{"id":71,"date":"2017-08-18T08:55:13","date_gmt":"2017-08-18T08:55:13","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/chemistry\/?page_id=71"},"modified":"2017-09-18T18:15:23","modified_gmt":"2017-09-18T18:15:23","slug":"quantum-mechanics-part-ii","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/chemistry\/quantum-mechanics-part-ii\/","title":{"rendered":"Quantum Mechanics Part II"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\">\n<p><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/chemistry\/quantum-mechanics-part-i\">\u2b05 Previous\u00a0Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/chemistry\/atomic-structure-periodicity-and-matter\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/chemistry\/the-nucleus-and-nuclear-reactions\">Next Lesson\u27a1<\/a><\/p>\n<\/div>\n<p><!-- UPDATE NEXT\/PREVIOUS ABOVE --><\/p>\n<p><!-- CONTENT STARTS HERE --><\/p>\n<h1 id=\"title\">Atomic Structure, Periodicity, and Matter:Quantum Mechanics Part II<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson we will review additional evidence leading to the quantum mechanical model of the atom.<\/p>\n<h4>Previously we covered&#8230;<\/h4>\n<ul>\n<li>As the 1900&#8217;s came to a close, four important phenomena remained unexplained by classical physics: black body\u00a0radiation, the photoelectric effect, the absorption and emission of light by atoms, and the structure and\u00a0stability of the atom.<\/li>\n<li>Planck introduced the energy quantum to explain black body radiation.<\/li>\n<li>Einstein explained the photoelectric effect by proposing that a light wave consists of discrete photons.<\/li>\n<li>Compton demonstrated the particle nature of light waves and de Broglie proposed that matter has wave\u00a0properties.<\/li>\n<li>Spectra of gaseous atoms consist of discrete lines. Lines in the hydrogen spectrum are described by a simple\u00a0formula.<\/li>\n<li>Patterns of ionization energies are the foundation of a simple empirical model of how electrons are arranged in\u00a0atoms, their electron configurations.<\/li>\n<li>The arrangement of electrons in atoms is closely related to the chemical properties of the elements and to the\u00a0structure of the periodic table.<\/li>\n<\/ul>\n<section>\n<h3>Early Theoretical Models of the Atom<\/h3>\n<p>J.J. Thomson, who discovered that the electron was a subatomic particle in 1897, proposed the \u201cplum pudding\u201d model of\u00a0the atom (1904) in which electrons were \u201cplums\u201d in a \u201cpudding\u201d of positive charge This model was proved wrong by\u00a0experiments done in 1909 by Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford, who interpreted\u00a0the results to mean that a large amount of the charge\u2014positive or negative\u2014in an atom was concentrated in a very\u00a0tiny nucleus at the center of the atom. In 1911 Rutherford proposed a model of the atom in which electrons orbit a\u00a0central positively charged nucleus.<\/p>\n<p>Rutherford\u2019s model of the atom had a problem. Consider the hydrogen atom. In it, the electron is acted upon by the\u00a0attractive <abbr title=\"The attractive (between unlike charges) or repulsive (between like charges) force between two point charges; Its magnitude and direction are given by Coulomb\u2019s law, F = , where q1 and q2 are the charges and r is the distance between them.\">electrostatic force<\/abbr> between it and the proton. In classical mechanics, the force acting on an object is equal\u00a0to the product of mass and acceleration, <em>F = ma<\/em>; this means that the electron in a hydrogen atom must be\u00a0accelerating. While this does not present a problem for analogous situations with electrically neutral objects\u2014a\u00a0planet orbiting the sun or a car accelerating down the road, for example\u2014classical electromagnetic theory predicts\u00a0that an accelerating charge must radiate electromagnetic energy. As it loses energy, the electron in a hydrogen atom\u00a0should spiral into the nucleus\u2014the atom should not be stable\u2014and the radiation emitted should be continuous. In\u00a0fact, hydrogen and all other atoms are electronically stable\u2014their electrons do not collapse into the nucleus\u2014and,\u00a0as we have seen in Part I of this lesson, hydrogen and all other gaseous atoms both absorb and emit radiation at\u00a0discrete frequencies.