In this section we are going to start very basic and finish slightly less basic, there is no
need to really worry too much about the depth of these first couple of lectures they are to
get us up to speed on the very basics.!
!
First a couple of definitions.!
Avagadros Constant / Number. The number of objects in a mole 6.023 x 10-23 .!
The charge on a electron is 1.6 X 10-19 C.!
!
!
Ok, here we go,!
The smallest building block we are going to concern ourselves with is the atom. Each
atom contains a nucleus which is pretty small (10-15) surrounded by a series of electrons. !
!
Each atom corresponds to a specific element and each element is defined by a unique
number corresponding to the number of protons in the atom. I.e it is the number of atoms
that define the element and as such the properties of the element. This is mainly (as far
as we are concerned) due to the fact that a neutral atom has the same number of
electrons as it has protons. The charge on the proton is the same as the charge on the
electron other that the sign.!
!
That’s not the whole story, the nucleus as I am sure you know contains neutrons, these
add to the mass of the atom such that we say that the atomic mass is defined as !
!
A=Z+N where Z is the atomic number and N is the number of neutrons.!
!
The mass in kg is defines such that 1 mole (6.023 X10-23 atoms) of Carbon 12 i.e. carbon
with 6 protons and 6 neutrons has a mass of 12g.!
!
Now your thinking hold on, if an element is defined by the number of protons and not
neutrons then could we have carbon 11 or carbon 13. Yes, these are the isotopes of
carbon. An isotope being a version of an element with a different atomic mass.!
!
Here is a good example, the atomic mass of Iron is 55.85 g/mol.!
!
I.e it’s not an integer.!
!
There are many isotopes of Iron!
!
The 55.85 is the average of them accounting for their percentage abundance
Isotope
Mass
Relative Abundance
54
54
0.05845
56
56
0.91754
57
57
0.02119
60
60
0.00282
!
!
!
Bohr new about the energy levels of atoms from the absorption spectrum of hydrogen. So
he devised a model in which the electrons orbited the nucleus at very specific distances,
these distances were dependent on an orbital number (which we now call the principle
quantum
this was
accomplished with a slight fudge factor. This worked for the
Atomic
Structure number)
and Interatomic
Bonding
electrons closets to the nucleus but as the electrons moved away there were problems!
0
0
"1.5
n=3
"3.4
n=2
3d
3p
3s
2p
2s
"5
"1
1 0 "1 8
"2
1 0 "1 8
Energy (J)
2 (a) The
ee electron
ates for the
ogen atom.
ron energy
e first three
f the wavel hydrogen
apted from
fatt, G. W.
d J. Wulff,
ructure and
f Materials,
cture, p. 10.
 1964 by
ey & Sons,
Reprinted
on of John
Sons, Inc.)
Energy (eV)
pter 2 /
Starting in the 20th century it was pretty obvious something was wrong with our view of
atoms. We had the Bohr model from scattering experiment which put the nucleus at the
centre and electrons orbiting around, but there were flaws in this model.!
"1 0
n=1
"13.6
"1 5
(a)
1s
(b)
!
Above is the absorption spectra as predicted using the Bohr model and via observation.
Thestates
Bohrdomodel
cancontinuously
not predictwith
theenergy;
2 n=2 that
states.
It also states
difficult
These
not vary
is, adjacent
areto see how it could as its
separated
by well
finitedefined
energies.orbitals.!
For example, allowed states for the Bohr hydrogen
built on
!
atom are represented in Figure 2.2a. These energies are taken to be negative,
whereas
zero reference
is the unbound
or free electron.
the single
The the
solution
to this came
in the quantum
modelOf
of course,
the atom,
the orbitals were still in place
electron associated with the hydrogen atom will fill only one of these states.
and
each
orbital
is
given
a
specific
quantum
number
n
(similar
to the bohr atom), this is the
Thus, the Bohr model represents an early attempt to describe electrons in atoms,
basics
energy
level.
To rebuild
theand
actual
youenergy
needlevels).
three other quantum numbers
in terms
of both
position
(electron
orbitals)
energyspectra
(quantized
and
these
will
be
dealt
with
extensively
in
your
Quantum
course.
This Bohr model was eventually found to have some significant limitations !
because of its inability to explain several phenomena involving electrons. A resolutionThe
was reached
with a wave-mechanical
in which
electron orientation
is considered of an electron orbital (or
size, shape
(i.e not alwaysmodel,
circular)
and the
spherical
to exhibit
both
wavelike
and
particle-like
characteristics.
