dominant +3 oxidation state of the lanthanides

Imagine a vending machine that always gives back exactly three coins, no matter which button you press. The lanthanides are chemically like that: whatever the element, the change you usually get is the +3 ion. Lanthanum, neodymium, gadolinium, lutetium — in solution and in most of their solids they appear as Ln3+, and that single shared habit is the backbone of all lanthanide chemistry.

Why three? Each lanthanide has an electron arrangement of roughly [Xe] 4f-n 5d-0-or-1 6s-2. When the atom forms a compound it gives up the two 6s electrons and one more (the single 5d electron, or one 4f electron), reaching a stable Ln3+ ion that keeps its 4f electrons tucked deep inside. Removing those first three electrons is comparatively easy, but tearing into the buried, low-energy 4f shell to make a +4 ion, or stopping early to leave a +2 ion, usually costs far more energy than chemistry can pay back. So +3 wins almost everywhere, and the resulting ions are hard Lewis acids that prefer oxygen and fluorine donors, form mostly ionic bonds, and have high coordination numbers (commonly 8 or 9 in water).

This relentless +3 sameness is exactly why the lanthanides are so hard to separate and so easy to substitute for one another in a crystal — a property exploited in everything from doped laser glass to mixed 'mischmetal' alloys. The honest framing: oxidation state is a bookkeeping number, not the literal charge on the metal (the real bonding is partly covalent), but as bookkeeping goes, +3 is an unusually reliable rule for this family. The famous exceptions, cerium(IV) and europium(II), are the ones worth memorizing precisely because everything else is +3.