Molecular Geometry & Bonding
what the shape decides
Series

Strain — the series

14 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The largest angle each ring size can have. The ceiling on a bond angle in a closed ring of equal bonds, 180°(n−2)/n, which is the interior angle of the regular planar polygon and follows from a closed curve having to turn through a full circle. The line at 109.47° is the tetrahedral angle: rings of 3, 4, 5 atoms cannot reach it at any geometry whatever, and every larger ring can, by leaving the plane.

    The angle a ring cannot have

    A closed ring of equal bonds has to turn through a full circle, so its bond angles cannot average more than 180°(n−2)/n. Three, four and five atoms are below the tetrahedral angle at every geometry whatever; six is above it, and reaches it only by leaving the plane.

    part 1 · shape
  2. A 6-ring at 111°: the twist-boat. A closed ring of 6 equal bonds meeting at 111°, drawn from the coordinates the closure conditions produce. Its torsions are 16.9°, -63.6°, 44.2° and repeat; its puckering amplitude is 0.508 bond lengths at a phase of 344°, with q₃ exactly zero, so it lies on the equator. Turning the ring changes none of those numbers, which is what makes them a description of the shape rather than of the view.

    The ring that cannot hold still

    Cyclohexane's chair is rigid and its boat is not, and that is a statement about the rank of a matrix rather than about strain. Hold every bond length and every bond angle fixed and count what is left: the chair has nothing, and the boat sits on a continuous loop of shapes with the same bonds and the same angles.

    part 2 · shape
  3. The twist the ring forces on the bond. For each ring size, the largest torsion about the double bond that the ring will close on — 180° being a flat trans arrangement and 90° being a π bond broken outright. The six-ring will not close at any torsion tested. The eight-ring reaches 139.3°, against 136° measured in trans-cyclooctene by diffraction: a model with bond lengths and bond angles in it and no energy anywhere agrees with the crystal to a few degrees.

    The double bond a ring cannot hold

    A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.

    part 3 · shape
  4. Flat rings: the angles, the torsions and what is measured. For each ring from three to eight, held flat: its interior angle, how far that is from tetrahedral, the angle strain that follows, the torsional strain of having every bond eclipsed, and the measured strain energy. The five-ring is the row the essay is about — its angles are almost ideal and it is strained.

    The strain that is not in the angles

    Cyclopentane's flat bond angles are 108°, a degree and a half from tetrahedral, and its angle strain computed from a standard bending constant is 0.4 kJ mol⁻¹. Its measured strain is twenty-six. The missing sixty kilojoules are torsional — one ethane barrier for every bond in the ring, which no account built on bond angles mentions.

    part 4 · shape
  5. The same angles, the same torsions, and not the same molecule. twelve closed conformers of a ring of 10 at a bond angle of 111.5 degrees. Every one has exactly the same bond angles, so an account built from angles and torsions places them all on the horizontal axis alone. The vertical axis is the closest approach of two atoms four or more bonds apart, which no term in that account mentions: two of these differ by 0.08 kilojoules in torsional energy and by 0.75 ångström in how close they come.

    The atoms that meet across a ring

    Twelve closed conformers of a ten-membered ring at one bond angle, so every one has identical angle strain by construction. Two of them differ by 0.026 kilojoules a mole in torsional energy and by 0.80 ångström in how close two atoms on opposite sides of the ring come — 2.331 against 3.131, where two carbons are in contact at about 3.4. An account built from angles and torsions calls those two structures the same.

    part 5 · shape
  6. The 6-ring at 111.5°: the alternating form and what a search finds. The alternating ring — every atom displaced above or below the plane in turn — with its dihedral angles and its two strain terms, beside the lowest-torsion member a search constrained only by bond angles and closure returns. Both satisfy every geometric constraint; only one of them is staggered.

    The explanation with the wrong sign

    Two methyl groups on one carbon make a ring easier to close, by up to eleven thousand-fold, and the textbook reason is that they compress the ring's internal angle. Computed from a standard bending term, that compression helps a three-ring and a four-ring and hinders every ring from five up — predicting a slowing of 0.754-fold for the five-ring measured to speed up 250-fold. The account with the right sign is about rotations rather than angles, and it has a ceiling of 36.5 that the measurement is already above.

    part 6 · shape
  7. The ceiling rises and the measurements fall, so they cross. The largest acceleration the rotamer account can produce, against the ring being closed, with the measured gem-dimethyl accelerations on the same axis. Closing a bigger ring means freezing more rotations, so the ceiling rises steeply; the measurements go the other way. The five-membered ring's 250-fold acceleration is above its own ceiling of 36.5 and the six-membered ring's tenfold one is far below its 121 — so the account is refused at one size and sufficient at the next.

    A ceiling that rises where the measurements fall

    The rotamer account of the gem-dimethyl effect has a largest possible acceleration, which looks like a limitation. It is a prediction: the ceiling is a closed form in the number of rotations a closure freezes, it rises steeply with ring size, and the measurements fall — so the account is refuted for the five-membered ring and more than sufficient for the six.

    part 7 · shape
  8. How far the rotor count would have to be wrong. The ceiling against the number of rotations a closure freezes, with the two measured accelerations drawn across it. The five-membered closure freezes three and its ceiling is 36.46; the ceiling does not reach the measured 250 until 5 rotors, so the count would have to be wrong by 2 on a ring that has three rotations to freeze. The six-membered closure freezes four at a ceiling of 120.88, and stays above its measured 10 down to 2 — so the refusal is airtight and the sufficiency is comfortable.

