A verdict inside its own error bar

The curve between two rows of a table — the factor a rotation contributes to a rotamer ceiling as it goes from free to frozen — shows where the six-membered closure’s sufficiency verdict gives way: at 4.09 kJ/mol of hindrance on three of its four rotors, or 2.70 on all four. With two hindered, no hindrance is enough, because the free part alone is 10.99 against a measured 10.

It then said what the nine per cent margin is worth. Nine per cent is inside the uncertainty of a rate ratio measured in the nineteen-sixties, and recomputing the whole comparison with an explicit uncertainty on the measured tenfold would say whether the six-membered case is a verdict at all.

It is not. And the uncertainty that decides it is not the one usually named.

The computed side has an error bar too #

Every rotamer ceiling is built on one number: how much a gauche arrangement about a carbon–carbon single bond costs relative to the anti one. It is quoted, at 3.8 kJ/mol, and the compilations give it to about a tenth of a kilocalorie — ±0.4 kJ/mol.

That uncertainty has never been carried through. Carried through, it moves the ceiling from 8.825 to 13.848: a band of ±22.8 per cent around 10.99, which is more than twice the margin the verdict was decided by.

The ceiling crosses the measurement at a gauche energy of 3.6300 kJ/mol — 0.425 of one standard deviation below the quoted 3.8. So the verdict is decided by less than half of the input’s own error bar.

The sensitivity has a closed form #

That is not a numerical accident and it does not need to be trusted from a sweep. The ceiling for $n$ free rotors is $(1 + 1/2x)^n$ with $x = \exp(-g/RT)$, so its logarithmic derivative in the gauche energy is

$$\frac{\mathrm{d}\ln C}{\mathrm{d}g} = \frac{n}{RT(2x+1)},$$

which is 0.563474 per kJ/mol for two rotors at 298 K.

A quarter of a per cent of the ceiling for every hundredth of a kilojoule, and the most quantitative comparison in the argument is a nine per cent margin. Put the other way round: the verdict is decided by the third significant figure of an energy that is quoted to two, and the ring whose closure this is about is a molecule whose own conformational energies were never measured to that precision. The two numbers were never compatible and nobody had put them side by side.

The same expression says how the exposure grows: four free rotors are twice as sensitive as two, so the five-membered case’s ceiling has a band of ±57 per cent rather than ±23. That case is refuted by a factor of seven, so it does not matter there — which is the point of computing it rather than assuming it.

What a nine per cent margin was worth #

It is worth being precise about what has and has not been shown, because the arithmetic is not in question.

Every number it computed is right. The ceiling for two free rotors at 3.8 kJ/mol and 298 K is 10.9945; the hindrance that takes four rotors below ten is 2.70 kJ/mol; the curve between the two rows is the curve. Nothing here recomputes any of that.

What changes is the status of one sentence. The six-membered case’s verdict was already known to be less robust than the five-membered one’s, for a reason — hindering lowers a ceiling, so it can destroy a sufficiency verdict and cannot rescue a refutation. That reason is sound and it is about the model. The reason found here is about the inputs, and it is stronger: even with no hindrance at all, the verdict is not decided.

So the tightest quantitative statement turns out to be the least decided one, and the two facts have the same cause. A margin of nine per cent is tight because the two numbers are close; two numbers that close cannot be told apart by a calculation whose input is known to a tenth. Tightness and decidedness are opposite properties here, and the first was reported while the second was offered.

And the temperature is not quoted either #

The ceiling is a Boltzmann quantity, so it falls as the temperature rises: 13.43 at 273 K, 10.99 at 298, 10.14 at 310, 9.34 at 323. It crosses the measurement at 312.1 K.

That is fourteen degrees above the temperature computed at, and it is inside the range ordinary kinetics is run over. The rate ratio being compared against was measured at some temperature, and that temperature is not carried with it anywhere — so a second unstated input decides the same verdict, in the same direction, by about as little.

One verdict survives and one does not #

The five-membered ceiling is 120.88 with a band of 77.9 to 191.8, against a measured acceleration of 250. Even at the top of its band it is a factor of 1.3 short, and it is the most generous version of the account — four free rotors, no hindrance, everything arranged to make the ceiling as high as it can be.