<\/p>\n<h3>A Quantized Model of the Hydrogen Atom: Bohr\u2019s Planetary Model<\/h3>\n<p>Between 1913 and 1915 Niels Bohr developed a planetary model of the hydrogen atom\u2014with the electron circling the\u00a0nucleus like a planet circling the sun\u2014that was based in part on classical mechanics and electromagnetic theory, but\u00a0incorporated a quantum postulate: that the angular momentum of the electron was restricted to integral multiples of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_002.gif\" width=\"26\" height=\"41\" \/>, in other words, it was quantized.<\/p>\n<p>Bohr\u2019s theory predicted that the energy of an electron in the <em>n<\/em><sup>th<\/sup> energy level of a hydrogen atom\u00a0is given by the expression<\/p>\n<p class=\"center\"><em>E<sub>n<\/sub> = <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_003.gif\" width=\"98\" height=\"44\" align=\"absmiddle\" \/>= <\/em>-13.6<br \/>\neV <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_004.gif\" width=\"40\" height=\"45\" align=\"absmiddle\" \/><\/p>\n<p>where <em>m <\/em>is the electron mass, <em>k<\/em> is Coulomb\u2019s constant\u00a0(-8.37 \u00d7 10<sup>-19<\/sup> J), <em>e<\/em> is the charge on an electron, <em>h<\/em> is Planck\u2019s constant (6.626 \u00d7\u00a010<sup>-34<\/sup> J<strong>\u00b7<\/strong>s) , and <em>n<\/em> is an integer, equal to or greater than one. Thus the energy\u00a0of an electron in the lowest energy level (<em>n<\/em> = 1) is -13.6 eV, where 1 eV(an energy unit often used when\u00a0dealing with atomic energy levels) is equivalent to 1.6022 \u00d7 10<sup>-19<\/sup> J. For an electron to move from the\u00a0<em>n<\/em><sup>th<\/sup>allowed energy level to the <em>m<\/em><sup>th<\/sup> allowed energy level of hydrogen, it\u00a0would have to absorb or emit a photon whose energy was equal to the difference in energy between the two levels:<\/p>\n<p class=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_016.gif\" width=\"428\" height=\"45\" \/><\/p>\n<p>If <em>m <\/em>is greater than <em>n<\/em>, the energy change is positive and a photon is absorbed; if <em>m <\/em>is\u00a0smaller than <em>n<\/em>, the energy change is negative and a photon is emitted.<\/p>\n<p>Since the wavelength of light is related to its frequency (the product of wavelength and frequency is equal\u00a0to the speed of light, <em>\u03bb\u03bd<\/em> = <em>c<\/em>), Bohr\u2019s equation for the energy of a photon can be rearranged to\u00a0give an expression equivalent to the Rydberg equation discussed in Part I of this lesson. Bohr found that the value\u00a0he predicted for <em>R<\/em>, the constant in the Rydberg equation, was equal (within the limits of experimental\u00a0uncertainties) to the experimental value for <em>R<\/em>, and the spectrum predicted by his theory was in complete\u00a0agreement with experimental values for the hydrogen spectrum.<\/p>\n<p>While Bohr\u2019s theory was successful with hydrogen and hydrogen-like ions, it could not be extended to atoms with more\u00a0than one electron, and it could not explain relative intensities of lines in the hydrogen spectrum. Other flaws in\u00a0the theory were that Bohr\u2019s quantum hypothesis was <em>ad hoc<\/em>\u2014there was no fundamental basis for this\u00a0assumption\u2014and that it was based on assumptions (most significantly, electron orbits and the validity of classical\u00a0mechanics and electromagnetism as applied to atoms) that could not be experimentally verified. Thus while Bohr\u2019s\u00a0theory was an important step, a completely correct theory of the atom remained to be discovered.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the wavelength of the photon emitted when an electron in the <em>n<\/em> = 5 energy level of hydrogen moves to the <em>n<\/em> = 2 energy level?<\/p>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<p class=\"q-reveal\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_008.gif\" width=\"596\" height=\"98\" \/><\/p>\n<p class=\"q-reveal\"><em>\u03bb<\/em> = <img loading=\"lazy\" decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_009.gif\" width=\"48\" height=\"46\" align=\"absmiddle\" \/> = <img loading=\"lazy\" decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_010.gif\" width=\"225\" height=\"44\" align=\"absmiddle\" \/> = 4.34 \u00d710<sup>-7<\/sup> m, which is in the violet region of the visible spectrum.