Download The Quantum Mechanical Model of the Atom

Chapter 2.2
Structure of the atom
The Quantum Mechanical Model
collide, some of the kinetic energy from one atom is transferred to the
other atom. The electron of the second hydrogen atom absorbs this energy
Figure 3.6 The electromagnetic spectrum and its properties
and is excited to a higher energy level.
When the electron of a hydrogen atom that has been excited to the
Thethird
visible
portion
of falls
the electromagnetic
spectrum
continuous
energy
level
to the second energy
level,isitcalled
emits alight
of certain
spectrum,
because
the
component
colours
are
indistinct.
They
appear
energy. Specifically, an electron that makes a transition from the third
“smeared”
together
into
a continuum
of colour.
tored
nineteenthenergy level
to the
second
energy level
emitsAccording
a photon of
light with
century
physics,
part
of
the
energy
emitted
by
electrons
should
be
a wavelength of 656 nm. Because line spectra result when atoms in an
observable
a continuous
spectrum.
Thistoisa not,
however,
the case.
excited as
state
emit photons
as they fall
lower
energy level,
these
Instead,
when
atoms
absorb
energy
(for
example,
when
they
are
exposed
spectra are also called emission spectra.
to an electric
you
pattern of that
discrete
(distinct), for the
Figurecurrent),
3.9 shows
theobserve
energy atransitions
are responsible
coloured
lines
separated
by spaces
of varying
length.Notice
See Figure
3.7.
coloured
lines
in hydrogen’s
emission
spectrum.
the use
ofYou
the
can symbol
also observe
this
line
spectrum
for
hydrogen,
and
for
other
atoms,
in
n to designate the allowed energy levels for the hydrogen atom:
Investigation
n = 1, n = 3-A.
2, and so on. This symbol, n, represents a positive integer (such
number. You will learn
more about
400 as 1, 2, or 3), and
500 is called a quantum
600
750 nm
the significance of quantum numbers in section 3.2.
The Quantum Mechanical Model of the Atom
Following Bohr’s Atomic Model, atoms have specific energy levels (or
electrons orbits). When excited, electrons change energy level by
emitting or absorbing a specific quantity of energy called quanta.
400
500
600
750 nm
Figure 3.7 The discrete, coloured lines of this spectrum are characteristic of
hydrogen atoms. No other atoms display this pattern of coloured lines.
n=6
n=5
n=4
n=3
Chapter 3 Atoms, Electrons, and Periodic Trends • MHR 123
e-
n=2
nn=3
=1
n=2
energy
n=2
nn=4
=3
n=4
n=5
n=5
n=6
n=6
A Note: Orbits are not drawn to scale.
n=1
B Note: In its unexcited state, hydrogen‘s electron is in
the energy level closest to the nucleus: n = 1. This is
the lowest-possible energy level, representing a state
of greatest stability for the hydrogen atom.
hydrogen
atom. suggested
As you can
seethere
in Figure
mercury
more spectral
lines, however,
that
were 3.12,
smaller
energyhas
differences
lines
hydrogen
Thewords,
same isscientists
true for other
many-electron
atoms
withinthan
energy
levels.does.
In other
hypothesized
that there
ar
Observations
likeeach
theseenergy
forcedlevel.
BohrEach
and other
scientists
to reconsider
sublevels within
of these
sublevels
has its ownthe
nature
energyelectrons
levels.
largeaspaces
between
the individual
colours
slightly
energy.Thethan
A mercury (Hg) atom
hasofdifferent
more
hydrogen
atom
and has more
suggested that there are energy differences between individual energy
spectral lines than hydrogen
does.
Hg
levels,
as stated in Bohr’s model. The smaller spaces between coloured
lines,
that550
there were
differences
400 however,
450 suggested
500
600 smaller
650energy700
750 nm
within energy levels. In other words, scientists hypothesized that there ar
H
Hydrogen spectrum sublevels
within each energy level. Each of these sublevels has its own
slightly
different
energy.
400
450
500
550
600
650
700
750 nm
How Bohr’s Atomic Model Explains the Spectrum
Mercury spectrum
The emission spectrum for mercury shows that it has more spectral lines
Hg the emission spectrum for hydrogen.
than
Figure 3.12
400
450
500
550
600
650
700
750 nm
It was fairly straightforward to modify Bohr’s model to include the idea
H
The large spaces of
between
individual
suggested
that there
energy the
sublevels
for colors
the hydrogen
spectrum
andare
for energy
atoms or ions with
400 one electron.
