Download Chapter 7 Quantum Theory of the Atom

Chapter 6: Electromagnetic Radiation
and the Electronic Structure of the
Atom
A wave is a continuously repeating change or oscillation in matter or in
a physical field.
Light is an electromagnetic wave, consisting of oscillations in electric
and magnetic fields traveling through space.
A wave can be characterized by its wavelength and frequency.
Wavelength, symbolized by the Greek letter lambda, l, is the distance
between any two identical points on adjacent waves.
Frequency, symbolized by the Greek letter nu, n, is the number of
wavelengths that pass a fixed point in one unit of time (usually a
second). The unit is 1/S or s-1, which is also called the Hertz (Hz).
Wavelength and frequency are related by the wave speed, which for
light is c, the speed of light, 2.998 x 108 m/s.
c = nl
The relationship between wavelength and frequency due to the
constant velocity of light is illustrated on the next slide.
When the
wavelength is
reduced by a factor
of two, the
frequency
increases by a
factor of two.
?
What is the wavelength of blue light with a frequency of 6.4 ×
1014/s?
What is the frequency of light having a wavelength of 681 nm?
?
The range of frequencies and wavelengths of electromagnetic radiation
is called the electromagnetic spectrum.
One property of waves is that they can be diffracted—that is, they
spread out when they encounter an obstacle about the size of the
wavelength.
In 1801, Thomas Young, a British physicist, showed that light could be
diffracted. By the early 1900s, the wave theory of light was well
established.
The wave theory could not explain the photoelectric effect, however
The photoelectric effect is the
ejection of an electron from the
surface of a metal or other
material when light shines on it.
Einstein proposed that light consists of quanta or particles of
electromagnetic energy, called photons. The energy of each photon is
proportional to its frequency:
E = hn
h = 6.626 × 10-34 J  s (Planck’s constant)
Einstein used this understanding of light to explain the photoelectric
effect in 1905.
Each electron is struck by a single photon. Only when that photon has
enough energy will the electron be ejected from the atom; that photon
is said to be absorbed.
Light, therefore, has properties of both waves and matter. Neither
understanding is sufficient alone. This is called the particle–wave duality
of light.
?
The blue–green line of the hydrogen atom spectrum has a
wavelength of 486 nm. What is the energy of a photon of this
light?
In the early 1900s, the atom was understood to consist of a positive
nucleus around which electrons move (Rutherford’s model).
This explanation left a theoretical dilemma: According to the physics of
the time, an electrically charged particle circling a center would
continually lose energy as electromagnetic radiation. But this is not the
case—atoms are stable.
In addition, this understanding could not explain the observation of line
spectra of atoms.
A continuous spectrum contains all wavelengths of light.
A line spectrum shows only certain colors or specific wavelengths of
light. When atoms are heated, they emit light. This process produces a
line spectrum that is specific to that atom. The emission spectra of six
elements are shown on the next slide.
In 1913, Neils Bohr, a Danish scientist, set down postulates to account for
1. The stability of the hydrogen atom
2. The line spectrum of the atom
Energy-Level Postulate
An electron can have only certain energy values, called energy levels.
Energy levels are quantized.
For an electron in a hydrogen atom, the energy is given by the following
equation:
E
RH
n2
RH = 2.179 x 10-18 J
n = principal quantum number
Transitions Between Energy Levels
An electron can change energy levels by absorbing energy to move to a
higher energy level or by emitting energy to move to a lower energy
level.
For a hydrogen electron the energy change is given by
ΔE  E f  E i
 1

1
ΔE  RH  2  2 
n

n
i 
 f
RH = 2.179 × 10-18 J, Rydberg constant
The energy of the emitted or absorbed photon is
related to DE:
E photon  ΔE electron  hn
h  Planck' s constant
We can now combine these two equations:
 1

1
hn   R H  2  2 
n

n
i 
 f
Light is absorbed by an atom when the electron transition is from lower
n to higher n (nf > ni). In this case, DE will be positive.
Light is emitted from an atom when the electron transition is from
higher n to lower n (nf < ni). In this case, DE will be negative.
An electron is ejected when nf = ∞.
Electron transitions for an
electron in the hydrogen
atom.
?
What is the wavelength of the light emitted when the electron in
a hydrogen atom undergoes a transition from n = 6 to n = 3?
 1

1
ΔE  RH  2  2 
n

n
i 
 f
Planck
Vibrating atoms have only certain energies:
E = hn or 2hn or 3hn
Einstein
Energy is quantized in particles called photons:
E = hn
Bohr
Electrons in atoms can have only certain values of energy. For
hydrogen:
E
RH
n2
RH  2.179 x 10 18 J, n  principal quantum number
Light has properties of both waves and particles (matter).
What about matter?
In 1923, Louis de Broglie, a French physicist, reasoned that particles
(matter) might also have wave properties.
The wavelength of a particle of mass, m (kg), and velocity, v (m/s), is
given by the de Broglie relation:
h
λ
mv
h  6.626 x 10  34 J  s
?
Compare the wavelengths of
(a) an electron traveling at a speed that is one-hundredth the
speed of light and (b) a baseball of mass 0.145 kg having a
speed of 26.8 m/s (60 mph).
Electron
me = 9.11 × 10-31 kg
v = 3.00 × 106 m/s
Baseball
m = 0.145 kg
v = 26.8 m/s
h
λ
mv
Electron
me = 9.11 × 10-31 kg
v = 3.00 × 106 m/s
6.63 x 10 34 J  s
λ


