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TAMU CHEM 101 - Chap3x-103
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Pr1and Pr2V' !2e2|Pr1|!2e2|Pr2|%e2|Pr2!Pr1|-3.1-Quantum numbers associated with electron spin Symbol Name Values s spin angular momentum quantum number ½ m spin magnetic quantum number ± ½s j total angular momentum quantum number |RRRR ± s|CHAPTER 3. MULTI-ELECTRON ATOMS AND PERIODIC PROPERTIESAtomic structure plays a major role in determining the chemical properties of the elementsand therefore the student of chemistry must have a fundamental understanding of the nature of multi-electron atoms. The quantum mechanical description of the hydrogen atom, discussed in thepreceding chapter, provides the basis for a description of the electronic structure of multi-electronatoms. In an atom with more than one electron, however, additional terms representing the electron-electron repulsions must be added to the potential energy function in the Schrödinger equation.Unfortunately, these terms also make it impossible to obtain an exact solution, but elaborateapproximation methods have been developed to provide ways of calculating atomic properties to ahigh degree of accuracy. An additional complication is that relativistic effects must be taken intoaccount to obtain a complete understanding of atomic spectra.Exercise: Write the potential energy function for a helium atom in terms of the vectors ,which extend from the nucleus to electrons 1 and 2, respectively..A. Quantum Numbers and their Physical SignificanceRecall that the solution of the Schrödinger equation for the one-electron atom introduced thethree quantum numbers n, R, and m . In 1928, P. A. M. Dirac developed the relativistic equivalentRof the Schrödinger equation and found that its solution introduced the concept of electron spin andthree more quantum numbers, which are defined below.-3.2-1. The principal quantum number - n As was the case for the hydrogen atom, the principal quantum number primarily determinesthe energy and average radius of the orbital. All orbitals with the same value of n have nearly thesame energy and are said to belong to an atomic shell (K-shell, L-shell, M-shell, etc.). In a multi-electron atom, however, the electron-electron interactions cause energy levels with the same valueof n, but different values of R and j to have slightly different energies. The ordering of the first fewenergy levels in a multi-electron atom is shown in the Fig. 3.1.Figure 3.1. Ordering of the energy levels in a multi-electron atom.2. The orbital angular momentum quantum number - RThe quantum number R is associated with the orbital angular momentum of the electron. Aclassical analogy is helpful in understanding the physical significance of this and the magneticquantum number. Consider an electron revolving about the nucleus in a circular orbit, as in theBohr model. Classically, we could represent the orbital angular momentum of the electron by avector directed along the axis of rotation. The quantum mechanical angular momentum associatedwith the orbital motion of an electron has a magnitude given by-3.3-.In addition to its direct connection with orbital angular momentum, the R quantum numberdetermines the shape of the angular part of the wavefunction, as was discussed in the precedingchapter.3. The magnetic quantum number - mRThe quantum mechanical angular momentum may only have z-components given by .In terms of our classical analogy, this means that the angular momentum vector can only point incertain directions relative to a z-axis and these directions are determined by the m quantum number.RSince a z-axis in an atom may be defined by applying a magnetic field, m is called the magneticRquantum number. The properties of the quantum mechanical angular momentum may be conciselysummarized using a vector diagram of the type shown for R = 1 in Fig. 3.2.Exercise: Construct a vector diagram for a 3d orbital showing the z-components of the angularmomentum vector.Exercise: From the number of allowed z-components of the angular momentum vector for R =1, 2, and 3, deduce a relationship between this number and the value of R.4. The spin quantum numbers - s and msThe classical analogy of electron spin is a rotation about an axis passing through the electron.If one considers the rotation of the earth about the sun as analogous to the orbital motion of anelectron, then the rotation of the earth about its own axis is analogous to electron spin. Quantummechanically, the magnitude of the spin angular momentum for a single electron is.-3.4-Figure 3.2. Vector representation of the angular momentum vector for R = 1. The allowedorientations of L are fixed by the requirement that its z-components have lengths of0 or ± 1 £.Furthermore, there are two allowed orientations of the spin angular momentum vector for a singleelectron corresponding to the two values of m . The projection of the spin angular momentum onsits rotation axis is given by .When m = ½, the electron is said to have spin up and when m = !½, the electron is said to haves sspin down. Classically, these two spin orientations would correspond to opposite directions ofrotation.5. The total angular momentum quantum number - jThe orbital and spin motions of the electron generate their own individual magnetic fields.These two fields interact with each other in a manner similar to the interaction of two bar magnets.Because of this magnetic interaction, the orbital and spin motions are not independent of each other(i.e., they are coupled). As a result, the orbital angular momentum vector and the spin angularJLS-3.5-No two electrons in an atom may have the same set of quantum numbers.momentum vectors form a resultant total angular momentum vector, as depicted in Fig. 3.3. Thetotal angular momentum quantum number j for the two spin orientations of a single Figure 3.3. The total angular momentum vector J is the resultant of the orbital angularmomentum vector L and the spin angular momentum vector S.electron are given by.The j quantum number is related to the magnitude of the total angular momentum vector J in thesame way that R is related to |L|;.B. Electron Configurations and the Aufbau PrincipleIn the simplest description of a multi-electron atom, the set of four quantum numbers (n, R,m , and m ) may be used to specify each orbital. According to the Pauli exclusion principle, eachR selectron in an atom must have a unique wavefunction. Therefore-3.6-The above rule restricts the number


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