One number decides which way it breaks

A verdict inside its own error bar put error bars on both sides of a comparison that had been quoted as a number against a number. The rotamer account’s ceiling for a six-membered closure is 10.99; the measured acceleration is tenfold; the margin is nine per cent, which was the tightest margin in the whole argument. Giving the gauche energy its quoted uncertainty made the ceiling a band from 8.4 to 13.5, and the verdict landed inside its own error bar.

It also drew the ceiling against temperature, because the ceiling is a Boltzmann quantity and falls as the temperature rises: 13.43 at 273 K, 10.99 at 298, 9.34 at 323. It crosses ten at 312.1 K, fourteen degrees above the temperature every comparison is made at, and inside the range ordinary solution kinetics is run over.

And then it said, carefully, in its own closing section, that this crossing is a statement about the ceiling and not a prediction that the ratio would be ten at that temperature — because a real rate ratio’s temperature dependence has an activation-energy difference in it that the model has no term for.

That caution is correct and it is not the end of the matter, because the term can be put in as a parameter even though its value is not known. What comes out is not a correction to 312.1 K. It is that the crossing does not have a value.

Both sides are Boltzmann quantities #

A rate ratio between a substituted closure and its parent is, to the extent that transition-state theory is being used at all, a ratio of two exponentials — which is the same population argument the rotamer account itself rests on, applied to transition states instead of to conformers. If the substituted transition state is lower by ΔΔ, the ratio goes as exp(ΔΔ/RT) and falls as the temperature rises — the same direction the ceiling falls in, for a related reason: both are counting how much of a Boltzmann advantage survives heating.

So the two sides of the comparison are not a curve and a line. They are two curves, and whether they cross depends on which falls faster.

The ceiling’s rate of fall has a closed form. Writing the ceiling as ((1+2x)/2x)ʳ with x = exp(−g/RT), and reading its slope on an Arrhenius plot in the usual way, gives r·g/(1 + 2x). For the two free rotors of the six-membered case at 298 K, with butane’s gauche energy, that is 5.308 kJ/mol.

It is worth pausing on why that expression has the shape it does. The ceiling is a ratio of populations, and raising the temperature flattens a Boltzmann distribution towards uniformity; the 1 + 2x in the denominator is the partition function of one rotation, so as x climbs towards one — as the gauche penalty stops mattering — the ceiling’s sensitivity to that penalty shrinks. Every part of the number is a statement about how much advantage is left to lose.

That number is the whole essay. It is the activation-energy difference a measured ratio would need in order to fall at exactly the ceiling’s rate. If the real difference is smaller, the ceiling falls faster and heating brings it down to meet the ratio. If it is larger, the ratio falls faster and heating pulls the two apart — so the crossing has to be looked for in the other direction.

The crossing has a pole #

At ΔΔ = 0 — the assumption that the measurement is a horizontal line — the crossing is at 312.1 K, exactly as before. This calculation contains that one, which is how it is checked.

At 1 kJ/mol it is 315.7 K. At 2, 321.7. At 3, 334.3. At 4, 388.6 — already past the boiling point of most solvents a closure like this is run in. The crossing is not drifting; it is running away, and it runs away because the two curves are becoming parallel.

At 5.308 they are parallel, and there is no crossing at all: no temperature between 150 K and 900 K reverses the verdict, because the margin between the two sides is the same at every temperature.

And then above the threshold the crossing comes back — at 273.3 K for a difference of 8 kJ/mol, at 288.0 K for 12 — on the other side of room temperature. Now the measured ratio is falling faster than the ceiling, so the account is sufficient when hot and refuted when cold. The experiment that would test the rotamer account is a cooling experiment rather than a heating one, and nothing about the numbers as quoted says which of those it is.

So the answer to “at what temperature does the verdict change” is not a temperature. It is a function with a pole in it, and the pole sits at a number of entirely ordinary size — about one and a half gauche interactions, well inside the range that separations of two transition states routinely have.

