John A. Schreifels

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?34Js.
• 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
Bohrs 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. E1ltE2ltE3.
• 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 2l1.
• 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