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+ + Cl− → NaCl
which would amount to -786 kJ/mol. Some older textbooks define lattice
energy with the opposite sign, i.e. the energy required to convert the crystal
into infinitely separated gaseous ions, atoms, or molecules in vacuum, an
endothermic process. Following this convention, 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 Born–Haber cycle.
The relationship between the molar lattice energy and the molar lattice
enthalpy is given by the following equation:
, where is the molar lattice energy, the molar lattice enthalpy and 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 .
Theoretical treatments = Born–Landé equation=
In 1918 Born and Landé proposed that the lattice energy could be derived from the
electric potential of the ionic lattice and a repulsive potential energy term.
where NA is the Avogadro constant;
M is the Madelung constant, relating to the geometry of the crystal;
z+ is the charge number of cation; z− is the charge number of anion;
qe is the elementary charge, equal to 6981160220000000000♠1.6022×10−19 C;
ε0 is the permittivity of free space, equal to 6988885400000000000♠8.854×10−12
C2 J−1 m−1; r0 is the distance to closest ion; and
n is the Born exponent, a number between 5 and 12, determined experimentally by
measuring the compressibility of the solid, or derived theoretically.
The Born–Landé equation gives a reasonable fit to the lattice energy.
From the Born–Landé equation it can be seen that the lattice energy of a
compound is dependent on a number of factors
as the charges on the ions increase the lattice energy increases,
when ions are closer together the lattice energy increases
Barium oxide, for instance, which has the NaCl structure and therefore the
same Madelung constant, has a bond radius of 275 picometers and a lattice
energy of -3054 kJ/mol, while sodium chloride has a bond radius of 283
picometers and a lattice energy of -786 kJ/mol.
= Kapustinskii equation= The Kapustinskii equation can be used as
a simpler way of deriving lattice energies where high precision is not
required. = Effect of polarisation=
For ionic compounds with ions occupying lattice sites with crystallographic
point groups C1, C1h, Cn or Cnv the concept of the lattice energy and the
Born–Haber cycle has to be extended. In these cases the polarization energy Epol
associated with ions on polar lattice sites has to be included in the
Born–Haber cycle and the solid formation reaction has to start from the already
polarized species. As an example, one may consider the case of iron-pyrite
FeS2, where sulfur ions occupy lattice site of point symmetry group C3. The
lattice energy defining reaction then reads
Fe2+ + 2 pol S− → FeS2 where pol S− stands for the polarized,
gaseous sulfur ion. It has been shown that the neglection of the effect led to
15% difference between theoretical and experimental thermodynamic cycle energy
of FeS2 that reduced to only 2%, when the sulfur polarization effects were
included. Notes for the Term
1. It is the gaseous ions that combine 2. The lattice energy is always
exothermic: the value of ΔH is always negative, because the definition
specifies the bonding together of ions, not the separation of ions.
See also Bond energy
Born–Haber cycle Chemical bond
Madelung constant Ionic conductivity
References