Chapter Three
2. Binding Energy in crystals
2.1. Bonding in Solids
• Atoms are held together in solids by interatomic attractive forces or
bonds.
• Some repulsive force also exists to keep the solid
uncompressed/uncollapsed.
• The attractive forces between the constituent particles in solids are
basically electrostatic in nature. The origin of these forces is electronic
and nuclear charges.
• According to the strength and directionality, chemical bonds are
grouped into primary and secondary.
• Primary bonds are interatomic bonds, whereas secondary bonds are
intermolecular bonds.
• The attractive forces in primary bonds are directly associated with the
valence electrons.
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• Depending on the positions assumed by the bond electrons during
the formation of the bond, there are three strong principal types of
primary bonds: ionic, covalent and metallic.
• van der Waals and hydrogen bonds are typical examples of
secondary bonds and they result from intermolecular attraction.
• An ion is an atom or group of atoms that has an electrical charge,
either positive or negative.(+ -)
• Bonding in solids
Ionic bond
Covalent bond
Metallic bond
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2.2. Ionic Bonding
• Electrons are transferred between valence shells of atoms
• Ionic compounds are made of ions
• ionic compounds are called Salts or Crystals
• Always formed between metals and non-metals
• Eg. Alkali halids: NaCl, LiF, etc.
• Ions are closed electronic shells.
• [ e.g. NaCl, Na (Is2 2s2 2p6 3s1) become Na+ (Is2 2s2 2p6)
• And the Cl (Is2 2s2 2p6 3s2 3p5) become Cl- (Is2 2s2 2p63s2 3p6 ).
• Example - Na+Cl
• Strong Coulomb interaction between a positive atom (lost an
electron, Na+) and a negative atom (an extra electron, Cl-)
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• Anion = negatively charged atom
• Cation = positively charged atom
• Ions are attracted by strong coulombic interaction
• Oppositely charged atoms attract
• An ionic bond is non-directional
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Properties of Ionic Compounds
Hard solid @ 22oC
High boiling point hard solids
Have high melting point because ionic bonds are relatively strong.
Nonconductors of electricity in solid phase
Good conductors in liquid phase or dissolved in water
Are brittle and tend to cleave rather than deform because bonds
are strong.
Form crystalline structures
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2.3. Covalent Bonding
• A covalent bond involves the mutual sharing of a pair of electrons
between a pair of atoms.
• The spins of the two electrons are oriented in opposite directions.
• In solid state, the most stable covalent bonds are formed between
non-metallic atoms like those of N, O, C, F and Cl.
• Some of the other elements that are well known to form covalent-
bonded crystals include Si, Ge, As and Se
• Between nonmetallic elements of similar electronegativity.
• Formed by sharing electron pairs
• Stable non-ionizing particles, they are not conductors at any state
• Examples; Cl2,O2, CO2, C2H6, H2O, SiC
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• The two non-metal atoms cannot form a bond by transferring
electrons from one to another. Instead, they share electrons.
• How is a covalent bond formed in hydrogen chloride (HCl, also
represented as H–Cl)?
• Hydrogen and chlorine both need one more electron to fill outer
shells. By sharing one electron each, they both have a stable outer
shell and a covalent bond is formed.
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• How do carbon and oxygen atoms form covalent bonds in a
molecule of carbon dioxide?(C of Z = 6 and O of Z = 8)
Properties of Covalent Bonds
Low melting point temp and boiling point temps
Relatively soft solids as compared to ionic compounds
Nonconductors of electricity in any phase
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2.4. Metallic Bonding
• Metallic bonding
– Occurs between like atoms of a metal in the free state
– Valence e- are mobile (move freely among all metal atoms)
– It’s the mobile electrons that enable metals to conduct electricity
– Positive ions in a sea of electrons
• It occurs in metals and alloys
• In metals the outer valence electrons of each atom form part of a collective
free-electron cloud or gas that floods the entire lattice.
• Individual electron-electron interaction is repulsive. However there is
sufficient electrostatic attraction between the free-electron gas and the
positive ion core (inner electrons and nucleus) to cause bond.
• Since bonding electrons are not localized between atoms, metals have non-
directional bonds.
• Bond found in metals; holds metal atoms together very strongly
• Formed between atoms of metallic elements
• Electron cloud around atoms
• Examples; Na, Fe, Al, Au, Co
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Properties of Metallic Bonding
• Metallic characteristics
– Good conductors at all states, lustrous, very high melting points
– High mp temps, ductile, malleable, shiny
– Hard substances
– Good conductors of heat and electricity
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2.5. PROPERTIES OF METALLIC CRYSTALS
(i). Bonding energies and melting temperatures are found very wide in these
bonding e.g., bonding may be weaker or stronger, energies range from 64 x
103 kJ/kmol (0.7 eV/atom) for mercury to 850 x 103 kJ/kmol (8.8 eV/atom)
for tungsten. Their respective melting points are -39 and 3410°C.
