Inorganic Chemistry
By
Dr. Khalil K. Abid
Lecture 10
Ionic Bonding
1
Crystal Lattices
When sodium and chlorine bond together to form sodium chloride, something very interesting
happens; a white crystal of sodium chloride forms. Look closely at that crystal:
Cl
NaCl
Ionic compound. Its atoms have exchanged electrons and now consist of oppositely charged
particles called ions. It is in these ions that the nature of a crystal is formed.
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The positive charge of the sodium ion is attracted to any charge that is negative. However, the
attraction is not limited to one particular ―place‖ on the ion. The positive charge emanates from the
ion in all directions. The chloride ion is doing the same thing but with its negative charge. The
negative charge is attracted to positive charges in all directions. When enough sodium and chloride
ions get together to form a crystal, the ions stack up alternatively into the cube-shaped arrangement
seen here:
If extrapolated out to the macroscopic world, we can now see
why salt crystals have the cubic shape we saw earlier. Of
course, all crystals are not cubic in shape. Depending on the
size of the ions and the ratio of the ions in the compound we
can get all sorts of different shapes. A few here are shown:
Alum, KAl(SO4)2
Rutile, TiO2
Aragonite, CaCO3
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The lattice energy of a crystalline solid is usually defined as the energy of formation of the crystal
from infinitely-separated ions, molecules, or atoms, and as such is invariably negative. The concept
of lattice energy was originally developed for rocksalt – structured and sphalerite – structured
compounds like NaCl and ZnS, where the ions occupy high-symmetry crystal lattice sites. In the
case of NaCl, the lattice energy is the energy released by the reaction
Na+ (g) + Cl− (g) → NaCl (s)
which would amount to -786 kJ/mol.
The lattice energy of NaCl would be +786 kJ/mol. The lattice energy for ionic crystals such as
sodium chloride, metals such as iron, or covalently linked materials such as diamond is
considerably greater in magnitude than for solids such as sugar or iodine, whose neutral molecules
interact only by weaker dipole – dipole or van der Waals forces. The precise value of the lattice
energy may not be determined experimentally, because of the impossibility of preparing an
adequate amount of gaseous ions or atoms and measuring the energy released during their
condensation to form the solid. However, the value of the lattice energy may either be derived
theoretically from electrostatics or from a thermodynamic cycling reaction, the so-called Born –
Haber cycle. The relationship between the molar lattice energy and the molar lattice enthalpy is
given by the following equation:
, where ΔGU is the molar lattice energy, ΔGH the molar lattice enthalpy and pΔVm the change of
the volume per mol. Therefore the lattice enthalpy further takes into account that work has to be
performed against an outer pressure .
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Ionic lattice energy is defined as the energy released when isolated gaseous ions come together
from infinite separation to form an ionic crystal. The model \lsed for calculating this energy is the
Born - Lande equation. Here we take the sodium chloride (6:6) crystal and show how the infinite
series giving the Madelung constant of the NaCl crystal lattice can be easily generated, relate it to the
more familiar series, discuss convergence and the alternative rapidly converging expressions
available for generating the constant. A method of approximation of the constant, based on simple
three dimensional geometry is also presented.
A large portion of the lattice energy (almost nine-tenth) is accounted for by electrostatic energy
arising from electrostatic interactions between the ions in the formed crystal. The Born – Lande
equation involves calculation of the electrostatic energy of a 'reference ion' resulting from
electrostatic interactions with an infinite number of ions surrounding it in the crystal lattice and the
Madelung constant plays a vital role in incorporating this infinite number of interactions into the
equation. The assumptions made are: ions are point charges, the crystal is infinitely large, the
distribution of ions is even and symmetric, so the environment around every ion in the crystal is the
same.
Therefore once the energy for the reference ion is determined, it is multiplied by the Avogadro
constant to obtain the electrostatic energy for one mole of ions. This forms the electrostatic
component of the Born – Lande equation.
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All stable ionic crystals have negative standard enthalpies of formation,ΔHof, and negative
standard free energies of formation, ΔGof.
