Chapter 8 – Modelling Atoms
and Their Electrons
The elements of the periodic table behave in a fashion
that demonstrates the “law of periodicity” – the properties
of the elements vary periodically with their atomic
number.
This allows us to group
the elements in families
where the chemistry
within a family has a
certain predictability
based on the other
members of the family.
Chapter 8
1
Strangely enough, after emphasizing that the periodic
table is constructed based on the periodicity of the
chemical properties of the elements, the textbook then
chooses to start this discussion with “melting points” and
“boiling points” of the main group elements.
Every element melts and every element boils – although in
some instances, this is hard to imagine. Gases, such as the
nitrogen in the air around us, must be very cold before
they will condense to give a liquid and near absolute zero
before they will freeze solid.
A metal, such as beryllium, will eventually melt and then
boil at high enough temperatures.
But trends in these properties are hard to see….
Chapter 8
2
Chapter 8
3
Similarly, we can not ascribe a particular property or
number to the concept of “metallicity”. However, it is
fairly easy to see that some elements have the properties
that we would ascribe to a metal – ductility, malleability,
conducting heat and electricity – and that others lack
these characteristics.
vs.
Chapter 8
4
Remember that we are talking about elements – in their
elemental form. Sodium, as a pure metal, looks very
different than sodium when it is combined with chlorine to
give sodium chloride. (This is a great source of confusion
for non-chemists!)
Chapter 8
5
So, what are the trends in electronegativity?
Maybe not surprisingly, these are the same trends for
oxidizing power.
Chapter 8
6
A common trend or trends in an elemental property would
suggest that some underlying cause or source. Something
about the atoms of the elements varies in a consistent
pattern.
Chapter 8
7
The size of the atoms of the elements vary in a fairly
predictable pattern – but first we have to define what we
mean by “the size of an atom”. There are three common
types of atomic radii in use (four if we include ions).
Covalent: measured as half of the distance between two
atoms of the same element when covalently bonded (i.e.
chlorine).
Metallic: measured as half the distance between the
atoms within a crystalline sample of the metal.
Van der Waals: measured as half the distance of closest
approach of two atoms – before bonding occurs.
“Atomic size” is a derived quantity and depends on the
method of measurement.
Chapter 8
8
Covalent radii of main group elements:
Chapter 8
9
Ionization energy is the minimum energy required to
remove an electron from a gaseous atom of an element
resulting in a more positively charged species.
Note:
X(g)
X+(g)
X+(g) + eX2+(g) + e-
Chapter 8
1st Ionization Potential
2nd I.P.
10
What is perhaps more interesting is to consider the trends
that become more apparent when we include more IP’s
than just the first one.
Chapter 8
11
For the main group elements, charges on the monoatomic
ions can be read directly from the periodic table.
Chapter 8
12
Ionic sizes are also a derived quantity - determined by
considering ionic compounds and rationalizing the
cations/anions to a self-consistent set of sizes.
By comparing the sizes of cations in
a variety of common compounds,
for example, we can determine
the size of the different ions.
Trends:
Chapter 8
13
One of the “trends” that we can see is that cations are
smaller than their parent atoms while anions are larger.
Chapter 8
14
Electronegativity and Electron Affinity are about an
elements ability to attract an electron. They are related
but different – in that “electronegativity” is a measure of
the ability of an atom to pull electrons to it within a
covalent interaction whereas “electron affinity” is
determined by the amount of energy released when a
gaseous atom acquires an electron.
Chapter 8
15
All of these trends observable across the periodic table why do they occur?
In the late 1800s, it was discovered that atoms were not
the smallest particles of matter. They contained structure.
Atoms are built from electrons, protons, and neutrons.
From the early 1900s, experiments and theoretical
understandings have allowed us to build a model of the
atom that recognizes that chemistry arises from the
electrons within an atom and, from this, an understanding
of the basis of the periodicity of the elements.
But first, a slight digression…..
Chapter 8
16
When “white light” is passed through a prism, the light is
split into all of the colours of the rainbow:
When “light” from a gas discharge tube, containing a
single element, is put through a prism it splits into lines of
specific wavelength:
Chapter 8
17
Every element has a unique line emission spectrum:
The more elements that were analyzed in the 1800s, the
more scientists realized (a) that each element was
uniquely defined by its line emission spectra (which was the
same regardless of the source of the element) and (b) that
there are some common relationships (susceptible to
mathematical analysis).
