The Size of Atoms

How large is an atom? It depends on the element, but generally atom sizes are in the Angstrom (10^-10 meter) range. Too small to see with the naked eye, or even a high-powered light microscope. You can see one with an electron microscope, provided you believe the images you’re seeing on the computer screen.

You could calculate (using quantum mechanics) the most probable distance of an electron from its nucleus in a hydrogen atom. We do this in P-Chem and the answer comes out to be 0.53 Angstroms or 0.053 nanometers. But the electron could be much further from the nucleus. How much further? Don’t know. The probability of finding the electron approaches zero asymptotically as a function of distance, so we know there’s little chance of finding it far away. You could pre-determine a cutoff, by arbitrarily defining an electron’s orbital as enclosing a 90% chance of finding it. (That’s what we tell G-Chem students.) While that’s easy to calculate for the hydrogen atom, it’s not so easy for other elements because the Schrodinger equation can no longer be solved exactly. (Despite beautiful pictures you might see!)

Yet in G-Chem, we happily discuss periodic trends, and the first one we tackle is atomic size! We discuss the factors that affect size. There’s inevitably a beautiful table in the textbook along with actual charts and actual numbers. Clearly some scientists must have measured these to two decimal places of precision! Some are simply misleading – a simple Google search yielded the following colorful graphic among many. There’s one very clear glaring error you should notice if you’ve taken G-Chem!

But if you read the fine print (and hopefully your chemistry textbook discusses this), measuring the atomic radius is tricky. That’s because, with the exception of the noble gases, it’s not easy to encounter atoms hanging out by their lonesome selves, for you to “catch” and measure. Most atoms form chemical bonds with other atoms, and how close they get depends on multiple factors. The whole business turns out to be rather complicated!

The latest iteration comes from a 2016 article by Rahm, Hoffmann, and Ashcroft (Chem. Eur. J. 2016, 22, 14625-14632). They provide high-level quantum calculations along with a rational cutoff for the electron density. Here’s the abstract.

After going through the history of previous charts and tables, they make the case for their version, and provide their own chart and table. I’ve re-created a version below. 

The most interesting thing that jumps out is that the largest atoms are (excepting Li) in Group II. In G-Chem students learn that atomic size decreases across a row so they would expect the Group I elements to be the largest. Why the difference in this study? Part of it has to do with the way cutoff density is defined, and part of it has to do with Pauli exchange and repulsion. The Li vs Be exception is likely related to limited shielding from the core 1s electrons.

There are other nice features of the paper. Their cutoff choice fits very well with experimental data for Van der Waals radii of noble gases. We see the expected contraction due to the transition metal block, and thus the small corresponding rise across the row when we get back into the main group p-block. There’s a particularly nice correlation to electron configurations to explain palladium’s smaller-than-expected radius (see figure below) coming from the relative differences in s versus d contraction.

There are several other interesting tidbits including trends and discussion of cations and anions. Overall, the paper is an interesting read and something I could potentially incorporate into a quantum or inorganic chemistry course. It’s also a reminder how much we don’t know, even about seemingly fundamental things, and that the periodic table is full of surprises. Kinky ones!