L> Quantum Numbers and Electronic StructureQuantum Numbers and Electronic StructureQuantum Numbers Table |Atomic Structure Slideshow |Quantum Chemistry QuizzesTo account for the behavior of an electron in an atom, it is not sufficientto describe the electron simply as a negatively charged particle; its wavelikeproperties must also be considered. The first such description of an electroncame in 1926 with the development of Schrodinger"s wave equation. The branchof science that deals with the solution of wave equations is called quantummechanics (or wave mechanics). Each solution to a wave equation is characterizedby three integers called quantum numbers. Each solution corresponds to adiscrete energy and defines a region of space about the nucleus (called anorbital) where an electron having that energy is generally found. A fourthquantum number is also necessary for a unique description of an equation.According to the quantum mechanical model, the allowed energy levels of anelectron are composed of one or more orbitals, and the distribution of electronsabout the nucleus is determined by the number and kinds of energy levelsthat are occupied. Therefore, in order to understand the way electrons aredistributed, we must first examine the energy levels. This is best accomplishedthrough a discussion of the four quantum numbers. The most important aspectsof each quantum number are presented below.The principal quantum number, n. The electronic energy levelsin an atom are arranged roughly into principal levels (or shells) as specifiedby n. The value of n gives an indication of the position of an electron inthe energy level relative to the nucleus; the larger the value of n, thegreater the average distance of an electron from the nucleus and the higherits energy. The principal quantum number may have values as follows: n= 1, 2, 3, 4, …..The azimuthal quantum number (also called subsidiary or secondary),l. Each principal energy level may be split into closely spacedsublevels (or sublevels) as specified by l. This quantum number maybe more aptly named the orbital shape quantum number, since each orbitalin a given type of sublevel (i.e., a given value of l) has the same"electron cloud" shape. For example, when l = 0, the orbital is spherical.For each principal energy level (designated by n) there are nsublevels (i.e., n values of l): l=0,1,2,3,…(n -1). Sublevels are commonly given letter designations. The l= 0,1,2,3,4,5, … sublevels are designated as s, p,d, f, g,…. sublevels, respectively. For known elementsno value of l higher than 3 (f sublevel) is necessary.Two quantum numbers (n and l) are required to specify a particularenergy sublevel.The magnetic quantum number, ml. Each orbital withina particular sublevel is distinguished by its value ofml. This quantum number may be more aptly named the orbitalorientation quantum number. In each energy sublevel (designated by l)there are 2l+1 possible independent orientations of the electron cloud.Each orientation is defined by a value of ml and is called an orbital.ml = l, (l -1), (l -2), … 0 …-( l -2), -( l -1), -l or ml = 0,±1, ±2, ±3, … ±1All orbitals in a given sublevel are of equal energy (they are degenerate).In the presence of a magnetic field their different orientations cause themto have different energies.Three quantum numbers (n, l, and ml) arerequired to specify a particular orbital.The spin quantum number, ms. An electron spins onits own axis as characterized by ms. There are two possibledirections of spin: ms = +1/2 or -1/2. Since a spinningcharge generates a magnetic field, an electron has a magnetic field associatedwith it. Two electrons in the same orbital are most stable when they haveopposite spins (+1/2 and -1/2) due to a magnetic attraction. Such electronsare said to be paired electrons or each other"s magnetic fields, but an unpairedelectron may be detected by magnetic measurements. In fact, elements withunpaired electron are attracted by magnetic fields; such elements are calledparamagnetic. Magnetic measurements have shown that electrons are distributedamong the orbitals of a sublevel in a way that gives the maximum number ofunpaired electrons with parallel spins (all ms values havethe same sign).Four quantum numbers (n, l, ml, andms) are required to specify a particular electron.The following list is a condensation of some of the most useful informationrelevant to quantum numbers and electronic structure. The four quantum numbers: n = 1, 2, 3, … l = 0, 1, 2, 3, …. (n-1) ml = 0, ±1, ±2, ±3, … ±l ms = +1/2 or -1/2 Sublevel Types QN l Type Orbitals Total Electrons QN l Type Orbitals Total Electrons 0 s 1 2 3 f 7 14 1 p 3 6 4 g 9 18 2 d 5 10 5 h 11 22 Principal energy level n contains: (a) n sublevels (b) n2 orbitals (c) 2n2 electrons maximum (see table above) Sublevel l contains: (a) 2l+1 orbitals (b) 2(2l+1) electrons maximum In the ground state electrons fill orbitals so that the total energy of the atom is minimized. Sublevel energies increase as: (a) n increases: 1sss…; 2ppp ….. etc. (b) l increases: 2sp; 3spd; 4spdf; etc. Each orbital can hold a maximum of two electrons; they must be paired. In a given sublevel electrons are distributed among the orbitals in a way that yields the maximum number of unpaired electrons with parallel spins. For a given value of l, orbital shape remains the same. For example, orbitals with l = 0 (1s, 2s, 3s, etc. sublevels) are all spherical. A given value of n designates a specific principal energy level. Given values of n and l designate a specific energy sublevel. Given values of n, l, and ml designate a specific orbital. Given values of n, l, ml, and ms designate a specific electron.To review the definitions and interrelationships of the 4 quantum numbers,study this Table. Careful study will show that the table is consistent withthe statements listed above. n l ml Subshell Number of orbitalsin subshell Number of electrons in subshell 1 0 0 1s 1 2 2 0 0 2s 1 2 2 1 -1, 0, +1 2p 3 6 3 0 0 3s 1 2 3 1 -1, 0, +1 3p 3 6 3 2 -2, -1, 0, +1, +2 3d 5 10 4 0 0 4s 1 2 4 1 -1, 0, +1 4p 3 6 4 2 -2, -1, 0, +1, +2 4d 5 10 4 3 -3, -2, -1, 0, +1, +2, +3 4f 7 14 There are two common methods of indicating the arrangement of electrons inan atom. Unless otherwise stated, the lowest energy state (ground state)is given. These methods are electron configurations and atomic orbital diagrams.An electron configuration shows the electrons distribution by sublevel usingthe quantum numbers n and l, where the notation for l is by its letterdesignation (s, p, d, f, etc.). For example,the notation 3d4 indicates 4 electrons in the dsublevel (l=2) of the n =3 principal energy level. You arereferred to Chapter 7 in the textbook where a complete discussion of electronconfiguration may be found.An atomic orbital diagram shows the electron distribution in an atom by meansof a diagram which accounts for the distribution by all four quantum numbers. An orbital is shown by a box, circle or line. An electron is shown by an arrow. The arrows also show the spin of the electron so that when two electrons are in the same orbital, the arrows point in opposite directions to represent their opposing spins. Sublevels are shown by a designation under the appropriate orbitals (see examples below).In the figure below, an atomic orbital diagram is used to illustrate theorder of filling for the first ten electrons as shown by the numbers enteredin the boxes. As an example, consider the electronic structure of sulfur.Since sulfur has 16 electrons, its electron configuration is1s2 2s2 2p63s2 3p4 Atomic orbital diagram for sulfur with two unpaired electrons: The chart below shows the order of orbital fill up by electrons.


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Rememberthat electrons ar put into the lowest available energy orbital before fillingup a higher energy orbital. In addition, you always put one electronin each of the degenerate orbitals of a particlar energy level before puttinga second electron in any of the orbitals of the same energy (Hund"s Rule).