450
500 was550
600
700 however.
750 nm The
only
There
a moreasfundamental
problem,
differences between
individual
energy
levels,
stated in 650
Bohr’s
model.
Figure 3.12
model
still could
not explain
spectrashows
produced
by more
many-electron
The emission
spectrumthe
for mercury
that it has
spectral lines
than
the emission
spectrum
for hydrogen.
atoms.
Therefore,
a simple
modification of Bohr’s atomic model was not
The smaller spaces between colored lines, however, suggested that there were
enough. The many-electron problem called for a new model to explain
smaller energy differences
within
energy
levels.
It
was fairly
straightforward
modify Bohr’s
model
to include
theanother
idea
spectra
of all
types of
atoms.to
However,
this was
not possible
until
of
energy sublevels
formatter
the hydrogen
spectrum and for atoms or ions with
important
property of
was discovered.
only one electron. There was a more fundamental problem, however. The
In other words
model
stillsub-levels
couldof
not
explain
theenergy
spectralevel”
produced
by many-electron
“there
are
within
each
.
The Discovery
Matter
Waves
atoms. Therefore, a simple modification of Bohr’s atomic model was not
Each of these sub-levels
itsearly
own energy
Byhas
the
1920s, it was standard knowledge that energy had matter-like
enough. The many-electron problem called for a new model to explain
properties. In 1924, a young physics student named Louis de Broglie
spectra of all types of atoms. However, this was not possible until another
The Discovery of Matter Waves
In 1924, the physics student Louis de Broglie stated an idea: “What if matter has wave-like
properties?”
He developed an equation to calculate the wavelength (λ) associated with any object (large,
small, or microscopic).
E.g.
A baseball with a mass of 142 g and moving with a speed of 25.0 m/s has a
wavelength of 2 × 10−34 m.
Objects that you can see and interact with, such as a baseball, have wavelengths so
small that they do not have any significant observable effect on the object’s motion.
For microscopic objects, such as electrons, the effect of
wavelength on motion becomes very significant.
E.g.
An electron moving at a speed of 5.9 × 106 m/s has a wavelength of 1 × 10−10
m. The size of this wavelength is greater than the size of the hydrogen atom
to which it belongs.
The Quantum Mechanical Model of the Atom
In 1926, an Austrian physicist, Erwin Schrödinger, used mathematics and statistics to combine
de Broglie’s idea of matter waves and Einstein’s idea of quantized energy particles (photons).
This resulted in the birth of Quantum Mechanics.
This is a branch of physics that uses mathematical equations to describe the wave
properties of sub-microscopic particles such as electrons, atoms, and molecules.
Schrödinger proposed a new atomic model: the quantum
mechanical model of the atom.
The model of the
atom in 1927. The fuzzy, spherical
Figure 3.13
Schrödinger and the orbital
Schrödinger’s quantum mechanical model describes atoms as having certain allowed quantities
of energy because of the wave-like properties of their electrons.
The volume surrounding the nucleus is indistinct because of a principle called “the uncertainty
principle”.
The physicist Werner Heisenberg proposed, using mathematics that it is impossible to know
precisely both the position and the momentum of an object. (recall: An object’s momentum is given
by its mass multiplied by its velocity).
E.g. According to the uncertainty principle, if you can know an electron’s precise path
around the nucleus (orbit), you cannot know with certainty its velocity.
Similarly, if you know its precise velocity, you cannot know with certainty its position.
Schrödinger used an equation called a wave equation to
define the orbital as the “probability of finding an
atom’s electrons at a particular point within
the atom”.
Chemists call these wave functions orbitals
The model of the
atom in 1927. The fuzzy, spherical
Figure 3.13
A level of probability is usually expressed as a percentage. Therefore,
Schrödinger
and
the3.14B
orbital
the contour line
in Figure
defines an area that represents 95 percent
of the probability graph. This two-dimensional shape is given three
Each orbital has its own associated energy, and each represents information about where,
dimensions in Figure 3.14C. What this means is that, at any time, there
inside the atom, the electrons would spend most of their time. However, orbitals indicate
is there
a 95 percent
chance ofoffinding
the electron within the volume defined
where
is a high probability
finding electrons
by the spherical contour.