Six of the ten assumptions swept put the crossing inside the range a solution reaction can actually be run over, and they are on both sides. Four of those six say heating refutes the account and two say cooling does. One puts it at 388.6 K, which is past the range without being unreachable in principle. Three say nothing does. That is the honest inventory, and the 312.1 K is one entry in it — the entry for the assumption that a rate ratio does not depend on temperature, which is the one assumption certain to be wrong.

Which way the margin moves #

The same fact is easier to see as the margin between the two sides rather than as the point where they meet.

Normalising the margin to one at 298 K and following it across the range: below the threshold it narrows on heating, above it, it widens, and the size of the movement is large in both directions. At ΔΔ = 0 the margin spans a factor of 2.24 across 253 to 373 K; at 16 kJ/mol it spans 5.15 the other way. Neither of those is a small correction to a nine per cent verdict.

This is the shape of the finding. The earlier error bar came from an input with a quoted uncertainty — the gauche energy, 3.8 ± 0.4 kJ/mol — and gave the ceiling a band of about a fifth. The temperature dependence of the measured side has no quoted uncertainty at all, because it has no quoted value, and it moves the comparison by considerably more than a fifth over a range a chemist would think nothing of.

It is also, unusually for this argument, a question with a short experimental answer. The activation-energy difference is what two Arrhenius plots would give — the substituted closure and its parent, each at a handful of temperatures — and the difference of their slopes is the number. Nothing about it is hard; it simply was not the number anybody was after when the tenfold was measured, and the comparison that has been built on that tenfold ever since needs it.

The ceiling is not an Arrhenius quantity #

There is one more thing in that picture, and it is the reason none of this can be tidied away by picking a good value for ΔΔ.

The ceiling’s apparent activation energy, r·g/(1 + 2x), is not a constant. It depends on temperature through x, and it falls from 5.719 kJ/mol at 253 K to 4.787 at 373 — a drift of nineteen per cent across the range. So the ceiling curves on an Arrhenius plot, and a rate ratio with a fixed activation-energy difference does not.

That means no ΔΔ makes the margin flat. Searching for the one that comes closest gives 5.271 kJ/mol, and even at that value the margin still moves by 1.8 per cent — a shallow U, 1.015, 1.004, 1.000, 1.003, 1.011, 1.021 across the six temperatures, with its minimum where the two were matched.

One point eight per cent is small against a nine per cent verdict, which is worth saying plainly: this residual is not what decides anything, and presenting it as though it did would be committing the error an explanation with the wrong sign commits in the other direction. It matters because of what it says about the kind of comparison being made. A ceiling from an equilibrium population and a ratio of two rate constants are not the same kind of quantity, and putting them on one axis and reading off where they cross treats them as though they were. The curvature is the model saying so.

A second unquoted temperature #

There is a further assumption buried in the arithmetic above and it should be stated rather than left implicit. The measured side is pinned at ten at 298.15 K, because that is where the ceiling is computed and where the tenfold has always been quoted alongside it. But the temperature at which the ratio was measured is not quoted either, and the model’s ratio passes through ten only at whatever temperature it is pinned to.

Shifting that reference translates the whole family: the ratio curve slides up or down, the ceiling does not move, and every crossing moves with it. So the picture above has two unquoted temperatures in it, not one — the difference in activation energies, which sets the shape, and the temperature of the original measurement, which sets the offset. Neither is recoverable from what has been published.

That is not a reason to distrust the arithmetic. It is a reason to read the threshold rather than the crossings: 5.308 kJ/mol is a property of the ceiling alone, and it does not move when the reference does. The crossings do.

What was computed, and how #

The ceiling is ((1 + 2x)/2x)ʳ with x = exp(−g/RT), r the number of free rotations, g butane’s gauche energy at 3.8 kJ/mol as quoted throughout. The rotor count is two, which is the count used for the six-membered closure in the original verdict — four rotations, two of them hindered by the substituent.

