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QUANTUM THEORY General Chemistry I (2012) Lecture by B. H. Hong Class Summary • Studies of black-body radiation led to Plank’s hypothesis of the quantization of electromagnetic radiation. • The photoelectric effect provides evidence of the particle nature of electromagnetic radiation. • Electrons (an matter in general) have both wavelike and particlelike properties. • The location and momentum of a particle are complementary; that is, the location and the momentum cannot both be known simultaneously with arbitrary precision. • When the wave equation is solved subject to the appropriate boundary conditions, it is found that the particle can possess only certain discrete energies. 45 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.7 Wavefunctions and Energy Levels 1) An electron is described by a wavefunction governed by the Schrödinger equation. 2) The nuclear model for an atom ⇒ To explain the ladder of energy levels suggested by spectroscopy THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.8 The Principal Quantum Number Constraints (boundary conditions) in solving the Schrödinger equation ⇒ Quantization of energies, discrete energy levels corresponding to a set of quantum numbers To solve the Schrödinger equation for ψ and energy levels of an electron in a hydrogen atom, the potential energy term V in the hamiltonian H is needed. The solution by Schrödinger (1927) THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.8 The Principal Quantum Number For other one-electron atoms such as He+, ···, C5+ with atomic number Z, Z2 dependence; 1) 2) nucleus of charge Ze Z times closer distance to the nucleus Energy levels of a hydrogen atom THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Atomic orbitals; wavefunctions of electrons in atoms (orbital; wave-like orbit) THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Tab 1.2 (needs correctio ns) THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Fig 1.36 Detailed solutions of the Schrödinger equation for a hydrogen atom require three quantum numbers. Three dimensional boundary conditions Principal quantum number (n); -> energy & size THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Shell; atomic orbitals of the same principal quantum number n As n increases, the energy (–1/n2) and the nucleus-electron distance (n2) increase. All the orbitals of a given shell of hydrogen-like atoms have the same energy; degenerate Orbital angular momentum quantum number (l); shape Angular momentum; THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals l = 0, 1, 2, ···, n – 1; n different values of l, n subshells Subshell; group of orbitals with the same value of l s (sharp); l = 0, d (diffuse); l = 2, p (principal); l = 1, f (fundamental); l = 3, g, h, ··· Orbital angular momentum of an electron in a subshell = ; measure of the rate at which the electron circulates around the nucleus THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Magnetic quantum number (ml); orientation ml = l, l – 1, ···, –l ; 2l + 1 different values of ml 3 p-orbitals; ml = –1, 0, 1 with l = 1 5 d-orbitals; ml = –2, –1, 0, +1, +2 with l = 2 The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field (Zeeman effect). THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Hierarchy of shells, subshells, and orbitals Tab 1.3 Fig 1.30 s-Orbitals; l = 0, no angular momentum Only one orbital in each subshell Spherically symmetric; Non-zero probability of being found at the nucleus; R2(0) ≠ 0 n – 1 radial nodes Figs 1.33 Figs 1.34 56 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals 57 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Fig 1.31 58 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Radial distribution function; probability density for finding an electron at a given radius summed over all directions P(r)δr; probability of finding an electron anywhere in a thin shell of radius r and thickness δr 59 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals For any s-orbitals of hydrogen-like atoms, Thus, Fig 1.32 P(r) of the 1s-orbital of hydrogen reaches a maximum at r = a0, the Bohr radius. 60 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals 61 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals 62 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals Boundary surface; a smooth surface that encloses most (typically 90%) of the electron cloud All s-orbitals have spherical boundary surfaces (with internal structures such as nodes). Figs 1.33, 1.35 63 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals p-Orbitals; 3 orbitals in each subshell; ⇒ The Boundary surface of each p orbital has two lobes and one nodal plane. ⇒ A p-electron will never be found at the nodal plane nor at the nucleus. Figs 1.36 64 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals d-Orbitals; 5 orbitals in each subshell; Figs 1.37 65 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals f-Orbitals; Figs 1.38 Total number of orbitals in a shell = n2 66 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.9 Atomic Orbitals 67 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.10 Electron Spin Box 1.1 Stern-Gerlach Experiment (1922, 1943) Unexpected splitting of an Hg atomic beam under an inhomogeneous magnetic field ⇒ Quantum mechanical electron spin 68 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.10 Electron Spin One of the biggest impacts on modern physics Hyperfine structures; Goudsmit and Uhlenbeck (1925) Nuclear magnetic resonance; Rabi (1937, 1944) Separated oscillatory fields; Ramsey (1989) 69 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.10 Electron Spin Spin angular momentum quantum number (s); Spin magnetic quantum number (ms); For an electron; s = ½ Fig 1.40 70 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong 1.11 The Electronic Structure of Hydrogen The state of an electron in a hydrogen atom is defined by a set of the four quantum numbers: {n, l, ml, ms} Ground state; 1s-electron, {n = 1, l = 0, ml = 0, ms = ½ or –½} Excited states; n > 1, higher energy, greater distance between the electron and the nucleus 71 THE HYDROGEN ATOM General Chemistry I (2012) Lecture by B. H. Hong Class Summary • The energy levels of a hydrogen atom are defined by the principal quantum number, n=1,2,…, and form a converging series. • The location of an electron in n atom is described by a wavefunction known as an atomic orbital; atomic orbitals are designated by the quantum numbers n, l, and ml and fall into shells and subshells. • An electron has the property of spin; the spin is described by the quantum number ms , which may have one of two values. • The state of an electron in a hydrogen atom is defined by the four quantum numbers n, l, ml , and ms ; as the value of n increases, the size of the atom increases. 72