{"id":373,"date":"2016-01-11T20:00:17","date_gmt":"2016-01-11T20:00:17","guid":{"rendered":"https:\/\/opentextbc.ca\/introductorychemistryclone\/chapter\/organization-of-electrons-in-atoms-2\/"},"modified":"2020-07-28T15:56:12","modified_gmt":"2020-07-28T15:56:12","slug":"organization-of-electrons-in-atoms","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introductorychemistryclone\/chapter\/organization-of-electrons-in-atoms\/","title":{"raw":"Organization of Electrons in Atoms","rendered":"Organization of Electrons in Atoms"},"content":{"raw":"[latexpage]\r\n
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

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  1. Learn how electrons are organized in atoms.<\/li>\r\n \t
  2. Represent the organization of electrons by an electron configuration.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nNow that you know that electrons have quantum numbers, how are they arranged in atoms? The key to understanding electronic arrangement is summarized in the [pb_glossary id=\"1729\"]Pauli exclusion principle[\/pb_glossary]: no two electrons in an atom can have the same set of four quantum numbers. This dramatically limits the number of electrons that can exist in a shell or a subshell.\r\n\r\nElectrons are typically organized around an atom by starting at the lowest possible quantum numbers first, which are the shells\u2013subshells with lower energies. Consider H, an atom with a single electron only. Under normal conditions, the single electron would go into the n<\/i> = 1 shell, which has only a single s<\/i> subshell with one orbital (because m<\/i>\u2113<\/sub> can equal only 0). The convention is to label the shell\u2013subshell combination with the number of the shell and the letter that represents the subshell. Thus, the electron goes in the 1s<\/i> shell\u2013subshell combination. It is usually not necessary to specify the m<\/i>\u2113<\/sub> or m<\/i>s<\/sub> quantum numbers, but for the H atom, the electron has m<\/i>\u2113<\/sub> = 0 (the only possible value) and an m<\/i>s<\/sub> of either +\u00bd or \u2212\u00bd.\r\n\r\nThe He atom has two electrons. The second electron can also go into the 1s<\/i> shell\u2013subshell combination but only if its spin quantum number is different from the first electron\u2019s spin quantum number. Thus, the sets of quantum numbers for the two electrons are {1, 0, 0, +\u00bd} and {1, 0, 0, \u2212\u00bd}. Notice that the overall set is different for the two electrons, as required by the Pauli exclusion principle.\r\n\r\nThe next atom is Li, with three electrons. However, now the Pauli exclusion principle implies that we cannot put that electron in the 1s<\/i> shell\u2013subshell because no matter how we try, this third electron would have the same set of four quantum numbers as one of the first two electrons. So this third electron must be assigned to a different shell\u2013subshell combination. However, the n<\/i> = 1 shell doesn\u2019t have another subshell; it is restricted to having just \u2113 = 0, or an s<\/i> subshell. Therefore, this third electron has to be assigned to the n<\/i> = 2 shell, which has an s<\/i> (\u2113 = 0) subshell and a p<\/i> (\u2113 = 1) subshell. Again, we usually start with the lowest quantum number, so this third electron is assigned to the 2s<\/i> shell\u2013subshell combination of quantum numbers.\r\n\r\nThe Pauli exclusion principle has the net effect of limiting the number of electrons that can be assigned a shell\u2013subshell combination of quantum numbers. For example, in any s<\/i> subshell, no matter what the shell number, there can be a maximum of only two electrons. Once the s<\/i> subshell is filled up, any additional electrons must go to another subshell in the shell (if it exists) or to higher-numbered shell. A similar analysis shows that a p<\/i> subshell can hold a maximum of six electrons. A d<\/i> subshell can hold a maximum of 10 electrons, while an f<\/i> subshell can have a maximum of 14 electrons. By limiting subshells to these maxima, we can distribute the available electrons to their shells and subshells.\r\n
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    Example 8.1<\/p>\r\n\r\n<\/header>\r\n

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    Problem<\/h1>\r\nHow would the six electrons for C be assigned to the n<\/i> and \u2113 quantum numbers?\r\n

    Solution<\/h2>\r\nThe first two electrons go into the 1s<\/i> shell\u2013subshell combination. Two additional electrons can go into the 2s<\/i> shell\u2013subshell, but now this subshell is filled with the maximum number of electrons. The n<\/i> = 2 shell also has a p<\/i> subshell, so the remaining two electrons can go into the 2p<\/i> subshell. The 2p<\/i> subshell is not completely filled because it can hold a maximum of six electrons.\r\n