Na(s) + ½ Cl2(g)
NaCl(s)
ΔHof = –410.9 kJ
ΔGof = –384.0 kJ
Cs(s) + ½ Cl2(g)
CsCl(s)
ΔHof = –442.8 kJ
ΔGof = –414.4 kJ
Mg(s) + ½O2(g)
MgO(s)
ΔHof = –385.2 kJ
ΔGof = –362.9 kJ
Ca(s) + C(s) + 3/2 O2(g)
CaCO3(s)
ΔHof = –1216.3 kJ ΔGof = –1137.6 kJ
– The exothermic and spontaneous formation of ionic solids can be understood in terms of a Hess's
Law cycle, called the Born-Haber cycle.
–The lattice energy is the most important factor in making the formation of ionic crystals exothermic
and spontaneous.
– Lattice energy, U, is defined as the enthalpy required to dissociate one mole of crystalline solid in
its standard state into the gaseous ions of which it is composed; e.g.,
NaCl(s)
Na+(g) + Cl–(g)
U = +786.8 kJ
Factors Favoring a More Stable Crystal Lattice
Large values of lattice energy, U, are favored by
1. Higher ionic charges
2. Smaller ions
3. Shorter distances between ions
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Selected Lattice Energies, Uo (kJ/mol) (Born-Haber Cycle Data)
F–
Cl–
Br–
I–
O2–
Li+
1049.0
862.0
818.6
762.7
2830
Na+
927.7
786.8
751.8
703
2650
K+
825.9
716.8
688.6
646.9
2250
Rb+
788.9
687.9
612
625
2170
Cs+
758.5
668.2
635
602
2090
Mg2+
2522
3795
Ca2+
2253
3414
Sr2+
2127
3217
All ionic compounds make a crystal shape. They are held together into a crystal lattice– an
arrangement of alternating positive and negative ions held together by opposite charges. Some
crystals are held together strongly, some are held together weakly. What are some determining
factors about the strength of these crystals? Here is a listing of various Lattice energies of different
ionic compounds. Lattice energy is essentially the strength of the crystal. The higher the lattice
energy, the stronger the crystal.
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Compound
Cation
Anion
Lattice Energy(kJ/mol)
NaF
Na+1
F-1
926
NaCl
Na+1
Cl-1
786
NaBr
Na+1
Br-1
752
NaI
Na+1
I-1
702
MgO
Mg+2
O-2
3800
CaO
Ca+2
O-2
3400
SrO
Sr+2
O-2
3200
BaO
Ba+2
O-2
3000
Al2O3
Al+3
O-2
15900
Ga2O3
Ga+3
O-2
15600
Notice the trend? In the first four examples, the lattice energies are only measured in the hundreds
of kJ/mol. In the second four examples, the lattice energies are measured in the thousands of kJ/mol.
What changed? The charges involved in the ionic compounds increased. In the first examples, the
ions are only charged +1 and –1. The second four are all charged +2 and –2; much stronger charges.
The result is that the lattice energy increased a great deal. Examine the last two examples.
Aluminum and gallium go to the +3 charge. When combined with oxygen suddenly the lattice energy
jumps to the tens of thousands of kJ/mol. As charge increases, so does the lattice energy of the
crystal; it becomes stronger.
Lattice energy α ionic charge
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Compound
Cation
Anion
Lattice Energy (kJ/mol)
NaF
Na+1
F-1
926
NaCl
Na+1
Cl-1
786
NaBr
Na+1
Br-1
752
NaI
Na+1
I-1
702
Is there another trend evident in that data, though? Examine just the first four. They are all +1 and –1
charges but their lattice energies steadily decrease. What is changing? If you examine the anion of
each compound you will notice that they are all in the same column of the periodic table, the halogens
of group VIIA.
We discussed last unit there are periodic trends moving down a column on the periodic table. The
ions get bigger as seen to the right. The fluoride ion has a size of 136 pm while the iodide is almost
twice as large at 216 pm. However, they all have the same charge. This means that the exact same —1
charge of the fluoride ion is packed closely together while the —1 charge of the iodide ion is diffuse
throughout a much larger space. This results in the fluoride ion being a much stronger attractor of
positive charges because of its smaller size. Thus we see that as the ions get bigger, the lattice
energies decrease.
Lattice energy α
1
ionic size
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To summarize, ionic compounds get stronger as the charge increases and the size decreases. The
strongest crystals are the ones with ions that are highly charged and small in size. To compare ionic
compounds, first compare the charges. If the charges are the same, then compare the sizes.
