I was interested in figuring out how deep a mole of peas would cover the land in all 50 states of the U.S. (The background to this comes from a previous blog post: https://sublimityofinfinity.wordpress.com/2015/10/30/picturing-a-mole-of-peas/.) The land area of the U.S., including Alaska and Hawaii, is 9,158,022 km^{2} (from Wikipedia). I counted out 285 peas that filled a 1/2-cup measuring cup. A cup is 236.6 cm^{3} and contains 2×285 = 570 peas. So, the number-density of a random pile of peas (not close-packed in a crystal) is 570 peas/236.6 cm^{3} = 2.41 peas/cm^{3}. The volume of a mole of peas, then, is 6.022 x 10^{23 }peas x (1 cm^{3}/2.41 peas) = 2.50 x 10^{23} cm^{3}. We already have the area of the bottom of the U.S.-sized pea container, so the only unknown variable is the height of that volume, h: h x 9,158,022 km^{2 }= 2.50 x 10^{23} cm^{3}. Converting cm^{3} to km^{3} uses 100 cm = 1 m = 0.001 km, or 1 cm = 1 x 10^{-5 }km, so 1 cm^{3} = 10^{-15 }km^{3} and 2.50 x 10^{23} cm^{3} = 2.50 x 10^{8} km^{3}. Solving for h: h = 2.50 x 10^{8} km^{3}/9,158,022 km^{2} = 27.3 km = 16.9 miles.

Now, it’s good to know the error on this number. I am willing to believe that there may be up to a 10% error on my pea count for 1/2 a cup. I was just putting them in a measuring cup and eyeballing when it was perfectly full. There were some peas poking above the rim of the measuring cup, and some were just below. But at most 28 peas one way or the other would get the exact number density, but I bet it was more like 5% or 14 peas. This 5% error would propagate right through the calculation, so that the height has a 5% error on it: 0.8 km. Even if it’s 10%, that’s only 1.7 miles out of 17 and the main point of the analogy is preserved: it’s a huge pile of peas.

Also, taking a cue from Randall Munroe, creator of xkcd and author of the awesome book What If?, I am assuming that the peas on the bottom of the 17 mile high pile won’t be smooshed and liquefied, reducing their volume, but will remain round and pea-like.

Loved What If. Keep them coming Pete.

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