<\/p>\n<\/section>\n<h3>Matrix Mechanics<\/h3>\n<p>In 1925, Werner Heisenberg developed what came to be known as <em>matrix mechanics<\/em>, a system of atomic mechanics\u00a0that uses only observable entities, such as spectral frequencies and intensities, as the basis for its calculations\u00a0and which was completely successful in its applications. In spite of its successes, matrix mechanics never gained\u00a0popularity, because it used mathematics that was unfamiliar to physicists and gave results that were not easily\u00a0visualized.<strong><br \/>\n<\/strong><\/p>\n<h3>Wave Mechanics<\/h3>\n<p>In the fall of 1925, Erwin Schr\u00f6dinger came across de Broglie\u2019s work, and began considering a theory of the electron\u00a0in an atom as a wave. Over the course of the following year, Schr\u00f6dinger published a series of papers in which he\u00a0presented his wave theory of atomic structure, completely different in form from Heisenberg\u2019s matrix mechanics, but\u00a0completely equivalent to it in content. While the two theories are equivalent, Schr\u00f6dinger\u2019s quantum mechanics was\u00a0more palatable to physicists and became the dominant model of the electron.<\/p>\n<h4>The Wave Model of Electrons in Atoms<\/h4>\n<p>While it seems quite odd that electrons, apples, and oranges should all behave like waves, we need to keep two things\u00a0in mind.<\/p>\n<p>First, the de Broglie wavelength is given by <em>\u03bb<\/em>=<img loading=\"lazy\" decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_011.gif\" width=\"16\" height=\"44\" \/>, which for an electron traveling at\u00a0speeds an electron in an atom might have\u2014say 1% of the speed of light\u2014will be of the order of 2\u00d710<sup>-10<\/sup> m,\u00a0or about the size of a typical atom. On the other hand, for apples and oranges traveling at ordinary apple and\u00a0orange speeds the wavelength is of the order of 1 \u00d710<sup>-34<\/sup> m or less\u2014a wavelength that is quite\u00a0unobservable!<\/p>\n<p>Second, a line spectrum of energy absorption is a characteristic of a wave that is confined in a bounded region, for\u00a0example, a wave on a string stretched between two fixed points. At certain frequencies, called <abbr title=\"Frequencies at which energy is absorbed efficiently and at which a standing wave is established\">resonant\u00a0frequencies<\/abbr>, the string efficiently absorbs energy and a stationary or <abbr title=\" A wave for which the locations of nodes, points at which the wave amplitude is near zero, and antinodes, points at which the wave amplitude is greatest, are fixed\">standing wave<\/abbr>\u2014a wave for which the locations of <abbr title=\"Points at which the wave amplitude is zero\">nodes<\/abbr>, points at which the wave amplitude\u00a0is zero, and <abbr title=\"Points at which the wave amplitude is maximal\">antinodes<\/abbr>, points\u00a0at which the wave amplitude is greatest, are fixed\u2014is established. At non-resonant frequencies, very little energy\u00a0is absorbed and wave amplitude is negligible. The line absorption and emission spectra of atoms suggest that a\u00a0standing wave phenomenon is involved, and since it is electrons in an atom that are involved in absorption or\u00a0emission of energy, the electron as wave seems plausible.<\/p>\n<h3>Chemistry and Quantum Mechanics<\/h3>\n<p>The details of quantum mechanical calculations need not concern us here, but the basic principles, concepts, and\u00a0results of quantum mechanics are fundamental to the contemporary chemist\u2019s understanding of molecular structure,\u00a0chemical properties, and chemical reactivity. Today, quantum mechanics serves as a predictive tool in the design of\u00a0new materials and drugs, and a wide variety of computational software makes quantum mechanical calculations and\u00a0molecular modeling accessible to anyone with a personal computer.