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Lattice energy

## Summary

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The lattice energy is the energy change on formation of one mole of an ionic compound from its constituent ions in the gaseous state. It is a measure of the cohesive forces that bind ions. Lattice energy is relevant to many practical properties including solubility, hardness, and volatility. The lattice energy is usually deduced from the Born–Haber cycle.[1]

## Lattice energy and lattice enthalpy

The formation of a crystal lattice is exothermic, i.e., the value of ΔHlattice is negative because it corresponds to the coalescing of infinitely separated gaseous ions in vacuum to form the ionic lattice.

Sodium chloride crystal lattice

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, lattice energy is the energy released by the reaction

Na+ (g) + Cl (g) → NaCl (s)

which would amount to -786 kJ/mol.[2]

The relationship between the molar lattice energy and the molar lattice enthalpy is given by the following equation:

${\displaystyle \Delta U=\Delta H-p\Delta V_{m}}$,

where ${\displaystyle \Delta U}$ is the molar lattice energy, ${\displaystyle \Delta H}$ the molar lattice enthalpy and ${\displaystyle \Delta V_{m}}$ the change of the volume per mole. Therefore, the lattice enthalpy further takes into account that work has to be performed against an outer pressure ${\displaystyle p}$.

Some textbooks [3] and the commonly used CRC Handbook of Chemistry and Physics[4] define lattice energy (and enthalpy) with the opposite sign, i.e. as the energy required to convert the crystal into infinitely separated gaseous ions 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.

## Theoretical treatments

The lattice energy of an ionic compound depends upon charges of the ions that comprise the solid. More subtly, the relative and absolute sizes of the ions influence ΔHlattice.

### Born–Landé equation

In 1918[5] 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.[2]

${\displaystyle E=-{\frac {N_{A}Mz^{+}z^{-}e^{2}}{4\pi \varepsilon _{0}r_{0}}}\left(1-{\frac {1}{n}}\right),}$

where

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;
e is the elementary charge, equal to 1.6022×10−19 C;
ε0 is the permittivity of free space, equal to 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.[6]

The Born–Landé equation shows that the lattice energy of a compound depends on a number of factors

• as the charges on the ions increase the lattice energy increases (becomes more negative),
• when ions are closer together the lattice energy increases (becomes more negative)

Barium oxide (BaO), 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 (NaCl) 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.[2]

### Effect of polarisation

For ionic compounds with ions occupying lattice sites with crystallographic point groups C1, C1h, Cn or Cnv (n = 2, 3, 4 or 6) the concept of the lattice energy and the Born–Haber cycle has to be extended.[7] 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+ (g) + 2 pol S (g) → FeS2 (s)

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.[8]

## Representative lattice energies

The following table presents a list of lattice energies for some common compounds as well as their structure type.

Compound Experimental Lattice Energy[1] Structure type Comment
LiF −1030 kJ/mol NaCl difference vs. sodium chloride due to greater charge/radius for both cation and anion
NaCl −786 kJ/mol NaCl reference compound for NaCl lattice
NaBr −747 kJ/mol NaCl weaker lattice vs. NaCl
NaI −704 kJ/mol NaCl weaker lattice vs. NaBr, soluble in acetone
CsCl −657 kJ/mol CsCl reference compound for CsCl lattice
CsBr −632 kJ/mol CsCl trend vs CsCl like NaCl vs. NaBr
CsI −600 kJ/mol CsCl trend vs CsCl like NaCl vs. NaI
MgO −3795 kJ/mol NaCl M2+O2- materials have high lattice energies vs. M+O. MgO is insoluble in all solvents
CaO −3414 kJ/mol NaCl M2+O2- materials have high lattice energies vs. M+O. CaO is insoluble in all solvents
SrO −3217 kJ/mol NaCl M2+O2- materials have high lattice energies vs. M+O. SrO is insoluble in all solvents
MgF2 −2922 kJ/mol rutile contrast with Mg2+O2-
TiO2 −12150 kJ/mol rutile TiO2 (rutile) and some other M4+(O2-)2 compounds are refractory materials