We hear it all the time: “Atoms are mostly empty space.” Sometimes, that factoid is even quantified: “Atoms are X % empty space.” And as part of my ongoing mission to exorcise nagging questions spawned by third-grade science books, I want to find out what X actually is. How much of an atom is actually empty space?

Well, atoms are just a nucleus of protons (and sometimes neutrons), with electrons sort of vaguely existing in their vicinity. They don’t really orbit. Because quantum mechanics is weird and terrifying, you can think of an electron as being smeared out in a haze of existence around the nucleus. There’s a probability cloud surrounding the nucleus, and at each point within that cloud, the density of the cloud determines how likely the electron is to be hanging out there. Depending on the electron’s energy level in the atom, the probability cloud might be spherical or it might be dumbbell-shaped or it might be a bit like an onion. But the probability cloud is always blurry.

Which means that, in record time, we’ve hit a damn stumbling block. Here’s an irrelevant-looking picture:

That’s the graph of the function exp(-x^2), which doesn’t have a lot to do with electrons or atoms, but is a simple analogy for the probability density in a spherical electron cloud. How wide would you say that bump is? 2 units? 4 units? Just like with swear words, no matter where the fuck you draw the line, somebody’s going to disagree with you. But sooner or later, you’ve gotta buckle down and decide that A is over the line and B isn’t. Luckily, science is built to be sensible and rigorous, so as long as we pick a defined point where the bump (or electron cloud) ends, and as long as we all work from the same definition (or tell each other if our definitions are different), we can at least have concrete numbers to work from.

So, to answer the question “How much of an atom is empty space?” I’m going to use the covalent radius for the atoms in question. This is the radius of the atom as deduced from how far it sits from other atoms when it forms covalent molecular bonds. There are other definitions that come closer to our intuitive idea of radius (van der Waals radius, for instance) but covalent radii are easier to measure, and are often known with higher precision.

So now we have a way to look up one parameter: the radius of an atom, and therefore, its volume. The smallest and least massive atom is hydrogen, with a radius of about 25 picometers (0.025 nanometers, or 20,000 times smaller than a bacterium). Hydrogen is a nice atom. It has one proton and one electron. That’s it. And the probability cloud for its single electron is a pleasant spherical shape (at least in the ground state). The largest atom is cesium, with a radius of 260 picometers (0.26 nanometers, about 2,000 times smaller than a bacterium). And the most massive naturally-occurring atom is (arguably) uranium, with a radius of 175 picometers. It’ll make sense why I included two different “largest” atoms in a moment.

To figure out what fraction of an atom is empty space, we need to know how much of it is not empty space. (The missile knows where it is because the missile knows where it isn’t…) Since I spent the start of this post talking about electrons (and since the answer is nice and simple), let’s ask the question: what’s the volume of an electron?

Well, as far as physics can tell (as of January 2021), the answer is zero. The electron has no substructure that we know of—it has no internal parts. There’s just this infinitesimal speck that has all the properties of an electron, and that’s as much as we know about them. Quantum physics and experimental evidence suggest an electron cannot be larger than 10^-18 meters—if it were, that’d cause observable effects. So, for our purposes, electrons are so small they’re not worth including.

That only leaves the nucleus. And hoo boy, if you thought the weird fuzziness of the electron cloud was frustrating, you ain’t seen nothin’ yet.

Let’s start with hydrogen, since it’s nice and simple. One zero-volume electron just sort of weirdly hanging out, in an unpleasant blurry (but spherically-symmetric) fashion in the vicinity of a single proton. Unlike the electron, the proton does have a measurable radius. It’s still a fuzzy, blurry, jittery thing that you can never quite pin down, but if you shoot, say, electrons at it and see how they bounce off, you can get an idea, and from that data, decide that the most sensible radius for a proton is 0.877 femtometers. That’s 0.000877 picometers, or 0.877 millionths of a nanometer. If a proton were the size of a 100-micron-diameter dust speck (right on the limit of naked-eye visibility; roughly the diameter of a hair), then a hair would be almost half the diameter of the earth. Did I mention that protons are really small? ‘Cause they are.

So a hydrogen atom is about 25 picometers in radius, and the proton, which is the only thing in it that takes up any space, has a radius of about 0.877 femtometers. The formula for volume of a sphere gives us a simple answer for “How much of a hydrogen atom is empty space?” 99.99999999999568%.

You guys know me, though—that’s too abstract a number. Too many digits. Let’s take, say, the United States. The USA is a big country. If it were a hydrogen atom, the whole thing would be empty space, except for a single patch about 24 inches (72 centimeters) across. Just big enough for an adult human to stand in. You probably wouldn’t be able to see it with the naked eye from an airplane. (I know I switched from volume to area here, but I used the same percentage to get the “area” of the proton, so the comparison is still valid, mathematically.)

For heavier elements, life gets more complicated. As I said, electrons are impossible to pin down for certain. They just exist in the nucleus’s general vicinity. Their existence is smeared out in a particular way around the nucleus. (That’s not exactly an accurate description of how it works, but I don’t know enough quantum mechanics to take you any deeper without the risk of misleading you.) The same is true for the nucleus, but because the protons and neutrons in a nucleus are much more massive, and because they’re so close together, and because they experience an additional very strong force (the strong nuclear force) that the electron doesn’t, their jittering is even more intense.

As a result, we know about the radii of atomic nuclei in the same vague way we know about the radius of a proton: we shoot particles at the nuclei and see how they bounce off. Most will just graze and barely deflect at all. Some will hit the nucleus closer to head-on, and some will hit it square enough to come back at you. By plotting how often electrons bounce off and at what angles, for a given electron speed, we can build up a pretty convincing picture of where all the matter in the nucleus is.

The radius of an atomic nucleus is roughly 1.5 femtometers times the cube-root of the element’s atomic number (for elements with atomic numbers above 20). For cesium, the largest atom by covalent radius, the nucleus has a radius around 5.7 femtometers. A cesium atom has a covalent radius of about 260 picometers, and therefore, is 99.99999999999895% empty space. If the United States were a cesium atom, the nucleus would be barely the size of one and a half sheets of standard printer paper. And uranium, the largest atom we’re concerned with (by mass) has a covalent radius of 175 picometers with a nuclear radius of 6.8 femtometers. 99.9999999999941% empty space. Compared to the United States, that’d be a circle about 34 inches (86 cm) across. Big enough to sit in, but not lie in comfortably.

You guys know me. I usually like to finish my posts with some clever coda. Some moral for the story. But there’s really not one this time. This time, the question was “How much of an atom is empty space,” and the answer is…well, it’s right up above.