{"id":7412,"date":"2021-06-08T21:56:09","date_gmt":"2021-06-08T21:56:09","guid":{"rendered":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/quantum-numbers-for-electrons\/"},"modified":"2023-03-30T22:03:50","modified_gmt":"2023-03-30T22:03:50","slug":"quantum-numbers-for-electrons","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/quantum-numbers-for-electrons\/","title":{"raw":"Quantum Numbers for Electrons","rendered":"Quantum Numbers for Electrons"},"content":{"raw":"[latexpage]\r\n
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

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  1. Explain what spectra are.<\/li>\r\n \t
  2. Learn the quantum numbers that are assigned to electrons.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nThere are two fundamental ways of generating light: either heat an object up so hot it glows or pass an electrical current through a sample of matter (usually a gas). Incandescent lights and fluorescent lights generate light via these two methods, respectively.\r\n\r\nA hot object gives off a continuum of light. We notice this when the visible portion of the electromagnetic spectrum is passed through a prism: the prism separates light into its constituent colours, and all colours are present in a continuous rainbow (part (a) in Figure 8.03 \"Prisms and Light\"). This image is known as a [pb_glossary id=\"8087\"]continuous spectrum[\/pb_glossary]. However, when electricity is passed through a gas and light is emitted and this light is passed though a prism, we see only certain lines of light in the image (part (b) in Figure 8.03 \"Prisms and Light\"). This image is called a line spectrum. It turns out that every element has its own unique, characteristic line spectrum.\r\n\r\n[caption id=\"attachment_389\" align=\"aligncenter\" width=\"600\"]\"Part Figure 8.03 \"Prisms and Light.\" (a) A glowing object gives off a full rainbow of colours, which are noticed only when light is passed through a prism to make a continuous spectrum. (b) However, when electricity is passed through a gas, only certain colours of light are emitted. Here are the colours of light in the line spectrum of Hg.[\/caption]\r\n\r\nWhy does the light emitted from an electrically excited gas have only certain colours, while light given off by hot objects has a continuous spectrum? For a long time, it was not well explained. Particularly simple was the spectrum of hydrogen gas, which could be described easily by an equation; no other element has a spectrum that is so predictable (Figure 8.04 \"Hydrogen Spectrum\"). Late-nineteenth-century scientists found that the positions of the lines obeyed a pattern given by the following equation:\r\n

    [latex]\\dfrac{1}{\\lambda}=(109,700\\text{ cm}^{-1})\\left(\\dfrac{1}{4}-\\dfrac{1}{n^2}\\right)[\/latex]<\/p>\r\nWhere n<\/i> = 3, 4, 5, 6,\u2026, but they could not explain why this was so.\r\n\r\n[caption id=\"attachment_391\" align=\"aligncenter\" width=\"600\"]\"Hydrogen Figure 8.04 \"Hydrogen Spectrum.\" The spectrum of hydrogen was particularly simple and could be predicted by a simple mathematical expression.[\/caption]\r\n\r\nIn 1913, the Danish scientist Niels Bohr suggested a reason why the hydrogen atom spectrum looked this way. He suggested that the electron in a hydrogen atom could not have any random energy, having only<\/em> certain fixed values of energy that were indexed by the number n<\/i> (the same n<\/i> in the equation above and now called a [pb_glossary id=\"8062\"]quantum number[\/pb_glossary]). Quantities that have certain specific values are called [pb_glossary id=\"8063\"]quantized[\/pb_glossary]. Bohr suggested that the energy of the electron in hydrogen was quantized because it was in a specific orbit. Because the energies of the electron can have only certain values, the changes in energies can have only certain values (somewhat similar to a staircase: not only are the stair steps set at specific heights but the height between steps is fixed). Finally, Bohr suggested that the energy of light emitted from electrified hydrogen gas was equal to the energy difference of the electron\u2019s energy states:\r\n