So the refutation is untouched, and the sufficiency verdict is not decided at all.

That asymmetry is sharper with error bars than without. The asymmetry was argued for a reason about direction — hindering a rotor lowers the ceiling, so it cannot rescue an account that is too low and can destroy one that is sufficient. This adds a second reason of a different kind: a refutation by a factor of seven survives a band of a fifth, and a sufficiency verdict by a tenth does not.

There is one more consequence for how the ceilings should be quoted. The five-membered case’s ceiling has four free rotors rather than two, so its logarithmic derivative is 1.127 per kJ/mol and its band is ±57 per cent — two and a half times the six-membered one’s. An argument that computes ceilings for closures of different sizes is computing quantities of very different precision, and quoting them in one table with the same number of decimal places is the presentation that made the nine per cent look comparable to the factor of seven.

The ceiling that rises where the measurements fall is where the comparison method comes from, and its whole finding — that the ceiling and the measurement move in opposite directions with ring size — is a statement about a trend rather than about any single comparison. A trend across six ring sizes is much better evidence than one nine per cent margin, and it is untouched by anything here.

It is worth being explicit about what “does not survive” means here, because it is weaker than a refutation and stronger than a doubt. The six-membered account is not shown to be wrong. It is shown to be undecided by this comparison: the ceiling’s band and the measurement’s band overlap, so a reader who wanted the account to be sufficient and a reader who wanted it to be insufficient can each find a point inside the bands that agrees with them, and nothing in the calculation prefers either. A verdict of that shape has no content, and the honest report of it is that the comparison was not sharp enough to make one.

What would make it sharp is visible from the same arithmetic, and it is one number. The crossing sits 0.170 kJ/mol below the quoted gauche energy, and that distance is fixed — it is a property of the ceiling and the measurement, not of anybody’s uncertainty. What is not fixed is the standard deviation it is being divided by, which is 0.4 kJ/mol and puts the crossing at 0.425 of one. Halve that standard deviation and the crossing is at 0.85 of one; quarter it and the crossing is at 1.7, which is a margin a reader would accept. The verdict is one input away from being decidable, the input is the gauche energy rather than the rate, and naming which one is the useful part of having computed the sensitivity at all.

That the gauche energy is the input to improve rather than the measurement is itself a result of the sensitivity, not an assumption. The measurement’s twenty per cent enters the comparison linearly and the gauche energy enters it exponentially, through a Boltzmann factor with a logarithmic derivative of 0.563 per kJ/mol — so the ±0.4 on the gauche energy is worth about ±25 per cent on the ceiling by itself, more than the whole stated uncertainty of the thing it is being compared against. A better rate constant would narrow the target and leave the ceiling exactly as wide.

What was computed, and how #

The ceiling is the closed form used throughout — three staggered states per rotation, the two gauche ones a butane gauche energy above the anti, and the ratio of the total to the closable one — evaluated at each gauche energy and temperature rather than at one.

The gauche energy and its uncertainty are quoted: 0.9 ± 0.1 kcal/mol from the standard compilations, converted. The rate ratio is quoted and its uncertainty is not: no error bar travels with it in the form used here, so the twenty per cent here is a stated hypothesis about what a rate constant of that vintage is worth, declared as such and not as a report.

The sensitivity is computed twice, once by a central difference and once from the closed form, and the two are required to agree to a part in a million. That is what makes it a derivative rather than a step size, and it is the only piece of numerical care the calculation needs.

The refusal is a closure with no rotors to freeze. Its ceiling is one whatever the gauche energy is, so the sensitivity must be exactly zero and no critical energy exists — a version of this that reported one there would be reporting its own arithmetic.

Why the input’s error bar is the larger one #

That the computed side is wider than the measured side is worth explaining, because it inverts the ordinary expectation.

A rate ratio is a measurement and carries an experimental uncertainty of perhaps a fifth. The ceiling is a closed form and carries none — until its input is given one, at which point the exponential does the work. A gauche energy enters through $\exp(-g/RT)$, and at room temperature $RT$ is 2.48 kJ/mol, so an uncertainty of 0.4 is a sixth of the natural energy scale of the exponential. An uncertainty of a sixth of $RT$ in an exponent is a much larger thing than an uncertainty of a fifth in a rate.

