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Text books say that a pure silicon atom has four electrons in its outermost orbit. How do the scientists know it has four electrons?

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One form of evidence is the ionization energies of silicon. Nth ionization energy is the energy needed to remove the nth electron. There is a big jump going from the 4th ionization energy (~4000 kJ/mol) to the 5th ionization energy (~16000 kJ/mol).

Another form of evidence is the compounds silicon makes. Silicon forms $\mathrm{SiH}_4$, $\mathrm{SiF}_4$, $\mathrm{SiCl}_4$ and other compounds that imply 4 valence shell electrons.

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We know that a bare silicon nucleus has electric charge +14$e$.

Electrons around an atomic nucleus don't follow any old orbit they wish, like planets or comets orbiting the sun, but occupy quantum-mechanical orbitals. Each orbital is like a "slot" that can hold two electrons with opposite spins; after the slot is filled, the next electron in the atom has to occupy a different orbital. The number of slots in each orbital depends on how much angular momentum the electrons in it carry: $$ \begin{array}{ccc} \text{Orbital name} & \text{Angular momentum }\ell & \text{Number of slots} \\ s & 0 & 1 \\ p & 1 & 3 \\ d & 2 & 5 \\ f & 3 & 7 \\ \vdots & n & 2n+1 \end{array} $$ The shape of the periodic table tells you which order the electrons prefer to fill up the orbitals. If you started with a bare silicon nucleus and added electrons one at a time,

  • the first two would fill up the 1s orbital, like in helium
  • the next two would fill up the 2s orbital, like in beryllium
  • the next six would fill up the 2p orbital, like in neon
  • the next two would fill up the 3s orbital, like in magnesium
  • the last two would live in the 3p orbital.

Most of the evidence for why the periodic table has this structure comes from chemistry, rather than physics. DavePhD gives a couple of good examples.

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Early, while the periodic table of the elements was explored by Mendeleev, it was already known that it can have nearly always 4 bonding. The source of this knowledge was at this time the analysis of the mass ratios of its compounds.

Later, after the Bohr model was developed, in the times of the modern quantum chemistry it is interpreted as it has 4 valency electrons. That means, 4 of its electrons is capable to participate in chemical bonds. Behind this knowledge are various approximations (f.e. this or this) to solve the very complex calculations of the electrons.

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