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Putting all the world’s water in buckets

Following this pair of tweets about water:

The obvious question is, at what point are the two numbers the same? Or,

If you put all the Earth’s water into containers of the same size so that each container carries as many atoms of water as there are containers, how big is each container?



We want this equation to hold:

\[ \textrm{atoms in }1l \textrm{ of water} \times \textrm{volume of container} = \frac{\textrm{litres of water on Earth}}{\textrm{volume of container}} \]

Rearrange that to get an expression for the volume of the container:

\[ \begin{align} \textrm{volume of container} &= \sqrt{\frac{ \textrm{litres of water on Earth} }{ \textrm{atoms in }1l \textrm{ of water}}} \\ &= \sqrt{ \frac{1.386 \times 10^{21}}{1 \times 10^{26}}} \\ &= \sqrt{1.386 \times 10^{-5}} \\ &= 0.00372290209\;l \\ &\approx 3.723\;ml \end{align} \]


The container is really, really tiny — just over half a teaspoon!



Qikipedia’s fact seems to be a pretty epic understatement, then. There are “more” atoms than buckets, but there are ten million times more. It says a lot about how hard it is for people to grasp the size of huge numbers or the teeniness of atoms that this statement looks so much more impressive than “there are more buckets of water in this reservoir than there are states in the USA”.