The strain that is not in the angles

Ring geometry sets a ceiling first. A closed ring of equal bonds has interior angles averaging 180(n−2)/n, so a three-ring’s angles are 60°, a four-ring’s 90° and a five-ring’s 108° — and the first two of those are a long way below the tetrahedral 109.47° at every geometry whatever.

That explains cyclopropane and cyclobutane, which are strained by about 110 kilojoules a mole each. It explains nothing about cyclopentane, whose angles are within a degree and a half of ideal and which is strained by twenty-six.

The term that is missing #

A flat ring has every substituent eclipsed. Across each carbon–carbon bond the two hydrogens above the plane eclipse each other, the two below eclipse each other, and the two ring bonds eclipse each other — three eclipsing pairs, which is exactly what ethane has when it is eclipsed.

So a flat ring of n carbons carries n ethane barriers of torsional strain: 12.1 kJ mol⁻¹ per bond, measured, quoted here rather than computed. For cyclopentane that is 60.5 kJ mol⁻¹, which is more than twice its measured total.

Two things follow at once. The torsional term is large — larger than the angle term for every ring from four to six — and it is the same size per bond for every ring, so it cannot be what distinguishes them. What distinguishes them is how much of it each ring can escape by leaving the plane.

What puckering buys and what it costs #

Leaving the plane rotates the bonds away from eclipsed and relieves the torsional term. It also bends the angles away from whatever the flat ring had, which costs angle strain.

For a six-ring the trade is a bargain. It can reach 111.5° — close to tetrahedral — and stagger every bond completely, so both terms come out at about a kilojoule and the total is 2.4 against a measured strain of exactly nothing.

For a five-ring the trade is partial. Puckering costs angle strain immediately, because its flat angles were already almost ideal, and it can only relieve part of the torsion. What is left is the twenty-six kilojoules that are measured — a residue of a competition rather than a property of either term.

For a three-ring there is no trade at all. Three points are coplanar whatever is done to them, so a cyclopropane cannot pucker, its dihedrals are zero by geometry, and it carries the full torsional term on top of an angle strain it also cannot escape.

The chair is a torsional result #

The sharpest consequence is about a molecule already familiar from the other side.

The ring that cannot hold still established that holding every bond length and every bond angle fixed leaves cyclohexane’s chair rigid and its boat sitting on a continuous family of solutions — a whole line of shapes that satisfy every geometric constraint equally. That is a statement about the rank of a matrix, and it says nothing about which member of the family a molecule adopts.

Seventeen times the torsional energy, at identical angle strain. So the chair is not what the bond angles pick out; it is what the torsional term picks out of what the bond angles allow.

That is worth stating plainly because the usual account has it the other way round. Cyclohexane is described as adopting the chair because its angles can reach 109.5° there — and so they can, but so they can in every twist form as well. The angles decide that a six-ring can be unstrained; the torsions decide which unstrained six-ring it is.

The five-ring’s own compromise #

Cyclopentane deserves its own section, because it is the case the whole essay is named for and its behaviour is the clearest demonstration that the two terms are separate.

Flat, it has 0.4 kilojoules of angle strain and 60.5 of torsion. Puckered, it trades some of the second for some of the first, and the best it can manage is a residue.

The molecule’s actual behaviour is more interesting than a single minimum. Cyclopentane’s puckering is not fixed to one set of atoms: the atom that is out of the plane moves round the ring in a continuous motion — the pseudorotation — so the molecule has no single conformer at all, and the twenty-six kilojoules is an average over a circuit. That is a fact a purely geometric model can see the beginning of, since every point on the circuit is a valid closed ring with the same angle strain, and it is the same continuous-family observation the chair result rests on.

The general shape of the result is worth extracting. A ring with bad angles is stiff and a ring with good angles is floppy. Cyclopropane has one geometry and no choices; cyclopentane has a circle of them; cyclohexane has two rigid ones and a soft family between. The determinant of the constraint matrix that a count of ring constraints computes is the same statement, arrived at from the geometry rather than from the energy.

A five-ring at 104 degrees, drawn from the coordinates a closure search returns rather than from a named conformer, has a small pucker and has already paid its angle strain. What the pucker buys is torsional: it takes the ring out of the eclipsed arrangement its planar form is stuck in, and the energy it saves by doing so is larger than the angle strain it spends.

