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! 100 CHAPTER 3 Quantum Theory and the Electronic Structure of Atoms 3.9 ELECTRON CONFIGURATIONS The hydrogen atom is a particularly simple system because it contains only one elec- tron. The electron may reside in the 1s orbital (the ground state), or it may be found in some higher-energy orbital (an excited state). With many-electron systems, we need to know the ground-state electron configuration—that is, how the electrons are dis- tributed in the various atomic orbitals. To do this, we need to know the relative ener- gies of atomic orbitals in a many-electron system, which differ from those in a one-electron system such as hydrogen. Energies of Atomic Orbitals in Many-Electron Systems Consider the two emission spectra shown in Figure 3.22. The spectrum of helium contains more lines than that of hydrogen. This indicates that there are more possible transitions, corresponding to emission in the visible range, in a helium atom than in a hydrogen atom. This is due to the splitting of energy levels caused by electrostatic interactions between helium’s two electrons. Figure 3.23 shows the general order of orbital energies in a many-electron atom. In contrast to the hydrogen atom, in which the energy of an orbital depends only on the value of n (see Figure 3.21), the energy of an orbital in a many-electron system Figure 3.22 Comparison of the emission spectra of H and He. Hydrogen Helium 700600500400 nm 700600500400 nm Student Annotation: “Splitting” of energy levels refers to the splitting of a shell into subshells of different energies, as shown in Figure 3.23. En er gy 1s 2s 3s 2p 2p 2p 3p 3p 3p 3d 3d 3d 3d 4p 4p 4p 4d 4d 4d 4d 4d 3d 4s 5s Figure 3.23 Orbital energy levels in many-electron atoms. For a given value of n, orbital energy increases with the value of ℓ. SECTION 3.9 Electron Configurations 101 depends on both the value of n and the value of ℓ. For example, 3p orbitals all have the same energy, but they are higher in energy than the 3s orbital and lower in energy than the 3d orbitals. In a many-electron atom, for a given value of n, the energy of an orbital increases with increasing value of ℓ. One important consequence of the splitting of energy levels is the relative energies of d orbitals in one shell and the s orbital in the next higher shell. As Figure 3.23 shows, the 4s orbital is lower in energy than the 3d orbitals. Likewise, the 5s orbital is lower in energy than the 4d orbital, and so on. This fact becomes important when we determine the order in which elec- trons in an atom populate the atomic orbitals. The Pauli Exclusion Principle According to the Pauli exclusion principle,17 no two electrons in an atom can have the same four quantum numbers. If two electrons in an atom have the same n, ℓ, and mℓ values (meaning that they occupy the same orbital), then they must have different values of ms; that is, one must have ms = +1 2 and the other must have ms = −1 2. Because there are only two possible values for ms, and no two electrons in the same orbital may have the same value for ms, a maximum of two electrons may occupy an atomic orbital, and these two electrons must have opposite spins. We can indicate the arrangement of electrons in atomic orbitals with labels that identify each orbital (or subshell) and the number of electrons in it. Thus, we could describe a hydrogen atom in the ground state using 1s1. 1s1 Denotes the number of electrons in the orbital or subshell Denotes the angular momentum quantum number ℓ Denotes the principal quantum number n We can also represent the arrangement of electrons in an atom using orbital diagrams in which each orbital is represented by a labeled box. The orbital diagram for a hydrogen atom in the ground state is 1s1 H The upward arrow denotes one of the two possible spins (one of the two possible ms values) of the electron in the hydrogen atom (the other possible spin is indicated with a downward arrow). Under certain circumstances, as we will see shortly, it is useful to indicate the explicit locations of electrons. The orbital diagram for a helium atom in the ground state is 1s2 He The label 1s2 indicates there are two electrons in the 1s orbital. Note also that the arrows in the box point in opposite directions, representing opposite electron spins. Generally, when an orbital diagram includes an orbital with a single electron, we repre- sent it with an upward arrow—although we could represent it equally well with a down- ward arrow. The choice is arbitrary and has no effect on the energy of the electron. The Aufbau Principle We can continue the process of writing electron configurations for elements based on the order of orbital energies and the Pauli exclusion principle. This process is based 17Wolfgang Pauli (1900–1958). One of the founders of quantum mechanics, Austrian physicist Pauli was awarded the Nobel Prize in Physics in 1945. Student Annotation: 1s1 is read as “one s one.” Student Annotation: The ground state for a many-electron atom is the one in which all the electrons occupy orbitals of the lowest possible energy. Student Annotation: 1s2 is read as “one s two,” not as “one s squared.”