An estimate that can be wrong by two

The rotamer account’s ceiling is a prediction rather than a limitation. The largest acceleration a gem-dimethyl group can buy through rotamer statistics alone is

$$\left(\frac{1 + 2x}{2x}\right)^{r}, \qquad x = e^{-g/kT},$$

with $r$ the number of internal rotations the closure has to freeze — 36.46 for three and 120.88 for four at butane’s gauche energy. The five-membered closure’s measured 250-fold acceleration is above its ceiling and the six-membered closure’s tenfold is well below, so the account is refuted where the effect is large and comfortable where it is small.

It then named the weakness in its own argument, and it is a real one. The count of $r = n - 2$ was taken as read. It is a reasonable estimate of how many rotations a closure freezes and it is not a calculation, and the ceiling is exponential in it: one rotor is a factor of 3.33, which is half the factor of 6.9 by which the five-ring’s measurement exceeds its ceiling.

So the refusal was probable rather than airtight. It can be made airtight, and the way to do so is not by counting better — which is the same move a fitted exponent needs, where the useful question turned out to be how far the answer moves with the window rather than what the answer is.

The question worth asking instead #

Counting the rotations a particular lactonisation freezes is a chemistry problem with a defensible answer and an arguable one, and settling it would produce a number that somebody could argue with. The question that cannot be argued with is:

How far would the count have to be wrong before the verdict changed?

That is an integer, it takes one sweep, and it needs no chemistry at all.

The five-membered closure’s nominal count is three, and the ceiling does not reach 250 until five. So the account survives only if a closure to a five-membered ring freezes five internal rotations — two more than the ring has bonds to make, on a tether whose whole length is four bonds.

The refusal survives an error of two rotors, which is not an error anybody could make. The explanation with the wrong sign — the other account of the same effect — was refuted by a sign rather than by a size, and between them the two refutations leave the five-membered case with no explanation of either kind.

The six-membered closure’s nominal count is four and its measurement is tenfold. The ceiling stays above ten down to two rotors, so the sufficiency survives losing two of the four. Both verdicts are robust and neither is robust for the same reason: one has a large margin because the measurement is far above the ceiling in the wrong direction, and the other because it is far below in the right one.

The measurement that was sitting there #

The table of gem-dimethyl accelerations has three rows and only two have been used. The third is the same closure with four methyl groups instead of two, at 11,000-fold.

The ceiling has no substituent in it. That is the first thing to notice about the closed form and it is worth restating: the substituent’s size, its shape and the size of the penalty it imposes have all gone, because the ceiling is what happens when the penalty is large enough that its value stops mattering. So a tetramethyl closure to a five-membered ring is measured against exactly the same 36.46.

It exceeds it by a factor of 302, and the ceiling does not reach 11,000 until eight rotors — five more than the closure has. The tetramethyl measurement is a second refutation, on the same tether, at a margin two and a half times the first.

That it went unused is worth noticing. It sat in the same table as the two numbers argued over at length, and nothing about it needed computing that the others did not; it is simply a larger version of the same statement, and a larger version is easy to overlook when the smaller one is already the finding.

The other variable, and the amount it would need #

The gauche energy is butane’s — quoted, and not the tether’s, which is the second caution about the ceiling. A substituted chain’s gauche interactions are not butane’s.

The ceiling rises with the gauche energy, so there is an energy at which the five-membered case would be rescued, and it is found by bisection rather than argued about:

$$g_{\text{needed}} = 5.85 \text{ kJ/mol}, \qquad \text{against butane's } 3.8.$$

Half again as large. A hydrocarbon torsion does not have a gauche energy of 5.85 kJ/mol — the largest ordinary values are around 3 to 4, and a value near 6 belongs to a chain carrying substituents so large that the rotamer picture is not the model to use.

And the direction matters. A larger gauche energy is what would help, and a substituted chain’s is expected to be larger — but a larger $x$ lowers the ceiling, which moves the conclusion the way it already points. The bisection above is the honest version of that: the energy would have to move by 54 per cent, and the sign of the plausible move is the wrong one.

What an ester tether actually does #

Both measured accelerations are lactonisations. A lactonisation closes a ring containing an ester, and the tether carries an oxygen — so the assumption that every bond frozen is a free three-state rotor is not automatic.