The measured side is modelled as 10 · exp((ΔΔ/R)(1/T − 1/298.15)), so that it passes through the quoted tenfold at the temperature the quotation implicitly assumes, and carries the activation-energy difference as its only parameter. The crossing solves the equality of the two by bisection over 150 to 900 K, with the bracket tested first.

The check covers six things. That a flat measurement returns 312.1 K, so this calculation contains the one it extends. That crossings are found on both sides of the threshold and that they sit on opposite sides of the reference temperature, which is the reversal. That each branch is ordered, and that the lower branch runs away rather than converging. That every crossing is a root rather than a bracket endpoint, to a part in 10⁹. And the refusal: that at and near the threshold the bracket fails, and reports both endpoints saying so, rather than the bisection handing back the edge of its range as though it were an answer.

That last is the one worth having. A bisection asked for a root that does not exist will return something, and here the something would be 900 K, which is not a temperature at which anything happens and would have looked like a finding.

Where the model stops #

The activation-energy difference is a parameter here and not a measurement, and there is no measured value to put in it. That is the third question in this argument whose answer needs a number nobody has quoted, and the response is the same as before: ask the question from the other end. What is computed is the threshold and the shape, not the answer.

Nor is a single activation-energy difference a good model of a real rate ratio over a hundred and twenty degrees. Activation energies drift with temperature too, for the same kind of reason the ceiling’s number does; the two-parameter Arrhenius form is itself an approximation, and treating it as exact while criticising the ceiling for curving would be helping oneself to the conclusion.

And the whole comparison still rests on the three-state model of a rotation and on the assumption that the rotations a closure freezes are independent, which is the estimate that can be wrong by a factor of two. A factor of two dwarfs everything computed here. What this adds is not a larger uncertainty; it is a direction that was not previously known to be undetermined. An uncertainty widens a band around an answer; an undetermined direction means the experiment that would test the account has not yet been specified, which is a different kind of gap and one that a ceiling quoted as a single number conceals particularly well.

The generalisation #

The transferable habit is this: when a computed quantity is compared against a measured one, ask whether both sides depend on the same variable, and if they do, put the dependence in as a parameter even when its value is unknown.

The threshold that comes out of doing that is often more informative than the answer would have been. This is the same inversion that an estimate can be wrong by two performed on the rotor count: not what is the value, but what would the value have to be for the conclusion to change. Here it converts a vague worry — the temperatures may not match — into a specific one: the verdict’s direction is decided by whether one unquoted number is above or below 5.308 kJ/mol. That is a question somebody could go and answer with two rate measurements, and it was not visible while the measured side was drawn as a horizontal line.

The same move applies wherever a model quantity meets a measurement on one axis. A ceiling that rises where the measurements fall is a comparison across ring size; this is the same comparison across temperature, and the two have the same defect available to them — an axis on which only one side was allowed to move.

Who found it, and when #

The gem-dimethyl effect and its rotamer explanation are old, and the tenfold acceleration this verdict turns on was measured in the nineteen-sixties. The ceiling, its closed form, its band and its crossing are original arithmetic, and so is everything above. Nothing here is a report of a published temperature study; no such study is known to report.

The error-bar argument noticed that the comparison had a temperature in it that nobody had stated. This one notices that it has two, and that only one of them was drawn.

Still open: the five-membered case, and the rotor count #

The obvious open question is the five-membered case, which is refuted by a factor of seven rather than by nine per cent. Nothing here threatens that verdict — a factor of seven survives a margin that moves by a factor of two — but the same arithmetic would say how far it survives, and a refutation with a stated temperature range is worth more than one without.

The nearer question is how many rotations a ring closure actually freezes, and it is not settled. One common count takes n − 2 for an n-membered ring, which gives the five-membered closure three free rotations and the six-membered four, while the verdict above used four for the five-ring and two for the six. Both counts are in use and they cannot both be right. Everything above was computed with the verdict’s own convention so that it extends that calculation rather than a different one — but which convention is correct changes the ceiling by a factor of three, which is thirty times the margin the verdict was decided by, and settling it deserves an essay of its own.