Chapter 8
18
By analyzing the line spectra of hydrogen, Johann Balmer
and later Johannes Rydberg were able to derive an
equation for the position of the lines in the spectrum:
E = R[1/22 - 1/n2] where n >2
Improvements in spectroscopy allowed for other lines to be
detected and the realization that the formula is actually:
E = RZ2[1/n12 - 1/n22] where n2 > n1
where R is the “Rydberg Constant” equal to 101,673 cm-1
or 2.179 x10-18 J or 1312.1 kJ/mol and Z is the atomic
number. (Note: Z = 1 for hydrogen)
Chapter 8
19
With the appropriate units, this formula calculates the
wavelengths of all of the emission lines for hydrogen.
Chapter 8
20
In 1910, Niels Bohr proposed an explanation for the
observations of the hydrogen line spectra. His model was
based on certain postulates:
- The energy of the electron can only have particular
“allowed” values, and then it is said to be in a “stationary
state”. This does not mean that the electron is stationary,
but that its energy in this state is a constant.
- While the electron is in any of its stationary states, it does
not radiate energy.
- The energy of the electron can be increased by the
absorption of a photon and can be decreased by the
emission of a photon. Shifts between stationary states
requires energy – either in or out.
Bohr’s model made sense of hydrogen’s line spectra.
Chapter 8
21
And could be applied to other atoms once Z was
included.
But the Bohr model of the atom is now “obsolete”.
- Electrons do not orbit the nucleus in circular orbits as
proposed by the Bohr model.
- Bohr’s model did not provide an explanation for why “n =
1,2,3,…” or why energy was quantized.
- The equations worked perfectly for hydrogen, but Bohr’s
model did not work so well for other atoms – and the
more electrons involved, the worse the results got.
It is important to realize that the Bohr model was the first
step in a long series that eventually has resulted in our
modern understanding of atoms – quantum theory. But
Bohr’s model was not “quantum theory”.
Chapter 8
22
In the early 1900s, scientists began to ask questions about
the nature of the light. Max Planck, for example, had
shown that light is quantized:
E = hv
where “h” is Planck’s constant – 6.6126 x10-34 J·s – and “v” is
the frequency of light.
Albert Einstein realized that this would
explain the “photoelectric effect”
wherein light striking a metal surface
results in the expulsion of electrons – but
only when the light is above a certain
threshold. Einstein proposed that light
occurred as particles, called “photons”.
Chapter 8
23
In 1925, a French graduate student, Louis de Broglie
proposed a rather startling idea. If light could occur as
both a wave and a particle, then particles should also be
able to occur as waves. Quantified, this gives:
λ = h / mv
where “m” is the mass of a free electron and “v” is its
velocity. Note the more mass, the shorter the wavelength.
It wasn’t long before scientists were able to measure the
diffraction of an electron by an object and demonstrate
that electrons exhibit what has been called “wave-particle
duality”. In the world of the atom, electrons, protons, and
neutrons can act as both particles and waves.
Chapter 8
24
Electrons, in atoms, travel at velocity that is a percentage
of the speed of light. At 0.01c, what would be the
wavelength of the electron?
Electron mass = 9.1091 x10-31 kg
Velocity = 0.01 x 2.998x108 m/s = 2.998x106 m/s
λ = h/mv = 6.626 x10-34 J·s/(9.1091x10-31kg x 2.998x106m/s)
= 2.43 x10-10 m
Note that the radius of the hydrogen atom has been given
as 37 pm or 37x10-12 m. This means that the circumference
of a hydrogen atom (if the electron was to orbit in circular
orbits….) would be 2πr or 2π(37x10-12) = 2.324x10-10 m –
about the same size as the wavelength of the electron at
this velocity.