A
B
+
C
+
+
Fig.3.14
A: probability
of finding
an probability
electron at any
point in
when theenergy
electron
is at in
thethe
lowest
Figure
Electron
density
graphs
forspace
the lowest
level
hydrogen
energy level (n = 1) of a hydrogen atom. Where the density of the dots is greater, there is a
atom. These
diagrams represent the probability of finding an electron at any point in this
higher probability of finding the electron.
energyFig.
level.
B defines an area that represents 95 percent of the probability graph. This two-dimensional
shape is given three dimensions in Fig. C
Quantum Numbers and Orbitals
Figure 3.14 showed electron-density probabilities for the lowest energy
level of the hydrogen atom. This is the most stable energy state for
hydrogen, and is called the ground state. The quantum number, n, for a
of the probability graph. This two-dimensional shape is given three
Quantum
Numbers and Orbitals
dimensions in Figure 3.14C. What this means is that, at any time, there
is a lowest
95 percent
ofhydrogen
findingatom
the electron
theis volume
The
energy chance
level of the
is the mostwithin
stable and
called the defined
ground
state.
by the spherical contour.
The quantum number, n, for a hydrogen atom in its ground state is 1.
A
B
+
C
+
+
Orb
When n=1 in the atom, its electron is associated with an orbital that has a characteristic energy
scienti
Figure
3.14In an
Electron
for with
the lowest
energy
level
hydrogen
and shape.
exciteddensity
state, theprobability
electron is graphs
associated
a different
orbital
withinitsthe
own
Orbitals have a variety of differ
characteristic
energy and
shape. the probability of finding an electron
atom.
These diagrams
represent
at any point in this
quantu
scientists
use
three
quantum
numb
The
figure
below
compares
the
sizes
of
hydrogen’s
orbitals
when
the
atom
is
in
its
ground
state
n
=
1
energy level.
(n=1) and when it is in an excited state(n=2
quantum number, n, describes
an o
quantu
n = 1 and n=3).
quantum number, l, describes an o
l , des
ml , describes an orbital’s m
orientatio
Quantum Numbers and Orbitals
numbers are described further
belo
numbe
Figure 3.14 showed electron-density probabilities for the lowest energy
afterward summarizes this informa
afterwa
level of the hydrogen atom. This
is the most stable
for number,
m
aboutenergy
a fourthstate
quantum
n=2
n=1
n=2
n=3
hydrogen, and is called the ground state. The
number,
n, for
a
about
a
electron
inside
an orbital.)
n =quantum
2
n=1
n=2
Orbitals have a variety of different possible shapes. Therefore,
scientists use three quantum numbers to describe an atomic orbital. One
quantum number, n, describes an orbital’s energy level and size. A second
quantum number, l, describes an orbital’s shape. A third quantum number,
ml , describes an orbital’s orientation in space. These three quantum
numbers are described further below. The Concept Organizer that follows
afterward summarizes this information. (In section 3.3, you will learn
about a fourth quantum number, ms, which is used to describe the
electron inside an orbital.)
The First Quantum Number:
Describing Orbital Energy Level and Size
Quantum Numbers and Orbitals
Non excited orbitals have different possible shapes. Three
quantum numbers are used to describe an atomic orbital.
1. One quantum number, n, describes an orbital’s energy
level and size.
2. A second quantum number describes the orbital’s shape.
3. A third quantum number describes the orbital’s
orientation in space.
The First Quantum Number:
Describing Orbital Energy Level and Size
The principal quantum number (n) is a positive number that specifies the energy
level of an orbital and its size.
The value of n, therefore, may be 1, 2, 3....
A higher value for n indicates a higher and larger energy level, with a higher probability of
finding an electron farther from the nucleus.
The maximum number of electrons that is possible in any energy level is 2n2.
value of “n”
maximum number of ein the energy level
1
2
2
8
3
18
The Second Quantum Number:
Describing Orbital Shape
The second quantum number, or angular quantum number (l) describes the orbital’s
shape and is a positive integer that ranges from 0 to (n−1).