With
this
model,
probability density) are defined by these three numbers.! an electron is no longer treated as a particle moving in a discrete orbital; but rather,
position is considered to be the probability of an electron’s being at various locations
around the nucleus. In other words, position is described by a probability distribution
or electron cloud. Figure 2.3 compares Bohr and wave-mechanical models for the
hydrogen atom. Both these models are used throughout the course of this book;
the choice depends on which model allows the more simple explanation.
!
QUANTUM NUMBERS
Using wave mechanics, every electron in an atom is characterized by four parameters
called quantum numbers. The size, shape, and spatial orientation of an electron’s
probability density are specified by three of these quantum numbers. Furthermore,
Bohr energy levels separate into electron subshells, and quantum numbers dictate
the number of states within each subshell. Shells are specified by a principal quantum
number n, which may take on integral values beginning with unity; sometimes these
shells are designated by the letters K, L, M, N, O, and so on, which correspond,
respectively, to n ! 1, 2, 3, 4, 5, . . . , as indicated in Table 2.1. It should also be
2.3 Electrons in Atoms
●
13
FIGURE 2.3 Comparison of
the (a) Bohr and (b) wavemechanical atom models in
terms of electron
distribution. (Adapted from
Z. D. Jastrzebski, The
Nature and Properties of
Engineering Materials, 3rd
edition, p. 4. Copyright 
1987 by John Wiley & Sons,
New York. Reprinted by
permission of John Wiley &
Sons, Inc.)
Probability
1.0
0
Distance from nucleus
Orbital electron
Nucleus
(a)
(b)
!
The first number n, relates the distance the electron is from the nucleus.!
!
The first new quantum we will introduce is the letter 𝓁, (known as the azimuthal qn) this
signifies a sub-shell and the value l is restricted by the value of n such that 𝓁=0,1,2,3.. (n-1)!
!
Table 2.1 The Number of Available Electron States in Some of the
Electron
Shellsof
and
Subshells
The number
available
energy states in each of our new sub
levels is given by our third
Principal number which is m. m is known as the magic number and has the values -𝓁,𝓁
quantum
Quantum
+1,... 𝓁.n
Number
1
!
Shell
Designation
K
Number
of States
1
Subshells
s
Number of Electrons
Per Subshell
Per Shell
2
2
s
2
2 we have L
finally
the number pof
electrons13 allowed in each
energy 8state is given by the
6
quantum number S which iss the spin state,
this can2take a value of either -½ or +½.!
1
!
So lets try this for n=1.!
!
M
3
4
n
1
!
n=2!
!
N
p
d
3
5
6
10
s
p
d
f
1
3
5
7
2
6
10
14
l
0
18
m
0
32
s
-½
+½
Probability
edition, p. 4. Copyright 
1987 by John Wiley & Sons,
New York. Reprinted by
permission of John Wiley &
Sons, Inc.)
n
0
2
l
m
s
0
0
-½
Distance from nucleus
+½
1
Orbital electron
Nucleus
(a)
(b)
!
-1
-½
-1
+½
0
-½
0
+½
1
-½
1
+½
We can now build up any atom in the periodic table, note that the Pauli exclusion principle
will only allow one electron per state.!
!
Table 2.1
The Number of Available Electron States in Some of the
The
shellsShells
and sub
have specific names.!
Electron
and shells
Subshells
Principal
Quantum
Number n
1
Shell
Designation
K
2
3
4
Number of Electrons
Per Subshell
Per Shell
2
2
Subshells
s
Number
of States
1
L
s
p
1
3
2
6
8
M
s
p
d
1
3
5
2
6
10
18
N
s
p
d
f
1
3
5
7
2
6
10
14
32
!
!
The outer most electrons, i.e. the electrons which do not fill a final shell, are called the
Valency electrons and it is these electrons which define the properties of a material as far
as we are concerned.!
!
The stable configuration is one where the outer shell is filled giving a noble gas
configuration. Some atoms which have few valency electrons will try to achieve a stable
state by loosing (or giving up) electrons. One with nearly filled shells will try to achieve a
stable state by acquiring electrons.!
!
So if we look at the noble gasses we can say that they occur at Z numbers 2, 10, 18.!
!
e.g. ! Argon has ! n=1 so K has 2 electrons (filled shell)!
!
!
!
n=2 has 2 in s and 6 in p (filled shell)!
!
!