Where the model fails, and it does #

Two constants are quoted and neither is fitted here: the ethane barrier and a bending force constant of about a millidyne-ångström per radian squared. Everything else is computed from geometry. The account should therefore be checked against the measurements it was not tuned to, and it fails in two places.

Cyclopropane’s angle strain comes out at 280 kJ mol⁻¹ against a measured total of 115. A harmonic bending term is being asked about a deviation of forty-nine degrees, and a harmonic term is a small-displacement approximation. It overestimates by a factor of two and a half, which is the honest thing for it to do and is why molecular mechanics force fields carry cubic and quartic bending terms.

Cyclooctane’s crown comes out at 60 against a measured 40. The alternating construction is not the conformer cyclooctane adopts — it takes a boat-chair, which the construction above cannot produce — so the number is an upper bound on a shape the molecule does not use.

Neither failure touches the central claim, which is about the five-ring and is a comparison between 0.4 and 26 rather than a fit.

Why the two terms are so often merged #

The habit of calling all of it “angle strain” or all of it “ring strain” has a reason, and the reason is that the two terms are correlated across the small rings where the subject is usually taught.

Cyclopropane and cyclobutane have terrible angles and fully eclipsed bonds, because they cannot pucker enough to help either. So both terms are large for both molecules, and either one alone gets the ordering of the small rings right. It is only at five and six that they part company — and five and six are the rings where the answer is interesting.

There is a second reason and it is about what is easy to draw. A bond angle is visible in a structural formula; a dihedral is not, and seeing one requires a picture of the molecule from an unhelpful direction. Newman’s projection exists precisely to make torsions visible, and it is taught as a separate topic from ring strain in most courses, which leaves the two ideas in different chapters.

The consequence is a rule of thumb that works for the rings it was formed on and fails immediately outside them. Cyclopentane is the standard counterexample and it is in every textbook as an aside — “cyclopentane is puckered to relieve torsional strain” — usually one sentence after an account that has explained ring strain entirely in terms of angles.

What the argument requires #

A flat five-ring’s angle strain is under a kilojoule and its measured strain is over twenty, which is the claim in one line and needs no model at all beyond the bending constant.

Ranking the rings by angle strain does not reproduce the measured ranking. Both orderings are computed and compared as lists rather than eyeballed.

A harmonic bending term overestimates cyclopropane on its own, before any torsion is added. The model is required to be seen failing rather than quietly repaired.

The torsional term is one ethane barrier per bond for every ring, checked as an identity across all six — because if it were not, it would be doing some of the work the angle term is supposed to do.

The constructed chair’s bonds are all equal to 10⁻¹⁵ without that having been imposed, and its dihedrals are all past 50°. Both are checks that the construction is the chair rather than something that resembles one.

What none of this includes #

No non-bonded repulsion. Two hydrogens across a ring can be close enough to push each other apart, and for the medium rings — seven through eleven — that transannular term is a substantial part of the strain. It is not here, and it is why cycloheptane’s measured 26 is not reproduced by anything above.

No bond-length relief. Every ring here has equal bonds by construction. A real strained ring lengthens its bonds slightly, which buys angle relief at a quadratic cost of its own.

One conformer per ring. A real ring interconverts between several, and what is measured is a thermal average over them weighted by their energies. The barrier between cyclohexane’s chairs is 45 kJ mol⁻¹ and it flips several thousand times a second at room temperature.

And the two constants are quoted. The essay’s conclusions are sensitive to the bending constant in exactly one place — the size of cyclopropane’s overestimate — and to the ethane barrier everywhere, because the torsional term is that barrier times a count.

There is a sharper contrast worth naming. Which ring sizes can close on a trans double bond is a question about geometry alone and has a yes-or-no answer at every size; the strain energies here are a question about a balance, and a balance has no such answer. The two kinds of claim are worth keeping apart, because the first is a theorem and the second is an arithmetic with parameters in it.

What a strain energy is measured against #

One more convention deserves naming, because the last column of the table is the only measured thing in it and it is a difference like every other quantity this thread collects.

A strain energy is the heat of combustion of the ring, per carbon, compared with the heat of combustion of a long unstrained chain, per carbon, times the number of carbons. The reference is therefore an acyclic hydrocarbon, and the number quoted for a ring is the excess over what its atoms would have released if they had been strung out in a line instead.

That is a good reference and it has two consequences worth stating. Cyclohexane’s strain of zero is not a statement that cyclohexane has no torsional or angle strain at all — it is a statement that it has no more than a long chain does, and a long chain in its all-staggered form has neither. And a strain energy contains everything: angle, torsion, transannular repulsion, and any change in the bonds themselves, with no way to separate them experimentally.