<\/p>\n<h4>Orbitals and Quantum Numbers<\/h4>\n<p>Schr\u00f6dinger\u2019s quantum mechanics gave results identical to the experimental results on which the electron shell model\u00a0was based. He showed that electrons with different energies occupy different regions about the nucleus called\u00a0<abbr title=\" A three-dimensional region about the nucleus in which a particular electron can be located; an orbital can contain only one or two electrons\">orbitals<\/abbr>.<\/p>\n<p>Each orbital in an atom has a unique set of three quantum numbers, <em>n<\/em>, <em>l<\/em>, and <em>m<\/em>, that\u00a0characterize, respectively, the distance of the orbital from the nucleus, the shape of the orbital, and its\u00a0orientation in space. The three quantum numbers are related to each other and to the electron shell model in the\u00a0following ways:<\/p>\n<ul>\n<li><em>n<\/em>, the <em>principal quantum number<\/em>, takes the integer values, 1, 2, 3,\u2026 and so on, and is the\u00a0same as the shell number.<\/li>\n<li><em>l<\/em>, the <em>angular quantum number<\/em>, can be any integer value between 0 and <em>n<\/em> \u2013 1. If\u00a0<em>n<\/em>\u00a0= 3, for example, <em>l<\/em> can be 0, 1, or 2.<\/li>\n<li><em>m<\/em>, the <em>magnetic quantum number<\/em>, can be any integer value between \u2013<em>l<\/em> and +<em>l<\/em>.\u00a0If <em>l<\/em>\u00a0= 2, <em>m<\/em> can be -2, -1, 0, 1, or 2.<\/li>\n<li>The larger the value of <em>n<\/em>, the higher the energy of that shell and the farther the average distance of\u00a0electrons in that shell from the nucleus.<\/li>\n<li>Orbitals that have the same principal quantum number form a shell.<\/li>\n<li>Orbitals with the same angular quantum number form a subshell.<\/li>\n<\/ul>\n<p>The value of <em>l<\/em> determines the type of orbital:<\/p>\n<table class=\"gas_law_table\" width=\"25%\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><strong><em>l<\/em><\/strong><\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>Orbital<\/strong><\/td>\n<td><em>s <\/em><\/td>\n<td><em>p <\/em><\/td>\n<td><em>d <\/em><\/td>\n<td><em>f <\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are orbitals corresponding to higher values of <em>l<\/em>, but we will consider only <em>s<\/em>, <em>p<\/em>,\u00a0<em>d<\/em>, and <em>f<\/em> orbitals, as only these are involved in ground state electron configurations.<\/p>\n<p>Each subshell has a specific value for <em>l<\/em> and within a subshell there is one orbital for each allowed value\u00a0of <em>m<\/em>. For example, in a subshell with <em>l <\/em>= 0 and <em>m <\/em>= 0, there is 1 single <em>s\u00a0<\/em>orbital; in a subshell with <em>l<\/em> = 1 and <em>m<\/em> = 1, 0, or -1, there are three <em>p<\/em> orbitals.\u00a0Orbitals are labeled with their principal quantum number and orbital type; for example the label for an orbital with<br \/>\n<em>n<\/em> = 2 and <em>l<\/em>\u00a0=\u00a01 is 2<em>p<\/em>. Table 1 lists the allowed combinations of quantum\u00a0numbers for <em>n<\/em> = 1, 2, 3, and 4, the orbital type and number of orbitals in a subshell, and the number of\u00a0electrons in a full subshell.<\/p>\n<table class=\"gas_law_table\" border=\"1\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<thead>\n<tr>\n<th>Shell (<em>n<\/em>)<\/th>\n<th>\n<div align=\"center\">Subshell (<em>l<\/em>)<\/div>\n<\/th>\n<th>\n<div align=\"center\">Orbital (<em>m<\/em>)<\/div>\n<\/th>\n<th>\n<div align=\"center\">Orbital Notation<\/div>\n<\/th>\n<th>\n<div align=\"center\">Number of Orbitals in Subshell<\/div>\n<\/th>\n<th>\n<div align=\"center\">Number of Electrons in Full Subshell<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<em>s <\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<em>s <\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">1, 0, -1<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<em>p <\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">6<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">3s<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">1, 0, -1<\/div>\n<\/td>\n<td>\n<div align=\"center\">3<em>p<\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">6<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">2, 1, 0, -1, -2<\/div>\n<\/td>\n<td>\n<div align=\"center\">3<em>d <\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">5<\/div>\n<\/td>\n<td>\n<div align=\"center\">10<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">0<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<em>s <\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">1, 0, -1<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<em>p <\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">6<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">2, 1, 0, -1, -2<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<em>d<\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">5<\/div>\n<\/td>\n<td>\n<div align=\"center\">10<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">3, 2, 1, 0, -1, -2, -3<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<em>f <\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\">7<\/div>\n<\/td>\n<td>\n<div align=\"center\">14<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"caption\">Table 1<\/p>\n<p>Below are sketches of <em>s<\/em>, <em>p<\/em>, and <em>d<\/em> orbitals. Each value of <em>m<\/em> corresponds to a\u00a0unique orbital orientation. For example, for <em>l<\/em> = 1, there are three <em>p<\/em> orbitals, one for each\u00a0allowed value of <em>m<\/em>, 1, 0, and -1. These are usually labeled as <em>p<sub>x<\/sub><\/em>,\u00a0<em>p<sub>y<\/sub><\/em>, and <em>p<sub>z<\/sub><\/em>, where the subscripts specify the shapes and orientations of the\u00a0orbitals. Thus, for example, an electron in a <em>p<sub>z<\/sub><\/em> orbital is confined to a region above and below\u00a0the <em>x-y<\/em> plane and centered around the <em>z-<\/em>axis. The shapes and orientations of <em>d <\/em>orbitals<br \/>\nare more complex. For example, a <em>d<sub>xy<\/sub><\/em> orbital has four lobes that lie in the <em>x-y<\/em> plane\u00a0between the <em>x<\/em> and <em>y<\/em> axes, and a <img loading=\"lazy\" decoding=\"async\" class=\"no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/s5_012.gif\" width=\"42\" height=\"25\" \/> orbital has four lobes that lie along the\u00a0<em>x-<\/em> and <em>y-<\/em>axes.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/quantummechanicsII6.sporbitals.png\" alt=\"Sketches of s, p, and d orbitals\" width=\"572\" height=\"569\" \/><\/center><\/p>\n<p class=\"figcaption\">Sketches of <em>s<\/em>, <em>p<\/em>, and <em>d<\/em> orbitals<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Without consulting Table 1, answer the following questions:<\/p>\n<ol>\n<li>What is the number of orbitals in the <em>n<\/em> = 4 shell?<\/li>\n<li>Which orbitals are in the <em>l<\/em> = 3 subshell?<\/li>\n<li>How many electrons can occupy the <em>l<\/em> = 3 subshell?<\/li>\n<\/ol>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>a) If <em>n<\/em> = 4, <em>l<\/em> can have values 3, 2, 1, or 0, so there will be one 4<em>s<\/em> orbital, three\u00a04<em>p<\/em> orbitals, five 4<em>d<\/em> orbitals, and seven 4<em>f<\/em> orbitals, for a total of sixteen\u00a0orbitals.<\/p>\n<p>b) The <em>l<\/em> =3 subshell is composed of seven <em>f<\/em> orbitals.<\/p>\n<p>c) The <em>l<\/em> = 3 subshell can hold fourteen electrons.<\/p>\n<\/div>\n<\/section>\n<h3>Electron Spin\u2014The Stern-Gerlach Experiment<\/h3>\n<p>At this point, the empirical electron shell model we developed in Part I of this lesson is essentially equivalent to\u00a0the electron shell model based on quantum mechanics. Though we now know that <em>p<\/em>, <em>d<\/em>, and <em>f\u00a0<\/em>subshells have multiple orbitals, we do not know how electrons in one of these subshells are distributed among these\u00a0orbitals. The key to understanding how electrons are arranged in the orbitals of a subshell involves a fourth,\u00a0two-valued quantum number <em>s<\/em>, introduced by Wolfgang Pauli in 1924 to explain the splitting of lines in\u00a0atomic spectra into closely-spaced pairs. The physical significance of this fourth quantum number was unknown at\u00a0this point.<\/p>\n<p>Otto Stern and Walter Gerlach had discovered in 1922 that a beam of silver atoms was split into two parts by an\u00a0inhomogeneous magnetic field, an effect that indicated that silver atoms acted like atomic magnets. The fact that\u00a0the beam was split into two parts rather than simply broadened, suggested that two quantum magnetic states were\u00a0involved.