Quantum Numbers and Energies of Electrons

EXAMPLE 4.6. Arrange the electrons in the foUowing list in order of increasing energy, lowest first  

Electron (b) has the lowest value of n +Z (2 + 1=3), and so it is lowest in energy of the four electrons. Electron (d) has the next-lowest sum ofn+/ (4+0 = 4) and is next in energy (despite the fact that it does not have the next-lowest n value). Electrons (a) and (c) both have the same sum of n + Z (4 + 1 = 3 + 2 = 5). Therefore, in this case, electron (c), the one with the lower n value, is lower in energy. Electron (a) is highest in energy. 

The Pauli exclusion principle states that no two electrons in the same atom can have the same set of four quantum numbers. Along with the order of increasing energy, we can use this principle to deduce the order of filling of electron shells in atoms. 

EXAMPLE 4.7. Use the Pauli principle and the n + l rule to predict the sets of quantum numbers for the 13 electrons in the ground state of an aluminum atom. 

We want all the electrons to have the lowest energy possible. The lowest value of n + Z will have the lowest n possible and the lowest Z possible. The lowest n permitted is 1 (Table 4-1). With that value of n, the only value of Z permitted is 0. With 1 = 0, the value of must be 0 (—0 + 0). The value of can be either — or +. Thus, the first 

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