Question

Why are electrons in the 3d orbital in a higher energy state although electrons in the 4s orbital are further out?

Aren't electrons in a higher energy state further out from the nucleus as seen in the previous orbitals? Why is it different when it comes the the 3d and 4s orbitals?

Answer 1

The `4s` orbitals are usually higher in energy than the `3d` (specifically, `"Sc"` through `"Kr"`), but some textbooks don't acknowledge or recognize that `"Ca"` is the border past which the orbital energies switch. This is discussed in detail here as well.

If you are interested, a detailed explanation for chromium's `3d^5 4s^1` configuration is here, and an argument that does NOT rely on a "half-filled subshell" sentiment can be found in the second half of this answer here.

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Given that orbitals are only as high or low energy as they are large (as nucleus-electron interaction goes as `1//r`), orbitals that extend farther out are often (but not always) higher in energy.

It's a result of two competing factors, namely

1. quantum level `n`.
2. angular momentum `l`.

The `3d` has lower `n` by `1` (which lowers energy) but higher `l` by `2` (which raises energy), but the effect of lower `n` predominates for first-row transition metals compared to greater `l` (meaning the `3d` is lower in energy).

This can be seen in the radial density distribution functions, which show that the `4s` does have the farther radial extent in the qualitative picture based on the hydrogen atom only:

This was graphed in the Excel sheet linked here (see "Combined RDFs" tab) using known hydrogen atom wave functions if you want to explore it.

The `4s` vs. `3d` comparison in potential energy is borderline, but Miessler, Fischer, and Tarr have cited their potential energies here (Appendix B.9), and I have graphed them below:

Apparently, the `4s` has been higher in energy all the way from `"Sc"` to `"Zn"` (and through `"Kr"` as well, `Z = 21 - 36`), NOT the other way around as even Tro might say.

The given trend suggests an ambiguity in how `"Sc"` (`Z = 21`) is ionized as well, and that is gone over here.