!
n=3 has 2 in s and 6 in p (filled shell) none in d!
!
We will not delve too deep in to the properties elements share down the table or the trends
across the table.!
!
So we have a little understanding of the properties of a single atom sitting on its own, in a
vacuum at infinity far from any friends.!
!
Lets bring a couple of atoms together, this is the start of our discussion on bonding. So
there is going to be attraction between the electrons and the nucleus of the opposing atom
and a repulsion between the two electron clouds.!
!
We can write the net force as!
!
Fn=Fa+Fr!
!
Where Fa is the attractive force and Fr is the repulsive force.!
!
We can also assume that the force is dependent on distance (from observation or more
frequently from our knowledge of coulomb interactions)!
!
Then we know that at some point in space the attractive and repulsive forces are equal,
such that Fa=Fr.!
!
2.5 Bonding Forces and Energies
+
Force F
Attraction
Attractive force FA
Repulsion
0
Interatomic separation r
r0
Repulsive force FR
Net force FN
●
19
FIGURE 2.8 (a) The
dependence of repulsive,
attractive, and net forces on
interatomic separation for
two isolated atoms. (b) The
dependence of repulsive,
attractive, and net potential
energies on interatomic
separation for two isolated
atoms.
(a)
+
Repulsion
Interatomic separation r
0
Attraction
Potential energy E
Repulsive energy ER
Net energy EN
E0
Attractive energy EA
(b)
!
which is also a function of the interatomic separation, as also plotted in Figure
2.8 a. When FA and FR balance, or become equal, there is no net force; that is,
FA ! F R " 0
(2.3)
Then a state of equilibrium exists. The centers of the two atoms will remain separated
So we can draw the attractive and repulsive forces and the sum of the two, here we have
guessed at the shape of the two force curves. In all honesty guess is not the correct term
we know the shape of these curves from experiment, we will get back to that later.!
!
We can write the force as a potential the Mie potential (proposed in 1903).!
!
!
!
We can use any numbers we want, the Mie potential is simply a way of adding two forces
together. This force curve can define the materials properties, for examples materials with
high values of E0 generally have high melting points. Also a deeper trough produces a
material with lower thermal expansion.!
!
!
!
Bonding!
So when we bring together two atoms (simple atoms like hydrogen) we have the
interaction of the electrons clouds and the electron to nucleus. This is not that easy to treat
but what we can see is that the out electron orbital has space for two electrons one spin
up and one spin down. This configuration is of a lower energy than two spin ups (which
also wouldn't be allowed due to the pauli exclusion principle)!
!
!
For heavier atoms the main repulsion is no longer from the nucleus interaction with the
other nucleus but rather it is due to the pauli exclusion principle not allowing the electrons
to penetrate. In these cases the the atoms / shells can be considered as fixed hard
spheres.!
!
We are going to talk now about interatomic bonding, we will look at covalent, ionic and
metallic bonding. All these types of bonding are dominated by the valency electrons.!
!
Lets first start with ionic bonding, which is always between a metal and a non metal,
because the metal gives up its electrons while the non metal collects the electrons.!
!
The constituent atoms become stable (i.e they end up with a full valency shell) to do this
they end up with a charge with the metals becoming positive and the non-metals becoming
negative. If we look at sodium chloride (table salt). The sodium atom ends up with a +Ve
charge of 1 and the outer shell is the same as stable neon. !
!
!
!
!
A very simple treatment of this situation would be write the electric field at a distance,!
!
ELECTRIC FIELD!
!
Volts m-1!
Two things to notice epsilon is the dieltric constant of vacuum (permitivity of free space)
and the dieletic constant of the material.!
!
So thats out one ION, if we bring in another ion we can write down the interaction force.!
!
!
!
!
Newtons!
!
!
!
!
!
Energy of the system, how much energy does it cost / produce (free energy) to bring these
two love ions together.!
!
Well that is simply the work done in taking the ions from infinity and bringing them together!
which is !
!
!
!
!
Which we can write as!
!
!
Joules!
!
`notes point out that the sub shells in the quantum model are additive’!
!
!
!
Just a note that this form is basically one of the Mie potentials but with the constant being
1/4pi(eplison(0)).*Z1eZ2e.!
!
!
!
!
!
!
Some notes, Z=+1 for a monovalent cation!
Z=-1 for a monovalent anion!
Z=2 for divalent ions!
!