What are the effects of a strong or a weak lattice energy? Lattice energy can be a great predictor of
melting point. When a solid melts the crystal structure has to be broken down. Logically speaking,
the stronger the crystal, the more difficult it will be to tear it apart. Consequently, it should have a
higher melting point.
Melting Point and Lattice energy
Compound Cation
Anion
Lattice Energy (kJ/mol)
Melting Point (oC)
Anion Radius (pm)
NaF
Na+1
F-1
926
993
119
NaCl
Na+1
Cl-1
786
801
167
NaBr
Na+1
Br-1
752
755
182
NaI
Na+1
I-1
206
702
660
In this table we see that they all have the same charges (+1/—1) but the anions get bigger as you
go down the periodic table. As the radius increases, the lattice energy decreases. As the crystal gets
weaker, the melting point goes down.
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Anion
Lattice Energy (kJ/mol)
Melting Point (oC)
Compound
Cation
KCl
K+1
Cl-1
717
775
CaF2
Ca+2
F-1
2805
1402
CaO
Ca+2
O-2
3400
2572
In this table we see that the charges get successive larger. KCl has a (+1/—1) charge, both weak.
Thus it has a low lattice energy and melting point. CaF2 has a (+2/-1) charge, stronger for the cation.
This is reflected in the lattice energy and melting point. CaO (+2/—2) has the highest charges and
therefore the highest lattice energy and melting point.
Keep in mind, though, that these trends are not perfect. Just like the periodic law there are any
number of exceptions and things that don’t fit. For instance, Al2O3 with a lattice energy of almost
16,000 kJ/mol should have an melting point larger than CaO at 2572 oC. It turns out that the melting
point of Al2O3 is only 2054 oC. Just because its not perfect doesn’t mean its not useful.
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Questions
1. Why do ionic compounds form crystal lattices when they are made?
2. What term is used to describe the strength of a crystal lattice?
3. How are lattice energy and ionic charge related?
4. How are lattice energy and ionic size related?
5. How is melting point related to lattice energy?
6. For each of the pairs below, circle the ionic compound that should have the stronger lattice energy:
A. CaCl2 vs. CaO
B. SrS vs. NaF
C. Al2O3 vs. AlCl3
D. Na3PO4 vs. K3PO4
E. LiF vs. NaF
F. K2O vs. Na2O
7. For each of the pairs below, decide which should have a higher melting point:
A. LiI vs. MgS
B. CaBr2 vs. BeBr2
C. Al2O3 vs. NaCl
D. CaSO4 vs. K2SO4
E. MgF2 vs. MgBr2
8. Arrange the following in order of increasing melting point (smallest to largest):
CaO, KCl, CaF2, BeO
9. Draw an example of a crystal lattice.
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What is lattice enthalpy?
Lattice enthalpy is a measure of the strength of the forces between the ions in an ionic solid. The
greater the lattice enthalpy, the stronger the forces. Those forces are only completely broken when the
ions are present as gaseous ions, scattered so far apart that there is negligible attraction between them.
You can show this on a simple enthalpy diagram.
For sodium chloride, the solid is more stable
than the gaseous ions by 787 kJ mol-1, and that
is a measure of the strength of the attractions
between the ions in the solid. Remember that
energy (in this case heat energy) is given out
when bonds are made, and is needed to break
bonds. So lattice enthalpy could be described
in either of two ways.
– You could describe it as the enthalpy change when 1 mole of sodium chloride (or whatever) was
formed from its scattered gaseous ions. In other words, you are looking at a downward arrow on the
diagram. In the sodium chloride case, that would be -787 kJ mol-1.
– Or, you could describe it as the enthalpy change when 1 mole of sodium chloride (or whatever) is
broken up to form its scattered gaseous ions. In other words, you are looking at an upward arrow on
the diagram. In the sodium chloride case, that would be +787 kJ mol-1.
Both refer to the same enthalpy diagram, but one looks at it from the point of view of making
the lattice, and the other from the point of view of breaking it up.
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You should talk about "lattice dissociation enthalpy" if you want to talk about the amount of
energy needed to split up a lattice into its scattered gaseous ions. For NaCl, the lattice dissociation
enthalpy is +787 kJ mol-1. You should talk about "lattice formation enthalpy" if you want to talk about
the amount of energy released when a lattice is formed from its scattered gaseous ions. For NaCl,
the lattice formation enthalpy is -787 kJ mol-1. To removes any possibility of confusion.
