How many molecules are there in a glass of water? (Natural Sciences, Cambridge)

A lot, is the simple answer. Molecules are so tiny and there are likely to be so many in a glass of water that it would be impossible to count them directly. So I’m going to have to come at the answer indirectly. In fact, the calculation is fairly simple, and the reason this is so takes us back to the very foundations of chemistry, and the first beginnings of the atomic theory of matter.

Indeed, it’s all down to a flash of insight by the man some called the father of chemistry, John Dalton. Back in the late 1700s, scientists knew about atoms but thought they were all the same size, and hadn’t quite twigged that each element has its own unique atom. While Dalton was doing some experiments with the gases of the air, he was surprised to see that pure oxygen will not absorb as much water vapour as pure nitrogen. He guessed that this is because oxygen atoms are bigger and heavier than nitrogen atoms, so leave less room. If this first guess was brilliant, his next was sheer genius.

The identity of atoms was all about weight ratios! And from that moment, weights have been at the heart of atomic and molecular chemistry – and they’re going to be the way to count our glassful of molecules.

Atoms are the ‘ultimate particles’ of each element, Dalton saw, and they combine to make compounds in very simple ratios. If that is so, he could work out the relative weight of an atom of any element just by measuring the total weight of the element involved in a compound. Simple and effective. Soon he’d worked out the relative atomic weights of each element then known.

He used hydrogen as a base, since it is the lightest gas, and gave it an atomic weight of 1. Because the amount of oxygen in water is seven times heavier than the amount of hydrogen, Dalton assigned oxygen an atomic weight of 7. Equally simple – or so he thought (it’s actually 16, or thereabouts).

Unfortunately, there was a flaw in Dalton’s method; he didn’t realise that atoms of the same element can combine. He always assumed that a compound of atoms, a molecule, had only one atom of each element. Of course, he was wrong.

This is where a near contemporary of Dalton’s comes in – probably very slowly, since his name is absurdly long and grand: Lorenzo Romano Amedeo Carlo Avogadro di Quaregna e di Cerreto. Avogadro, as he’s usually known, was an aristocratic Italian scientist. It had already been shown by Gay-Lussac that when two gases react together to form a third, they always combine in simple whole-number ratios. But for this to be true, Avogadro realised, equal volumes of any two gases at the same temperature and pressure must hold an equal number of particles. If so, then the ratios must mean a molecule can combine varying numbers of atoms. What matters in working out ratios, then, is molecular proportions. Over the next half-century, scientists realised that Avogadro’s idea of using molecular proportions would allow them to calculate atomic weights correctly.

Avogadro went on to show that equal volumes of a gas (at a given temperature and pressure) always contain an equal number of atoms or molecules. In other words, the relationship between volume and the particle number is always the same, and since 1909 it’s been called Avogadro’s constant.

What Avogadro’s constant does is tell us how many particles there are in a particular amount of a substance. The numbers involved, of course, are so huge and unmanageable that a special unit has been devised for this: the mole (a word which owes its origin to molecules, not infiltrators or blind burrowers).

Although Avogadro came up with the principle in the early 1800s, it wasn’t until 1910 that Robert Millikan was finally able to give a number to the mole. In the same way as I will with our glassful of molecules, Millikan came at it indirectly. He simply measured the total electric charge in a particular mass of carbon-12 then divided it by the recently discovered charge on a single electron. That way he could work out how many electrons the mass contained. The numbers were of course absolutely gigantic. In every 12g of carbon-12, there are 6.022 × 10²³ atoms! Some very keen chemists celebrate 10/23 (23 October) as Mole Day every year …

The number has since been refined,¹ but that figure will do for my estimate. A mole is the amount of a substance that contains this number of particles, whether it be molecules, electrons or atoms. Since the atomic mass of hydrogen is about 1, a twelfth that of carbon-12, it contains this number of particles in just a twelfth of this mass – that is, 1g. A mole of hydrogen is 1g. Oxygen has an atomic mass of about 16, so a mole of oxygen is 16g. So a mole of water, H₂O, its molecular mass, is 1g + 1g + 16g, that is, 18g.

So the key to my calculations is mass, as it was with John Dalton two centuries ago. I can’t count the molecules in a glass of water, but I can have a good guess at the water’s mass. I’m going to guess that there’s a fifth of a litre of water in the glass – so 200g of water.

Since the molecular mass of water is 18g, this means the glass contains just over eleven moles (200 divided by 18). So there you have it: there are approximately 11 × 6.022 × 10²³ or 6.624 × 1024 molecules in a glass of water. That’s just about 6 trillion trillion.

This is only an approximate figure of course, but the method works, and if I was able to measure the weight of water accurately and used the very precise figures for atomic weights now known, I could work out the number of molecules in a glass precisely. And I was right first time. It’s a lot …

Footnote

1 6.02214129(27) × 10²³