<\/p>\n<p>In 1925, Ralph Kronig, a graduate student at Columbia University, and George Uhlenbeck and Samuel Goudsmit, graduate\u00a0students at the University of Leiden, independently explained the Stern-Gerlach results by suggested that electrons<br \/>\nact like rotating spheres of electric charge which behave like magnets and which, in an inhomogeneous magnetic\u00a0field, will be in one of two energy levels. In an atom with an even number of electrons, these tiny magnets would\u00a0form pairs that would not be affected by passing through a magnetic field. In atoms like silver, which have an odd\u00a0number of electrons, there would be one unpaired electron, and a magnetic field would separate a beam of atoms into\u00a0two parts.<\/p>\n<h3>The Pauli Exclusion Principle, the Aufbau Rule, Hund\u2019s Rule, and the Arrangement of Electrons in Atoms<\/h3>\n<p>Later in 1925, Pauli showed that his fourth quantum number <em>s<\/em> identified the spin state of the electron,\u00a0electron spin up with <em>s<\/em> = +\u00bd or electron spin down with <em>s<\/em> = -\u00bd, and developed his exclusion\u00a0principle to explain a variety of experimental results. The exclusion principle specifies that each electron in an\u00a0atom must have a unique set of four quantum numbers. If two electrons occupy a single orbital, each has the same\u00a0values of <em>n<\/em>, <em>l<\/em>, and <em>m<\/em>, so one electron must have<em> s<\/em>=+\u00bd and the other <em>s<\/em><br \/>\n=\u2013\u00bd , (i.e. with spins \u201cpaired\u201d) which means that no more than two electrons can occupy the same orbital.<\/p>\n<p>With the exclusion principle and the assumption that as electrons are added to an atom they fill the lowest energy\u00a0orbitals first (the \u201cbuilding up\u201d or aufbau rule), we can predict the order in which electrons will go into\u00a0orbitals. For example, the first and second electrons added to a carbon atom will go into the 1<em>s<\/em> orbital,\u00a0filling it. The third and fourth electrons fill the 2<em>s<\/em> orbital, and the next two electrons have a choice of\u00a02<em>p<sub>x<\/sub><\/em>, 2<em>p<sub>y<\/sub><\/em>, and 2<em>p<sub>z<\/sub><\/em>, orbitals to occupy.<\/p>\n<p>However, while the exclusion principle specifies that no more than two electrons can be in the same orbital, it does\u00a0not tell us which arrangement of valence electrons is most stable. Thus, we cannot decide on the basis of the\u00a0exclusion principle or the aufbau rule whether a carbon atom with two 2<em>p<\/em> electrons will have both in the same 2<em>p<\/em> orbital with paired spins, or in separate 2<em>p<\/em> orbitals with paired spins, or in separate\u00a0orbitals with unpaired spins. To answer this question, we must apply a rule devised by Friedrich Hund, also in 1925.<\/p>\n<p>Hund\u2019s rule says (1) that there must one electron in each orbital of a subshell before any electrons are paired in\u00a0any orbital of that subshell, and (2) that all electrons in a subshell have the same spin until each orbital in the\u00a0subshell has at least one electron. For example, in the second shell of carbon, the 2s orbital has two paired\u00a0electrons, but the two 2p electrons occupy separate 2p orbitals and are unpaired.<\/p>\n<p>The usual explanation of Hund\u2019s rule is that by occupying separate orbitals in a subshell electron-electron repulsion\u00a0is minimized, and this lowers the energy of these electrons. In general, this is not correct, as very accurate\u00a0calculations show that in most cases the average distance between unpaired electrons in separate orbitals is\u00a0actually slightly less than that between paired electrons in a single orbital. This means that electron-electron\u00a0repulsion is actually slightly greater between unpaired electrons in different orbitals than between paired\u00a0electrons in a single orbital.<\/p>\n<p>What does account for the stability of unpaired electrons is that screening of the nuclear charge by electrons in a\u00a0subshell is reduced when they occupy separate orbitals, so the attractive forces between electrons and the nucleus\u00a0are greater when electrons are in different orbitals. It is this increase in the nuclear attraction that allows\u00a0unpaired electrons occupying separate orbitals in a subshell to be closer to the nucleus, making them more\u00a0stable.