    Test Yourself<\/h1>\r\nHow would the 11 electrons for Na be assigned to the n<\/i> and \u2113 quantum numbers?\r\n

    Answer<\/h2>\r\nTwo 1s<\/i> electrons, two 2s<\/i> electrons, six 2p<\/i> electrons, and one 3s<\/i> electron.\r\n\r\n<\/div>\r\n<\/div>\r\nNow that we see how electrons are partitioned among the shells and subshells, we need a more concise way of communicating this partitioning. Chemists use an [pb_glossary id=\"1730\"]electron configuration [\/pb_glossary] to represent the organization of electrons in shells and subshells in an atom. An electron configuration simply lists the shell and subshell labels, with a right superscript giving the number of electrons in that subshell. The shells and subshells are listed in the order of filling.\r\n\r\nFor example, an H atom has a single electron in the 1s<\/i> subshell. Its electron configuration is:\r\n\r\n\\[\\ce{H}\\text{: }1s\\]\r\n\r\nHe has two electrons in the 1s<\/i> subshell. Its electron configuration is:\r\n\r\n\\[\\ce{He}\\text{: }1s^2\\]\r\n\r\nThe three electrons for Li are arranged in the 1s<\/i> subshell (two electrons) and the 2s<\/i> subshell (one electron). The electron configuration of Li is:\r\n\r\n\\[\\ce{Li}\\text{: }1s^22s^1\\]\r\n\r\nBe has four electrons, two in the 1s<\/i> subshell and two in the 2s<\/i> subshell. Its electron configuration is:\r\n\r\n\\[\\ce{Be}\\text{: }1s^22s^2\\]\r\n\r\nNow that the 2s<\/i> subshell is filled, electrons in larger atoms must go into the 2p<\/i> subshell, which can hold a maximum of six electrons. The next six elements progressively fill up the 2p<\/i> subshell:\r\n

    \\(\\begin{array}{rl}\r\n\\ce{B}\\text{:}&1s^22s^22p^1 \\\\\r\n\\ce{C}\\text{:}&1s^22s^22p^2 \\\\\r\n\\ce{N}\\text{:}&1s^22s^22p^3 \\\\\r\n\\ce{O}\\text{:}&1s^22s^22p^4 \\\\\r\n\\ce{F}\\text{:}&1s^22s^22p^5 \\\\\r\n\\ce{Ne}\\text{:}&1s^22s^22p^6\r\n\\end{array}\\)<\/p>\r\nNow that the 2p<\/i> subshell is filled (all possible subshells in the n<\/i> = 2 shell), the next electron for the next-larger atom must go into the n<\/i> = 3 shell, s<\/i> subshell.\r\n

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    Example 8.2<\/p>\r\n\r\n<\/header>\r\n

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    Problem<\/h1>\r\nWhat is the electron configuration for Na, which has 11 electrons?\r\n

    Solution<\/h2>\r\nThe first two electrons occupy the 1s<\/i> subshell. The next two occupy the 2s<\/i> subshell, while the next six electrons occupy the 2p<\/i> subshell. This gives us 10 electrons so far, with 1 electron left. This last electron goes into the n<\/i> = 3 shell, s<\/i> subshell. Thus, the electron configuration of Na is 1s<\/i>2<\/sup>2s<\/i>2<\/sup>2p<\/i>6<\/sup>3s<\/i>1<\/sup>.\r\n

    Test Yourself<\/h1>\r\nWhat is the electron configuration for Mg, which has 12 electrons?\r\n

