Concept

Boltzmann distribution — where it appears

The rule that the population of a state falls exponentially with its energy, at a rate set by the temperature. It turns an energy difference into a ratio of populations, and because the relation is exponential a small uncertainty in the energy becomes a large one in the ratio.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

shape · Strain

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.

shape · Strain

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.

shape · Strain

Named alongside it

The objects these essays reach for when they reach for this one.

Approximation Closed form Conformer Model limit Reference state Ring strain Rotamer Temperature Torsion Underdetermination Convention Gem-dimethyl effect

All concepts