    An estimate that can be wrong by two

    The ceiling on the gem-dimethyl effect is exponential in the number of rotations a closure freezes, and that number was taken as n − 2 without counting — which left the refutation at five rings probable rather than airtight. It is airtight. The ceiling does not reach the measured 250 until five rotors, on a ring that has three, and no hindering of the tether can raise it.

    part 8 · shape
  9. The two rows of a table, and the curve between them. The rotamer ceiling of a closure freezing 4 rotations, against how hard each of a stated number of them is to turn. The top curve is the free-rotor row and the right-hand end of the bottom curve is the absent-rotor row; everything else is what a real hindered tether does. The measured acceleration of 10-fold is drawn across it. It falls below that line when 3 of the 4 are hindered by 4.09 kJ/mol, or when 4 of the 4 are hindered by 2.70 kJ/mol.

    The curve between two rows

    A rotation a ring closure has to freeze was treated as free or as absent, and the two answers sat in adjacent rows of a table. A real hindered rotor is neither. The factor one rotor contributes runs from 3.316 to one along a curve nobody had drawn, and where it matters is between one and eight kilojoules a mole — which is a torsional barrier rather than a conformational preference.

    part 9 · shape
  10. The two sides of a verdict, with error bars. The computed ceiling for the six-membered closure — 10.9945 — with the band a gauche energy of 3.8 ± 0.4 kJ/mol puts on it, against the measured tenfold rate ratio with an assumed 20 per cent uncertainty. The two bands overlap over most of their length, and the nine per cent margin the verdict was decided by sits inside both of them.

    A verdict inside its own error bar

    The tightest comparison in the rotamer argument is a computed 10.99 against a measured 10 — a nine per cent margin, offered as a verdict. The computed side is built on one quoted energy known to ±0.4 kJ/mol, and that alone puts a band of twenty-three per cent on it. The verdict turns on four tenths of one standard deviation of a number nobody had put an error bar on.

    part 10 · shape
  11. The measurement is not a horizontal line. The rotamer ceiling against temperature, with the measured tenfold drawn four ways: flat, and as a rate ratio whose two activation energies differ by 3, 5.31 and 8 kJ/mol. Flat, it is crossed at 312.1 K, which is the flat-rotor answer. At 5.31 kJ/mol the two curves are parallel and never meet. At 8 they meet on the other side, and it is cooling rather than heating that refutes the account.

    One number decides which way it breaks

    The six-membered verdict gives way at 312.1 K, and that is a statement about the ceiling rather than a prediction about the ratio — because a real rate ratio has a temperature dependence the model has no term for. Putting that term in moves the crossing, and at 5.308 kJ/mol it removes it entirely.

    part 11 · shape
  12. The five-membered ring would need more rotors than its chain has bonds. For each measured acceleration, the least rotor count whose ceiling reaches it, against the most rotors any convention gives that ring and against the number of bonds the chain has at all. The five-membered cases need five and eight rotors from a chain with four bonds, so they are refused by supply rather than by a margin — a refutation that needs no convention to be chosen, because every convention is below the requirement.

    The count that was never written down

    The rotamer ceiling for a ring closure needs a count of frozen rotations, and the counts in use disagree in direction: one gives the six-membered ring more frozen rotations than the five, another fewer. Every measured verdict survives all three defensible conventions unchanged — and the margins move by a factor of eleven, which is why a verdict decided by ten per cent was worth being nervous about.

    part 12 · shape
  13. The ceiling moves by 8.7-fold across the gauche energy's own reported range. The rotamer ceiling against butane's gauche energy, for the rotor counts each convention assigns to a five-membered and a six-membered closure, with the three measured accelerations drawn as horizontal lines. Across the reported range the ceiling moves by up to 8.70-fold, and two of the nine verdicts cross a measurement during the sweep. The natural prediction is that this input would be the smaller lever of the two; it is the larger.

    The lever that was supposed to be smaller

    Sweeping the rotor conventions leaves every verdict unmoved, and suggests that the other quoted input will be a smaller lever. It is a larger one. Sweeping butane's gauche energy over its reported range moves the ceiling by 8.7-fold, flips the six-membered verdict at 3.630 kilojoules a mole — inside the quoted error bar — and takes the five-membered refutation with it at 4.422.

    part 13 · shape
  14. Two sweeps, and one of them covers 28 per cent less ground than the other. What each of the two sweeps reaches, measured in the one variable the rotamer ceiling has: g/RT. The temperature sweep lies entirely inside the gauche sweep, so it explored no arrangement the other does not. Their widths are 0.581 and 0.807, their union is 0.807, and treating them as independent would have credited them with 1.387.

    Two sweeps and one lever

    The rotamer ceiling depends on the gauche energy and on the temperature only through their ratio, so a sweep of either traces the same curve. Measured in that one variable, the temperature sweep lies entirely inside the gauche sweep — it covered no ground the other does not, and the whole reported spread in one energy is worth a temperature swing from 196 to 353 kelvin.

    part 14 · shape

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