It has been said (but I don’t know who originally said it) that every mouthful of London water drunk has already been drunk seven times.

Wrong.

One day, sitting in a pub in Dorset and watching the rain lashing down over the hills, I wondered what proportion of the world’s water was in my drink – and so how many molecules of it had already been drunk by someone else.

The answer is pretty startling, and a consequence of the very large magnitude of the Avogadro Constant.

The total volume of the world’s water is said to be 1.385 x 10⁹ km³ (Greenwood & Earnshaw, ‘Chemistry of the Elements’: Pergamon 1984). This is 1.385 x 10¹⁸ m³, or 1.385 x 10¹⁸ tonnes.

Suppose that the ‘average’ human excretes during the ‘average’ 70-year lifespan 2 dm³, that is 2 kg, of urine a day. The total mass of water excreted in a lifetime is thus

70 a x 365.25 d a^-1 x 2 kg d^-1 » 50,000 kg = 50 tonnes.

The fraction of the world’s water that every (deceased) human has urinated on average is therefore

50 tonnes/1.385 x 10¹⁸ tonnes = 3.61 x 10^-17

Now, I suppose that you have had a mug of tea, or coffee, or a Diet Coke, sometime today. You will perhaps wonder how many of those molecules have already been drunk by others…..

I make the assumption that any excreted water comes into equilibrium with the whole of the world’s water sooner or later. This is probably not true, but even if it is two orders of magnitude adrift it doesn’t make much difference to the conclusion.

Your mug of tea is roughly 330 cm³, that is 330 g/18 g mol ^–1 or roughly 18.3 moles of water. This contains 18.3 mol x 6.023 x 10²³ molecules mol^-1 or 1.1 x 10²⁵ molecules.

If every human is responsible for 3.61 x 10^-17 of the world’s water, then the number of molecules in your drink that have already been drunk by any person you care to name is

3.61 x 10^-17 x 1.1 x 10²⁵ = 3.98 x 10⁸.

Enjoying your tea?