The number of possible values for l in a given energy level is the same as the value of n.
e.g. if n = 2, then there are only two types of orbital shapes at this energy level.
value of “n”
values of “l”
1
0
2
0 or 1
3
0, 1 or 2
4
0,1,2 or 3
The Second Quantum Number:
Describing Orbital Shape
Each value for l is given a letter: s, p, d, or f
•
•
•
•
The l=0 orbital has the letter s
The l=1 orbital has the letter p
The l=2 orbital has the letter d
The l=3 orbital has the letter f
To identify an energy type of orbital, you combine the value of n with the letter of the
orbital shape.
e.g.
The orbital with n=3 and l=0 is called the 3s orbital.
The orbital with n=2 and l=1 is the 2p orbital
The Third Quantum Number:
Describing Orbital Orientation
The magnetic quantum number (m) is an integer with values ranging from -l to +l,
including 0. This quantum number indicates the orientation of the orbital in the space.
The value of m is limited by the value of l. If l=0, m can be only 0.
e.g.
If l=1, m may have one of three values: -1, 0, or +1.
If l=2, m may have one of five values: -2,-1, 0, +1 or +2.
In other words, for a given value of n, there is one orbital s (l=0) and three orbitals of p
(l=1). Each of these p orbitals has the same shape and energy, but a different orientation
around the nucleus.
value of “n” values of “l”
values of “m”
1
0
0
2
0
0
1
-1, 0, +1
2
-2,-1, 0, +1, +2
Quantum Numbers
The total number of possible orbitals for
any energy level n is given by n2 .
For example, if n = 2, it has a total of 4 (22) orbitals (one s orbital and three p orbitals)
n
1
Quantum number
l
m
0
0
type
s
Orbital
name
1s
quantity
1
2
0
1
0
-1, 0, +1
s
p
2s
2p
1
3
3
0
1
2
0
-1, 0, +1
-2,-1, 0, +1, +2
s
p
d
3s
3p
3d
1
3
5
4
0
1
2
3
0
-1, 0, +1
-2,-1, 0, +1, +2
-3,-2,-1,0,+1,+2,+3
s
p
d
f
4s
4p
4d
4f
1
3
5
7
Sample Problems
Determining Quantum Numbers
If n=3, what are the allowed values for l and m?
What is the total number of orbitals in this energy level?
The allowed values for l range from 0 to (n−1). The allowed values for m are integers ranging from −l to +l
including 0. Since each orbital has a single m value, the total number of values for m gives the number of
orbitals.
• To find l from n:
If n=3, l may be either 0, 1, or 2.
• To find ml from l:
If l=0, m=0
If l=1,m maybe -1,0,+1
If l=2, m maybe -2,-1,0,+1,+2
• Since there are a total of 9 possible values for m , there are 9 orbitals when n=3 (32).
What are the possible values for m if n=5 and l=1?
What kind of orbital is described by these quantum numbers? How many orbitals can
be described by these quantum numbers?
Determine the type of orbital by combining the value for n with the letter used to identify l. You can find
possible values for m from l, and the total of the m values gives the number of orbitals.
•
•
•
To name the type of orbital:
l=1, which describes a p orbital
Since n=5, the quantum numbers represent a 5p orbital.
To find m from l:
If l=1, m maybe -1, 0, +1
Therefore, there are 3 possible 5p orbitals.
measured, and trajectories that can be photographed. They exist in
the physical universe.
Orbitals are
mathematical
descriptions
electrons. They
do not
An orbital is •associated
with
a size, a shape
and anoforientation
around
thehave
nucleus.
measurable
physical
properties
such as
or temperature.
Theya exist
Together, the size,
shape and
position
represent
themass
probability
of finding
specific
in the imagination.
Shapes of Orbitals
electron
around the nucleus of an atom.
The figure shows the probability shapes associated with the s, p, and d orbitals
s orbitals
n=1
!=0
p orbitals
z
z
z
y
y
n=2
!=1
x
m =-1
d orbitals
z
y
y
x
x
n=3
!=2
m =-1
y
n=2
!=1
x
m =0
z
z
n=3
!=2
m =-2
n=3
!=0
n=2
!=0
n=3
!=2
m =0
n=2
!=1
x m =+1
z
z
y
y
x
x
n=3
!=2
m =+1
y
n=3
!=2
m =+2
x
Shapes of the s, p, and d orbitals. Orbitals in the p and d sublevels are oriented
along or between perpendicular x, y, and z axes.
Figure 3.16
Notice that the overall shape of an atom is a combination of all its orbitals. Thus, the overall
shape of an atom is spherical.