# Lattice energy

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In chemistry, the lattice energy is the energy change upon formation of one mole of a crystalline ionic compound from its constituent ions, which are assumed to initially be in the gaseous state. It is a measure of the cohesive forces that bind ionic solids. The size of the lattice energy is connected to many other physical properties including solubility, hardness, and volatility. Since it generally cannot be measured directly, the lattice energy is usually deduced from experimental data via the Born–Haber cycle. [1]

## Lattice energy and lattice enthalpy

The concept of lattice energy was originally applied to the formation of compounds with structures like rocksalt (NaCl) and sphalerite (ZnS) where the ions occupy high-symmetry crystal lattice sites. In the case of NaCl, lattice energy is the energy change of the reaction

Na+ (g) + Cl (g) NaCl (s)

which amounts to −786 kJ/mol. [2]

Some chemistry textbooks [3] as well as the widely used CRC Handbook of Chemistry and Physics [4] define lattice energy 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. Both sign conventions are widely used.

The relationship between the lattice energy and the lattice enthalpy at pressure ${\displaystyle P}$ is given by the following equation:

${\displaystyle \Delta U_{lattice}=\Delta H_{lattice}-P\Delta V_{m}}$,

where ${\displaystyle \Delta U_{lattice}}$ is the lattice energy (i.e., the molar internal energy change), ${\displaystyle \Delta H_{lattice}}$ is the lattice enthalpy, and ${\displaystyle \Delta V_{m}}$ the change of molar volume due to the formation of the lattice. Since the molar volume of the solid is much smaller than that of the gases, ${\displaystyle \Delta V_{m}<0}$. The formation of a crystal lattice from ions in vacuum must lower the internal energy due to the net attractive forces involved, and so ${\displaystyle \Delta U_{lattice}<0}$. The ${\displaystyle -P\Delta V_{m}}$ term is positive but is relatively small at low pressures, and so the value of the lattice enthalpy is also negative (and exothermic).

## Theoretical treatments

The lattice energy of an ionic compound depends strongly upon the charges of the ions that comprise the solid, which must attract or repel one another via Coulomb's Law. More subtly, the relative and absolute sizes of the ions influence ${\displaystyle \Delta H_{lattice}}$. London dispersion forces also exist between ions and contribute to the lattice energy via polarization effects. For ionic compounds made of molecular cations and/or anions, there may also be ion-dipole and dipole-dipole interactions if either molecule has a molecular dipole moment. The theoretical treatments described below are focused on compounds made of atomic cations and anions, and neglect contributions to the internal energy of the lattice from thermalized lattice vibrations.

### 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]

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

where

M is the Madelung constant, relating to the geometry of the crystal;
z+ is the charge number of the cation;
z is the charge number of the 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 nearest-neighbor distance between ions; 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 above shows that the lattice energy of a compound depends principally on two 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. The bond radii are similar but the charge numbers are not, with BaO having charge numbers of (+2,−2) and NaCl having (+1,−1); the Born–Landé equation predicts that the difference in charge numbers is the principal reason for the large difference in lattice energies.

Closely related to this widely used formula is the Kapustinskii equation, which can be used as a simpler way of estimating lattice energies where high precision is not required. [2]

### Effect of polarization

For certain ionic compounds, the calculation of the lattice energy requires the explicit inclusion of polarization effects. [7] In these cases the polarization energy Epol associated with ions on polar lattice sites may be included in the Born–Haber cycle. As an example, one may consider the case of iron-pyrite FeS2. It has been shown that neglect of polarization led to a 15% difference between theory and experiment in the case of FeS2, whereas including it reduced the error to 2%. [8]

