Isotopes

Chlorine's atomic weight is 35.45, and no chlorine atom weighs anything close to that. About three quarters of natural chlorine weighs almost exactly 35 u and the remaining quarter almost exactly 37 u. The tabulated figure is a population average over a mixture, and once you see that, the strangest thing about the periodic table — that its masses are decimals when atoms are made of whole particles — stops being strange.

Isotopes are atoms of the same element with different numbers of neutrons. Same proton count, same chemistry, different mass. If you need the counting arithmetic that connects mass number, atomic number and neutron number, that is on the protons, neutrons and electrons page; this one is about what the differences do.

The word came from a dinner party

Frederick Soddy needed a name in 1913 for something awkward: substances that were chemically inseparable but radioactively distinct, and which therefore had to occupy the same square of the periodic table. Margaret Todd, a physician he met at a family dinner in Glasgow, suggested isotope, from the Greek for "same place". Soddy took the word and the 1921 Nobel Prize in Chemistry.

The evidence that ordinary, non-radioactive elements are also mixtures came from J. J. Thomson in 1913, who found that a beam of neon split into two — mass 20 and mass 22 — and from Francis Aston, whose mass spectrograph turned that observation into systematic measurement across the periodic table.

Where 35.45 comes from

The standard atomic weight is an abundance-weighted mean. Each isotope's exact mass is multiplied by the fraction of atoms that are that isotope, and the products are added.

Chlorine has two stable isotopes:

Convert the percentages to decimals and multiply:

34.96885 × 0.7576 = 26.4924
36.96590 × 0.2424 =  8.9605
                    -------
                    35.4529

Which rounds to 35.45, exactly as printed.

The weighting is the whole point. A simple average of 35 and 37 would give 36, which is nearly a whole unit out. Two mistakes account for almost every wrong answer here: forgetting to convert percentages to decimals, which inflates the answer by a factor of a hundred, and averaging the mass numbers rather than the exact masses, which is close enough to look right and wrong in the third figure.

A useful instant check is that the result must always fall between the lightest and heaviest isotope, and must lie nearer the more abundant one. 35.45 is between 35 and 37 and much closer to 35, which is what a 76% majority should produce.

Running the average backwards

The harder version of the question gives you the average and asks for the abundances. It is one linear equation.

Boron has two stable isotopes, boron-10 at 10.0129 u and boron-11 at 11.00931 u, and an atomic weight of 10.81. Let x be the fraction that is boron-10; then the fraction that is boron-11 must be (1 − x), because the two together account for all of it.

10.0129x + 11.00931(1 − x) = 10.81

Expand and collect:

11.00931 − 0.99641x = 10.81
0.99641x = 0.19931
x = 0.2000

So natural boron is about 20.0% boron-10 and 80.0% boron-11. The measured values are 19.9% and 80.1%, and the small discrepancy is real rather than arithmetical: boron is one of the elements whose isotope ratio varies measurably between sources, so IUPAC quotes a range for it in place of one fixed value.

The trick that makes this manageable is refusing to use two unknowns. Writing the second abundance as (1 − x) rather than y turns a pair of simultaneous equations into one.

One isotope or ten

The number of stable isotopes an element has varies enormously, and it has consequences you can see on the periodic table.

Twenty-six elements have exactly one stable isotope — fluorine, sodium, aluminum, phosphorus, manganese, cobalt, gold and others. Their atomic weights are known with extraordinary precision, because there is no mixture to average and no natural variation to worry about. Fluorine's is 18.998403163.

Tin holds the record at ten stable isotopes. Its atomic weight, 118.710, is a compromise between masses spread from 112 to 124, and it corresponds to no actual tin atom whatsoever.

At the other extreme, some elements have no stable isotope at all. Everything from polonium upwards is radioactive, which is unsurprising. What is odd is that two elements in the middle of the table, technetium (43) and promethium (61), also have none — both are hemmed in by neighbours that do. The reason is a quirk of nuclear pairing rules: for those two proton counts, every possible neutron number produces a nucleus that can decay into a more tightly bound neighbour with the same mass number. Technetium's absence from nature left a hole in Mendeleev's table that was not filled until it was made artificially in 1937.

The line that stopped being true in 2003

For most of the twentieth century, every textbook named bismuth-209 as the heaviest stable nuclide. In 2003 a group at the Institut d'Astrophysique Spatiale in Orsay detected its alpha decay and measured a half-life of about 2 × 10¹⁹ years — roughly a billion times the age of the universe.

Bismuth is therefore radioactive, and also, for every practical purpose, not. A kilogram of it undergoes a handful of decays per minute. The episode is a good illustration of what "stable" means in this field: many nuclides are observationally stable, meaning no decay has ever been detected, which is a statement about the sensitivity of experiments rather than about the nucleus. Lead-204, tellurium-128 and several others are in the same category.

Heavy water is not just water

Courses teach that isotopes are chemically identical. That is a good approximation and it is not true, and the exceptions matter.

Replacing hydrogen with deuterium doubles the mass of the atom, which is the largest proportional mass change available anywhere in chemistry. Heavy water, D₂O, freezes at 3.8 °C rather than 0 °C, boils at 101.4 °C, and is about 10% denser than ordinary water. Reactions in which a hydrogen–carbon bond breaks in the rate-determining step can run up to about seven times slower when that hydrogen is deuterium — the kinetic isotope effect, which chemists use deliberately to work out reaction mechanisms, and which the pharmaceutical industry now uses to slow the metabolism of drugs by deuterating them at the position the body attacks first.

For heavier elements the effect fades to nothing. Uranium-235 and uranium-238 differ in mass by 1.3%, and their chemistry is indistinguishable by any ordinary means — which is precisely why separating them is one of the hardest industrial problems there is.

Two half-lives for carbon-14

Radiocarbon dating carries a discrepancy that is often mistaken for an error.

Willard Libby's original 1949 determination gave carbon-14 a half-life of 5568 years. Better measurements in the 1960s put the true value at 5730 ± 40 years, about 3% higher. Radiocarbon laboratories nonetheless still report conventional ages calculated with Libby's 5568-year figure, by international agreement, so that ages published decades apart remain directly comparable. The correction is folded into the calibration curve instead.

So a laboratory reporting an age is knowingly using a number it believes to be wrong, for the sake of consistency, and saying so in the footnotes. Uranium–lead dating has no such convention, which is one reason Clair Patterson's 1956 determination of the Earth's age at 4.55 billion years from meteorite lead isotopes has needed so little revision since.