    [latex]E_{\\text{light}}=h\\nu = \\Delta E_{\\text{electron}}[\/latex]<\/p>\r\nThis means that only certain frequencies (and thus, certain wavelengths) of light are emitted. Figure 8.05 \"Bohr\u2019s Model of the Hydrogen Atom\" shows a model of the hydrogen atom based on Bohr\u2019s ideas.\r\n\r\n[caption id=\"attachment_392\" align=\"aligncenter\" width=\"400\"]\"Bohr's Figure 8.05 \"Bohr\u2019s Model of the Hydrogen Atom.\" Bohr\u2019s description of the hydrogen atom had specific orbits for the electron, which had quantized energies.[\/caption]\r\n\r\nBohr\u2019s ideas were useful but were applied only to the hydrogen atom. However, later researchers generalized Bohr\u2019s ideas into a new theory called [pb_glossary id=\"8064\"]quantum mechanics[\/pb_glossary], which explains the behaviour of electrons as if they were acting as a wave, not as particles. Quantum mechanics predicts two major things: quantized energies for electrons of all atoms (not just hydrogen) and an organization of electrons within atoms. Electrons are no longer thought of as being randomly distributed around a nucleus or restricted to certain orbits (in that regard, Bohr was wrong). Instead, electrons are collected into groups and subgroups that explain much about the chemical behaviour of the atom.\r\n\r\nIn the quantum-mechanical model of an atom, the state of an electron is described by four quantum numbers, not just the one predicted by Bohr. The first quantum number is called the [pb_glossary id=\"8065\"]principal quantum number[\/pb_glossary]. Represented by n<\/i>, the principal quantum number largely determines the energy of an electron. Electrons in the same atom that have the same principal quantum number are said to occupy an [pb_glossary id=\"8066\"]electron shell[\/pb_glossary] of the atom. The principal quantum number can be any nonzero positive integer: 1, 2, 3, 4,\u2026.\r\n\r\nWithin a shell, there may be multiple possible values of the next quantum number, the [pb_glossary id=\"8067\"]angular momentum quantum number[\/pb_glossary], represented by \u2113. The \u2113 quantum number has a minor effect on the energy of the electron but also affects the spatial distribution of the electron in three-dimensional space \u2014 that is, the shape of an electron\u2019s distribution in space. The value of the \u2113 quantum number can be any integer between 0 and n<\/i> \u2212 1:\r\n

    [latex]\\ell=0,1,2,\\dots,n-1[\/latex]<\/p>\r\nThus, for a given value of n<\/i>, there are different possible values of \u2113, as shown in Table 8.1.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Table 8.1 Possible Values of \u2113<\/caption>\r\n
    If n<\/i> equals<\/th>\r\n\u2113 can be<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    1<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
    2<\/td>\r\n0 or 1<\/td>\r\n<\/tr>\r\n
    3<\/td>\r\n0, 1, or 2<\/td>\r\n<\/tr>\r\n
    4<\/td>\r\n0, 1, 2, or 3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nand so forth. Electrons within a shell that have the same value of \u2113 are said to occupy a [pb_glossary id=\"8068\"]subshell[\/pb_glossary]\u00a0in the atom. Commonly, instead of referring to the numerical value of \u2113, a letter represents the value of \u2113 (to help distinguish it from the principal quantum number):\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Table 8.2 Atomic Subshells as Defined by the Value of \u2113<\/caption>\r\n
    If \u2113 equals<\/th>\r\nThe subshell is<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    0<\/td>\r\ns<\/i><\/td>\r\n<\/tr>\r\n
    1<\/td>\r\np<\/i><\/td>\r\n<\/tr>\r\n
    2<\/td>\r\nd<\/i><\/td>\r\n<\/tr>\r\n
    3<\/td>\r\nf<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe next quantum number is called the [pb_glossary id=\"8069\"]magnetic quantum number[\/pb_glossary], represented by m<\/em>\u2113. For any value of \u2113, there are 2\u2113 + 1 possible values of m<\/i>\u2113<\/sub>, ranging from \u2212\u2113 to \u2113:\r\n

    [latex]\\begin{array}{c}\r\n-\\ell \\varleq m_\\ell \\varleq \\ell \\\\ \\\\\r\n\\text{or} \\\\ \\\\\r\n|m\\ell | \\varleq \\ell\r\n\\end{array}[\/latex]<\/p>\r\nThe following explicitly lists the possible values of m<\/i>\u2113<\/sub> for the possible values of \u2113:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Table 8.3 Possible Values of m<\/em>\u2113<\/span><\/caption>\r\n
    If \u2113 equals<\/th>\r\nThe m<\/i>\u2113 values can be<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    0<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
    1<\/td>\r\n\u22121, 0, or 1<\/td>\r\n<\/tr>\r\n
    2<\/td>\r\n\u22122, \u22121, 0, 1, or 2<\/td>\r\n<\/tr>\r\n
    3<\/td>\r\n\u22123, \u22122, \u22121, 0, 1, 2, or 3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe particular value of m<\/i>\u2113<\/sub> dictates the orientation of an electron\u2019s distribution in space. When \u2113 is zero, m<\/i>\u2113<\/sub> can be only zero, so there is only one possible orientation. When \u2113 is 1, there are three possible orientations for an electron\u2019s distribution. When \u2113 is 2, there are five possible orientations of electron distribution. This goes on and on for other values of \u2113, but we need not consider any higher values of \u2113 here. Each value of m<\/i>\u2113<\/sub> designates a certain [pb_glossary id=\"8070\"]orbital[\/pb_glossary]. Thus, there is only one orbital when \u2113 is zero, three orbitals when \u2113 is 1, five orbitals when \u2113 is 2, and so forth. The m<\/i>\u2113<\/sub> quantum number has no effect on the energy of an electron unless the electrons are subjected to a magnetic field \u2014 hence its name.\r\n\r\nThe \u2113 quantum number dictates the general shape of electron distribution in space (Figure 8.06 \"Electron Orbitals\"). As shown in part (a), any s<\/i> orbital is spherically symmetric, and there is only one orbital in any s<\/i> subshell. Part (b) shows that any p<\/i> orbital has a two-lobed, dumbbell-like shape; because there are three of them, we normally represent them as pointing along the x<\/i>-, y<\/i>-, and z<\/i>-axes of Cartesian space. In part (c), we see that d<\/i> orbitals are four-lobed rosettes; they are oriented differently in space (the one labelled dz<\/sub><\/i>2<\/sup> has two lobes and a torus instead of four lobes, but it is equivalent to the other orbitals). When there is more than one possible value of m<\/i>\u2113<\/sub>, each orbital is labelled with one of the possible values. It should be noted that the diagrams in Figure 8.06 are estimates of the electron distribution in space, not surfaces electrons are fixed on.\r\n\r\n[caption id=\"attachment_393\" align=\"aligncenter\" width=\"600\"]\"Shapes Figure 8.06 \"Electron Orbitals.\" (a) The lone s<\/em> orbital is spherical in distribution. (b) The three p<\/em> orbitals are shaped like dumbbells, and each one points in a different direction. (c) The five d<\/em> orbitals are rosette in shape, except for the dz<\/sub><\/i>2<\/sup> orbital, which is a \u201cdumbbell +\u00a0torus\u201d combination. They are all oriented in different directions.[\/caption]\r\n\r\nThe final quantum number is the [pb_glossary id=\"8071\"]spin quantum number[\/pb_glossary], represented by\u00a0m<\/i>s<\/sub>. Electrons and other subatomic particles behave as if they are spinning (we cannot tell if they really are, but they behave as if they are). Electrons themselves have two possible spin states, and because of mathematics, they are assigned the quantum numbers +\u00bd and \u2212\u00bd. These are the only two possible choices for the spin quantum number of an electron.\r\n
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    Example 8.3<\/p>\r\n\r\n<\/header>\r\n