That is a general feature of any quantity computed from a Boltzmann factor, and it is why conformational arguments are so much less quantitative than they look. Two conformers within a kilojoule of each other are within half an $RT$, and half an $RT$ is a factor of 1.5 in population — so a population computed from an energy known to a tenth of a kilojoule is known to five per cent, and one computed from an energy known to half a kilojoule is known to about a quarter.

The same exponential sensitivity is what makes a Peierls amplitude easier to read backwards than forwards, and the arithmetic is identical: an exponential turns a small uncertainty in an energy into a large one in a population, and a large uncertainty in a population into a small one in the energy.

Where the model stops #

The band computed here is the band from one input. There are others: the three-state model of a rotation is a caricature of a torsional potential, the gauche energy of a substituted chain is not butane’s, and the assumption that every rotation a closure freezes is independent is the estimate that can be wrong by two. That last one is the largest of them: a factor of two dwarfs a band of a fifth, and it applies to the count rather than to the energy, so the two do not combine in any simple way. What is computed here is the band from the input that has a quoted uncertainty; the ones that do not have quoted uncertainties are worse and cannot be put in a band at all. Each of those would widen the band further, and none of them narrows it.

The temperature dependence is the model’s, not a measurement’s. A real rate ratio’s temperature dependence has an activation-energy difference in it that this model has no term for, so the crossing at 312 K is a statement about the ceiling and not a prediction that the ratio would be ten at that temperature.

And the measurement’s uncertainty is assumed. If it were quoted at five per cent rather than twenty the bands would still overlap, because the computed side is the wider of the two — which is the one conclusion here that does not depend on the assumption.

The generalisation #

A verdict is a comparison of two numbers, and it is only as sharp as the wider of their two error bars. That is obvious and it is routinely skipped, because a computed number arrives without one: nothing in a closed form suggests an uncertainty, and the inputs it was evaluated at usually arrive as bare values.

The practical rule that follows is cheap. For every computed quantity, differentiate it with respect to its worst-known input and multiply by that input’s own error bar. Here that is one line of algebra and it turns the tightest result into an undecided one — which is a loss of a result and a gain of a true statement about what can be settled.

The same shape appears in two other places. A discordant pair whose measurements differ by less than their uncertainties is not a refutation of anything, and four electronegativity tables disagree about one molecule by more than the molecule is worth. In all three the arithmetic is exact, the inputs are not, and the conclusion was drawn at a margin smaller than the inputs allow.

Who found it, and when #

The butane gauche energy has been measured since the nineteen-forties and its modern value — 0.9 kcal/mol, with the anti rotamer favoured — is from a long series of spectroscopic and thermodynamic determinations. The uncertainty of a tenth of a kilocalorie is what the compilations carry; individual determinations differ by more.

Rate accelerations for ring closures are Illuminati and Mandolini’s territory and the classic numbers are from the nineteen-sixties and seventies. They are usually quoted as effective molarities rather than as ratios, and the compilations do carry uncertainties for them — which makes the assumption used above conservative rather than generous.

That the strain a ring carries is not all in its angles is a caution about what these comparisons are comparisons of, and it is a larger one than any error bar here.

Propagating an uncertainty through a closed form is not anybody’s discovery. What is worth recording is that nine calculations went by without doing it, and that the first time it is done the tightest result goes.

Still open: a band on every ceiling #

The obvious open question is the other inputs. The gauche energy is one of at least four numbers the ceiling depends on — the others being the number of rotations a closure actually freezes, the assumption that they are independent, and the three-state model of each. Each has a sensitivity with a closed form and each has an uncertainty that could be stated, and the four together would give the ceiling a band rather than a value. That is what every ceiling should be quoted with, and it is one afternoon of algebra.

The nearer question is the direction of the temperature effect. The ceiling falls with temperature and a real rate ratio rises or falls with it depending on the difference in activation energy between the two reactions being compared — so the two sides of this comparison move with temperature in ways that are not obviously the same. Measuring the ceiling’s temperature coefficient against the ratio’s, where the ratio’s is published, would say whether the comparison is even being made at the same temperature, which nothing here has checked.