If the enegy (w(r)) is positive then the force is repulsive, we have to put energy in to put
them together, and if its negative its a release of energy so the will spontaneously form a
bond and we have to put energy in to break up the bond.!
!
w is positive if the charge is the same.!
!
!
!
Example Sodium Chloride, the combined atomic radius of sodium and chlorine ions is
0.276 nm.!
!
W(r)= (1-)*(1)(1x10^-19)^2!
---------------------------- !
!
!
4*pi*(8.85X10^-12)*(0.276x10^-9)!
!
= -8.4 x 10 ^-19 Joules!
!
lets compare this with the energy available at room temperature!
!
E=KT (K is the boltzman constant)!
!
=1.38X10^-23 x 300 = 4.1X10^-21 Joules, so the bonding energy of the salt is much
higher than room temp.!
!
If you workout the energy at the actual melting temperature of salt you will see its very
different but we will come to that.!
!
We know the energy required to break the bond, but we can also calculate the force
needed to pull the ion pair apart. which is just F=w(r)/r where r is the bond length. gives
about 3nN.!
!
Calculate the bond strength in water.!
!
We have learnt a few things here, mainly the bond strength of an ionic bond depends on
the valency of the ions and the dielectric constant of the surrounding material.!
!
One thing we should also think about is that the ionic bond is non directional so that all the
ions around will be attracted in to a crystal this crystal will effect the bonding strength.
Simplictically we treat this summing up the contributions of the ions. Each sodium ion has
6 close cl ions at a distance r =0.276nm, then it has 12 Na+ ions at √2r, 8 more Cl- ions at
√3r and so on.!
!
writing this out we get!
!
!
U=
!
This new constant we have just created is called the Madelung constant, it varies from
1.638 to 1.763 for different crystals.!
!
Any system in which the over all charge is zero will have a net attraction between
constituents.!
!
!
Which is larger than a single ion pair. (note the lack of epsilon) the lack of the epsilon is
due to fact that we assumed our ions were in a vacuum, obviously this is not the case
when they are in a crystal, we can not really just change our dielectric because then we
change the original reference state.!
!
We have calculated the amount of energy for a single ion, we can also calculate the
energy of a mole of lattice ions, this is known as the molar lattice energy of the cohesive
energy and it is simply U=NoUi where No is avagadroes constant and Ui is the energy per
ion!
for sodium chloride it is 880 KJ mol-1!
!
We know that when we have an energy associated with the stable crystal but we also
know that energies are fairly relative in materials, for example would you get all of that
energy back if you split up the crystal, well we defined the ions as originally being at infinity
but we didn’t really consider how we made the ions.!
!
A few points, actually the lowest state when isolated would not really be the ion that we
assumed but it would actually be the atom, the trouble is the coulomb method that we
have just used would not work for atoms. If you did bring the two atoms together what you
would see is that at about 1nm the electron would just from the sodium to the chlorine.
This is because the electronic configuration of system would be more favorable, the triplet
state is lower energy on the return the ionic attraction energy would be reduced by the
electron jumping back. This is know as the harpooning effect.!
!
When we had that ion at infinity it still had an energy associated with it, evern though its
not interacting with anything it still has an electrostatic free energy associated with it.!
!
This energy is known as the Born Energy of an ion and its going to be pretty useful.!
!
Born energy!
!
Imagine an atom sitting all alone it has a radius of ,a, and con be considered a sphere.!
We bring charges from infinity to ,r=a, using our original coulomb equation we have Q1=q
and Q2=dq where dq is the addition of charge.!
!
!
The total free energy of the systems would be!
!
!
!
!
!
!
!
!
!
!
\delta w=\int _{ 0 }^{ Q }{ \frac { q\delta q }{ 4\pi { \varepsilon }_{ 0 }\varepsilon a } } ={ \mu
}^{ i }=\frac { Q^{ 2 } }{ 4\pi { \varepsilon }_{ 0 }\varepsilon a } =\frac { (ze)^{ 2 } }{ 4\pi
{ \varepsilon }_{ 0 }\varepsilon a } !
!
!
!
So that is the definition of the Born energy!
!
We can easily now work out what happens to an ions energy in different dielectric
constants.!
!
!
Transferring an ion from a low dielectric constant to a high one is energetically favorable.!
!
!
For a mole of ions we can write!
!
!
The Born energy of an ion is the energy of taken to build up an ion from its constituent
components.!
!
So we can now form an ion !
!
It changes depending on the dielectric constant !
!
!
!
!