<\/p>\n<h3>Electron Configurations Revisited<\/h3>\n<p>With our rules for placing electrons in atoms we can now predict electron configurations that include the number of\u00a0electrons in orbitals. We can use the order-of-filling sequence we presented in Part I of this lesson.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/chemistry\/wp-content\/uploads\/sites\/3\/2017\/08\/quantummechanics10.orderoffillingsubshells.gif\" width=\"650\" height=\"20\" \/><\/center>We can now apply the exclusion principle and Hund\u2019s rule to distribute the electrons among the\u00a0available orbitals. In using this procedure to predict electron configurations, one should keep two things in\u00a0mind:<\/p>\n<ul>\n<li>While the order-of-filling scheme above is helpful in predicting electron configurations, it does <em>not\u00a0<\/em>give the correct sequence of energy levels. In scandium, for example, the most weakly held electron is a\u00a04<em>s<\/em> electron with an ionization energy of 633 kJ\/mol, not the 3<em>d<\/em> electron, for which the\u00a0ionization energy is 770 kJ\/mol.<\/li>\n<li>There are many exceptions to expected electron configurations among the transition metal elements. For example,\u00a0the electron configuration predicted for copper using our rules and filling scheme is\u00a0[Ar]3<em>d<\/em><sup>9<\/sup>4<em>s<\/em><sup>2<\/sup>, but spectroscopic studies show that it is actually [Ar]3<em>d<\/em><sup>10<\/sup>4<em>s<\/em><sup>1<\/sup>,\u00a0which has a filled set of <em>d<\/em> orbitals. As we mentioned in Part I, half full and completely full\u00a0<em>d<\/em> and <em>f <\/em>subshells are more stable than other configurations.<\/li>\n<\/ul>\n<p>If a periodic table is handy you need not memorize this scheme for filling subshells, because it can be deduced from\u00a0the structure of the periodic table. For example, moving across the first row of the periodic table the 1<em>s <\/em>subshell\u00a0is filled; moving across the second and third rows the 2<em>s<\/em> and 2<em>p<\/em> and 3<em>s<\/em> and 3<em>p\u00a0<\/em>subshells, respectively, are filled. With the fourth row, the 4<em>s<\/em> subshell is filled, then the 3<em>d\u00a0<\/em>subshell, and lastly the 4<em>p<\/em> subshell. This is the same order of filling outlined in the scheme above.<\/p>\n<p>A few examples should serve to make clear what is involved:<\/p>\n<blockquote>\n<h4>The Ground State Electron Configuration in Oxygen<\/h4>\n<p>The ground state electron configuration of an oxygen atom can be written\u00a0[He]2<em>s<\/em><sup>2<\/sup>2<em>p<sub>x<\/sub><\/em><sup>2<\/sup>2<em>p<sub>y<\/sub><\/em><sup>1<\/sup>2<em>p<sub>z<\/sub><\/em><sup>1<\/sup>.\u00a0The assignment here of two electrons to a 2<em>p<sub>x<\/sub><\/em> orbital and one each to 2<em>p<sub>y\u00a0<\/sub><\/em>and 2<em>p<sub>z<\/sub><\/em> orbitals is arbitrary; what is important is that one 2<em>p<\/em> orbital is full and\u00a0two are half full.<\/p>\n<h4>Ground State Electron Configurations in Vanadium and Chromium<\/h4>\n<p>The electron configuration of vanadium can be written\u00a0[Ar]4<em>s<\/em><sup>2<\/sup>3<em>d<sub>xy<\/sub><\/em><sup>1<\/sup>3<em>d<sub>yz<\/sub><\/em><sup>1<\/sup>3<em>d<sub>xz<\/sub><\/em><sup>1<\/sup>.\u00a0As with oxygen, the choice of orbitals is arbitrary; the significant point is that there are three 3<em>d\u00a0<\/em>orbitals that are singly occupied. More commonly, the electron configuration in vanadium is written\u00a0[Ar]3<em>d<\/em><sup>3<\/sup>4<em>s<\/em><sup>2<\/sup>, which reflects the fact that the 4<em>s<\/em> electrons are\u00a0least tightly held and lie above the 3<em>d<\/em> electrons energetically.<\/p>\n<p>The next element in the periodic table is chromium, for which we might expect a ground state electron\u00a0configuration [Ar]3<em>d<\/em><sup>4<\/sup>4<em>s<\/em><sup>2<\/sup>. Spectroscopic results show that the ground\u00a0state electron configuration is actually [Ar]3<em>d<\/em><sup>5<\/sup>4<em>s<\/em><sup>1<\/sup>, with a half filled\u00a0set of <em>d<\/em> orbitals.