## Representative lattice energies

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

CompoundExperimental Lattice Energy [1] Structure typeComment
LiF−1030 kJ/molNaCldifference vs. sodium chloride due to greater charge/radius for both cation and anion
NaCl−786 kJ/molNaClreference compound for NaCl lattice
NaBr−747 kJ/molNaClweaker lattice vs. NaCl
NaI−704 kJ/molNaClweaker lattice vs. NaBr, soluble in acetone
CsCl−657 kJ/molCsClreference compound for CsCl lattice
CsBr−632 kJ/molCsCltrend vs CsCl like NaCl vs. NaBr
CsI−600 kJ/molCsCltrend vs CsCl like NaCl vs. NaI
MgO−3795 kJ/molNaClM2+O2- materials have high lattice energies vs. M+O. MgO is insoluble in all solvents
CaO−3414 kJ/molNaClM2+O2- materials have high lattice energies vs. M+O. CaO is insoluble in all solvents
SrO−3217 kJ/molNaClM2+O2- materials have high lattice energies vs. M+O. SrO is insoluble in all solvents
MgF2−2922 kJ/molrutilecontrast with Mg2+O2-
TiO2−12150 kJ/molrutileTiO2 (rutile) and some other M4+(O2-)2 compounds are refractory materials

## Related Research Articles

Ionic bonding is a type of chemical bonding that involves the electrostatic attraction between oppositely charged ions, or between two atoms with sharply different electronegativities, and is the primary interaction occurring in ionic compounds. It is one of the main types of bonding, along with covalent bonding and metallic bonding. Ions are atoms with an electrostatic charge. Atoms that gain electrons make negatively charged ions. Atoms that lose electrons make positively charged ions. This transfer of electrons is known as electrovalence in contrast to covalence. In the simplest case, the cation is a metal atom and the anion is a nonmetal atom, but these ions can be of a more complex nature, e.g. molecular ions like NH+
4
or SO2−
4
. In simpler words, an ionic bond results from the transfer of electrons from a metal to a non-metal in order to obtain a full valence shell for both atoms.

An intermolecular force (IMF) is the force that mediates interaction between molecules, including the electromagnetic forces of attraction or repulsion which act between atoms and other types of neighbouring particles, e.g. atoms or ions. Intermolecular forces are weak relative to intramolecular forces – the forces which hold a molecule together. For example, the covalent bond, involving sharing electron pairs between atoms, is much stronger than the forces present between neighboring molecules. Both sets of forces are essential parts of force fields frequently used in molecular mechanics.

Ferroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are also piezoelectric and pyroelectric, with the additional property that their natural electrical polarization is reversible. The term is used in analogy to ferromagnetism, in which a material exhibits a permanent magnetic moment. Ferromagnetism was already known when ferroelectricity was discovered in 1920 in Rochelle salt by Joseph Valasek. Thus, the prefix ferro, meaning iron, was used to describe the property despite the fact that most ferroelectric materials do not contain iron. Materials that are both ferroelectric and ferromagnetic are known as multiferroics.

In chemistry, solubility is the ability of a substance, the solute, to form a solution with another substance, the solvent. Insolubility is the opposite property, the inability of the solute to form such a solution.

In chemistry, an ionic compound is a chemical compound composed of ions held together by electrostatic forces termed ionic bonding. The compound is neutral overall, but consists of positively charged ions called cations and negatively charged ions called anions. These can be simple ions such as the sodium (Na+) and chloride (Cl) in sodium chloride, or polyatomic species such as the ammonium (NH+
4
) and carbonate (CO2−
3
) ions in ammonium carbonate. Individual ions within an ionic compound usually have multiple nearest neighbours, so are not considered to be part of molecules, but instead part of a continuous three-dimensional network. Ionic compounds usually form crystalline structures when solid.

The standard enthalpy of reaction for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. This can in turn be used to predict the total chemical bond energy liberated or bound during reaction, as long as the enthalpy of mixing is also accounted for.

The Born–Haber cycle is an approach to analyze reaction energies. It was named after the two German scientists Max Born and Fritz Haber, who developed it in 1919. It was also independently formulated by Kasimir Fajans and published concurrently in the same issue of the same journal. The cycle is concerned with the formation of an ionic compound from the reaction of a metal with a halogen or other non-metallic element such as oxygen.

