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Lattice Energy
Discussion Questions

How is lattice energy estimated using BornHaber cycle?

How is lattice energy related to crystal structure?
Lattice Energy
The Lattice energy, U, is the amount of energy requried to
separate a mole of the solid (s) into a gas (g) of its ions.
M_{a}L_{b}(s) ® a M^{b+}(g) +
b X^{a} (g) U kJ/mol
This quantity cannot be experimentally determined directly, but it can be
estimated using Hess Law in the form of BornHaber cycle. It can also be
calculated from the electrostatic consideration of its crystal structure.
As defined, the lattice energy is positive, because energy is always required
to separate the ions. For the reverse process, the energy released is
called energy of crystallization, E_{cryst}.
a M^{b+}(g) + b X^{a} (g) ® M_{a}L_{b}(s)
E_{cryst} kJ/mol
Therefore, U =  E_{cryst}
Values of lattice energies for various solids have been given in literature,
especially for some common solids. Some are given here.
Comparison of Lattice Energies (U in kJ/mol) of Some Salts
Solid  U  Solid  U  Solid  U
 Solid  U


LiF  1036  LiCl  853  LiBr  807  LiI  757

NaF  923  NaCl  786  NaBr  747  NaI  704

KF  821  KCl  715  KBr  682  KI  649



MgF_{2}  2957  MgCl_{2}  2526
 MgBr_{2}  2440  MgI_{2}  2327

The following trends are obvious at a glance of the data above:

As the ionic radii of either the cation or anion increase, the lattice
energies decrease.

The solids consists of divalent ions have much larger lattice energies than
solids with monovalent ions.
How is lattice energy estimated using BornHaber cycle?
Estimating lattice energy using the BornHaber cycle has been discussed in
Ionic Solids.
For a quick review, the following is an example that illustrate the estimate
of the energy of crystallization of NaCl.
H_{sub} of Na = 108 kJ/mol (Heat of sublimation)
D of Cl_{2} = 244 (Bond dissociation energy)
IP of Na(g) = 496 (Ionization potential or energy)
EA of Cl(g) = 349 (Electron affinity of Cl)
H_{f} of NaCl = 411 (Enthalpy of formation)
The BornHaber cycle to evaluate E_{lattice} is shown below:
Na^{+} + Cl(g)

 349
496+244/2 ¯
 Na^{+}(g) + Cl^{}(g)
 
Na(g) + 0.5Cl_{2}(g) 

108 
 E_{cryst}= 788
Na(s) + 0.5Cl_{2}(l) 
 
411 
¯ ¯
 NaCl(s) 
E_{cryst} = 411(108+496+244/2)(349) kJ/mol
= 788 kJ/mol.
Discussion
The value calculated for U depends on the data used.
Data from various sources differ slightly, and so is the result.
The lattice energies for NaCl most often quoted in other texts is
about 765 kJ/mol.
Compare with the method shown below
Na(s) + 0.5 Cl_{2}(l) ® NaCl(s)
  411
 H_{f}