<\/p>\n<p>Variation in the electron configurations for the transition metal elements in rows 5-7 of the periodic table is\u00a0greater, and no simple pattern can be used to predict actual electron arrangements.<\/p><\/blockquote>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Predict the electron configurations for the following ions:<\/p>\n<ol>\n<li>Cu<sup>+1<\/sup><\/li>\n<li>Cu<sup>+2<\/sup><\/li>\n<li>Fe<sup>+2<\/sup><\/li>\n<\/ol>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>a) The electron configuration for Cu<sup>+1<\/sup> is [Ar]3<em>d<\/em><sup>10<\/sup>. Copper, [Ar]3<em>d<\/em><sup>10<\/sup>4<em>s<\/em><sup>1<\/sup>,\u00a0loses its single 4<em>s <\/em>electron to form the Cu<sup>+1<\/sup> ion.<\/p>\n<p>b) The electron configuration for Cu<sup>+2<\/sup> is [Ar]3<em>d<\/em><sup>9<\/sup>. Copper loses its 4<em>s\u00a0<\/em>electron and one 3<em>d<\/em> electron to form the Cu<sup>+2<\/sup> ion.<\/p>\n<p>c) The electron configuration for Fe<sup>+2<\/sup> is [Ar]3<em>d<\/em><sup>6<\/sup>. Iron,\u00a0[Ar]3<em>d<\/em><sup>6<\/sup>4<em>s<\/em><sup>2<\/sup>, loses both 4<em>s<\/em> electrons to form the Fe<sup>+2\u00a0<\/sup>ion.<\/p>\n<\/div>\n<\/section>\n<hr \/>\n<h3>Why Doesn\u2019t the Atom Collapse? Heisenberg\u2019s Uncertainty Principle<\/h3>\n<p>We conclude our lesson on quantum mechanics with a brief discussion of one last concept, Heisenberg\u2019s uncertainty\u00a0principle. In classical physics it is, in theory, possible to determine precisely the initial state of a system, and\u00a0from this information predict the behavior of the system as far into the future as desired. However, there is a\u00a0fundamental limit to the precision with which we can determine the location or <abbr title=\" Measurement expressing the motion of a system. The product of mass (m) and velocity (v), and denoted by the letter p. p = mv\">momentum<\/abbr> of any particle. Heisenberg made a careful analysis of the relationship between the uncertainty in the momentum of a\u00a0particle, \u0394<em>p<\/em>, and the uncertainty in its location, \u0394<em>x<\/em>, and concluded that the product of the two\u00a0was of the order of Planck\u2019s constant <em>h<\/em>:<\/p>\n<p class=\"center\">\u0394<em>p\u00b7<\/em>\u0394<em>x \u2248 h <\/em><\/p>\n<p>The fact that atoms do not collapse can be considered a consequence of the uncertainty principle. Forcing the\u00a0electrons closer to the nucleus means that \u0394<em>x<\/em> is reduced, which causes a larger uncertainty \u0394<em>p<\/em> in\u00a0the momentum. While a proof is beyond the scope of this lesson, it can be shown that applying the uncertainty\u00a0principle leads to a correct prediction of the dimensions of an atom.<\/p>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<p><!-- UPDATE NEXT\/PREVIOUS BELOW --><\/p>\n<div class=\"advance\">\n<p><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/chemistry\/quantum-mechanics-part-i\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/chemistry\/atomic-structure-periodicity-and-matter\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/chemistry\/the-nucleus-and-nuclear-reactions\">Next Lesson\u27a1<\/a><\/p>\n<\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous\u00a0Lesson\u00a0Workshop Index\u00a0Next Lesson\u27a1 Atomic Structure, Periodicity, and Matter:Quantum Mechanics Part II Objective In this lesson we will review additional evidence leading to the quantum mechanical model of the atom. Previously we covered&#8230; As the 1900&#8217;s came to a close, four important phenomena remained unexplained by classical physics: black body\u00a0radiation, the photoelectric effect, the absorption [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-71","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/pages\/71","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/comments?post=71"}],"version-history":[{"count":11,"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/pages\/71\/revisions"}],"predecessor-version":[{"id":814,"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/pages\/71\/revisions\/814"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/chemistry\/wp-json\/wp\/v2\/media?parent=71"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}