We want to think about putting our salt in to water, because it seems that it shouldn’t
dissolve as the melting temperature of salt is much higher than room temp.!
!
!
!
!
!
On a very simple level we can write the energy of separation of a Na Cl pair can be written
as,!
!
Example.!
!
to go from the gas phase to the solid phase for NaCl!
!
Gives -1.46x10^-18 J per ion.!
!
for a mole of Na Cl it is 880 KJ mole^-1.!
!
!
!
The mole fraction is given as!
!
!
!
!
!
fraction
amount
!
is defined as the amount of a constituent
of all constituents in a mixture
[1!
!
divided by the total
!
!
there is an error in this formula the 4 should be an 8!
!
!
for our system!
!
!
!
!
!
!
!
!
!
!
Working this out for room temperature T=298K, E=78 (a++a_)= 0.276 nm!
!
we get Xs=e-2.6 => 0.075 mole fraction.!
!
The actual value is 0.11 mole / mole.!
!
We used a really simple model here but it works ok, it shows a trend!
that is !
!
e^-(constant/E) where E is the dieltric constant!
!
!
A plot of log X against E should be constant which it is from the data!
!
!
Notes, The solubility of an electrolyte (liquid or gel that contains ions) is strictly determined
by the difference in energy of the free ions in the solid lattice to that in solution.!
!
SO you should approach the problem by taking the ions and dissociate them in a vacuum,
work out that energy, then transfer the ions from vacuum to your solution and work out the
change in energy !
!
!
!
!
from a gas to a liquid!
!
!
e1=1 e2=78.!
!
!
!
!
we use 0.14 because it the average of the two ionic radi, the 2 comes from the fact that we
have both Na and Cl ions (they are separate)!
!
this gives -973 KJ mol^-1!
!
!
!
!
!
We don’t need to worry about the number in front of e as they are one and one, we make it
positive so we know we are putting energy in to the system. A+ and A- are the ionic radius.!
!
Finally ionic bonding is between a metal and a non metal!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
W
849 (203)
8.8
3410
van der Waals
Ar
Cl2
7.7 (1.8)
31 (7.4)
0.08
0.32
"189
"101
Hydrogen
NH3
H2O
35 (8.4)
51 (12.2)
0.36
0.52
"78
0
!
Covalent bonding!
!
2.3 contains bonding energies and melting temperatures for several ionic materials.
Ionic materials
are stable
characteristically
and brittle and,
electrically
In covalent
bonding
electron hard
configurations
arefurthermore,
assumed by
the sharing of electrons
and thermally insulative. As discussed in subsequent chapters, these properties are
between
adjacent
atoms.ofTwo
atoms
that are covalently
bonded
will
each
contribute at least
a direct
consequence
electron
configurations
and/or the nature
of the
ionic
bond.
one electron to the bond, and the shared electrons may be considered to belong to both
COVALENT BONDING
atoms. Covalent bonding is schematically illustrated in Figure below for a molecule of
In covalent bonding stable electron configurations are assumed by the sharing of
methane
(CH4between
). The carbon
valence
whereas
electrons
adjacentatom
atoms.has
Twofour
atoms
that areelectrons,
covalently bonded
willeach
each of the four
contribute
leasta single
one electron
to the
bond, and
thehydrogen
shared electrons
may acquire
be
hydrogen
atomsathas
valence
electron.
Each
atom can
a helium
considered to belong to both atoms. Covalent bonding is schematically illustrated
electron
configuration (two 1s valence electrons) when the carbon atom shares with it one
in Figure 2.10 for a molecule of methane (CH4 ). The carbon atom has four valence
electron.
The carbon
four
shared
one from
each hydrogen, for
electrons,
whereasnow
each has
of the
fouradditional
hydrogen atoms
haselectrons,
a single valence
electron.
Each
hydrogen
atom
can
acquire
a
helium
electron
configuration
(two
1
s
valence
a total of eight valence
!
!
electrons) when the carbon atom shares with it one electron. The carbon now has
four additional shared electrons, one from each hydrogen, for a total of eight valence
FIGURE 2.10 Schematic representation of
covalent bonding in a molecule of methane
(CH4 ).
H
Shared electron
from carbon
Shared electron
from hydrogen
H
C
H
H
!
!
!
Calculate the amount of IONic bonding for a bond!
!
Metallic bonding!
!
In metallic bonding the metal give up their outer electrons to achieve a filled electron shell,
these free electrons are now