended arrangements expensive raises it.

Two methyls on a middle carbon do exactly that. They do not move a bond angle to speak of; what they do is meet the rest of the chain when the chain is extended, in the way the two ends of syn-pentane meet. The penalty is on the anti arrangements, and it pushes the population towards the gauche ones the ring needs.

The sign is right by construction, and the magnitude is of the right order for a penalty of the right size. That is two things the angle account has neither of.

And its ceiling #

The rotamer account is not the whole story either, and the honest way to say so is to compute its limit rather than to gesture at one.

However large the penalty, all it can do is drive every rotation gauche. Once every rotation is gauche there are still $2^3 = 8$ sign patterns and only two of them curl the chain the right way, so the closable population cannot exceed a quarter — and the factor cannot exceed a quarter divided by the plain chain’s 0.006858, which is 36.456.

The measured 250-fold acceleration for gem-dimethyl is seven times the ceiling. The 11,000-fold for tetramethyl is three hundred times it.

So the rotamer account has the right sign, produces the right order of magnitude for a modest penalty, and cannot reach the measurement. Something else is contributing — the transition state is not the ring, the substituents affect the entropy of activation as well as the population, and a real acceleration is a ratio of rate constants rather than of populations.

Why an exponential is doing the work #

There is one piece of arithmetic in the rotamer account that is worth separating out, because it explains the most striking number in the measurements.

Gem-dimethyl accelerates the five-ring closure by 250. Tetramethyl — two substituted carbons rather than one — accelerates it by 11,000. That is not twice, and it is not four times; it is forty-four times.

In the rotamer account that follows without adjustment. Two substituted carbons put the penalty on twice as many rotations, so the energy is doubled and the population is an exponential of it, which squares the factor. At a penalty of eight kJ/mol the model gives 28.0 for one substituted carbon and 36.1 for two — held down by the ceiling — but away from the ceiling the relation is a squaring.

In the angle account it does not follow at all. Two substituted carbons give twice the relief in kilojoules, which is again an exponential in the factor, so that half would also predict a squaring — and it predicts a squaring of a number below one, which makes the tetramethyl compound slower than the dimethyl by a factor of a hundred and eighty.

So the two accounts agree about the arithmetic of doubling and disagree about what is being doubled, and the measurement separates them by a factor of nearly two thousand.

What the substituent argument is about is not the closed ring but the open chain before it closes, and the difference between the two is a count of freedoms rather than a shape. A ring is a chain with one constraint added, and the constraint is what makes some of the chain’s cheap arrangements unavailable.

Which ring sizes can hold a double bond is the same shape of question — a geometric constraint priced against a computed term — and it has a sharp yes-or-no answer where this one has a balance. Keeping the two kinds of claim apart is most of the discipline here.

A five-membered ring at its geometric interior angle of a hundred and eight degrees is within a degree and a half of the tetrahedral preference, which is why its angle strain is 0.4 kJ/mol. Any account that attributes the five-ring’s behaviour to angle strain is attributing it to a term that is almost exactly zero.

What the two accounts have in common #

Both are one-term accounts of a many-term problem, and the terms can be assembled.

The angle a ring cannot have is the bending term. The strain that is not in the angles added the torsional term and found it dominant for cyclopentane — 26 kJ/mol measured against 0.4 of angle strain. The atoms that meet across a ring added the transannular term and found two conformers with identical angle and torsion strain and very different room across the ring.

The gem-dialkyl effect turns out to sit in the second of those, not the first — and the received explanation places it in the first. That is the same misattribution found for cyclopentane, arriving in a different question: the angle term is the one everybody reaches for and it is rarely the one that decides.

What survives, and what a chemist should say instead #

It would be a mistake to read all this as saying the effect is not real or not steric. It is both. What is wrong is the sentence that explains it.

The effect is a conformational one: substituents restrict where a chain can be, and restricting a chain to the arrangements that can close it makes closing more likely. That is a statement about entropy rather than about strain, and it is the reason the effect is largest for the flexible chains and smallest for the rigid ones.

The angle compression is also real — two methyls do open the angle between them, and the ring’s angle does close in response — and it is simply the wrong size and the wrong sign for the rings in question. In a three- or four-membered ring it would be a large help, and there the measurements are dominated by the ring strain itself and the substituent effect is not what anybody studies.

So the received explanation is a correct mechanism applied to the wrong rings, and it survives because both it and the true mechanism predict an acceleration for the compounds people actually make. Which angles are symmetry and which are the model is the same distinction in a neighbouring subject: two accounts agreeing on the observed cases and differing on the ones nobody has looked at.

What this cannot say #

There is no transition state. A rate is a property of a barrier and everything here is a property of a ground-state population or a ground-state strain. Converting either into a rate assumes the substituent affects the reactant and not the transition state, which is exactly the assumption a proper treatment would test. It is the same assumption, and the same gap, that separates a computed strain from a measured heat of formation throughout ring-strain arguments.