Na(g) ® Na(s)
  108
 H_{sub}

Na^{+}(g) + e ® Na(g)
  496
 IP

Cl(g) ® 0.5 Cl_{2}(g)
  0.5 * 244
 0.5*D

Cl^{}(g) ® Cl(g) + 2 e
 349
 EA

Add all the above equations leading to

Na^{+}(g) + Cl^{}(g) ® NaCl(s)
 788 kJ/mol = E_{cryst}

How is lattice energy related to crystal structure?
There are many other factors to be considered such as covalent character
and electronelectron interactions in ionic solids. But for simplicity,
let us consider the ionic solids as a collection of positive and negative ions.
In this simple view, appropriate number of cations and anions come together
to form a solid. The positive ions experience both attraction and repulson
from ions of opposit charge and ions of the same charge.
As an example, let us consider the the NaCl crystal. In the following
discussion, assume r be the distance between Na^{+} and
Cl^{} ions. The nearest neighbors of Na^{+} are 6
Cl^{} ions at a distance
Ö1r, 12 Na^{+} ions at
a distance Ö2r, 8 Cl^{}
at Ö3r, 6 Na^{+}
at Ö4r, 24 Na^{+}
at Ö5r, and so on.
Thus, the energy due to one ion is
z^{2}e^{2} 6 12 8 6 24
E =   [    +    +  ...]
4pe_{o}r Ö1 Ö2 Ö3 Ö4 Ö5
where z is the number of charges of the ions, (=1 for NaCl);
e is the charge of an electron (= 1.6022x10^{19} C);
4pe_{o} = 1.11265x10^{10} C^{2}/(J m)
and the series in the [ ] is called the Madelung constant, M.
The above discussion is valid only for the sodium chloride (also called
rock salt) structure type. This is a geometrical factor, depending on
the arrangement of ions in the solid. The Madelung constant depends on
the structure type, and its values for several structural types
are given below:
Solid  M  A : C  Type


NaCl  1.747558  6 : 6  Rock salt

CsCl  1.747558  8 : 8  CsCl type

CaF_{2}  2.51939  8 : 4  Fluorite

TiO_{2}  2.408  6 : 3  Rutile

Al_{2}O_{3}  4.1719  6 : 4  Corundum

A is the number of anions coordinated to cation
and C is the numbers of cations coordinated to anion.
Madelung constants
for a few more types of crystal structures are available from the
Handbook Menu.
Madelung Energy discuss further the lattice energy of ionic crystals.
There are other factors to consider for the evaluation of energy of
crystallization, and the treatment by M. Born led to the formula for
the evaluation of crystallization energy E_{cryst},
for a mole of crystalline solid:
N z^{2}e^{2} 1
E_{cryst} =   ( 1  )
4pe_{o}r n
where N is the Avogadro's number (=6.022x10^{23}), and n
is a number related to the electronic configurations of the ions involved.
The n values and the electronic configurations (e.c.) of the corresponding
inert gases are given below:
n =  5  7  9  10  12

e.c.  He  Ne  Ar  Kr  Xe

The following values of n have been suggested for some common solids:
n =  5.9  8.0  8.7  9.1  9.5

e.c.  LiF  LiCl  LiBr  NaCl  NaBr

Example 1
Estimate the energy of crystallization for NaCl.
Solution
Using the values giving in the discussion above, the estimation is given by
6.022x10^{23} /mol (1.6022^{19})^{2} * 1.747558
E_{cryst} =   ( 1  1/9.1)
4p * 8.854x10^{12} C^{2}/m * 282x10^{12} m
=  766376 J/mol
=  766 kJ/mol
Discussion
Much more should be considered in order to evaluate the lattice energy
accurately, but the above calculation leads you to a good start.
When methods to evaluate the energy of crystallization or lattice energy
lead to reliable values, these values can be used in the BornHable
cycle to evaluate other chemical properties, for example the electron
affinity, which is really difficult to determine directly by experiment.
Confidence Building Questions

Which one of the following has the largest lattice energy?
LiF, NaF, CaF_{2}, AlF_{3}

Which one of the following has the largest lattice energy?
LiCl, NaCl, CaCl_{2}, Al_{2}O_{3}
Discussion 
Corrundum Al_{2}O_{3} has some covalent character in
the solid as well as the higher charge of the ions.

Lime, CaO, is know to have the same structure as NaCl and the
edge length of the unit cell for CaO is 481 pm. Thus, CaO distance
is 241 pm. Evaluate the energy of crystallization,
E_{cryst} for CaO.
Skill 
Evaluate the lattice energy and know what values are needed.

Assume the interionic distance for NaCl_{2} to be the same as
those of NaCl (r = 282 pm), and assume the structure to be of the
fluorite type (M = 2.512). Evaluate the energy of crystallization,
E_{cryst}
Discussion 
This number has not been checked. If you get a different value, please let
me know.
©cchieh@uwaterloo.ca