Chapter 7: Quantum Theory and Atomic Structure
7.1 The Nature of Light
7.2 Atomic Spectra
7.3 The Wave-Particle Duality
of Matter and Energy
7.4 The Quantum-Mechanical
Isaac Newton
(1642 – 1727)
Max Plank
(1858 – 1947)
Model of the Atom
Erwin Schrödinger
(1887-1961)
Albert Einstein
(1879 –1955)
Electromagnetic Radiation: the wave model
Christiaan Huygens and Isaac Newton
The Wave Nature of Light
Visible light is a type of electromagnetic radiation.
The wave properties of electromagnetic radiation are described by three
variables:
- frequency (𝝂), cycles per second
- wavelength (λ), the distance a wave travels in one cycle
- amplitude, the height of a wave crest or depth of a trough.
The speed of light is a constant:
c=𝝂xλ
= 3.00 x 108 m/s in a vacuum
The reciprocal relationship of frequency and wavelength.
Differing amplitude (brightness, or intensity) of a wave.
Regions of the electromagnetic spectrum
Interconverting Wavelength and Frequency
Problem 1.
Calculate the frequency of each wavelength of electromagnetic radiation:
a. 632.8 nm (wavelength of red light from helium–neon laser)
b. 503 nm (wavelength of maximum solar radiation)
c. 0.052 nm (a wavelength contained in medical X-rays)
Problem 2.
Calculate the wavelength of each frequency of electromagnetic radiation:
a. 100.2 MHz (typical frequency for FM radio broadcasting)
b. 1070.0 kHz (typical frequency for AM radio broadcasting)
(assume four significant figures)
c. 835.6 MHz (common frequency used for cell phone communication)
Wavelike properties of light




Refraction
Dispersion
Interference
Diffraction
Refraction
Dispersion
Different behaviors of waves and particles:
Formation of a diffraction pattern.
Diffraction
Particle-like properties of light




Blackbody radiation
Photoelectric effect
Compton effect
Emission spectra
The birth of Quantum Theory (1900)
The Quantum Theory of Energy
Any object (including atoms) can emit or absorb only
certain quantities of energy.
Energy is quantized; it occurs in fixed quantities, rather
than being continuous. Each fixed quantity of energy is
called a quantum.
An atom changes its energy state by emitting or
absorbing one or more quanta of energy.
E = nhn where n can only be a whole number.
The Photoelectric Effect (1905)
1. The electrons were emitted immediately - no time lag!
2. Increasing the intensity of the light increased the number of photoelectrons,
but not their maximum kinetic energy!
3. Red light will not cause the ejection of electrons, no matter what the intensity!
4. A weak violet light will eject only a few electrons, but their maximum kinetic
energies are greater than those for intense light of longer wavelengths!
hν = Ek + Φ
Problem 3.
Calculate the energy of a photon of electromagnetic radiation at the following
wavelengths:
632.8 nm (wavelength of red light from helium–neon laser);
503 nm (wavelength of maximum solar radiation);
0.052 nm (a wavelength contained in medical X-rays).
Problem 4.
A laser pulse with wavelength 532 nm contains 3.85 mJ of energy. How many
photons are in the laser pulse?
Atomic Spectra
Element
Abundance (% )
Hydrogen
91.2
Helium
8.7
Oxygen
0.078
Carbon
0.043
Nitrogen
0.0088
Silicon
0.0045
Magnesium
0.0038
Neon
0.0035
Iron
0.030
Sulfur
0.015
Exciting Gas Atoms to Emit Light with Electrical Energy
Identifying Elements with Flame Tests
Na
K
Li
Ba
Three series of spectral lines of atomic hydrogen.
Rydberg equation
1
l
=
R
1
2
n1
1
n22
R is the Rydberg constant = 1.096776x107 m-1
for the visible series, n1 = 2 and n2 = 3, 4, 5, ...
Johannes Rydberg (1854 – 1919)
The Bohr Model of the Hydrogen Atom (1913)
Bohr’s atomic model postulated the following:
 The H atom has only certain energy levels, which Bohr called
stationary states.
– Each state is associated with a fixed circular orbit of the electron
around the nucleus.
– The higher the energy level, the farther the orbit is from the
nucleus.
– When the H electron is in the first orbit, the atom is in its lowest
energy state, called the ground state.
 The atom does not radiate energy while in one of its stationary states.
 The atom changes to another stationary state only by absorbing or
emitting a photon.
– The energy of the photon (hn) equals the difference between the
energies of the two energy states.
– When the E electron is in any orbit higher than n = 1, the atom is
in an excited state.
Three series of spectral lines of atomic hydrogen.
The visible spectrum
A tabletop analogy for the H atom’s energy.
1
l
1
=
1
R
n1
2
-
n22
1
E = -2.18x10-18 J
DE = Efinal – Einitial =
-2.18x10-18
J
1
n2final
n2
1
n2initial
Problem 7.
A hydrogen atom absorbs a photon of UV light and its electron enters the
n = 4 energy level. Calculate:
a) the change in energy of the atom and
b) the wavelength (in nm) of the photon.
