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Example 6.8: Maximum Number of Electrons

Calculate the maximum number of electrons that can occupy a shell with (a) \(n\) equal to 2, (b) \(n\) equal to 5, and (c) \(n\) as a variable. Note you are only looking at the orbitals with the specified \(n\) value, not those at lower energies.

Solution

(a) When \(n\) is 2, there are four orbitals (a single 2s orbital, and three orbitals labeled 2p). These four orbitals can contain eight electrons.
(b) When \(n\) is 5, there are five subshells of orbitals that we need to sum:
1 orbital labeled 5s
3 orbitals labeled 5p
5 orbitals labeled 5d
7 orbitals labeled 5f
9 orbitals labeled 5g
25 orbitals total
Again, each orbital holds two electrons, so 50 electrons can fit in this shell.
(c) The number of orbitals in any shell \(n\) will be \(N_{\mathrm{orbital}} = {n}^{2}\) . There can be up to two electrons in each orbital, so the maximum number of electrons will be \(N_{\mathrm{electrons}} = 2 \cdot {n}^{2}\)