 31
6 m
9.11 x 10 kg  3.00 x 10

s



2.43 × 10-10 m
Baseball
m = 0.145 kg
v = 26.8 m/s
6.63 x 10 34 J  s
λ

m

0.145 kg  26.8 
s

1.71 × 10-34 m
Building on de Broglie’s work, in 1926, Erwin Schrödinger devised a
theory that could be used to explain the wave properties of electrons in
atoms and molecules.
The branch of physics that mathematically describes the wave
properties of submicroscopic particles is called quantum mechanics or
wave mechanics.
Quantum mechanics alters how we think about the motion of particles.
In 1927, Werner Heisenberg showed how it is impossible to know with
absolute precision both the position, x, and the momentum, p, of a
particle such as electron.
h
(Δx )(Δp) 
4π
Because p = mv this uncertainty becomes more significant as the mass
of the particle becomes smaller.
Quantum mechanics allows us to make statistical
statements about the regions in which we are most
likely to find the electron.
Solving Schrödinger’s equation gives us a wave
function, represented by the Greek letter psi, y,
which gives information about a particle in a given
energy level.
Psi-squared, y 2, gives us the probability of finding
the particle within a region of space.
The wave function for the
lowest level of the hydrogen
atom is shown to the left.
Note that its value is greatest
nearest the nucleus, but rapidly
decreases thereafter. Note also
that it never goes to zero, only
to a very small value.
Two additional views are shown on the next slide.
Figure A illustrates the probability density for an electron in hydrogen.
The concentric circles represent successive shells.
Figure B shows the probability of finding the electron at various
distances from the nucleus. The highest probability (most likely)
distance is at 50 pm.
According to quantum mechanics, each electron is
described by four quantum numbers:
1.
2.
3.
4.
Principal quantum number (n)
Angular momentum quantum number (l)
Magnetic quantum number (ml)
Spin quantum number (ms)
The first three define the wave function for a
particular electron. The fourth quantum number
refers to the magnetic property of electrons.
A wave function for an electron in an atom is called
an atomic orbital (described by three quantum
numbers—n, l, ml). It describes a region of space
with a definite shape where there is a high
probability of finding the electron.
We will study the quantum numbers first, and then
look at atomic orbitals.
Principal Quantum Number, n
This quantum number is the one on which the
energy of an electron in an atom primarily
depends. The smaller the value of n, the lower the
energy and the smaller the orbital.
The principal quantum number can have any
positive value: 1, 2, 3, . . .
Orbitals with the same value for n are said to be in
the same shell.
Shells are sometimes designated by uppercase
letters:
Letter
n
K
1
L
2
M
3
N
4
...
Angular Momentum Quantum Number, l
This quantum number distinguishes orbitals of a
given n (shell) having different shapes.
It can have values from 0, 1, 2, 3, . . . to a
maximum of (n – 1).
For a given n, there will be n different values of l,
or n types of subshells.
Orbitals with the same values for n and l are said
to be in the same shell and subshell.
Subshells are sometimes designated by lowercase
letters:
l
Letter
0
s
1
p
2
d
3
f
n≥
1
2
3
4
...
Not every subshell type exists in every shell. The
minimum value of n for each type of subshell is
shown above.
Magnetic Quantum Number, ml
This quantum number distinguishes orbitals of a given n and l—that is,
of a given energy and shape but having different orientations.
The magnetic quantum number depends on the value of l and can have
any integer value from –l to 0 to +l. Each different value represents a
different orbital. For a given subshell, there will be (2l + 1) values and
therefore (2l + 1) orbitals.
Let’s summarize:
When n = 1, l has only one value, 0.
When l = 0, ml has only one value, 0.
So the first shell (n = 1) has one subshell, an s-subshell, 1s. That
subshell, in turn, has one orbital.
When n = 2, l has two values, 0 and 1.
When l = 0, ml has only one value, 0. So there is a 2s subshell with one
orbital.
When l = 1, ml has only three values, -1, 0, 1. So there is a 2p subshell
with three orbitals.
When n = 3, l has three values, 0, 1, and 2.
When l = 0, ml has only one value, 0. So there is a
3s subshell with one orbital.
When l = 1, ml has only three values, -1, 0, 1. So
there is a 3p subshell with three orbitals.
When l = 2, ml has only five values, -2, -1, 0, 1, 2.
So there is a 3d subshell with five orbitals.
We could continue with n =4 and 5. Each would gain an additional
subshell (f and g, respectively).
In an f subshell, there are seven orbitals; in a
g subshell, there are nine orbitals.
Table 7.1 gives the complete list of permitted values for n, l, and ml up
to the fourth shell. It is on the next slide.
The figure shows
relative energies
for the hydrogen
atom shells and
subshells; each
orbital is indicated
by a dashed-line.
Spin Quantum Number, ms
This quantum number refers to the two possible
orientations of the spin axis of an electron.
It may have a value of either +1/2 or -1/2.
?
Which of the following are permissible sets of quantum
numbers?
n = 4, l = 4, ml = 0, ms = ½
n = 3, l = 2, ml = 1, ms = -½
n = 2, l = 0, ml = 0, ms = ³/²
n = 5, l = 3, ml = -3, ms = ½
Atomic Orbital Shapes
An s orbital is spherical.
A p orbital has two lobes along a straight line through the nucleus, with
one lobe on either side.
A d orbital has a more complicated shape.
The cross-sectional view of a 1s
orbital and a
2s orbital highlights the difference
in the two orbitals’ sizes.
The cutaway diagrams
of the 1s and 2s orbitals
give a better sense of
them in three
dimensions.
The next slide illustrates p orbitals.
Figure A shows the general shape of a p orbital.
Figure B shows the orientations of each of the three p orbitals.
The complexity of the d orbitals can be seen below