Counting the Uncountable
We can count parts, this being Part 1
Just before Christmas, we lost the writer John le Carré. Not lost in the sense of being misplaced in a sock drawer, euphemistically lost. The Brits thrive on the euphemism: we have scores of them for those words and subjects which polite company whisper. I could have written that le Carré had kicked the bucket, or had shuffled off this mortal coil, or had been called home. Enough of this brown bread, le Carré was nearly 90, so he had a good innings. For many of those years at bat, he carried with aplomb the spy novel genre. He started his innings playing after the great openers of Joseph Conrad and W. Somerset Maugham. Alongside Eric Ambler, Graham Greene and Len Deighton, he formed a formidable top order and with his passing we are now in the capable middle order hands of Mick Herron and Charles Cumming. Stumped?
Cricket is England (and Wales). Apparently, the Good Lord gave the English test match cricket so that we could experience eternity before we arrived at the Pearly Gates. Le Carré’s spy novels were likewise quintessentially British. That’s about as tenuous as I can get a link between le Carré and cricket to make the opening salvo work. I’ve read many obituaries and his memoir, and could not find out whether Mr. le Carré enjoyed the sound of leather upon willow. So upon appeal, I get to bat on. You can have plenty of fun writing with English (as I hope you have appreciated up to here with my euphemisms and cricketing idioms - count them if you like) and if you are good with your English writing, you may even slip (still there with the cricket terms) a word or two into the common vernacular. Mr. le Carré added a couple of spy terms to our already rich language:
to come in from the cold - after a period of isolation to be welcomed back into the community again.
honey trap - whereby an attractive (usually female) person is used to lure another (usually male) into revealing sensitive or secret information.
Perhaps his most lasting word is mole.
It’s interesting looking up the word mole in the dictionary. Thanks to the OED, you can see their full definition of the word. You thought it only meant talpa europaea. The rarely seen mammal with spade like paws, adapted to work well in its below ground environment. Digging tunnels to find its food of preference worms, it simultaneously aerates the soil, thus improving soil quality. This enables more plants to grow and more insects to thrive. It also improves soil drainage, reducing flooding and local puddling. A wonderful environmentally helpful animal, which I have been lucky enough to see twice. Alas, both times they had become worm food.
At school, I didn’t do well at chemistry. I remember being taught about moles. The chemical mole burrowed its way to us with the help of the eighteenth century Italian chemist, Amedeo Avogadro. The chemical mole derives, strangely, from the German molekül shortened to mol. I guess German was the lingua franca of the eighteenth century chemistry world. Our friends at the Bureau International des Poids et Mesures give us a definition of the mole as being:
The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 x 10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol⁻¹ and is called the Avogadro number.
The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.
Clear as water, right? I may not be the best person to embellish what this means, as I had a somewhat sticky wicket with my chemistry exams. What I know is, in layperson’s terms, for a mass of a chemical substance (i.e. good old H₂O), you can determine how many molecules (i.e. the number of hydrogen atoms and oxygen atoms) are contained within that mass using Avogadro’s law and constant.
By way of example (Disclaimer: Please, feel free to dispute all the calculations and numbers within this post. I’ve checked them to the best of my capabilities and they are ball-park. I am merely bowling out big numbers and small sizes to embellish a point. Nothing more, nothing less):
Two metric tablespoons of water (30ml) contain approximately 1,002,798,750,000,000,000,000,000 molecules:
Atomic mass of Hydrogen is 1.008g/mol
Atomic mass of Oxygen is 16g/mol
Atomic mass of H₂O is 18.016g/mol
Number of moles in 30ml (assuming normal temperatures and pressures, 30ml of water is 30g) of water = 30 x 1/18.016 = 1.6651865
If 1 mole contains 6.02214076 x 10²³ elementary entities then 30ml/30g of water contains:
1.6651865 x (6.02214076 x 10²³) = 1,002,798,750,000,000,000,000,000 molecules.
In researching this post, the foregoing has thrown up many questions that I must find answers for at some stage. However, for this post, let’s just say that 1,002,798,750,000,000,000,000,000 (or 1.00279875 x 10²⁴) is a big number and molecules of water are small. But exactly how big and how small?
The following is a crude table of volumes (in km³) and assumes, amongst many other things to make it easy, that all the objects below, except for the Milky Way, are perfect spheres.
Water molecule 2.991627 x 10⁻³⁸
Virus 1.022654 x 10⁻³⁰
Grain of sand 4.597231 x 10⁻¹⁸
Football 5.547487 x 10⁻¹²
Earth 1.087800 x 10¹²
Sun 1.412270 x 10¹⁸
Flat Milky Way 6.700000 x 10⁵¹
From these figures, you can appreciate just how small molecules are. You can get 153,669,926,097,070,256,419 molecules of water into one grain of sand. But you can only get 1,206,702 grains of sand into a football. Counting at 1 grain per second for 8 hours each day, it would take you 41 days to fill that football with sand. Using similar for trying to fill the grain of sand 1 water molecule at a time, you’d still be filling it if you started out at the same time as we believe the universe formed (about 13.8 billion years ago). It would only be 0.1% full: at that pace, it would take another 14,618,524,172,095 years to fill it.
As we go from the microscopic to the macro, we still encounter some enormous numbers. You’d need about 1.3 million earths to fill the sun and 4,744,150,682,950,400,000,000,000,000,000,000 suns to completely fill the Milky Way. Even though the Milky Way could accommodate such a number, it only has 100,000,000,000 or so stars. That’s a lot of ‘nothing’ out there. The Milky Way is but one galaxy of the universe. We have many, many more! About 2,000,000,000,000 of them fill our universe. Ready to hit your wicket yet? If not, why not consider the multiverse?
We can count the small. We can count the medium. We can also count the large. We can count the birds and the bees. We count runs, wickets, centuries and caps. We count books, authors, and words. We count Counts.
Admittedly, some of those counts are best guesses and use assumptions, but you get the point. As a species we like to count things, we like to compare things; we like to understand the relationships between all of this. Quantify and qualify. We enjoy putting things in compartments and studying them each and every way. Then we dig a little further.
However, for all the tremendous advances we have made with our counting, thinking and the aid of computing power, we still cannot count age, height or width. We can’t count Hydrogen, Oxygen, rain or water. Knowledge, gratitude and wisdom. No way to count that. Nonsense? No, can’t count that nor hope nor beauty. We can’t even count pronunciation. As for espionage? Yes, you guessed it: impossible to count… or is it?