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## Chapter 7

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**Chapter 7**Quantum Theory of the Atom**Overview**• Light Waves, Photons, and the Bohr Theory • Wave Nature of Light • Quantum Effects and Photons • Bohr Theory: Hydrogen and Hydrogen-like atoms • Quantum Mechanics and Quantum Numbers • Quantum Mechanics • Quantum Numbers and Atomic Orbitals**The Wave Nature of the Light**• Atomic structure elucidated by interaction of matter with light. • Light properties: characterized by wavelength, , and frequency,. • Light = electromagnetic radiation, a wave of oscillating electric and magnetic influences called fields. • Frequency and wavelength inversely proportional to each other. c = where c = the speed of light = 3.00x108 m/s; units = s1, = m E.g. calculate the frequency of light with a wavelength of 500 nm. E.g.2 calculate the frequency of light if the wavelength is 400 nm.**Electromagnetic Radiation and Atomic Spectra - 2**• Line spectra result from the emission of radiation from an excited atom. • Spectrum: characteristic pattern of wavelengths absorbed (or emitted) by a substance. • Emission Spectrum: spontaneous emission of radiation from an excited atom or molecule. • Line Spectrum: spectrum containing only certain wavelengths. • Balmer : hydrogen has a line spectrum in the visible region with wavelengths of 656.3 nm, 486.1 nm, 434.0 nm, 410.1 nm. • Balmer equation: where n = 3.**Quantized Energy and Photons**• Light = wave arriving as stream of particles called "photons". • Each photon = quantum of energy where h (Planck's constant) = 6.63x1034J*s. • An increase in the frequency = an increase in the energy • An increase in the wavelength gives an decrease in the energy of the photon. • E.g. determine the energies of photons with • wavelengths of 650 nm, 700 nm and • frequencies 4.50x1014 s1, 6.50x1014 s1 • Photoelectric effect: E = h where = constant • the energy of the electron is directly related to the energy of the photon. • the threshold of energy must be exceeded for electron emission. • The total energy of a stream of particles (photons) of that energy will be: where n = 1, 2, …(only discrete energies).**Bohr’s Model of the Hydrogen Atom**• Line spectra in other spectral regions also were observed: • Lyman series ultraviolet • Paschen, Brackett, Pfund infrared • Balmer-Rydberg equation predicted the wavelengths of emission. where m = 1, 2, 3,… and n = 2, 3, …(always at least m + 1 • Longest wavelength observed when n = m + 1. • Shortest wavelength observed when n = . E.g. Determine the wavelength of emission for the first line in the Paschen series (m = 3, n = 4). E.g. Determine the shortest wavelength in the Paschen series (m = 3 and n = ).**Bohr model of the Hydrogen Atom II**• Duality of matter led to the hypothesis that electrons behave as waves. • Bohr model assumed • Only circular orbits around the nucleus and that the angular momentum around the atom must be quantized. • Stable orbital where constructive interference occurs. • Assumption led to the conclusions: • Radius of an orbital: rn = n2r1. • Energy of an orbital: En = E1/n2 = 21.93x1019J/n2 where E1 = energy of the most stable hydrogen orbital. E1<E2<E3. • Most stable state E1,r1 = ground state. • Higher energy states = excited states. • A photon is emitted when the electron moves from a higher energy state to a lower one. • Photon energy equals the difference in energy of the two states.**Bohr model of the Hydrogen Atom III**• If Ei = the initial state energy and Ef = final state energy, then the energy of the transition would be: E = Ei Ef. R = Rydberg constant = 1.097x107m1. • Theory and experiment agree for hydrogen and hydrogen-like particles.**Wave Nature of Matter**• Light behaves like matter since it can only have certain energies. • Light had both wave- and particle-like properties matter did too. • Einstein equation helps describe the duality of light: • E = mc2 Particle behavior • E = h Wave behavior • Wave and particle behavior • Duality of matter expressed by replacing the speed of light with the speed of the particle to get: where called the de Broglie wavelength of any moving particle. • E.g. determine the de Broglie wavelength of a person with a mass of 90 kg who is running 10 m/s.**Quantum Mechanics: Hydrogen**• Bohr model did not work with multielectron atoms, i.e. line spectra not predicted. • Quantum mechanics provides universal description of the electron distribution in atoms. • Heisenberg uncertainty principle = impossible to determine the position and momentum with absolute precision or (position uncertainty)(momentum uncertainty) • Schroedinger used wave concepts to derive the wave equation.Electrons allowed to be in anywhere. • Solution of the Schroedinger three dimensional wave equation, , led to the discrete energy levels of the hydrogen atom. • Lowest level is spherical. • Predicts distribution of electrons in other elements.**Quantum Mechanics and Atomic Orbitals**• The first orbital of all elements is spherical. • Other orbitals have a characteristic shape and position as described by 4 quantum numbers: n,l,ml,ms. All are integers except ms • Principal Quantum Number (n): an integer from 1... Total # e in a shell = n2. • Angular quantum number (l). (permitted values l = 0 to n1): the subshell shape. • Common usage for l = 0, 1, 2, 3, 4, and use s, p, d, f, g,... respectively. • Subshell described as 1s, 2s, 2p, etc. • Magnetic quantum number,ml, (allowed l to +l ) directionality of an l subshell orbital. • Total number of possible orbitals is 2l+1. • E.g. s and p subshells have 1 & 3 orbitals, respectively. • Spin quantum number,ms (allowed values 1/2). Due to induced magnetic fields from rotating electrons. • Pauli exclusion principle: no two electrons in an atom can have the same four quantum numbers.**Orbital Energies of Multielectron Atoms**• All elements have the same number of orbitals (s,p, d, and etc.). • In hydrogen these orbitals all have the same energy. • In other elements there are slight orbital energy differences as a result of the presence of other electrons in the atom. • The presence of more than one electron changes the energy of the electron orbitals (click here)