The Avogadro Constant N_A is well-known as a fairly large quantity – in some parts of the world it is called Loschmidt’s number, L, after the Austrian chemist who first estimated its value. But how large is it?

About 20 years ago a colleague in my then department rang me; he wanted to get a pile of sand with N_A (or L) sand grains in it to show to his pupils. I said that if he could find a merchant to supply it then I would guarantee that the school would pay for it. Perhaps he would let me know how much he needed…..

For the sake of this order-of-magnitude calculation let’s assume that each grain is a cube 0.1mm on a side, and that the grains pack perfectly with no spaces. Since it makes the calculation marginally easier, let’s also deal with 10²⁴ grains – this is the same as the number of molecules in 30cm³ or a couple of tablespoonsful of water.

• If each grain is 0.1mm on a side, using our assumptions how many grains are there in a cubic metre?

There are (10⁴)³ = 10¹² sand grains m^-3.

• What volume of sand is therefore needed to produce 10²⁴ grains?

The volume required is 10²⁴ grains/10¹² grains m^-3 = 10¹² m³.

• If the sand is to be delivered by lorry, each one will carry 10 m³ of sand. How many lorry-loads will be needed?

 

The number of loads will be 10¹² m³/10 m³ load ^–1 = 10¹¹ loads.

• Now we know how much sand we need, we can order it. The builders’ merchant would like some sort of schedule, so let’s have one lorry-load every 10 minutes, day and night throughout the year. How many loads per year is this?

This comes out to be 6 loads hr ^-1 x 24 hr d ^–1 x 365.25 d a ^–1 = 52596 loads a ^-1.

• How long will it take to deliver the sand?

It will take 10¹¹ loads/52596 loads a ^–1 = 1.9 x 10⁶ a.

Now you know why I guaranteed to buy it.

Archimedes had some interest in the number of grains of sand in the universe; and the number of sand-grains on Coney Island has been estimated at 10²⁰, well short of what we would need.

To get a sense of the effects of powers of ten, there is no better book than ‘Powers of Ten’, from Scientific American. It has a series of photos each taken at 10 times the distance of the previous one, and is pretty startling.

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