Quantum Theory and the Electronic Structure of Atoms, Lecture notes of Physics

Download Quantum Theory and the Electronic Structure of Atoms and more Lecture notes Physics in PDF only on Docsity! 96 CHAPTER 3 Quantum Theory and the Electronic Structure of Atoms 3.8 ATOMIC ORBITALS Strictly speaking, an atomic orbital does not have a well-defined shape because the wave function characterizing the orbital extends from the nucleus to infinity. In that sense, it is difficult to say what an orbital looks like. On the other hand, it is certainly useful to think of orbitals as having specific shapes. Being able to visualize atomic orbitals is essential to understanding the formation of chemical bonds and molecular geometry, which are discussed in Chapters 5 and 7. In this section, we will look at each type of orbital separately. s Orbitals For any value of the principal quantum number (n), the value 0 is possible for the angular momentum quantum number (ℓ), corresponding to an s subshell. Furthermore, when ℓ = 0, the magnetic quantum number (mℓ) has only one possible value, 0, cor- responding to an s orbital. Therefore, there is an s subshell in every shell, and each s subshell contains just one orbital, an s orbital. Figure 3.18 illustrates three ways to represent the distribution of electrons: the probability density, the spherical distribution of electron density, and the radial probability distribution (the probability of finding the electron as a function of distance from the nucleus) for the 1s, 2s, and 3s orbitals of hydrogen. The boundary surface (the outermost surface of the spherical representation) is a com- mon way to represent atomic orbitals, incorporating the volume in which there is about a 90 percent probability of finding the electron at any given time. All s orbitals are spherical in shape but differ in size, which increases as the principal quantum number increases. The radial probability distribution for the 1s orbital exhibits a maximum at 52.9 pm (0.529 Å) from the nucleus. Interestingly, this distance is equal to the radius of the n = 1 orbit in the Bohr model of the hydrogen atom. The radial probability distribution plots for the 2s and 3s orbitals exhibit two and three maxima, respectively, with the greatest probability occurring at a greater distance from the nucleus as n increases. Between the two maxima for the 2s orbital, there is a point on the plot where the probability drops to zero. This corresponds to a node in the electron density, where the standing wave has zero amplitude. There are two such nodes in the radial probability distribution plot of the 3s orbital. Although the details of electron density variation within each boundary surface are lost, the most important features of atomic orbitals are their overall shapes and relative sizes, which are adequately represented by boundary surface diagrams. p Orbitals When the principal quantum number (n) is 2 or greater, the value 1 is possible for the angular momentum quantum number (ℓ), corresponding to a p subshell. And, when ℓ = 1, the magnetic quantum number (mℓ) has three possible values: −1, 0, and +1, each corresponding to a different p orbital. Therefore, there is a p subshell in every shell for which n ≥ 2, and each p subshell contains three p orbitals. These three p orbitals are labeled px, py, and pz (Figure 3.19), with the subscripted letters indicating the axis along which each orbital is oriented. These three p orbitals are identical in size, shape, and energy; they differ from one another only in orienta- tion. Note, however, that there is no simple relation between the values of mℓ and the x, y, and z directions. For our purpose, you need only remember that because there are three possible values of mℓ, there are three p orbitals with different orientations. The boundary surface diagrams of p orbitals in Figure 3.19 show that each p orbital can be thought of as two lobes on opposite sides of the nucleus. Like s orbit- als, p orbitals increase in size from 2p to 3p to 4p orbital and so on. Student Annotation: The radial probability distribution can be thought of as a map of “where the electron spends most of its time.” Student data indicate you may struggle with the characteristics of p orbitals. Access the SmartBook to view additional Learning Resources on this topic. Student Hot Spot SECTION 3.8 Atomic Orbitals 97 Pr ob ab ili ty d en si ty (ψ 2 ) 0 r (10−10 m) n = 1 ℓ = 0 Pr ob ab ili ty d en si ty ( ψ 2 ) n = 2 ℓ = 0 0 6 84242 r (10−10 m) Pr ob ab ili ty d en si ty ( ψ 2 ) n = 3 ℓ = 0 0 6 8 10 12 1442 r (10−10 m) R ad ia l p ro ba bi lit y di st rib ut io n (s um o f a ll ψ 2 ) 0 r (10−10 m) n = 1 ℓ = 0 R ad ia l p ro ba bi lit y di st rib ut io n (s um o f a ll ψ 2 ) R ad ia l p ro ba bi lit y di st rib ut io n (s um o f a ll ψ 2 ) n = 2 ℓ = 0 0 6 84242 r (10−10 m) n = 3 ℓ = 0 0 (a) 6 8 10 12 1442 r (10−10 m) 1s orbital (b) 2s orbital (c) 3s orbital Figure 3.18 From top to bottom, the probability density and corresponding relief map, the distribution of electron density represented spherically with shading that corresponds to the relief map above, and the radial probability distribution for (a) the 1s, (b) the 2s, and (c) the 3s orbitals of hydrogen. z zz z yy yy x xx x (a) (b) pzpx py Figure 3.19 (a) Electron distribution in a p orbital. (b) Boundary surfaces for the px, py, and pz orbitals. d Orbitals and Other Higher-Energy Orbitals When the principal quantum number (n) is 3 or greater, the value 2 is possible for the angular momentum quantum number (ℓ), corresponding to a d subshell. When ℓ = 2, the magnetic quantum number (mℓ) has five possible values, −2, −1, 0, +1, and +2,