Molecular Geometry & Bonding
what the shape decides
Figure

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 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.

One of the figures on where the atoms go: Repulsion minimised on a sphere, the angles that fall out of it, and the arrangements that are not all alike.

Three essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

The angle a ring cannot have

The ceiling for each ring size, with the tetrahedral angle drawn across it. Three, four and five atoms are below the line at every geometry available to them; six is above it with a margin of ten degrees, which is what the chair spends.

The residuals. At the ceiling and just below it a ring exists and the search finds it to eleven or fifteen decimal places; three degrees above, nothing closes. The “2° below” column is the surprise and is the next section.

The received explanation, with the wrong sign. The standard account attributes the small ring’s behaviour to the angle its carbons are forced into, and the arithmetic of the closure says the effect goes the other way — which is the reason this essay exists and the reason the repair has to come from somewhere other than the angles.

The ring that cannot hold still

Every torsion of the six-ring, against the puckering phase that labels the conformer. Thirty-six solutions, all with the same six bonds and the same six angles to a residual below 10⁻¹¹, and torsions running over 120°. Six of them have a torsion at zero — those are the boats — and the twist-boats sit between them.

The small-ring table, for the sizes this essay leaves out. Three and five refuse to close below their ceilings as well as above them, because there the conditions outnumber the freedoms; four escapes by symmetry. The rigidity question only becomes interesting once a ring can exist at more than one shape.

The double bond a ring cannot hold

The largest bond angle a ring of each size can have, which is a theorem rather than a search: a closed curve turns through at least a full circle, so the angles cannot exceed 180°(n−2)/n. The small rings’ angle strain is this bound, and it is a separate constraint from the torsional one — the two act together in a cyclopentene and independently in a cyclodecene.

Every figure · Every orbital, by what it encloses · All essays