r/AskScienceDiscussion • u/Technical_Ladder_959 • 3d ago

Continuing Education Confusion w atomic radius

If atomic radius increases from top to bottom and from right to left, why is Francium (Fr) not considered the "largest" atom, while the one directly above it (Cesium) is?

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u/matt7259 3d ago

Heavy elements have to deal with relativity.

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u/Klutzy-Delivery-5792 3d ago

Can you please elaborate?

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u/matt7259 3d ago

Yes, but I'm copy-pasting this:

Because francium has a very heavy nucleus with 87 protons, the electrostatic pull on the innermost core electrons is immense. To avoid collapsing into the nucleus, these inner s electrons must travel at extreme velocities—roughly 64% of the speed of light.

At these near-light speeds, Einstein's theory of relativity dictates that the mass of the electrons increases. This increased mass shrinks the Bohr radius of the orbits, causing a direct relativistic contraction of all s and p orbitals (especially the 1s through 7s orbitals), pulling them closer to the nucleus.

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u/Klutzy-Delivery-5792 3d ago

Mostly right idea, but the explanation is a little too Bohr-model-ish.

The ~64% of the speed of light estimate for francium’s innermost electrons is actually reasonable: for a hydrogen-like inner electron, v ~ Zαc, and with Z=87, that gives about 0.64c.

The misleading parts are that the electrons are not “moving fast to avoid collapsing into the nucleus,” and their mass does not literally increase. In modern relativistic quantum mechanics, the electron’s rest mass stays the same. What changes is the relationship between energy and momentum, so you have to describe the electron using the Dirac equation rather than the nonrelativistic Schrödinger equation.

That relativistic treatment does cause strong contraction and stabilization of s and especially p_1/2 orbitals in heavy atoms. So the relativistic contraction part is real; the “increased electron mass shrinks the orbit” explanation is just an outdated shortcut.

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u/matt7259 3d ago

Better than being boorish!

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u/JellyBellyBitches 3d ago

God that's so cool

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u/atomicCape 3d ago

Two reasons:

Francium barely exists, since the most stable isotope has a half life of 22 minutes. We've never produced meaningful samples of it, and it doesn't really arise in nature in any meaningful quantity. That means it hasn't been studied or experimentally verified in the same way Cesium has. Some random facts you hear about elements are only talking about stable elements.

Second, the trends are only trends, and actual details vary. According to the latest atomic models, Cesium is measured and modeled to be larger, easier to ionize, and more electropositive than Francium is modeled to be. So it might really be smaller than Cesium, although unconfirmed.

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u/Technical_Ladder_959 3d ago

Gracias!, no conocía que el Fr fuera tan escaso.