Worked example: silicon's electron configuration

Given silicon's atomic number $Z = 14$ and valence count of 4, sketch the electron configuration and identify which shell participates in bonding. Then compare with germanium ($Z = 32$, 2-8-18-4) and carbon ($Z = 6$, 2-4).

1. Given. Silicon, atomic number $Z = 14$. We need to fit 14 electrons into shells K, L, M (n = 1, 2, 3) using the shell-capacity rule, and identify the valence shell.
2. Equation. Each shell of principal quantum number $n$ holds at most $N_{\max} = 2n^2$ electrons. So $K = 2,\; L = 8,\; M = 18$ and so on, filling inner shells first.
3. Substitution. Place 14 electrons starting at K:
   $$\underbrace{2}_{K} + \underbrace{8}_{L} + \underbrace{4}_{M\,\text{(valence)}} = 14$$
   The K and L shells fill completely (2 + 8 = 10 electrons used), leaving 4 to go into the M shell — well below its capacity of 18.
4. Result. Silicon's configuration is 2-8-4. The M shell, with its 4 unpaired electrons, is the valence shell — the one that participates in bonding. Each Si atom uses those 4 valence electrons to form 4 covalent bonds with neighbouring atoms.
5. Sanity-check. Compare with germanium (Z = 32, 2-8-18-4) and carbon (Z = 6, 2-4). Both also end with 4 in the outermost shell, just like silicon — the inner shells fill up but the valence count stays at four. The total electron count balances Z in each case ($2 + 4 = 6$ for C; $2 + 8 + 18 + 4 = 32$ for Ge).