Molecules and the Void

Are there more stars in the universe, molecules in a glass of water, or grains of sand on Earth? Many of us have heard this question. The last time I heard it was 5 years ago in a trivia game at Boda Borg (think escape room meets Chuck E Cheese for large children who convince themselves they’re adults). I thought, well it’s obviously stars in the universe, the universe is infinite… I got it wrong. The nerd in me was embarrassed. The science educator let his team down that day.

So do you know the answer? If not, pause and take a guess. Can you justify your guess?

Read on so you can out-nerd the science teachers in your next game of trivia. I’ll equip you with back-of-the-envelope calculations that you can bust out at your next cocktail party. I love back-of-the-envelope calculations; I love the name, the idea, the conversations they foster… Just thinking of them makes me giddy, like the way one of my students visibly trembles with excitement at the site of IB physics problems. The steps to a good back-of-the-envelope calculation are: 1.) Memorize a couple important numbers; 2.) Reasonably justify those numbers; 3.) Do some quick order of magnitude mental math approximations. Before we dive in, let’s set something straight.

I got the question wrong because I was uneducated about cosmology at the time, but also because it was poorly worded. We need to qualify the first quantity as the number of stars in the observable universe. The observable universe consists of the regions that we can probe with the current suite of available instruments. The universe has a cosmic horizon, much like the horizon observed from a ship out at sea. What lies beyond the edge is the subject of philosophical musing, cosmology, and the banter of best friends bonding over “sandwiches” and Pink Floyd in their freshmen dorm rooms. Maybe it’s just more stars. Maybe it’s more universes. Maybe it's dragons. Or maybe it’s nothing. But that nothing would sure be something!

Ok let’s start with stars in the observable universe. This quantity is a million dollar question obscured by a wealth of factors centered on physical limitations of measurement. A scientist could spend their entire career chipping away at the uncertainty. For the purpose of this exercise, I will simplify the problem by estimating the number of stars in the average galaxy and multiplying by the estimated number of galaxies in the universe.

In 1995 scientists pointed the Hubble Space Telescope at a remarkably uninteresting and seemingly vacant region of the sky nestled in between stars of the Big Dipper constellation. The telescope collected several images over ten days and the results were shocking. The famous “Hubble Deep Field” image revealed roughly 3,000 galaxies in just one 22-millionth of the entire sky! The discovery is akin to Antoni Van Leeuwenhoek’s serendipitous discovery of bacteria in 1676. Imagine sticking a cup of water under your microscope and finding thousands of tiny microbes – or “animalcules” as he called them – doing the backstroke.

Photo credit: R. Williams (STScI), the Hubble Deep Field Team and NASA

Ok so 3,000 stars in one 22-millionth of the entire sky. This suggests that there are approximately 66 billion galaxies in the universe if, of course, this image portrays a representative sample of the universe. More recent surveys from the past 25 years put this estimate anywhere between 100 billion and 10 trillion galaxies.

The Milky Way is estimated to host about 100-200 billion stars (as determined by meticulous telescope surveys and physical measurements from which we can infer the total mass of the galaxy). For the sake of our back-of-the-envelope exercise, let’s go with 1 trillion galaxies and 100 billion stars per galaxy to give a conservative estimate with easy numbers for mental math. This yields our first key quantity:

The number of stars in the observable universe is approximately: 100 billion stars per galaxy x 1 trillion galaxies = 100 billion trillion or 1 x 10^23 stars.

(Quick tip: When multiplying multiples of ten, add all the zeros and tally them in powers of ten – the power is the “order of magnitude.” e.g. 1,000 x 10,000 = 10^3 x 10^4 = 10^7)

Now for the more straightforward problem of estimating the number of molecules in a glass of water. The central idea here is to multiply the mass of the water in our glass by the number of water molecules per gram of water. As it turns out, this is pretty easy.

Remember the mole? No not Leonardo DiCaprio in The Departed (sorry for the spoiler but honestly it’s kind of on you if you haven’t seen it yet). And no, not the furry little mammal either. I’m talking about the counting unit of particles you learned about in chemistry class. Maybe you remember Avogadro’s number, 6.022 x 10^23. That’s how many particles are in a “mole” of particles, the same way there are 12 eggs in a dozen. Dozen is to mole as egg is to any small particle like an atom or molecule.

The periodic table of elements is a catalogue of molar masses among other things. In other words, it lists the mass of one mole of atoms for each element in grams. Water, H2O, contains two hydrogen atoms and one oxygen atom per molecule. The molar masses of H and O are very close to 1 and 16 respectively. Thus the mass of one mole, or 6.022 x 10^23 molecules, of H2O is 18 grams. Let's call it 20 grams for the sake of mental math. We'll say the average glass of water is a pint, 16 oz (stay hydrated!). That’s approximately 500 mL. The density of water is conveniently 1 g/mL. So the average glass of water is 500 g. No we multiply 500 g of water by 1 mole per 20 g by 6 x 10^23 molecules per mole:

That’s 2 orders of magnitude greater than stars in the observable universe. There are a hundred times more molecules in your glass of water than there are stars in the observable universe. There are a hundred universes in every glass of water, 10 billion galaxies in every drop!

Now for the final quantity: grains of sand on every beach in the world. Unfortunately I don’t have any more thought experiments or absurd numbers to offer you. Instead, let’s trust the Hawaiians on this one. Who would know better, right? According to David Blatner’s book Spectrums, researchers at the University of Hawaii estimate 7.5 billion billion, or 7.5 x 10^18 grains of sand on Earth. To compare orders or magnitude, let's call it 1 x 10^19 grains of sand.

There you have it. There are approximately 10,000 times more stars in the observable universe than there are grains of sand in the world. And there are approximately a million times more molecules in your glass of water than there are grains of sand.

Let’s summarize. There are an estimated 1 trillion galaxies in the observable universe, each containing an estimated 100 billion stars. Thus there are approximately 1 trillion x 100 billion = 1 x 10^23 stars in the observable universe. Your average glass of water is 500 mL, or 500 g, and 6.022 x 10^23 molecules of water have a mass of about 20 g. Thus there are approximately (500 g/20 g) 6 x 10^23 molecules/g = 1.5 x 10^25 molecules in a glass of water.

What do you take away from this? To me, it means the smallest of the small is smaller than the biggest of the big is big. I think most of us have contemplated the vastness of the universe. All we need to do is look up to see the void. But how often do you think about the expansiveness of a glass of water? Or your own body? The average adult human contains nearly 100 glasses of water. That’s 100 glasses x 100 universes per glass = 10,000 universes.

Your body is a multiverse.

References:

Gest, H. The discovery of microorganisms by Robert Hooke and Antoni Van Leeuwenhoek, fellows of the Royal Society. Notes Rec R Soc Lond. 2004 May; 58(2):187-201.

NASA. (1996). Hubblesite. Retrieved from https://hubblesite.org/image/388/gallery

Blatner, D. (2012). Spectrums: Our Mind-boggling Universe from Infinitesimal to Infinity. Bloomsbury USA.

#cosmology #backoftheenvelope #galaxy #astronomy #chemistry #atomsandthevoid #moleculesandthevoid #multiverse #hubble #democritus #universe #hubbledeepfield