So the split this essay makes is a split of a computed model, checked against an unsplittable measurement. Which is what a model is for — but it means the two columns cannot be validated separately, only their sum, and a model that got both wrong in compensating directions would look right.

What the two constants are worth #

The whole account rests on two quoted numbers, and it is worth saying how much each is carrying.

The ethane barrier carries the entire torsional column, which is n times it. Nothing about the ring enters, so a reader who prefers a different value can rescale that column in their head — and the central claim, that a flat five-ring’s torsional strain is two orders of magnitude larger than its angle strain, survives any value between eight and sixteen kilojoules a mole.

The bending constant carries the angle column, quadratically in the deviation. It is the number the cyclopropane overestimate is sensitive to, and it is the one a molecular mechanics force field would replace with a series rather than a constant. Halving it would bring cyclopropane’s angle strain to 140 against a measured 115, which is a much better number obtained by fitting rather than by computing — which is why it has not been done here.

So the two quoted numbers are doing different jobs: one sets a scale that every ring shares, and the other sets a term that the small rings are dominated by and the middle ones are not. Neither was adjusted to make anything agree.

Which term wins, in closed form #

The table gives the two columns for six ring sizes. The comparison behind it can be done once, for every $n$ at a stroke, because both terms have closed forms for a flat ring.

Per bond, the torsional term is the ethane barrier — the same 12 kJ mol⁻¹ whatever the ring is, since a flat ring eclipses every bond and an eclipsed bond knows nothing about how many others there are. Per angle, the bending term is $\tfrac{k}{2}\,d(n)^2$ with

$$d(n) = \frac{180°(n-2)}{n} - 109.47°$$

and $k/2 = 0.0381$ kJ mol⁻¹ deg⁻², which is the constant the table already uses. The two are compared per bond because a ring has as many of each.

Setting them equal gives $d = \sqrt{12 / 0.0381} = 17.75°$, and solving the closure formula for that deviation on each side gives the two crossings:

$$n = 4.08 \quad\text{and}\quad n = 6.82.$$

Below four and above seven, angle strain is the larger term. Between them, torsional strain is. That single line is the whole of why the textbook account works where it works and fails where it fails: cyclopropane and cyclobutane are angle-strained, and the account taught for all of them is an account of those two.

The reason the failure is worst at five rather than merely present there is visible in the same formula. The angle deviation $d(n)$ passes through zero at

$$n = \frac{360°}{180° - 109.47°} = 5.104,$$

so a flat ring of about five atoms has no angle strain at all — it is the one ring size whose planar polygon happens to present the tetrahedral angle. Cyclopentane sits a tenth of an atom away from that zero, which is why its angle term is 0.4 kJ mol⁻¹ and not merely small. The torsional term has no zero anywhere; it is $n$ times a constant. So the ratio between the two terms does not merely become unfavourable near five, it diverges there, and an account carrying only the angle term predicts cyclopentane to be the one unstrained ring.

Two further readings come free.

The upper crossing at 6.82 is why cyclohexane is the exception rather than the rule. Six is the only integer strictly between the two crossings on the upper side, and it is also — as the chair result shows — the ring that can relieve its torsion completely. Every ring from seven up is angle-strained as a flat ring and relieves that by puckering, which costs torsion back.

And the flat-ring angle term grows without bound. At twelve atoms it is 62.6 kJ mol⁻¹ per bond, five times the torsional term, and at large $n$ it tends to 189. No large ring is flat, and this is the arithmetic that says why: staying planar would cost more than a bond.

Still open: the transannular term, and larger rings #

The obvious open question is the transannular term, and it needs something deliberately left out here: a distance-dependent repulsion between atoms that are not bonded. That is a force field, its parameters are fitted, and adding one would replace a model whose two constants are measured quantities with one whose parameters are chosen — which is the trade a model with measured constants declines to make for the same reason.

The nearer question is the one the chair result raises. If the torsional term is what selects a conformer out of a geometric family, then the same should be true of every ring large enough to have a family — and the families get rapidly larger with ring size. Cyclooctane has several conformers within a few kilojoules and cyclodecane more, so the selection stops being a choice between two shapes and becomes a search over a landscape. That is where conformational analysis stops being geometry and starts being a minimisation, and a geometric account has reached its boundary.