The lattice dissociation enthalpy is the enthalpy change needed to convert 1 mole of solid crystal
into its scattered gaseous ions. Lattice dissociation enthalpies are always positive.
The lattice formation enthalpy is the enthalpy change when 1 mole of solid crystal is formed from
its scattered gaseous ions. Lattice formation enthalpies are always negative.
Factors affecting lattice enthalpy
The two main factors affecting lattice enthalpy are the
charges on the ions and the ionic radii (which affects the
distance between the ions). The charges on the ions
Sodium chloride and magnesium oxide have exactly the same
arrangements of ions in the crystal lattice, but the lattice
enthalpies are very different.
You can see that the lattice enthalpy of magnesium oxide is
much greater than that of sodium chloride. That's because in
magnesium oxide, 2+ ions are attracting 2- ions; in sodium
chloride, the attraction is only between 1+ and 1- ions.
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The radius of the ions : The lattice enthalpy of magnesium oxide is also increased relative to sodium
chloride because magnesium ions are smaller than sodium ions, and oxide ions are smaller than
chloride ions. That means that the ions are closer together in the lattice, and that increases the
strength of the attractions. You can also see this effect of ion size on lattice enthalpy as you go down
a Group in the Periodic Table. For example, as you go down Group 7 of the Periodic Table from
fluorine to iodine, you would expect the lattice enthalpies of their sodium salts to fall as the negative
ions get bigger - and that is the case:
Attractions are governed by the distances between the centeres of the oppositely charged ions,
and that distance is obviously greater as the negative ion gets bigger. And you can see exactly the
same effect if as you go down Group 1. The next bar chart shows the lattice enthalpies of the Group 1
chlorides.
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Calculating lattice enthalpy
It is impossible to measure the enthalpy change starting from a solid crystal and converting it into
its scattered gaseous ions. It is even more difficult to imagine how you could do the reverse - start
with scattered gaseous ions and measure the enthalpy change when these convert to a solid crystal.
Instead, lattice enthalpies always have to be calculated, and there are two entirely different ways in
which this can be done. You can use a Hess's Law cycle (in this case called a Born-Haber cycle)
involving enthalpy changes which can be measured. Lattice enthalpies calculated in this way are
described as experimental values.
Or you can do physics-style calculations working out how much energy would be released, for
example, when ions considered as point charges come together to make a lattice. These are
described as theoretical values. In fact, in this case, what you are actually calculating are properly
described as lattice energies.
Standard atomization enthalpies: ΔH°a.
The standard atomization enthalpy is the enthalpy change when 1 mole of gaseous atoms is formed
from the element in its standard state. Enthalpy change of atomization is always positive.
All of the following equations represent changes involving atomization enthalpy:
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Calculating Lattice Energy
U In principle, the lattice energy for a crystal of known structure can be calculated by summing all the
attractive and repulsive contributions to the potential energy.
– For a pair of gaseous ions:
U=
where Z+, Z– = ionic charges,
ro = distance between ions
e = electronic charge = 1.602 × 10–19 C,
4πεo = vacuum permittivity = 1.11 × 10–10 C2.J–1.m–1
– Potential energy is negative for the attraction of oppositely charged ions and positive for repulsion
of like-charged ions.
– The potential energy arising from repulsions and attractions acting on one reference ion can be
calculated.
– Scaled up to a mole of ion pairs (and with a change of sign) this should equal the lattice energy of
the crystal.
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For a mole of ion pairs (N), using the Madelung constant (M), the expression for the potential energy
of an NaCl-type lattice due to Coulombic interactions is:
U=
– For the NaCl-type lattice M = 1.74756, and for NaCl ro = 280 pm. Using these values,
UNaCl = – 867 kJ/mol, which is too negative .
• Discrepancy arises from assuming ions are point charges.
• Electron clouds of adjacent ions repel each other as they approach one another.
– Born proposed that the repulsive (positive) contribution to the potential energy is given by
Urep =
where B is a constant specific to the ionic compound and n is a power in the range 6 - 12.
– Adding the Born repulsion correction to the Coloumbic term gives:
U=
+
Born-Landé Equation
Uo=
The value of n can be calculated from measurements of compressibility or estimated from theory.
– For NaCl, n = 9.1 from experiment, and the Born-Landé equation gives Uo = -771 kJ/mol.
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