Problem 8.
Calculate the wavelength of the light emitted or absorbed when an electron
in a hydrogen atom makes each transition and indicate the region of the
electromagnetic spectrum (infrared, visible, ultraviolet, etc.) where the light
is found.
a) n = 2 → n = 1
b) n = 4 → n = 2
c) n = 4 → n = 3
d) n = 5→ n = 6
Emission vs. Absorption Spectra
Spectra of Mercury
Infrared Spectroscopy
The Wave-Particle Duality of Matter and Energy
The Wave-Particle Duality of Matter and Energy
All matter exhibits properties of both particles and
waves. Electrons have wave-like motion and therefore
have only certain allowable frequencies and energies.
Louis de Broglie
l=
h
mu
m = mass
u = speed in m/s
Problem 9.
The smallest atoms can themselves exhibit quantum mechanical
behavior. Calculate the de Broglie wavelength (in pm) of a hydrogen atom
traveling 475 m/s.
Problem 10.
A proton in a linear accelerator has a de Broglie wavelength of 122 pm.
What is the speed of the proton?
Heisenberg’s Uncertainty Principle
The paradox solver
Heisenberg’s Uncertainty Principle states that it is not
possible to know both the position and momentum of a
moving particle at the same time.
D x∙m D u ≥
h
4p
x = position
u = speed
The more accurately we know the speed, the less
accurately we know the position, and vice versa.
Problem 11.
a) An electron moving near an atomic nucleus has a speed 6x106 m/s ±
1%. What is the uncertainty in its position (Dx)?
b) How accurately can an umpire know the position of a baseball (mass
= 0.142 kg) moving at 100.0 mi/h ± 1.00% (44.7 m/s ± 1.00%)?
The Quantum-Mechanical Model of the Atom
Newton’s laws are deterministic
Quantum mechanics laws are
indeterministic
The Quantum-Mechanical Model of the Atom
The Schrödinger wave equation allows us to solve for
the energy states associated with a particular atomic
orbital.
The square of the wave function gives the probability
density, a measure of the probability of finding an
electron of a particular energy in a particular region of the
atom.
Electron probability density in the ground-state H atom.
A radial probability distribution of electrons in hydrogen atom
The Schrödinger wave equation solution
The radial equation
The colatitude equation
The azimuthal equation
Taken from www.hyperphysics.phy-astr.gsu.edu
Where did Quantum Numbers beam from?
Taken from www.hyperphysics.phy-astr.gsu.edu
1s, 2s and 3s hydrogen atom orbitals
Taken from www.hyperphysics.phy-astr.gsu.edu
The 1s, 2s, and 3s orbitals.
2p and 3p hydrogen atom orbitals
Taken from www.hyperphysics.phy-astr.gsu.edu
The 2p orbitals.
3d hydrogen atom orbitals
Taken from www.hyperphysics.phy-astr.gsu.edu
The 3d orbitals.
The 3d orbitals.
The 4fxyz orbital, one of the seven 4f orbitals.
Quantum Numbers and Atomic Orbitals
An atomic orbital is specified by three quantum numbers.
The principal quantum number (n) is a positive integer.
The value of n indicates the relative size of the orbital and therefore its
relative distance from the nucleus.
The angular momentum quantum number (l) is an integer from 0 to
(n -1). The value of l indicates the shape of the orbital.
The magnetic quantum number (ml) is an integer with 2 x l +1 values,
from –l to +l . The value of ml indicates the spatial orientation of the
orbital.
The angular momentum quantum number (l) is an integer from 0 to
(n -1). The value of l indicates the shape of the orbital.
L=rxp
The magnetic quantum number (ml) is an integer with 2 x l +1 values, from
–l to +l . The value of ml indicates the spatial orientation of the orbital.
The Hierarchy of Quantum Numbers for Atomic Orbitals
Energy shells (levels) and subshells (sublevels)
 Energetic levels (shells)
 Energetic sublevel (subshells)
 Orbitals
 Main Quantum Number (n) - SHELLS
 Secondary Quantum Number (l) – SUBSHELLS
(l = 0 s subshell; l = 1 p subshell; l = 2 d subshell; l = 3 f subshell)
Magnetic Quantum Number (m) – ORBITALS
(m = 0 s orbital; m = -1,0,1 p orbitals;
m = -2,-1,0,1,2 d orbitals, m = -3,-2,-1,0,1,2 ,3 f orbitals)
Problem 12.
What values of the angular momentum (l) and magnetic (ml) quantum
numbers are allowed for a principal quantum number (n) of 3? How many
orbitals are allowed for n = 3?
Problem 13.
What are the possible values of l for each value of n?
a. 1 b. 2 c. 3 d. 4
Problem 14.
What are the possible values of m for each value of l?
a. 0 b. 1 c. 2 d. 3
Problem 15.
Which set of quantum numbers cannot occur together to specify an orbital? In
case the set is plausible, give the name of the atomic orbital
a. n = 2, l = 1, ml = -1
b. n = 3, l = 2, ml = 0
c. n = 3, l = 3, ml = 2
d. n = 4, l = 3, ml = 0