Sunday, June 6, 2010

Burning questions of a Saturday morning.

So there I was lying in bed this morning, still half-asleep, when I got to pondering the fact that I, like the atoms that compose me, am made up of 99.99% empty space. I then idly contemplated the fact that most of this mass was in the nuclei of my atoms, at which point I was seized by a burning desire to find out how much all the electrons in all the atoms of my body put together weighed.


I know that last I checked, I had a mass of approximately 60 kg (my weight is left as an exercise for the reader), and a little searching on the 'net produced a table with the percent-by-mass composition of an average person. Starting with the first entry in the list, I have around \(61.35\text{%}\cdot 60\ \text{kg} = 36.81\ \text{kg}\) of oxygen in me. That's 36,810 grams of oxygen. A mole of oxygen being 16.00 grams, that's almost exactly 2,300 moles of oxygen in me.

(A mole, in chemistry, is simply a very large number \(-\ 6.022\times10^{23}\), to be exact. The conversion between grams and moles is pretty easy; if you have an amount of a substance equal in grams to the atomic weight of one atom (or molecule) of that substance, you have one mole of that substance, i.e., \(6.022\times10^{23}\) atoms (or molecules) of the substance. Since the atomic weight of oxygen is 16.00 atomic mass units, one mole of oxygen is 16.00 grams of oxygen. One more of oxygen at \(0^\circ\)C and normal atmospheric pressure would occupy about 22.4 liters, or just under 6 gallons.)

Now, each oxygen atom comes with 8 electrons in tow, \(8\) electrons \(\times\) \(2,300\) moles \(\times\) \(6.022\times10^{23}=1.11\times10^{28}\) electrons.

That...is a very large number. (11.1 billion billion billion).

And that's just the first element on the list! (Although, by virtue of its nature, it is likely to provide the largest number of electrons.) Next up is carbon, which, at 22.83%, comprises 13.70 kg of me. Dividing by 12.00 grams per mole, and multiplying by 6 electrons to each carbon atom, we arrive at \(3.09\times10^{27}\) electrons from carbon, a factor of 10 less than from oxygen (not surprisingly).

Since the procedure is relatively boring, and quite simple, I will spare you further talk and simply list the amounts in the table below:


oxygen        36.81 kg  \(1.11\times10^{28}\) electrons
carbon        13.70 kg  \(3.09\times10^{27}\) electrons
hydrogen      6.00 kg   \(3.57\times10^{27}\) electrons
nitrogen      1.54 kg   \(4.64\times10^{26}\) electrons
calcium       0.86 kg   \(2.58\times10^{26}\) electrons
phosphorus    670 g     \(1.94\times10^{26}\) electrons
potassium     120 g     \(3.51\times10^{25}\) electrons
sulfur        120 g     \(3.61\times10^{25}\) electrons
sodium        84 g      \(2.42\times10^{25}\) electrons
chlorine      84 g      \(2.43\times10^{25}\) electrons
magnesium     18 g      \(5.35\times10^{24}\) electrons
iron          6.0 g     \(1.68\times10^{24}\) electrons
fluorine      2.4 g     \(6.85\times10^{23}\) electrons
zinc          1.8 g     \(5.00\times10^{23}\) electrons
silicon       0.60 g    \(1.80\times10^{23}\) electrons
rubidium      0.60 g    \(1.56\times10^{23}\) electrons
strontium     0.30 g    \(7.84\times10^{22}\) electrons
bromine       0.24 g    \(6.33\times10^{22}\) electrons
lead          0.12 g    \(2.86\times10^{22}\) electrons
copper        0.06 g    \(1.65\times10^{22}\) electrons
aluminum      60 mg     \(1.74\times10^{22}\) electrons
cadmium       60 mg     \(1.54\times10^{22}\) electrons
cerium        60 mg     \(1.50\times10^{22}\) electrons
barium        18 mg     \(4.42\times10^{21}\) electrons
iodine        18 mg     \(4.53\times10^{21}\) electrons
tin           18 mg     \(4.57\times10^{21}\) electrons
titanium      18 mg     \(4.98\times10^{21}\) electrons
boron         18 mg     \(5.01\times10^{21}\) electrons
nickel        12 mg     \(3.45\times10^{21}\) electrons
selenium      12 mg     \(3.11\times10^{21}\) electrons
chromium      12 mg     \(3.34\times10^{21}\) electrons
manganese     12 mg     \(3.29\times10^{21}\) electrons
arsenic       6.0 mg    \(1.59\times10^{21}\) electrons
lithium       6.0 mg    \(1.56\times10^{21}\) electrons
cesium        5.4 mg    \(1.35\times10^{21}\) electrons
mercury       5.4 mg    \(1.30\times10^{21}\) electrons
germanium     4.2 mg    \(1.11\times10^{21}\) electrons
molybdenum    4.2 mg    \(1.10\times10^{21}\) electrons
cobalt        2.4 mg    \(6.62\times10^{20}\) electrons
antimony      1.8 mg    \(4.54\times10^{20}\) electrons
silver        1.8 mg    \(4.72\times10^{20}\) electrons
niobium       1.2 mg    \(3.19\times10^{20}\) electrons
zirconium     0.60 mg   \(1.58\times10^{20}\) electrons
lanthanum     0.60 mg   \(1.48\times10^{20}\) electrons
gallium       0.60 mg   \(1.61\times10^{20}\) electrons
tellurium     0.60 mg   \(1.47\times10^{20}\) electrons
yttrium       0.54 mg   \(1.42\times10^{20}\) electrons
bismuth       0.42 mg   \(1.00\times10^{20}\) electrons
thallium      0.42 mg   \(1.00\times10^{20}\) electrons
indium        0.36 mg   \(9.25\times10^{19}\) electrons
gold          0.18 mg   \(4.35\times10^{19}\) electrons
scandium      0.18 mg   \(5.06\times10^{19}\) electrons
tantalum      0.18 mg   \(4.37\times10^{19}\) electrons
vanadium      0.12 mg   \(3.26\times10^{19}\) electrons
thorium       0.060 mg  \(1.40\times10^{19}\) electrons
uranium       60 μg     \(1.40\times10^{19}\) electrons
samarium      42 μg     \(1.04\times10^{19}\) electrons
beryllium     30 μg     \(8.02\times10^{18}\) electrons
tungsten      18 μg     \(4.36\times10^{18}\) electrons


(if you're wondering about the units, conveniently, 0.06 g = 60 mg and 0.06 mg = 60 μg)

If you add all those electrons up, you come up with a total of \(1.88\times10^{28}\) electrons. That's a lot of electrons -- 18.8 billion billion billion, to be exact. But how much mass do they have? And how much do they weigh?

Finding the mass is easy. The mass of an electron is \(9.11\times10^{-31}\) kg so \(1.88\times10^{28}\) of them together have a mass of 17.1 grams, which has a weight of approximately 1/25 of a pound.

Wow. All the electrons in my body make up a measly 0.0000029% of my mass. Everything else is concentrated in the nuclei of my atoms. Wild, huh?

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