The interesting part is which way the correction goes, and it is the opposite of what a reader expects from hindered.

A hindered rotor is not a rotation the closure has to freeze. An ester linkage sits in one conformer, its alternative being some 20 kJ/mol higher and unpopulated; the open chain is already confined along that coordinate, so the closure gains nothing by confining it. Removing it from the count lowers $r$, and lowering $r$ lowers the ceiling.

No amount of hindering rescues the five-membered case — every count makes it worse. And the six-membered case survives two of its four being locked, at 10.99 against a measured 10, which is a margin of nine per cent and is the tightest number here.

The one place the argument is tight #

That nine per cent is worth dwelling on, because everything else here has a factor in it.

If a δ-valerolactone closure has three of its four frozen rotations hindered rather than free, its ceiling is 3.32 and the account fails on the six-membered ring too — at which point there is no ring size at which the rotamer explanation works, and the pattern of a refutation beside a success becomes a uniform failure rather than a crossing.

So the whole of the ceiling’s finding — that the account is refuted where the effect is large and sufficient where it is small — rests on the six-membered lactonisation having at least two genuinely free rotations in its tether. A five-atom chain from a hydroxyl to a carboxyl has three carbon–carbon bonds and one carbon–oxygen bond; three of the four are ordinary hydrocarbon torsions.

That is the assumption, it is stated, and it is the only one here with less than a factor of three behind it.

What a kineticist should take from it #

A margin is worth more than a better estimate, in the same way that a tie refutes a model without any fitting. The count could have been refined by an afternoon of thinking about tethers, producing a number somebody could dispute; the margin took one sweep and cannot be disputed, because it says how much dispute the conclusion can absorb.

The exponential is what makes the margin computable, and it is the same property a rotamer ceiling has as a prediction rather than a bound. A quantity that goes as a power of an integer has a verdict that changes at an integer, so how wrong could the integer be has an answer that is itself an integer. A model with a smooth dependence on a fitted parameter has no such statement.

And a hindered coordinate does not help. The instinct is that anything reducing conformational freedom aids a closure; the arithmetic is that a coordinate already confined is not one the closure gets credit for confining. The gem-dimethyl group’s whole mechanism is taking freedom away that the chain had, so a chain that never had it gains nothing.

Four assumptions, and what each is worth #

The whole conclusion about the gem-dimethyl effect now rests on a short list, and each item has a number beside it.

None of them is a factor smaller than about three, and one of them is nine per cent. That is the honest summary, and it is a more useful thing to hand a reader than a refined count would be — a count comes with an error bar somebody has to believe, and a margin comes with an integer.

The one at nine per cent is the six-membered lactonisation’s tether, and it is the only place the conclusion could still be overturned by a fact about a molecule — the kind of single tight number a measurement’s own precision decides rather than an argument. Everything else would need an error nobody could make.

That asymmetry is itself worth carrying: the refutation is safe and the sufficiency is not. A reader who wants to attack the argument should attack the claim that the rotamer account works for the six-membered ring, not the claim that it fails for the five — which is the opposite of where the argument feels vulnerable, since the failure is the surprising half.

What is quoted, and what is computed #

Four things are quoted and all four are measurements: the butane gauche energy of 3.8 kJ/mol, the two gem-dimethyl accelerations of 250-fold and tenfold, and the temperature.

Everything else is computed: the ceiling in closed form, the sweep across rotor counts, the bisection for the gauche energy that would reproduce the measurement, and the hindered-tether counts. The tetramethyl acceleration of 11,000-fold is the third quoted measurement and is the one not used until now.

The closed form is checked against an enumeration over $3^r$ conformations at both nominal counts, and the two agree to about $2 \times 10^{-10}$. That check is why the sweep can use the closed form alone: the enumeration at ten rotors is sixty thousand states and the closed form is three operations, and the sweep runs to eleven.

What this cannot say #

The rotor count is still not calculated. What is established is how far it could be wrong, not what it is. A closure through a tether nobody has drawn could freeze any number of rotations, and the margins above are what protect the conclusion rather than a substitute for the count.

