Chapter 7

1. Chapter 7 Quantum Theory of the Atom

2. 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

3. 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  = s1,  = 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.

4. 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.

5. Quantized Energy and Photons • Light = wave arriving as stream of particles called "photons". • Each photon = quantum of energy where h (Planck's constant) = 6.63x1034J*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 s1, 6.50x1014 s1 • 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).

6. 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 = ).

7. 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 = 21.93x1019J/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.

8. 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 = 1.097x107m1. • Theory and experiment agree for hydrogen and hydrogen-like particles.

9. 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.

10. 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.

11. 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 n1): 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.

12. Permissible Quantum States

13. Figure 7.23: Orbital energies of the hydrogen atom.

14. 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)

15. Shape of 1s Orbital

16. Shape of 2p Orbital

17. Shape of 3d Orbitals