Quantum Physics

28.9 The Exclusion Principle and the Periodic Table 917this curve. First, the curve peaks at a value of r 0.052 9 nm, the Bohr radius forthe first (n 1) electron orbit in hydrogen. This means that there is a maximumprobability of finding the electron in a small interval centered at that distancefrom the nucleus. However, as the curve indicates, there is also a probability offinding the electron in a small interval centered at any other distance from the nucleus.In other words, the electron is not confined to a particular orbital distancefrom the nucleus, as assumed in the Bohr model. The electron may be found atvarious distances from the nucleus, but the probability of finding it at a distancecorresponding to the Bohr radius is a maximum. Quantum mechanics alsopredicts that the wave function for the hydrogen atom in the ground state is sphericallysymmetric; hence the electron can be found in a spherical region surroundingthe nucleus. This is in contrast to the Bohr theory, which confines the positionof the electron to points in a plane. The quantum mechanical result is often interpretedby viewing the electron as a cloud surrounding the nucleus. An attempt atpicturing this cloud-like behavior is shown in Figure 28.13. The densest regions ofthe cloud represent those locations where the electron is most likely to be found.If a similar analysis is carried out for the n 2, 0, state of hydrogen, a peakof the probability curve is found at 4a 0 . Likewise, for the n 3, 0 state, thecurve peaks at 9a 0 . Thus, quantum mechanics predicts a most probable electrondistance to the nucleus that is in agreement with the location predicted by theBohr theory.P1s(r)a 0 = 0.0529 nmFigure 28.12 The probability perunit length of finding the electronversus distance from the nucleusfor the hydrogen atom in the 1s(ground) state. Note that the graphhas its maximum value when r equalsthe first Bohr radius, a 0 .yr28.9 THE EXCLUSION PRINCIPLEAND THE PERIODIC TABLEEarlier, we found that the state of an electron in an atom is specified by four quantumnumbers: n, , m , and m s . For example, an electron in the ground state of hydrogencould have quantum numbers of n 1, 0, m 0, and m s 1 2. As itturns out, the state of an electron in any other atom may also be specified by thissame set of quantum numbers. In fact, these four quantum numbers can be usedto describe all the electronic states of an atom, regardless of the number of electronsin its structure.How many electrons in an atom can have a particular set of quantum numbers?This important question was answered by Pauli in 1925 in a powerful statementknown as the Pauli exclusion principle:No two electrons in an atom can ever have the same set of values for the set of quantumnumbers n, , m , and m s .zFigure 28.13 The sphericalelectron cloud for the hydrogenatom in its 1s state. The Pauli exclusion principlexThe Pauli exclusion principle explains the electronic structure of complex atomsas a succession of filled levels with different quantum numbers increasing in energy,where the outermost electrons are primarily responsible for the chemicalproperties of the element. If this principle weren’t valid, every electron would endup in the lowest energy state of the atom and the chemical behavior of the elementswould be grossly different. Nature as we know it would not exist—and wewould not exist to wonder about it!As a general rule, the order that electrons fill an atom’s subshell is as follows:once one subshell is filled, the next electron goes into the vacant subshell that islowest in energy. If the atom were not in the lowest energy state available to it, itwould radiate energy until it reached that state. A subshell is filled when it contains2(2 1) electrons. This rule is based on the analysis of quantum numbersto be described later. Following the rule, shells and subshells can contain numbersof electrons according to the pattern given in Table 28.3.The exclusion principle can be illustrated by an examination of the electronicarrangement in a few of the lighter atoms.Hydrogen has only one electron, which, in its ground state, can be described by1either of two sets of quantum numbers: 1, 0, 0, 2 or 1, 0, 0, 1 . The electronicconfiguration of this atom is often designated as 1s 1 2. The notation 1s refers to aTIP 28.3 The ExclusionPrinciple is More GeneralThe exclusion principle stated here isa limited form of the more generalexclusion principle, which states thatno two fermions (particles with spin1/2, 3/2, . . .) can be in the samequantum state.