Since we’ve done the hard work of counting those grains of salt, let’s play with this a bit more.   A thousand grains of sand corresponds to a cm³ pile.  There would therefore be 6 x 10²⁰ such piles in an Avogadro’s number pile of sand.  If we take a skyscraper of 100 stories, like the Empire State Building, as our standard, how big a box would our sand pile fill?  100 stories is about 1 thousand feet, which is about 300 meters or 30,000 cm.  30,000 goes into 6 x 10²⁰ 2 x 10¹⁶ times, which corresponds to a square 1.4 x 10⁸ cm on a side.  That corresponds to 1.4 million meters, or 1.4 thousand kilometers, or about 875 miles on a side.  This is an expanse larger than Alaska, all covered by sand up to the height of a skyscraper!

Now then to put a cap to the magic of Avogadro’s number, take out a small graduated cylinder and fill it with 18 milliliters of water.  That is a mole’s worth of water molecules. Do you see why?  Water is H₂O.  The relative atomic weight of H is 1, so H₂ is 2.  The relative atomic weight of O is 16, and 2 + 16 = 18.  The relative molecular weight of water is 18, and a mole of these molecules would weigh 18 grams. Now we know a grain of sand is tiny, but think of how much smaller a molecule of water must be if you can pack 6 x 10²³ in the small space of the cylinder, when that many grains of sand would more than cover the entire state of Alaska up to the height of the Empire State building.

I enjoy this sort of play, and think it is really valuable for students…especially in an age dominated by calculators and computers.  It gives students a facility at making estimates and working at the issue of scale.  I like to say that most of the time, for our purposes, π = 3 and π² = 10. That seems to capture the sort of play in this approach. The great physicist, Enrico Fermi, was famous for devising and solving such problems.  He was able, for example, to estimate the power of the first atomic explosion in the New Mexico desert by seeing how far bits of paper were blown by the blast.  If you are interested there are many such problems online.

Finally, since students are often uncertain about exponents, it’s also useful to stop and play with this a bit.  If you ask how much is one half of, say, 8 x 10⁶, they will often answer 4 x 10³; instead of 4 x 10⁶…even though they know that half of 8 million is 4 million and not 4 thousand.