Effective Atomic Number (EAN) Rule

The Effective Atomic Number (EAN) rule, proposed by Sidgwick, is a simple electron-counting rule that offers a rough but useful check on the stability of a coordination entity. It proposes that a complex tends to be especially stable when the total number of electrons "effectively" surrounding the central metal — its own remaining electrons after accounting for its oxidation state, plus all the electrons donated to it by every ligand — exactly equals the electron count of the noble gas that follows the metal in the periodic table.

The EAN is calculated as:

where $Z$ is the atomic number of the neutral metal atom, $x$ is the oxidation state (positive charge) of the metal in the complex, and $\text{CN}$ is the coordination number (each donor atom is assumed, by convention, to donate a pair of electrons, hence the factor of 2).

Working through $[\text{Fe}(\text{CN})_6]^{4-}$: the six $\text{CN}^-$ ligands each carry a $-1$ charge, contributing $-6$ overall, so for the complex ion to have its actual $-4$ charge, iron itself must be present as $\text{Fe}^{2+}$ (i.e. $x = 2$). Iron's atomic number is $Z = 26$, and the coordination number here is 6. So $\text{EAN} = 26 - 2 + 2(6) = 26 - 2 + 12 = 36$. This is exactly the atomic number of krypton, the noble gas immediately following iron — so $[\text{Fe}(\text{CN})_6]^{4-}$ satisfies the EAN rule and is, as observed experimentally, a very stable complex.

The same value of 36 recurs strikingly often among stable complexes of the first-row transition metals in this region of the periodic table, precisely because it is easy to reach: $[\text{Co}(\text{NH}_3)_6]^{3+}$ (cobalt, $Z=27$, $x=3$, $\text{CN}=6$): $\text{EAN} = 27 - 3 + 12 = 36$. $[\text{Ni}(\text{CO})_4]$ (nickel, $Z=28$; here nickel is in the zero oxidation state since $\text{CO}$ is a neutral ligand, so $x = 0$; $\text{CN} = 4$): $\text{EAN} = 28 - 0 + 2(4) = 28 + 8 = 36$. All three examples — despite involving three different metals, three different oxidation states, and two different coordination numbers — converge on the same "36-electron", noble-gas-matching total, which is why this specific check is sometimes informally called the "36-electron rule" or, in the broader context of organometallic chemistry, the "18-electron rule" (18 valence electrons in the metal's outer $n$s, $n$p, and $(n-1)$d shells combined, which is the same physical idea expressed slightly differently). …