Chapter 8
25
In 1927, Werner Heisenberg proposed a “thought
experiment” in which he showed that measuring an
electron altered the energy of the electron. The
consequence is that certain properties of electrons (and
all other particles) do not “commute”. This means that we
can not measure the position and the momentum of an
electron simultaneously:
Δx·Δmv ≥ h/4π
This means that the electron can not be described as a
particle but as a probability. This is one of the most
profound implications of quantum mechanics and
resulted in the development of the wave model of the
electron and atoms.
Chapter 8
26
Around the same time as DeBroglie and Heisenberg were
working on their ideas, Erwin Schrödinger was working
towards a theory of atoms involving simple harmonic
oscillators. He developed a model that has come to be
called “quantum mechanics” in which the electron is
treated as a 3-dimensional standing wave, not just a
particle.
This naturally introduces the
idea of a quantum number
– the parameter “n” which
arises from the requirement
that the number of
wavelengths for a standing
must be an integer.
(λ = 2L/n)
Chapter 8
27
The Schrödinger Equation is a generalized mathematical
equation describing the 3-dimensional standing “electron
waves” or “probability function” around an atom’s
nucleus. It can be written a number of ways:
HΨ = EΨ
where E is the energy of the electron in the particular
standing wave, Ψ is the “wave function” which describes
the orbital that holds the electron, and H is the
“Hamiltonian Operator” which represent a set of functions
that operate on the wave function to generate the
energy of the electron.
Chapter 8
28
The solutions to the Schrödinger Equation lead to a set of
orbitals that can be categorized using three quantum
numbers:
- The principal quantum number, n, which must have an
integer value (n = 1,2,3 …. ), and which – in many
ways – fits with the Bohr model of the atom.
- An angular momentum quantum number, ℓ, that is
defined as any integer from 0 to n-1 in value (i.e. if n = 2,
then ℓ = 0 or 1)
- A magnetic momentum quantum number, mℓ, which
can have a value from -ℓ to +ℓ (including “0”). That is, if
ℓ = 1, then mℓ has values of -1, 0, and 1.
One way that we could label the atomic orbitals would
be to tag them with their quantum numbers – i.e. Ψ2,1,-1.
Chapter 8
29
However, for a number of reasons, the orbitals are actually
labelled using the letters s, p, d, and f (which stand for
“sharp”, “principal”, “diffuse”, and “fundamental”)
according to the table:
This means that an orbital Ψ2,1,-1 would be a p-orbital with
an n-value of 2. This would be designated as the “2pz”
orbital where the subscript “z” arises from mℓ.
Chapter 8
30
The term “orbital” refers to a mathematical function (Ψ)
which is one part of the solution to the wave equation for
each stationary state of an electron in an atom (the other
part is the energy, E). For practical purposes, it is useful to
think of an orbital as the non-uniform distribution of the
electron’s matter over the space surrounding the nucleus.
The term “orbital” was coined to avoid confusion with the
term “orbit”. Electrons do not follow an “orbit” or a
prescribed circular path around the nucleus.
Chapter 8
31
Orbitals of an atom that have been derived by use of the
same value of n in the Schrödinger’s equation are said to
be in the same shell. For example, the 2s orbital (n = 2, ℓ =
0) and the three 2p orbitals (n = 2, ℓ = 1) are all part of the
n = 2 shell. But there are four separate orbitals in this shell.
In a given shell, each set of orbitals derived by use of a
particular value of ℓ in the Schrödinger’s equation are said
to be in the same sub-shell. For example, with n = 3, we
have the 3s sub-shell which consists of the 3s orbital, the
3p sub-shell which consists of the three 3p orbitals (px, py,
and pz or mℓ = 1, 0, and -1), and the 3d sub-shell (dz2, dx2-y2,
dxy, dxz, and dyz or mℓ = 2, 1, 0, -1, and -2).
Chapter 8
32
For any solution of the Schrödinger equation (i.e. n = 2 and
ℓ = 1), it is common to say that the electrons occupies that
orbital (i.e. a 2p orbital). It is important to realize that this is
just shorthand as there isn’t a region of space, called an
orbital, that an electron can enter or exit.
It is common to say that an orbital has a certain energy,
when in fact, we mean that the electron in that particular
standing waveform has that energy.
If just one electron in an atom has a particular standing
waveform, then we call it an “unpaired” electron. If two
electrons have the same standing waveform, we call them
an “electron pair” or “paired electrons”. That is, orbitals
can be either singly occupied (unpaired) or doubly
occupied (paired) or empty.