(ii). Due to the symmetrical arrangements of the positive ions in a space
lattice, metals are crystalline.
(iii). Metallic bonds being weak, metals have a melting point moderate to
high, i.e., the melting points of metallic crystals are lower than those of the
electrovalent crystals.
(iv). Since a large number of free electrons are available, metallic crystals
have high electrical conductivity.
(v). Metallic crystals have higher thermal conductivity because of the
availability of large number of free electrons which act as carriers of heat.
(vi). Metals are opaque to light since the light energy is absorbed by free
electrons. Copper, sodium, silver, aluminium, etc. are examples of metallic
crystals.
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2.6. Cohesive energy
• The energy required to disassemble it into its constituent parts (ions,
atoms, or molecules), i.e., its binding energy
• The attractive and repulsive forces two atoms exerted on each other
can be defined as
…….. ………….(1)
• A, B, M and N are constants. M is usually 2 as per Coulomb's law
and the value of N is usually 7 to 10.
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• The interaction energy is obtained by integrating equation (1)
------------------(3)
--------(4)
-----------------------------(5)
Where
Figure 12/14/2022
3.4 Variation of cohesive forces and potential (cohesive) energy 13
• The particle form stable lattice if the energy U(r) is minimum which
occurs at equilibrium distance ro
--------------------------------------------(6)
and,
dU am bn
dr m 1
n 1
0 ---------------------------------------------------------(7)
r ro ro ro
Thus,
--------------------------------------------------------------(8)
Thus substituting this in equation (6), we get
12/14/2022 ---------------------------------------------------------------(9) 14
Calculation of Lattice Energy of Ionic crystals
• The cohesive energy of an ionic crystal is the energy that would be
liberated by the formation of the crystal from individual neutral atoms.
• The lattice energy is that energy evolved when a crystal is formed
from individual ions, rather than from individual atoms(about 5-10 eV
per molecule).
Figure12/14/2022
3.5 Schematic representation of lattice energy of ionic crystals as a function of interatomic spacing15
For two ions of charges Z1e and Z2e separated by a distance r, the attractive energy is and therefore
this energy is if both the atoms are respectively monovalent, divalent and
trivalent. For the whole crystal the Coulomb potential energy may be written as . This term
represents the net Coulomb potential energy of anyone ion due to the presence of all other similar and dissimilar
ions present in the crystal. The minus sign shows that the net Coulomb energy is attractive. The constant A is
known as Madelung constant.
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the total energy of one ion due to the presence of all others is given by
----------------------------------------(10)
For the univalent alkali halides (Z1=Z2=1)
---------------------------------(11)
The total energy per kmol of the crystal is
---------------------------------(12)
The potential energy will be minimum at the equilibrium spacing r0 .
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------------------------------------------(13)
Or
-------------------------------(14)
Substituting this value of B in equation (12) and putting r = r0 we get the total equilibrium energy per kmol of the
crystal.
Thus,
--------------------------------(15)
Where U1 and U2 represent net Coulomb attraction energy and repulsive potential
Thus,
-------(16)
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CALCULATION OF MADELUNG CONSTANT OF IONIC CRYSTALS
Madelung constant represents the effect of a specific geometrical array of ions on the electrostatic potential
energy. It is a property of the crystal structure and depends on the lattice parameters, anion-cation distances, or
molecular volume of the crystal. The calculation of Madelung constant in a linear chain of ions of alternate signs,
as in Fig. 3.6, is explained as follow.
Figure3.6. A hypothetical one-dimensional NaCl lattice.
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Let us pick up a positive ion for reference, and let r0 be the shortest distance between adjacent ions. This ion has
two negative ions as its neighbours on either side at a distance r0 . Now the attractive Coulomb energy due to the
nearest neighbours is
------------------------------------------(17)
Similarly the repulsive energy due to the two positive ions at a distance of 2r0 is and the attractive
Coulomb energy due to the two next neighbours at a distance 3r0 is and so on. Thus the total
energy due to all the ions in the linear array is
--------(18)
Thus (21og2 ) is the Madelung constant per molecule of the ionic solid. Hence (2NA1og2 ) is the Madelung
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constant per kmol of the ionic solid.
