Brad Collins
Electron Arrangement - Part 1
Chapter 8
Some images Copyright © The McGraw-Hill Companies, Inc.
Properties of Waves
Wavelength (λ) is the distance between identical
points on successive waves.
Amplitude is the vertical distance from the midline
of a wave to the peak or trough.
8.1
Properties of Waves
(λ)
Frequency (ν) is the number of waves that pass through a particular point in
1 second (Hz = 1 cycle/s).
The speed (u) of the wave = λ x ν
8.1
Maxwell (1873), proposed that visible light consists of
electromagnetic waves.
Electromagnetic radiation
is the emission and
transmission of energy in
the form of electromagnetic
waves.
Speed of light (c) in vacuum = 3.00 x 108 m/s
ALL electromagnetic radiation:
c = λ∙ν
8.1
Types of Electromagnetic (EM)
Radiation
8.1
A photon has a frequency of 6.0 x 104 Hz. Convert
this frequency into wavelength (nm). Does this frequency
fall in the visible region?
λ
c = λ∙ν
λ=c÷ν
λ = 3.00 x 108 m/s ÷ 6.0 x 104 s–1
λ 5.0 x 103 m
1m = 1 x 109 nm
λ = 5.0 x 103 m × 109 nm/m
λ = 5.0 × 1012 nm
ν
8.1
Particle Behavior of Light
• Max Planck (1900) found that energy is
related to wavelength.
• Shorter wavelengths have higher
frequencies and higher energy
• Expressed as E = constant × frequency (ν)
• Planck’s constant (h) is the smallest unit of
energy
• Energy exists in discrete units with
energies that are integer multiples of hν.
• Energy is gained or lost in integer (n)
multiples of h∙ν
• ∆E = n × h∙ν
Energy (light) is emitted or
absorbed in discrete units
(quanta).
8.1
Photoelectric Effect
• The photoelectric effect is the ability of light to dislodge
electrons from a metal surface.
• If light shines on the surface of a metal, there is a point at
which electrons are ejected from the metal.
• The electrons will only be ejected once the threshold
frequency is reached.
• Below the threshold frequency, no electrons are
ejected.
• Above the threshold frequency, the number of
electrons ejected depends on the intensity of the light.
• The photoelectric effect provides evidence for the particle
nature of light -- “quantization”.
8.1
Particle Nature of Light
“Photoelectric Effect”
Solved by Einstein, 1905
h
Light has dual nature
• Wave nature, c = λ∙ν
• Particle nature, E = h∙ν
KE
A photon is a “particle” or “quanta” of light
A photon has energy, E = h∙ν
Electrons are held in the metal with binding energy, BE
The kinetic energy (KE) of the ejected electron:
KE(electron) = hν(photon) – BE(electron)
To eject an electron, a photon must have
more energy than the BE of the electron.
8.1
The Bohr Model
Emission Spectra of Elements
• Radiation composed of only one wavelength is
called monochromatic.
• Radiation that spans a whole array of different
wavelengths is called continuous.
• White light can be separated into a continuous
spectrum of colors.
• Elements do not emit light in a continuous
spectrum
8.2
Continuous Spectrum of Light
Note that there are no dark
regions in the continuous
spectrum that would
correspond to different lines.
White Light
8.2
Line Spectrum of Elements
Line Emission Spectrum of Hydrogen Atoms
8.2
8.2
The Rydberg Equation
• 1888 - Johannes Rydberg found hydrogen
emission lines could be described using an
equation:
!
!
1
λ
=
(
1
nA2
–
1
nA2
)
• where RH is a constant = 1.097 x 107 m–1
• and nA and nB are two whole number
integers
• Based on measurements - empirical
• Why does this equation work?
8.2
Bohr Model of the Atom
• Neils Bohr - 1913
• Believed electron energy is quantized
• Electrons can only have specific,
quantized energy values, En
• *En = – B/n2
• B is constant = 2.18 x 10–18 J
• n is one of the quantized energy
levels, n = 1, 2, 3, etc.