In chemistry, bond energy (BE), also called the mean bond enthalpy or average bond enthalpy is a measure of bond strength in a chemical bond. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy for all bonds of the same type within the same chemical species.

The Madelung constant is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges. It is named after Erwin Madelung, a German physicist.

Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. Although neither atoms nor ions have sharp boundaries, they are treated as if they were hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice. Ionic radii are typically given in units of either picometers (pm) or angstroms (Å), with 1 Å = 100 pm. Typical values range from 31 pm (0.3 Å) to over 200 pm (2 Å).

In chemistry, a non-covalent interaction differs from a covalent bond in that it does not involve the sharing of electrons, but rather involves more dispersed variations of electromagnetic interactions between molecules or within a molecule. The chemical energy released in the formation of non-covalent interactions is typically on the order of 1–5 kcal/mol. Non-covalent interactions can be classified into different categories, such as electrostatic, π-effects, van der Waals forces, and hydrophobic effects.

A Schottky defect is an excitation of the site occupations in a crystal lattice leading to point defects named after Walter H. Schottky. In ionic crystals, this defect forms when oppositely charged ions leave their lattice sites and become incorporated for instance at the surface, creating oppositely charged vacancies. These vacancies are formed in stoichiometric units, to maintain an overall neutral charge in the ionic solid.

Kröger–Vink notation is a set of conventions that are used to describe electric charges and lattice positions of point defect species in crystals. It is primarily used for ionic crystals and is particularly useful for describing various defect reactions. It was proposed by F. A. Kröger and H. J. Vink.

The Kapustinskii equation calculates the lattice energy UL for an ionic crystal, which is experimentally difficult to determine. It is named after Anatoli Fedorovich Kapustinskii who published the formula in 1956.

In thermodynamics, the enthalpy of mixing is the enthalpy liberated or absorbed from a substance upon mixing. When a substance or compound is combined with any other substance or compound, the enthalpy of mixing is the consequence of the new interactions between the two substances or compounds. This enthalpy, if released exothermically, can in an extreme case cause an explosion.

An ion is an atom or molecule with a net electrical charge.

The Born–Landé equation is a means of calculating the lattice energy of a crystalline ionic compound. In 1918 Max Born and Alfred Landé proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.

In chemistry, ion association is a chemical reaction whereby ions of opposite electric charge come together in solution to form a distinct chemical entity. Ion associates are classified, according to the number of ions that associate with each other, as ion pairs, ion triplets, etc. Ion pairs are also classified according to the nature of the interaction as contact, solvent-shared or solvent-separated. The most important factor to determine the extent of ion association is the dielectric constant of the solvent. Ion associates have been characterized by means of vibrational spectroscopy, as introduced by Niels Bjerrum, and dielectric-loss spectroscopy.

A metal ion in aqueous solution or aqua ion is a cation, dissolved in water, of chemical formula [M(H2O)n]z+. The solvation number, n, determined by a variety of experimental methods is 4 for Li+ and Be2+ and 6 for most elements in periods 3 and 4 of the periodic table. Lanthanide and actinide aqua ions have higher solvation numbers (often 8 to 9), with the highest known being 11 for Ac3+. The strength of the bonds between the metal ion and water molecules in the primary solvation shell increases with the electrical charge, z, on the metal ion and decreases as its ionic radius, r, increases. Aqua ions are subject to hydrolysis. The logarithm of the first hydrolysis constant is proportional to z2/r for most aqua ions.

The Born–Mayer equation is an equation that is used to calculate the lattice energy of a crystalline ionic compound. It is a refinement of the Born–Landé equation by using an improved repulsion term.

## 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  . doi:10.1007/BF01313054. S2CID   122527743.
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. S2CID   250815717.