Chapter 8
33
Chapter 8
34
In the 1930s, a series of experiments demonstrated that
there was one final quantum property – electron spin –
that must be taken into account. It is the reason why there
is a maximum of two electrons allowed in an orbital – there
is only one electron of each spin-type allowed.
There is a quantum number, ms or electron spin that has a
value of either +1/2 or -1/2.
Since this is a magnetic property, it can be observed in a
number of ways.
Spin is a property
of many subatomic particles,
including the
proton.
Chapter 8
35
Electron spin accounts for one of the magnetic properties
of matter. Most substances have all of their electrons
paired and as a consequence, do not have any net
“magnetism” – they are “diamagnetic” and actually
repelled slightly from a magnetic field. But some
substances have “unpaired electron density” and
attracted into a field. These substance are called
“paramagnetic” or “magnetic”, in common parlance.
Chapter 8
36
The wave function has the property that it can be
separated into a “radial function” – the distance from the
nucleus – and an “angular function”. (Ψr,θ,Φ = Ψr Ψθ,Φ)
For the hydrogen 1s orbital (Ψ1,0,0), the angular component
is “1” so the wave function is just the radial function:
Ψ1,0,0 = - (π)-1/2(1/ao)3/2e-(r/a )
o
In point of fact, we don’t
plot Ψ but Ψ2 which is the
probability of finding the
electron at any point in
space or the electron
density function.
Chapter 8
37
There are many ways of drawing Ψ2 but the most common
is to draw “3-D” shapes which encompass 90% of the
electron density. Note that the wave function is an
exponential function and therefore extends to infinity (it is
asymptotic with zero) so it would actually be impossible to
draw a surface containing 100% of the electron density.
Note that there are
“nodes” (surfaces of
zero electron density)
within the orbitals
giving rise to the
shape of the wave
function.
Chapter 8
38
Chapter 8
39
Drawing orbital pictures is an important part of chemistry –
and understanding molecules. However, they are not
always convenient and for some purposes, we use an
alternative notation where the orbital is represented as a
“box” and the electrons as either “up” or “down” arrows
(representing the two spin states).
Chapter 8
40
The Pauli Exclusion Principle is an important restraint in the
filling (and drawing) of orbitals:
No two electrons in an
atom can have the same
values of the four quantum
numbers (n, ℓ, mℓ, and ms).
This is just another way of
saying that each orbital is
unique in an atom and
that they can only hold
two electrons.
Chapter 8
41
From the Pauli Exclusion
Principle, and the fact that
an orbital can only hold two
electrons, we can develop
the Aufbau Principle which
prescribes the order in which
all of the orbitals are filled.
With this, we can start to see
the structure of the Periodic
Table developing.
Hund’s Rule tells us that for a degenerate set of orbitals
(such as the 2p orbitals), the orbitals are filled singly before
pairing occurs. For carbon:
Chapter 8
42
So, we have a picture of the atom where electrons
surround the nucleus by occupying orbitals – which have
unique quantum numbers. How does this lead to the
periodic table and the periodicity of the elements?
Chemical properties are a result of the “outermost” or
valence electrons. These are the electrons that are closest
to other atoms and to the surroundings. They are distinct
from the core electrons which are all of the electrons
below the valence layer. (We often represent the core
electrons with the next lowest noble gas configuration.)
For the main group elements, the valence shell are the
ns/np orbitals with the highest value of “n” while the core
electrons are then any electron with an “n” value of “n-1”
or less.
Chapter 8
43
When we put it all
together, the atoms of
elements in the same
group have a similar
ground-state valence
electron configuration.
Chapter 8
44
The nucleus does play a role in the properties of the atom.
Remember that, at its heart, the attraction between the
nucleus and the electron is the attraction between a
positive charge and a negative charge. It is governed by
Coulomb’s Law:
F = q1q2/(4πεor2)
One of the charges (q) is simply the electron. The other is
the charge on the nucleus and one would assume that it is
determined by the number of protons – each of which is
carrying a positive charge.
