Lattice energy

The lattice energy of a crystalline solid is a measure of the energy released when ions are combined to make a compound. 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]

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	M^2+O^2- materials have high lattice energies vs. M^+O^−. MgO is insoluble in all solvents
CaO	−3414 kJ/mol	NaCl	M^2+O^2- materials have high lattice energies vs. M^+O^−. CaO is insoluble in all solvents
SrO	−3217 kJ/mol	NaCl	M^2+O^2- materials have high lattice energies vs. M^+O^−. SrO is insoluble in all solvents
MgF2	−2922 kJ/mol	rutile	contrast with Mg^2+O^2-
TiO2	−12150 kJ/mol	rutile	TiO2 (rutile) and some other M^4+(O^2-)2 compounds are refractory materials

Lattice energy and lattice enthalpy

The formation of a crystal lattice is exothermic, i.e., the value of ΔH_lattice 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 ΔH_lattice.

Born–Landé equation

Main article: 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]

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

where

N_A 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;

q_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, C_1h, C_n or C_nv (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 E_pol 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]

See also

• Bond energy
• Born–Haber cycle
• Chemical bond
• Madelung constant
• Ionic conductivity
• Enthalpy of melting
• Enthalpy change of solution
• Heat of dilution

Notes

References

1. Atkins; et al. (2010). Shriver and Atkins' Inorganic Chemistry (Fifth ed.). New York: W. H. Freeman and Company. ISBN 978-1-4292-1820-7.
2. David Arthur Johnson, Metals and Chemical Change, Open University, Royal Society of Chemistry, 2002,ISBN 0-85404-665-8
3. Zumdahl, Steven S. (1997). Chemistry (4th ed.). Boston: Houghton Mifflin. pp. 357–358. ISBN 978-0-669-41794-4.
4. Haynes, William M.; Lide, David R.; Bruno, Thomas J. (2017). CRC handbook of chemistry and physics : a ready-reference book of chemical and physical data. Boca Raton, FL: CRC Press, Taylor & Francis Group. pp. 12–22 to 12–34. ISBN 9781498754293.
5. I.D. Brown, The chemical Bond in Inorganic Chemistry, IUCr monographs in crystallography, Oxford University Press, 2002, ISBN 0-19-850870-0
6. Cotton, F. Albert; Wilkinson, Geoffrey; (1966). Advanced Inorganic Chemistry (2d Edn.) New York:Wiley-Interscience.
7. M. Birkholz (1995). "Crystal-field induced dipoles in heteropolar crystals – I. concept". Z. Phys. B. 96 (3): 325–332. Bibcode:1995ZPhyB..96..325B. CiteSeerX 10.1.1.424.5632. doi:10.1007/BF01313054.
8. M. Birkholz (1992). "The crystal energy of pyrite". J. Phys.: Condens. Matter. 4 (29): 6227–6240. Bibcode:1992JPCM....4.6227B. doi:10.1088/0953-8984/4/29/007.