    Answer<\/h2>\r\n1s<\/i>2<\/sup>2s<\/i>2<\/sup>2p<\/i>6<\/sup>3s<\/i>2<\/sup>\r\n\r\n<\/div>\r\n<\/div>\r\nFor larger atoms, the electron arrangement becomes more complicated. This is because after the 3p<\/i> subshell is filled, filling the 4s<\/i> subshell first actually leads to a lesser overall energy than filling the 3d<\/i> subshell. Recall that while the principal quantum number largely dictates the energy of an electron, the angular momentum quantum number also has an impact on energy; by the time we get to the 3d<\/i> and 4s<\/i> subshells, we see overlap in the filling of the shells. Thus, after the 3p<\/i> subshell is completely filled (which occurs for Ar), the next electron for K occupies the 4s<\/i> subshell, not the 3d<\/i> subshell:\r\n\r\n\\[\\ce{K}\\text{: }1s^22s^22p^63s^23p^64s^1\\text{, \\textbf{not} }1s^22s^22p^63s^23p^63d^1\\]\r\n\r\nFor larger and larger atoms, the order of filling the shells and subshells seems to become even more complicated. There are some useful ways to remember the order, like that shown in Figure 8.1 \"Electron Shell Filling Order.\" If you follow the arrows in order, they pass through the subshells in the order that they are filled with electrons in larger atoms. Initially, the order is the same as the expected shell\u2013subshell order, but for larger atoms, there is some shifting around of the principal quantum numbers. However, Figure 8.1 \"Electron Shell Filling Order\" gives a valid ordering of filling subshells with electrons for most atoms.\r\n
    \r\n\r\n[caption id=\"attachment_367\" align=\"aligncenter\" width=\"400\"]\"\"<\/a> Figure 8.1 \"Electron Shell Filling Order.\" Starting with the top arrow, follow each arrow. The subshells you reach along each arrow give the ordering of filling of subshells in larger atoms. The n = 5 and higher shells have more subshells, but only those subshells that are needed to accommodate the electrons of the known elements are given.[\/caption]\r\n\r\n
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    Example 8.3<\/p>\r\n\r\n<\/header>\r\n

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    Problem<\/h1>\r\nWhat is the predicted electron configuration for Sn, which has 50 electrons?\r\n

    Solution<\/h2>\r\nWe will follow the chart in Figure 8.1 \"Electron Shell Filling Order\" until we can accommodate 50 electrons in the subshells in the proper order:\r\n\r\n\\[\\ce{Sn}\\text{: }1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^24d^{10}5p^2\\]\r\n\r\nVerify by adding the superscripts, which indicate the number of electrons: 2 +\u00a02 +\u00a06 +\u00a02 +\u00a06 +\u00a02 +\u00a010 +\u00a06 +\u00a02 +\u00a010 +\u00a02 = 50, so we have placed all 50 electrons in subshells in the proper order.\r\n

    Test Yourself<\/h1>\r\nWhat is the electron configuration for Ba, which has 56 electrons?\r\n

    Answer<\/h2>\r\n\\(1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^24d^{10}5p^66s^2\\)\r\n\r\n<\/div>\r\n<\/div>\r\nAs the previous example demonstrated, electron configurations can get fairly long. An [pb_glossary id=\"1731\"]abbreviated electron configuration[\/pb_glossary]\u00a0uses one of the elements from the last column of the periodic table, which contains what are called the noble gases<\/i>, to represent the core of electrons up to that element. Then the remaining electrons are listed explicitly. For example, the abbreviated electron configuration for Li, which has three electrons, would be:\r\n\r\n\\[\\ce{Li}\\text{: }[\\ce{He}]2s^1\\]\r\n\r\nWhere [He] represents the two-electron core that is equivalent to He\u2019s electron configuration. The square brackets represent the electron configuration of a noble gas. This is not much of an abbreviation. However, consider the abbreviated electron configuration for W, which has 74 electrons:\r\n\r\n\\[\\ce{W}\\text{: }[\\ce{Xe}]6s^24f^{14}5d^4\\]\r\n\r\nThis is a significant simplification over an explicit listing of all 74 electrons. So for larger elements, the abbreviated electron configuration can be a very useful shorthand.\r\n
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    Example 8.4<\/p>\r\n\r\n<\/header>\r\n

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    Problem<\/h1>\r\nWhat is the abbreviated electron configuration for P, which has 15 electrons?\r\n

    Solution<\/h2>\r\nWith 15 electrons, the electron configuration of P is:\r\n\r\n\\[\\ce{P}\\text{: }1s^22s^22p^63s^23p^3\\]\r\n\r\nThe first immediate noble gas is Ne, which has an electron configuration of 1s<\/i>2<\/sup>2s<\/i>2<\/sup>2p<\/i>6<\/sup>. Using the electron configuration of Ne to represent the first 10 electrons, the abbreviated electron configuration of P is:\r\n\r\n\\[\\ce{P}\\text{: }[\\ce{Ne}]3s^23p^3\\]\r\n