• Light is emitted when an electron
moves from a higher energy value to
a lower one.
Excited State
Ground State
*Sign is negative to indicate that electrons in the atom
have lower energy than free electrons (infinitely far
from the nucleus)
8.2
E = hν
E = hν
8.2
Bohr Model of the Atom
E(photon) = ∆E = Ef – Ei
Ef = –B/n2f
ni = 3
ni = 3
Ei = –B/n2i
∆E = –B/n2f – (–B/n2i)
ni = 2
nf = 1
nf = 2
∆E = B × (1/n2i – 1/n2f)
8.2
Calculate the wavelength (in nm) of a photon emitted by
a hydrogen atom when its electron drops from the n = 5
state to the n = 3 state.
Energy of Photon
E(photon) = ∆E = B × (1/n2i – 1/n2f)
E(photon) = 2.18 × 10–18 J × (1/25 – 1/9)
E(photon) = 2.18 × 10–18 J × (–0.0711)
E(photon) = –1.55 × 10–19 J
Wavelength of Photon
E(photon) = h∙ν, ν = c/λ, so:
E(photon) = h∙c/λ
λ = h∙c/E(photon)
λ = (6.63 × 10–34 J∙s × 3.00 × 108 m/s)/1.55× 10–10 J
λ = 1.280 × 10–6 m * 109 nm/m
λ = 1280 nm
8.2
The Dual Nature of Electrons
Why is electron energy quantized?
Louis de Broglie, 1924
• Electron is both a particle and a
wave
• Electrons behave as standing
waves
• Circumference (2∙π∙r) of the
electron’s standing wave is
given by:
• 2∙π∙r = n∙λ = n∙h/m∙u
• where
• u = velocity of electron
• m = mass of electron
8.3
Practice Problem
What is the de Broglie wavelength (in nm)
associated with a 2.5 g Ping-Pong ball
traveling at 15.6 m/s?
h in J•s m in kg u in (m/s)
λ = h/mu
λ = 6.63 x 10-34 / (2.5 x 10-3 x 15.6)
λ = 1.7 x 10-32 m = 1.7 x 10-23 nm
8.3
The Schrödinger Wave Equation
• Heisenberg’s Uncertainty Principle: on the mass scale of atomic
particles we cannot determine the exact location AND momentum
of a particle simultaneously
• If Δx is the uncertainty in position and Δmv is the uncertainty in
momentum, then:
∆x × ∆mv = h/4∙π
• If the position, x is known precisely (∆x is small), then the
momentum (mv) value will be known less precisely (∆mv will
be large).
• Uncertainty + wave nature means we cannot describe an electron’s
‘orbit’ the same way we would a satellite or planet.
• Instead we describe the ‘probability of finding an electron’ in a
region around the nucleus. (Schrödinger Wave Equation)
8.3
The Quantum Model
• Schrödinger’s wave equation (Ψ) describes wave and particle
nature of electrons
• Ψ is a function of 4 properties:
• Ψ = fn(n, l, ml, ms)
• where n, l, ml, ms are called ‘quantum numbers’
• The 4 quantum numbers are the ‘address’ of the electron
• Schrödinger’s wave equation describes the energy of an
electron, and the probability of finding an electron in a region
of space outside the nucleus.
• Schrödinger’s wave equation can only be solved exactly for
the hydrogen atom.
• For multi-electron atoms the solution must be approximated.
8.4
Quantum Numbers
• n is the Principal Quantum Number
• Same as Rydberg’s ’n’ and Bohr’s ’n’
• Defines energy level (shell) of the electron
• n = 1, 2, 3, etc.
• n corresponds to the distance from the
nucleus
Shells are sometimes given letter-names
8.4
Quantum Numbers
• l is the Angular Momentum Quantum Number
• l describes the number and type of sub-levels (subshells) in a given
shell and the shape of the volume of space that the electron occupies.
