Sunday, 9 May 2010

What if everyone were a nucleus?

In a series of tweets this evening Jim Al-Khalili pointed out that there has been a small disagreement between him in his Atom series and Michael Mosley in the Story of Science. They each illustrated the ratio of occupied space to empty space in atoms by saying that if all the empty space were taken out of the entire human population then we would occupy a volume the size of an apple (Jim's calculation) or a sugarcube (Michael's calculation).

In either case, the analogy makes clear that atoms have a whole lot of empty space, but in terms of volume, there's a fair bit of difference between a sugarcube (around 1 cm3) and an apple (around 200 cm3). So who is right?

The population of the world is around 7 billion. The average mass of people is usually taken to be around 70kg so the mass of the human population is 7×109×70 kg = 490 000 000 000 kg. Let's call that 5×1011 kg. Now, if all the space were taken out of all these atoms, we would essentially be left with an enormous nucleus (as Jim says, a pulsar). The density of nuclear matter (i.e. of the inside of an enormous nucleus) is 0.16 nucleons per cubic femtometer. A nucleon weighs 1.7×10-27 kg.

So: The number of nucleons in total is 5×1011 / 1.7×10-27 = 3×1038 nucleons, giving a volume of 3×1038/0.16 = 2×1039 fm3 = 2 cm3.

Looks like I agree with Michael, more or less... unless I've guessed the size of a sugarcube wrongly. I mean, I haven't seen a sugarcube for years.


  1. Wilton Catford9 May 2010 at 09:57

    The size of a sugar cube turns out to be the least well defined part of the calculation... but according to the data on this link, the standard 48/cup sugar cube comes out at 2.9 cm3, whilst one correspondent has a packet of them at 1.8 cm3... so 1.83 rounded to 2 is sustainable ;) unless you're talking about Bjork; she is apparently 5'4" in one dimension (but I don't have the others to hand)

  2. I feel the guests at a Mad Hatters Tea Party would disagree. Relativity dear fellows, relativity!

  3. Wilton (inadvertently) brings up a good point - how much do the order of magnitude calculations disagree when based on mass or volume?