There is still no transition state. Everything here is the population of closable conformations, which is a ground-state statistic, and a rate is about a barrier — the boundary every argument about populations works inside and one not crossed here.

The transannular term is still exactly zero here. A ring of five, six or seven has no pair of atoms four bonds apart, so the objection about crowding the closed ring cannot apply to either measurement — a counting fact with no parameter in it, and unaffected by anything here.

And the ester’s hindering is treated as complete. A rotor is either free with three states or removed entirely; a real ester linkage has a small population of its higher conformer, so the truth lies between two of the rows above. That interpolation would move the six-membered case’s 10.99 upwards, which is the direction that helps the only tight number here.

What was checked #

The account is refused at five rings and sufficient at six, at the nominal counts — the ceiling’s finding, reproduced as the starting point rather than assumed.

The refusal at five survives an error of one rotor and of two, checked as an integer margin.

The sufficiency at six survives the loss of two, likewise.

The closed form and the enumeration agree at both nominal counts, which is what licenses using the closed form alone across the sweep.

The gauche energy would have to be more than 1.4 times butane’s to rescue the five-membered case, computed by bisection rather than estimated.

And the tetramethyl measurement on the same closure is refused by a wider margin still — five more rotors rather than two — checked as a comparison between the two margins, because a second refutation that was not wider would mean the ceiling was somehow responding to the substituent, which it cannot.

And hindering a rotor lowers the ceiling at every closure — checked as a monotone sequence, because the direction is the counter-intuitive part and a calculation with the sign the other way round would make every lactonisation easier to explain rather than harder.

What the excess is, in energy rather than in factors #

The two excesses are quoted as factors, and a factor is the wrong unit for asking whether substituents act independently. Converted at 298 K they become energies, and the arithmetic is immediate.

The dimethyl closure’s unexplained factor of 6.86 is 4.77 kJ mol⁻¹. The tetramethyl closure’s 301.7 is 14.15. Two more methyl groups have therefore bought not twice the unexplained energy but 2.97 times it — very nearly three.

Independence would have given two. So the departure is a factor of about one and a half in energy, and it is what the factor of 302 against an independent prediction of 47.0 looks like once the exponential is taken out.

Fitting the two numbers to the simplest model that can hold a cross term — a contribution $a$ from each methyl and $b$ from each pair of them — gives $a = 1.81$ and $b = 1.15$ kJ mol⁻¹. Two numbers fitted to two measurements is not a test of the model, and it is a statement of what the model would have to carry: the pairwise term is 24% of the unexplained energy at two substituents and 49% at four, because four methyls make six pairs where two make one.

That is a sharper constraint than “superadditive” and it is a checkable one. A pairwise term of about 1.2 kJ mol⁻¹ per pair of substituents is the size of a single gauche interaction, not of a bond or of a large steric clash — so whatever the missing mechanism is, it is not large per interaction and it is being amplified by the combinatorics of how many interactions there are. A hexamethyl measurement would have fifteen pairs and, on this model, an unexplained energy of 28 kJ mol⁻¹ — a factor of 90,000 above the ceiling, which is the prediction the model makes and the way to refuse it.

Still open: what the superadditive excess is #

The obvious open question is what the tetramethyl measurement’s excess is a measurement of. Two methyls buy a factor of 250 where the statistics allow 36.5, and four buy 11,000 where the statistics still allow 36.5 — so the part the rotamer account cannot explain grows from a factor of 6.9 to a factor of 302 when two more substituents are added. Whatever the missing term is, it is strongly superadditive in the substituents, which is a much sharper constraint than anything available before, and it points at a mechanism that involves the substituents interacting with each other rather than each with the chain. The section above puts a size on that cross term — about 1.2 kJ mol⁻¹ per pair — and names the hexamethyl measurement that would refuse it.

The nearer question is the interpolation the last caution names. A rotor is treated here as free or absent, and a real hindered rotor is neither: its higher conformers carry a Boltzmann weight, and a rotor with three states of unequal energy has a ceiling factor between 3.33 and 1. Computing that factor as a function of the hindering energy would replace the two rows of the table with a curve, and it would say exactly how hindered the six-membered lactonisation’s tether can be before the only tight verdict in the argument gives way.