However, this would have several consequences that are
not consistent with observation (i.e. atoms would get
smaller as the nucleus gets bigger).
Chapter 8
45
The valence electrons experience a repulsion induced by
the other electrons in the atom – specifically, the core
electrons. The effect is to reduce the charge that the
valence electrons experience from the nucleus. This
shielding or screening of the nucleus by the core electrons
leads to an “effective nuclear charge” (Z*).
We have already established that Z increases linearly and
continuously throughout the periodic table – it is the
defining property of an element – but Z* increases in a
periodic fashion.
We calculate Z* from Z by subtracting the shielding:
Z* = Z - σ
Chapter 8
46
How big is the shielding constant?
A simple approximation is that it is the sum of the core
electrons. For example, for fluorine, chlorine, and bromine:
Fluorine: [He]2s22p5
Chlorine: [Ne]3s23p5
Bromine: [Ar]3d104s24p5
Z = 9; σ =2;
Z = 17; σ =10;
Z = 35; σ =28;
Z* = 9 – 2 = 7
Z* = 17 – 10 = 7
Z* = 35 – 28 = 7
In fact, it is a little more subtle. Slater’s Rules are used for
calculating the shielding constant:
- shielding due to other valence electrons = 0.35/electron
- shielding due to the n-1 shell = 0.85/electron
- shielding due to other core electrons = 1.0/electron
Chapter 8
47
When we consider Slater’s rules, a periodic pattern in Z*
becomes apparent:
The same scheme can be applied to both the cations and
anions formed for the main group elements – and explains
the Aufbau Principle.
Chapter 8
48
Quantum mechanics is our best understanding of atoms –
and provides a theoretical underpinning for the periodic
table. It can explain our observations.
Sizes of atoms:
- Across the periodic table, Z* increases resulting in a
stronger attraction for the valence shell electrons and a
smaller atoms
- While Z* increases a small amount down a group, each
row adds another shell of electrons pushing the electrons
further from the nucleus. The two forces approximately
balance to give increasing atomic radii as we go down
a group.
Note that the same arguments can be presented for the
size of ions.
Chapter 8
49
Ionization Energies:
Since the Rydberg Equation defines the energy difference
between any two energy levels, we can use it to
calculate the Ionization Energy of an atom. When ionized
an electron has essentially reached an “infinite” energy
level. Thus, for hydrogen, the ionization energy is given by:
EIP = R(1)2(1/12 – 1/
2)
= R = 1312 kJ/mol
More generally, the Ionization Energy is given by:
EIP = RZ* 2/n2
So, it is possible to calculate an approximately value for
the Ionization Energy or Potential for all the elements.
Chapter 8
50
It should now be obvious where the trends in ionization
energy come from:
- across a period, ionization energy increases – as does Z*
- down a group, ionization energy decreases – Z* stays
roughly the same but the value of “n” increases with
each row
- second, third, and so on ionization energies increase – as
does Z* when electrons are removed from an atom
(roughly 0.35 units per electron)
- the ionization energy increases dramatically after the
group oxidation number – which corresponds to a
change in both Z* and n
Chapter 8
51
Charges on Monoatomic Ions of the Elements:
Atoms are stabilized by having empty orbital sub-shells,
half-filled orbital sub-shells, or filled orbital subshells. From
this, it is not hard to realize that:
- elements on the left-side of the periodic table can
achieve an empty orbital sub-shell by losing electrons
and forming cations.
- elements on the right-side of the periodic table can
achieve a full orbital sub-shell by gaining electrons and
forming anions.
- elements in the middle are ambivalent and can go
either way (or not at all, as in carbon compounds)
Chapter 8
52
Electronegativities and Electron Affinity:
Both of these can be explained using the concept of an
effective nuclear charge by considering the effective
nuclear charge on the atoms in a bond (electronegativity)
or the atom and its corresponding anion (electron affinity).
In addition, the value of n influences both properties as the
further away from the nucleus, the larger the value of “r”
and the smaller the force of attraction according to
Coulomb’s Law.
Chapter 8
53
The Quantum Mechanical model provides a theoretical
explanation for all of the observable properties of the
elements. It is the best model that we have for atoms –
and has been tested, without ever failing, thousands of
times.
Chapter 8
54