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    Problems<\/h1>\r\nOf the set of quantum numbers {n<\/i>, \u2113, m<\/i>\u2113<\/sub>, m<\/i>s<\/sub>}, which are possible and which are not allowed?\r\n
      \r\n \t
    1. {3, 2, 1, +\u00bd}<\/li>\r\n \t
    2. {2, 2, 0, \u2212\u00bd}<\/li>\r\n \t
    3. {3, \u22121, 0, +\u00bd}<\/li>\r\n<\/ol>\r\n

      Solutions<\/h2>\r\n
        \r\n \t
      1. The principal quantum number n<\/i> must be an integer, which it is here. The quantum number \u2113 must be less than n<\/i>, which it is. The m<\/i>\u2113<\/sub> quantum number must be between \u2212\u2113 and \u2113, which it is. The spin quantum number is +\u00bd, which is allowed. Because this set of quantum numbers follows all restrictions, it is possible.<\/li>\r\n \t
      2. The quantum number n<\/i> is an integer, but the quantum number \u2113 must be less than n<\/i>, which it is not. Thus, this is not an allowed set of quantum numbers.<\/li>\r\n \t
      3. The principal quantum number n<\/i> is an integer, but \u2113 is not allowed to be negative. Therefore, this is not an allowed set of quantum numbers.<\/li>\r\n<\/ol>\r\n

        Test Yourself<\/h1>\r\nOf the set of quantum numbers {n<\/i>, \u2113, m<\/i>\u2113<\/sub>, m<\/i>s<\/sub>}, which are possible and which are not allowed?\r\n
          \r\n \t
        1. {4, 2, \u22122, 1}<\/li>\r\n \t
        2. {3, 1, 0, \u2212\u00bd}<\/li>\r\n<\/ol>\r\n

          Answers<\/h2>\r\n
            \r\n \t
          1. Spin must be either +\u00bd or \u2212\u00bd, so this set of quantum number is not allowed.<\/li>\r\n \t
          2. allowed<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
            \r\n

            Chemistry Is Everywhere: Neon Lights<\/h1>\r\nA neon light is basically an electrified tube with a small amount of gas in it. Electricity excites electrons in the gas atoms, which then give off light as the electrons go back into a lower energy state. However, many so-called \u201cneon\u201d lights don\u2019t contain neon!\r\n\r\nAlthough we know now that a gas discharge gives off only certain colours of light, without a prism or other component to separate the individual light colours, we see a composite of all the colours emitted. It is not unusual for a certain colour to predominate. True neon lights, with neon gas in them, have a reddish-orange light due to the large amount of red-, orange-, and yellow-coloured light emitted. However, if you use krypton instead of neon, you get a whitish light, while using argon yields a blue-purple light. A light filled with nitrogen gas glows purple, as does a helium lamp. Other gases \u2014 and mixtures of gases \u2014 emit other colours of light. Ironically, despite its importance in the development of modern electronic theory, hydrogen lamps emit little visible light and are rarely used for illumination purposes.\r\n\r\n[caption id=\"attachment_7411\" align=\"aligncenter\" width=\"300\"]\"A Figure 8.07 \"Neon.\" The different colours of these \u201cneon\u201d lights are caused by gases other than neon in the discharge tubes.[\/caption]\r\n\r\n<\/div>\r\n
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            Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

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