How Long Is A Spoonful Of Water?

I can't quite remember how the teaspoon question arose except to recall that it came up in conversation with my son, a physics student.

First let me clarify the question. To state it more precisely: if you took all the molecules in a teaspoon of water and laid them end to end, how far would that thin aqueous line stretch? 

The first step is to figure out how many water molecules we have. 

A typical teaspoon holds about 5 millilitres (mL), which weights 5 grams. To find out how many water molecules there are in 5 grams you need to know that the molecular weight of water is 18 — the sum of the weights of one oxygen atom (16) and two hydrogen atoms (1 a piece) in H2O. What that means is 18 grams of water contains one mole of water molecules. Students of chemistry also know the mole to be a defined number of atoms or molecules (which relates the arbitrarily set scale of atomic weights — hydrogen = 1, helium = 2, and so on — to the actual weights of atoms in grams). It's rather big: one mole is 6.022 x 1023 in scientific notation. That's

602,200,000,000,000,000,000,000

So in 5 grams of water there would be 5/18ths of this number which is:

167,300,000,000,000,000,000,000

In other words: a lot.

But water molecules are very small; each one is only about 0.3 nanometers wide. That's 0.0003 micrometres, or 0.0000003 millimetres or 0.0000000003 metres. These are bizarre numbers — we have no real experience of them so it's hard to get much sense of scale. But let's plough on anyway.

If we lay down 167,300,000,000,000,000,000,000 water molecules end to end, the total length of the line is:

167,300,000,000,000,000,000,000 molecules x 0.0000000003 metres per molecule.

Which is 50,190,000,000,000 metres.

Or 50,190,000,000 km (that's 31,368,750,000 miles for older readers).

Which is 50 billion km. (How good was your guess?)

That's over 10 times the width of the solar system. From a teaspoon. 

Just think how far you could go with a bucket of water. 

Stephen Curry is a Professor of Structural Biology at Imperial College.