• l = 0, 1, … n-1
!
!
!
!
!
n
1
2
3
4
Allowed l’s
0
0, 1
0, 1, 2
0, 1, 2, 3
Translation
one subshell in first shell
two subshells in second shell
three subshells in third shell
four subshells in fourth shell
• Types of subshells l = 0, 1, are given letter names
!
!
!
l
symbol!
0
s
1
p
2
d
3
f
4
g
5
h
etc
…
• Subshell nomenclature:
• Call l = 0 orbital type in the first (n=1) energy shell, 1s
8.4
Subshell Shapes
• When l = 0, shape is spherical
l = 0 (s orbitals)
8.4
Subshell Shapes
• When l = 1, shape is ‘dumbbell’
l = 1 (p orbitals)
8.4
Subshell Shapes
• When l = 2, shape is (mostly) ‘cloverleaf’
l = 2 (d orbitals)
8.4
Quantum Numbers
• ml is the Magnetic Quantum Number
• Describes the number of orbitals in a subshell and
their orientation in space
• ml can have values of –l…0…+l
subshell
s (l = 0)
p (l = 1)
d (l = 2)
f (l = 3)
Allowed ml’s
0
–1, 0, 1
–2, –1, 0, 1, 2
–3, –2, –1, 0, 1, 2, 3
Translation
one s-orbital
three p-orbitals
five d-orbitals
7 f-orbitals
8.4
ml = -1
ml = -2
ml = 0
ml = -1
ml = 0
ml = 1
ml = 1
ml = 2
8.4
Quantum Numbers and Orbitals
8.4
Quantum Numbers
• ms is the Electron Spin Quantum Number
• Describes the 2 possible ‘spins’ of an electron
• Spin gives rise to a magnetic field
• 2 possible fields:
• Values of ms = +½, –½
ms = +½
ms = -½
8.4
Schrödinger
Consequences
• Each set of 4 quantum numbers describes
the existence and energy of each electron in
an atom.
• Pauli Exclusion Principal: No 2 electrons in an
atom can have the same 4 quantum numbers.
• In a stadium each seat is identified by section, row,
and seat numbers.
S1, R42, S15
Only 1 person per seat
Only 1 electron per set of 4 quantum numbers
8.4
Schrödinger Summary
• Electrons with the same n, occupy the same shell
• Electrons with the same n and l, occupy the same subshell
• Electrons with the same n, l, and ml, occupy the same
orbital
Practice Problem
How many electrons can one orbital hold?!
If n, l, and ml are fixed, ms can be +½ or –½
So, electrons in a single orbital can have the quantum
numbers:
n, l, and ml, +½ or n, l, and ml, –½
Answer: 2 electrons
8.4
Practice Problem
How many 2p orbitals are there in an atom?
n=2
2p
l=1
‘p’-type orbitals mean l = 1, so ml = –1, 0 or +1
Answer: 3 orbitals
8.4
Practice Problem
How many electrons can be placed in the 3d
subshell?
n=3
3d
l=2
If l = 2, then ml = –2, –1, 0, +1, +2
• 5 orbitals, each with 2 electrons
• Total = 10 electrons
8.4
Bohr’s Model Revisited
Hydrogen (1 electron)
n=4
4s
4p
4p
4p
4d
4d
4d
4d
4d
n=3
3s
3p
3p
3p
3d
3d
3d
3d
3d
2s
2p
2p
2p
n=2
Orbitals on the same level are degenerate (all types have the same energy)
n=1
1s
8.4
Bohr’s Model Revisited
Multi-electron
n=4
4p
4p
4d
4d
4d
4d
3d
3d
3d
3d
3d
4p
4s
n=3
4d
3p
3p
3p
2p
2p
2p
3s
n=2
2s
• Transitions from 3s or
3p have different ∆E
values.
Orbitals on the same level are split (types have different energy)
• Split energy levels lead
to more emission lines.
n=1
1s
8.4