    Test Yourself<\/h1>\r\nWhat is the abbreviated electron configuration for Rb, which has 37 electrons?\r\n

    Answer<\/h2>\r\n\\([\\ce{Kr}]5s^1\\)\r\n\r\n<\/div>\r\n<\/div>\r\nThere are some exceptions to the rigorous filling of subshells by electrons. In many cases, an electron goes from a higher-numbered shell to a lower-numbered but later-filled subshell to fill the later-filled subshell. One example is Ag. With 47 electrons, its electron configuration is predicted to be:\r\n\r\n\\[\\ce{Ag}\\text{: }[\\ce{Kr}]5s^24d^9\\]\r\n\r\nHowever, experiments have shown that the electron configuration is actually:\r\n\r\n\\[\\ce{Ag}\\text{: }[\\ce{Kr}]5s^14d^{10}\\]\r\n\r\nThis, then, qualifies as an exception to our expectations. At this point, you do not need to memorize the exceptions; but if you come across one, understand that it is an exception to the normal rules of filling subshells with electrons, which can happen.\r\n

    Electron Configuration Energy Diagrams<\/h1>\r\nWe have just seen that electrons fill orbitals in shells and subshells in a regular pattern, but why does it follow this pattern? There are three principles which should be followed to properly fill electron orbital energy diagrams:\r\n
      \r\n \t
    1. The aufbau principle<\/li>\r\n \t
    2. The Pauli exclusion principle<\/li>\r\n \t
    3. Hund\u2019s rule<\/li>\r\n<\/ol>\r\nThe overall pattern of the electron shell filling order emerges from the [pb_glossary id=\"1733\"]aufbau principle[\/pb_glossary] (German for \u201cbuilding up\u201d): \u00a0electrons fill the lowest energy orbitals first. Increasing the principle quantum number, n<\/i>, increases orbital energy levels, as the electron density becomes more spread out away from the nucleus. In many-electron atoms (all atoms except hydrogen), the energy levels of subshells varies due to electron-electron repulsions. The trend that emerges is that energy levels increase with value of the angular momentum quantum number, l<\/i>, for orbitals sharing the same principle quantum number, n<\/i>. This is demonstrated in Figure 8.2, where each line represents an orbital, and each set of lines at the same energy represents a subshell of orbitals.\r\n\r\n[caption id=\"attachment_368\" align=\"aligncenter\" width=\"400\"]\"Diagram<\/a> Figure 8.2 \"Generic Energy Diagram of Orbitals in a Multi-Electron Atom.\" [Long Description]<\/a>[\/caption]As previously discussed, the Pauli exclusion principle states that we can only fill each orbital with a maximum of two electrons of opposite spin. But how should we fill multiple orbitals of the same energy level within a subshell (e.g., the three orbitals in the 2p<\/i> subshell)? Orbitals of the same energy level are known as degenerate orbitals, and we fill them using [pb_glossary id=\"1734\"]Hund\u2019s rule[\/pb_glossary]: place one electron into each degenerate orbital first, before pairing them in the same orbital.Let\u2019s examine a few examples to demonstrate the use of the three principles.Boron is atomic number 5, and therefore has 5 electrons. First fill the lowest energy 1s<\/i> orbital with two electrons of opposite spin, then the 2s <\/i>orbital with 2 electrons of opposite spin and finally place the last electron into any of the three degenerate 2p<\/i> orbitals (Figure 8.3).\r\n\r\n[caption id=\"attachment_369\" align=\"aligncenter\" width=\"400\"]\"1s<\/a> Figure 8.3 \"Boron Electron Configuration Energy Diagram.\"[\/caption]\r\n\r\nMoving across the periodic table, we follow Hund\u2019s rule and add an additional electron to each degenerate 2p<\/i> orbital for each subsequent element (Figure 8.4). At oxygen we can finally pair up and fill one of the degenerate 2p<\/i> orbitals.\r\n\r\n[caption id=\"attachment_370\" align=\"aligncenter\" width=\"600\"]\"Electron<\/a> Figure 8.4 \"Electron Configuration Energy Diagrams for Carbon, Nitrogen, and Oxygen.\" [Long Description]<\/a>[\/caption]\r\n
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      Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

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