The *New York Times* recently had an article on adding peas to your guacamole; peas reduce the number of costly avocados needed, thereby also reducing the calories and fat, all without substantially changing the taste. (This was not without controversy!). Well, this post fits in nicely, though I’m writing here about a “mole” of peas and not a “guaca-mole” of peas. (By the way, I hate peas, but I am willing to try this recipe because I know how healthy they are).

Those of you who have had chemistry at some point in your life know that a mole is the main unit of quantity for a chemist, just like a dozen is the main unit of quantity for a baker. A mole is defined as the number of carbon-12 atoms in exactly 12 grams of carbon-12. (The ’12’ designation in carbon-12 refers to one specific isotope of carbon, the one with 6 neutrons….naturally occurring carbon has a mixture of isotopes, including mostly carbon-12 and 1% carbon-13 (7 neutrons), so a mole of naturally occurring carbon atoms weighs 12.011 grams). The original designation of a mole was the number of hydrogen atoms in exactly 1 gram of hydrogen, since hydrogen is the lightest element, but for various reasons the official description was switched to carbon-12.

Once the existence of atoms finally became widely accepted among scientists around 1900, they naturally began to wonder just how big these atoms were. How many atoms were in that mole of carbon-12 that weighed 12 grams? Through a variety of means it was found that this number, which is called Avogadro’s number, is incredibly huge: 6.0 x 10^{23}. (That 10^{23} notation means we have to scoot the decimal over 23 spaces, i.e. 600,000,000,000,000,000,000,000.) One mole of water molecules weighs around 18 grams and occupies 18 mL (mL is ‘milliliter’ which equals 1 cc or ‘cubic centimeter’), which is about 1.2 tablespoons of water. So, we know there are 6.0 x 10^{23} water molecules in 1.2 tablespoons of water.

How can we possibly picture that, though? Avogadro’s number is so huge that we have great difficulty wrapping our mind around its immensity; it’s difficult to truly appreciate how large it is. Similarly, we also have difficulty fathoming just how small individual atoms are! So, just as we always do to increase understanding, we turn to analogies. There are all sorts of analogies out there for visualizing how big a mole is, such as marshmallows or marbles, but I like using peas the best.

The question is this: what would a mole of peas look like? How much volume would they take up? Could we put them in a semi-truck? Peas are a good example, because we have everyday experience with them. They’re also pretty small, but not too small, and we typically see them all piled up. The question, then, is how big a pile is 6.0 x 10^{23} peas?

Here’s the answer: **a mole of peas would cover all the land in the United States to a depth of 17 miles**. This is not too difficult a number to arrive at: the calculation is here. I just took some garden-variety peas (pun intended) and counted how many filled up 1/2 a cup. My family looked on with a mixture of amusement and horror as I counted out 285 peas while they ate their dinners (I think I remember my 12 year old snidely asking “How do you spell ‘nerd’?”). Given the known land area of the U.S. from Wikipedia and knowing the number-density of that 1/2-cup of peas, etc., etc., I arrived at a depth of 17 miles.

So, if each water molecule in 1.2 tablespoons of water were enlarged to the size of a humble pea, *the entire U.S. would be covered to a depth of 17 miles*! It boggles the mind. I told my 15-year old daughter this, who is in her first chemistry class right now, and she just shook her head and said “I don’t believe that…I mean…<*shaking head*>…I know it’s 23 zeros for a mole, but I just don’t think it’s that many peas…I mean…<*gutteral sound of incredulity*>…”

This is how it goes when we encounter the sublime: “I know…but I don’t believe…I don’t see how…”

## 2 thoughts on “Picturing a Mole of Peas”