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Quantum Numbers and Electron Configurations

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Example: Quantum Numbers and Electron Configurations

What is the electron configuration and orbital diagram for a phosphorus atom? What are the four quantum numbers for the last electron added?

Solution

The atomic number of phosphorus is 15. Thus, a phosphorus atom contains 15 electrons. The order of filling of the energy levels is 1s, 2s, 2p, 3s, 3p, 4s, . . . The 15 electrons of the phosphorus atom will fill up to the 3p orbital, which will contain three electrons:

This figure provides the electron configuration 1 s superscript 2 2 s superscript 2 2 p superscript 6 3 s superscript 2 3 p superscript 3. It includes a diagram with two individual squares followed by 3 connected squares, a single square, and another connected group of 3 squares all in a single row. The first square is labeled below as, “1 s.” The second is similarly labeled, “2 s.” The first group of connected squares is labeled below as, “2 p.” The square that follows is labeled, “3 s,” and the final group of three squares is labeled, “3 p.” All squares except the last group of three squares has a pair of half arrows: one pointing up and the other down. Each of the squares in the last group of 3 contains a single upward pointing arrow.

The last electron added is a 3p electron. Therefore, n = 3 and, for a p-type orbital, l = 1. The ml value could be –1, 0, or +1. The three p orbitals are degenerate, so any of these ml values is correct. For unpaired electrons, convention assigns the value of \(+\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\) for the spin quantum number; thus, \({m}_{s}=+\frac{1}{2}.\)

The periodic table can be a powerful tool in predicting the electron configuration of an element. However, we do find exceptions to the order of filling of orbitals that are shown in the previous lesson. For instance, the electron configurations (shown in the previous lesson) of the transition metals chromium (Cr; atomic number 24) and copper (Cu; atomic number 29), among others, are not those we would expect. In general, such exceptions involve subshells with very similar energy, and small effects can lead to changes in the order of filling.

In the case of Cr and Cu, we find that half-filled and completely filled subshells apparently represent conditions of preferred stability. This stability is such that an electron shifts from the 4s into the 3d orbital to gain the extra stability of a half-filled 3d subshell (in Cr) or a filled 3d subshell (in Cu). Other exceptions also occur. For example, niobium (Nb, atomic number 41) is predicted to have the electron configuration [Kr]5s24d3.

Experimentally, we observe that its ground-state electron configuration is actually [Kr]5s14d4. We can rationalize this observation by saying that the electron–electron repulsions experienced by pairing the electrons in the 5s orbital are larger than the gap in energy between the 5s and 4d orbitals. There is no simple method to predict the exceptions for atoms where the magnitude of the repulsions between electrons is greater than the small differences in energy between subshells.

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