The penalty is a parameter. Its value is not computed; what is computed is the factor at a stated value and the ceiling that no value can pass. The ceiling is the part that carries the argument.

Three rotations is a choice. A chain closing to a five-ring has about that many, and a longer chain has more and a much larger ceiling. So the ceiling result applies to small-ring closures and not to macrocyclisation.

Nothing is compared with a computed rate. The measured factors are quoted from the lactonisation studies and the model produces populations and strains. What a lone pair is worth is the standing example of a knob fitted to observations rather than derived, and both parameters here are of that kind.

No conformer enumeration. The rotamer count is a combinatorial idealisation — three minima per rotation, all extended arrangements equally penalised — where a real chain’s minima are neither equally spaced nor equally deep. The ring that cannot hold still is where that is treated properly.

And the angle shrink is a parameter too, at six degrees. The sign of the angle account does not depend on it, which is the whole reason that half of the essay works without measuring it.

What was checked #

With no shrink there is no relief, at every ring size, to twelve decimal places — the tripwire that the effect is the substituent’s rather than the arithmetic’s.

The relief is positive below five and negative above six, and falls monotonically with ring size, which is the sign change the account turns on.

Two substituted carbons are worth twice one in energy and more than twice in factor, because one is linear and the other is an exponential.

The refusal: the angle account predicts a slowing for every ring where an acceleration is measured, checked case by case against the quoted factors — so the received explanation is refused by three comparisons rather than by an argument.

The rotamer account has no effect at no penalty, exactly, and rises monotonically with it.

And it has a ceiling, reached rather than assumed: driving the penalty to four hundred kJ/mol reproduces the arithmetic limit to a part in a million, and the measured five-ring acceleration is above it.

The closed ring is made worse, and by a ring-size-dependent amount #

Both accounts above are about the open chain: the angle account says the chain’s terminal angle is compressed, the rotamer account says fewer of its shapes are unclosable. Neither says anything about what the substituent does once the ring has closed, and it does something, because a closed ring has places to put a methyl group and they are not equivalent.

The measurement that settles the size of it is the one conformational analysis quotes most often. A methyl group on a cyclohexane ring prefers the equatorial position by about 7.3 kJ mol⁻¹ — its A-value — and the reason is two contacts across the ring, to the axial hydrogens three carbons away. Those are transannular contacts of exactly the kind computed for medium rings, and they are already measured.

A gem-dimethyl group cannot avoid them. Whichever chair the ring adopts, one of the two methyls is axial and the other equatorial, and flipping the ring exchanges them without helping. So relative to a ring carrying its substituents where they would rather be, gem-dimethyl costs one full A-value, and at 298 K that is a factor of

$$e^{7.28/2.479} = 18.9$$

against the closure rather than for it.

That is the term both accounts leave out, and it has the opposite sign to the effect they are explaining.

It is also strongly ring-size dependent, which is the useful part. The A-value is a six-ring quantity: it exists because a chair has a clean axial direction pointing at two specific hydrogens across the ring. Cyclopentane has no such direction — it pseudorotates, its substituent positions interconvert continuously, and the analogous penalty is a couple of kilojoules rather than seven, worth a factor of about three. A three- or four-ring has no transannular anything, by the counting argument for transannular contacts.

So the opposing term is negligible at three and four, small at five, and largest at six — which is the same ordering as the discrepancy the measurements show. The five-ring accelerates by 250 and the six-ring by 10, a ratio of twenty-five, and the difference in this one opposing term is worth about six.

That does not explain the ratio and it removes the need to explain all of it. The rotamer account supplies a ceiling that behaves one way with ring size; this term supplies a floor that behaves the other way; and the measured factor is what is left when both have acted. An account that computes only the open chain is computing a rate constant’s numerator and calling it the rate.

The general shape of the lesson is one that keeps recurring. A substituent changes both ends of an equilibrium, and an explanation that follows it into the starting material and stops there will get the magnitude wrong even when it gets the sign right — which is a milder failure than the angle account’s, and is a failure of the same kind.

Still open: the transannular penalty computed, and the ceiling’s rise #

The section above takes the term both accounts leave out as far as a measured A-value will carry it: the closed ring is made worse, by about seven kilojoules at six atoms and by much less at five. What it does not do is compute that penalty rather than quote it, and the method for computing it is the transannular contact sum, which need only a substituted ring to run on. The number to check against is the one quoted here.

The nearer question is the ceiling’s. A factor of 36.5 for three rotations becomes a much larger number for four, and the closable population falls as $2/2^r$ while the plain chain’s falls faster than that. So the ceiling rises with chain length, and the acceleration a substituent can produce should therefore be larger for a bigger ring — which is the opposite of what the measurements say, since the six-ring’s factor of 10 is smaller than the five-ring’s 250. Working out which of the two effects wins, and where they cross, would turn the ceiling from a limitation into a prediction.