How Many Subatomic Particles Are In That Dandelion?

A lot of septillions.

Phineas and Ferb is a show about a bunch of supergeniuses. It is also awesome. In one episode, the characters build a helmet that makes people smarter, and use it on their smartest supergenius. To make sure it worked, they ask him to find the number of subatomic particles in a dandelion. It takes him a few seconds, and he covers the fence in math-like squiggles.

Here’s how to estimate it simpler and with less effort than the intelligence-enhanced fictional supergenius, and do it in your head.

We’re just going for an order-of-magnitude estimation. That basically means figuring out the number of zeroes on the end of the number. This kind of estimate is not very precise, but is still useful and much easier. In high school I was taught that mathematicians are lazy and always do things the easy way. Never bothered confirming that.

First thing: What’s the mass of the dandelion? Phineas didn’t specify if he meant just the flower or the whole plant, but the calculation is similar either way.

Using xkcd’s phrasing: Picture a bundle of ten dandelion plants. That’s something you can pick up in your hand and throw, something you can pick up and throw is one pound, one pound is one kilogram. (That emphasizes that we’re really only estimating, but 100 grams per plant is probably in the right ballpark.) The point is that it’s probably a better estimate than either 10g or 1000g.

What is all that mass made of? Anything smaller than an electron has negligible mass. Those 100 grams are all protons and neutrons. Those have the same mass (well, they don’t, but the difference is too small to matter). What’s the mass? I don’t know. I can never remember that. But we can find it. There is a number called Avogadro’s number, since in science everything needs to have a name. This is so that outsiders know when to be scared. It says that there are 6.022×10^23 atoms in one mole. (Mole as in unit, not mole as in small furry cat food.) And hydrogen has a mass of one gram per mole and a hydrogen atom is just a proton and an electron. So our 100-gram dandelion contains the mass of 100*6*10^23 protons.

But the question wasn’t about mass; it was about the number of subatomic particles. There isn’t enough information so far to find that. Maybe it’s only made of one giant dandelion-shaped particle. (It’s a dandelion, right? So it’s made of dandelion. Obviously.)

Well, we know what dandelions are made of. Carbon, hydrogen, nitrogen, oxygen, stuff like that. All of those atoms except for hydrogen have the same numbers of protons and neutrons. Carbon is six of each, oxygen is eight, etc. So that tells us that it is something like 3*10^25 protons and 3*10^25 neutrons. It’ll be slightly toward the proton side because of the hydrogen, which is a proton without a neutron, but that won’t throw it off much.

The obvious next thing we’re missing is the electron. Since the dandelion is grounded, it most likely has net zero electrical charge. So there are as many electrons as there are protons, meaning 3*10^25 of each. This is where it starts to matter that there are more protons than neutrons. If the dandelion were made entirely of hydrogen, there’d be 12*10^25 particles (half protons, half electrons). If it were entirely made of those other elements with no hydrogen at all, it’d be 9*10^25 (one third protons, one third neutrons, one third electrons). The actual value will be somewhere in between, but it’ll be much closer to that last number. In terms of composition by mass, hydrogen is heavily outweighed (heh) by neutron-containing elements.

The tally so far: 3*10^25 protons, 3*10^25 neutrons, 3*10^25 electrons.
But each proton and neutron is made of three quarks. A proton is two up quarks and a down quark, a neutron is two down one up, but the point is that it’s three each. (This also makes the hydrogen thing from earlier even less important.) So it’s 9*10^25 quarks in protons, 9*10^25 quarks in neutrons, and 3*10^25 electrons. Luckily, none of those particles are made up of smaller ones. We’re probably supposed to count both the protons and the quarks making them up, because they are both subatomic particles. Anyway, the total comes to 2.7*10^26 particles, or two hundred seventy septillion. I promise “quark” is a real word.

I’m not counting gluons. That’s because I have no idea how many gluons are in a nucleus at any given time. (After Googling it, I’m not even sure if it’s a meaningful question.) We’re talking about light elements, so it’ll probably be relatively low, but I have no frame of reference for what that means. Well, we just wanted an order of magnitude, and unless the gluons significantly outnumber everything else (unlikely) the true number is somewhere on the hundreds-of-septillions scale. Plus or minus a couple of orders of magnitude. I never said it was a very precise estimate.

Start with the mass in grams, multiply by 6*10^23 to find the number of nucleons, multiply by three halves to include the number of electrons. Of those nine hundred sextillions, six are made of three quarks each. So multiply those by four (so we count the quarks as well as the nucleons. If you’re asked for the number of elementary particles, multiply by three instead) and add the electrons. You get 6*4+3=27*10^23. Since questions like this come up a lot, here’s how to do it in your head, as promised. For ease of remembering (and mental calculation), round it off to two and a half septillion subatomic particles per gram. Ta-da!

How well this works depends on what it’s made of. Light elements (except hydrogen) work best. This approximation would overestimate stuff with a lot of hydrogen in it; for heavier atoms it’ll underestimate it unless you want to factor in the proton-neutron ratios. Nobody wants to do that, but it won’t matter too much. It’ll always stay within a factor of 1.5 or so and we’re just aiming for an order of magnitude.

Turns out, and I did not know this in advance, this is the same answer, right down to two significant figures, as the Phineas and Ferb writers had Baljeet say. (Page search this transcript for “Heisenberg.”) This is probably not coincidence. In other words, HOLY FLYING FISH, THEY ACTUALLY DID THE MATH.

UPDATE: I emailed Swampy Marsh, and he confirmed that they did indeed do the math. He didn’t say who specifically, but Doing The Research as standard practice is even better. I told you they were awesome.


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