# Does a drop of water contain one molecule per litre of water on Earth?

## Question

The molecules in a single drop of water diluted evenly in the Earth’s oceans would result in the density of one molecule per litre of sea water. In other words, if you made one little drop of water go into the sea somewhere, given enough time, there’s enough molecules in one drop to spread out across the whole of the Earth’s oceans so there was one molecule per litre...

## Answer

The answer is that this is true and here is the calculation to prove it:

There are about 24,000 droplets in one litre. That gives you a droplet volume of about 0.03cm cubed.

The relative molecular mass of water is 18g per mole. A mole is the number of molecules; so 18g is the mass in grams of the molecules in one mole. You need that to calculate the next bit of the equation.

One litre of water has a mass of 1000 grams, and in one litre there must be 1000/18 moles.

We know from Avogadro's constant there are 6.022 10^{23} molecules in a mole of something. That means that, in a litre of water, there must be 1000/18 X 6.022 X 10^{23} molecules.

That means, per droplet, you have to divide that number by 24,000 because there are 24,000 droplets in a litre. So there must be 1.39 X 10^{21} molecules of water in an individual droplet of water, which is an amazing number of molecules.

The volume of the Earth's oceans is 1.37 X 10^{9} cubic km. That's a reasonably well-understood figure. If you need to convert that into metres cubed you have to times it by 1000 cubed because there's a thousand metres in a kilometre. That's 10^{9}.

To turn that into litres you've got to times it by 1000 cubed again. So that's 10^{12}. That means that, on Earth, there are 1.37 X 10^{21} litres of water. That's nearly the same number as there are molecules in the droplet of water. If you put one into the other you have almost one molecule of water per litre of water on Earth!

## Comments

## why

you did not answer the question

## Err - yes we did!

If you follow through the calculation you'll see that we do answer the question.

## is the number so large it can

is the number so large it can only be expressed exponentially?

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