Gauging the size of a molecule of water - an exercise in using maths in science teaching

There's an interesting proposition that gives a staggering indication of just how tiny a molecule of water is.  It is that there are more molecules in a glass of water than there are glasses of water available in all the oceans, seas, lakes and rivers of the world.  Can this be true?  And what's more, can we actually test this bold claim in the classroom?  

Spoiler alert - yes, it is true, and I'll demonstrate below how you can show this.  Before I do though, it would be worth thinking about when you might incorporate this little gem into your teaching, and how.  The maths is relatively straightforward, by which I mean it involves nothing but the basic operations such as plus, minus, times and divide, but I've seen undergraduates and postgraduates struggle with it, so you shouldn't assume it's particularly easy.  There are two things that make it difficult: one is that it involves very large numbers that inevitably have to be written in standard form - something children will probably learn about at a certain point during secondary school, but which might be a bit tricky if they've never encountered it.  The second thing is that there is an inevitable interconversion of units of volume, which sounds easy enough but has some common pitfalls.  For instance, most people would know there are 1,000 metres in a kilometre.  However, ask them how many cubic metres there are in a cubic kilometre and you'll often hear the same answer (1,000) - because people forget that they need to 'cube' the 1,000.  The actual answer is 1 billion, which is 1,000 x 1,000, x 1,000.  

Normally when I teach about this I want to convey the seeming impossibility of the claim.  I ask my students if they've ever stood on a beach (just in case they haven't, although usually all have).  I share my own perception of how vast the sea is, just from looking at it, and then I recollect that in my head I'm normally only thinking of the North Sea (as I'm usually standing on a UK East Coast beach in my head), and I invite them to contemplate the gargantuan size of the Atlantic Ocean, and then the incomprehensibly bigger Pacific Ocean.  This is to get them trying to imagine how many glasses we could fill with all that water - it must be trillions of trillions...  This is to then pose the question, how could it be feasible that the number of molecules in the glass could be bigger than that huge number we've just been imagining...?     

The mathematical demonstration

The first step is to work out how many glasses of water there are available on Earth.  For that we need to know the volume of all the water on Earth.  A quick internet search gives a total volume for all the Earth's seas and oceans as 1.386 x 10⁹ km³

This is something we cannot easily confirm for ourselves, so we do just have to accept that it's a close enough figure (albeit an estimate).  To look for yourself, go to hypertextbook.com/facts/2001/SyedQadri.shtml

It's worth pointing out that the amount that is in lakes and rivers is vanishingly small compared to this amount, so we don't really need to worry that it would make a big different to the calculation.

Now, we would not measure the volume of a glass of water in cubic kilometres - we'll use litres, and we'll suggest our glass has a volume of half a litre, which is 500ml, or about a pint (if any of you prefer imperial units).  The first challenge therefore is to convert the 1.386 x 10⁹ km³of the Earth's water into litres.

Step 1:  Convert cubic kilometres into cubic metres

1 km = 1,000 m = 10³ m

1 km³ = 10³ m x 10³ m x 10³ m  = 10⁹ m³  

\ 1.386 x 10⁹ km³ = 1.386 x 10⁹ x 10⁹ m³ = 1.386 x 10¹⁸ m³

Step 2:  Convert cubic metres into millilitres

1 millilitre = 1 cm³

1 m = 100 cm

1 m³ = 100 cm x 100 cm x 100 cm  = 10⁶ cm³  =  10⁶ millilitres

\ Volume of Earth's water = 1.386 x 10⁹ km³ = 1.386 x 10¹⁸ m³ = 1.386 x 10¹⁸ x 10⁶ millilitres = 1.386 x 10²⁴ millilitres

Step 3:  Work out how many 500 millilitre glasses that is

This is just dividing the volume of water (in millilitres) by 500:

1.386 x 10²⁴/ 500  = 2.772 x 10²¹ glasses  (A)

Step 4:  Work out how many molecules in the 500 millilitre glass

For this we'll make use of the density of water, the molar mass of water and the Avogadro constant.

1 mole of water (its molar mass) = 18 g     (For anyone needing a reminder, this is the relative molecular mass in grams)

Density of water at 'normal' temperature = 1 g/cm³

\ Volume of 1 mole of water = 18 cm³

Number of moles in 500 millilitres of water = 500/ 18 = 27.8 moles

1 mole contains 6.022 x 10²³ molecules

\ 27.8 moles = 27.8 x 6.022 x 10²³  = 1.674 x 10²⁵ molecules  (B)

Step 5:  Compare the two numbers calculated (A and B)

B/A = 1.674 x 10²⁵ / 1.386 x 10²¹ = 2.416 x 10⁴ = 24,160

In other words, there are approximately 24,000 times more molecules of water in the pint glass than there are pints of water on Earth.

Now that, in my view, is amazing!

It means that you would need 24,000 Earths before the number of pints of water matched the number of molecules in one glass.  This gives some sense of how tiny those molecules are.

As an aside, if you wanted to do the comparison for smaller bodies of water, I found the following data: Lake Windermere volume 314.3 x 10⁶ m³ = 3.143 x 10⁸ m³ (https://en.wikipedia.org/wiki/List_of_lakes_of_the_Lake_District)

Loch Ness volume = 7.5 km³  (https://en.wikipedia.org/wiki/Loch_Ness)