lattice energy

File:Sodium-chloride-3D-ionic.png

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.David Arthur Johnson, Metals and Chemical Change, Open University, Royal Society of Chemistry, 2002,{{ISBN|0-85404-665-8}}

Some chemistry textbooks{{cite book|last=Zumdahl|first=Steven S.|title=Chemistry|year=1997|publisher=Houghton Mifflin|location=Boston|isbn=978-0-669-41794-4|pages=357–358|edition=4th}} as well as the widely used CRC Handbook of Chemistry and Physics{{Cite book|title=CRC handbook of chemistry and physics : a ready-reference book of chemical and physical data|last1=Haynes|first1=William M.|last2=Lide|first2=David R.|last3=Bruno|first3=Thomas J.|publisher=CRC Press, Taylor & Francis Group|year=2017|isbn=9781498754293|location=Boca Raton, FL|pages=12–22 to 12–34}} 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 $\backslash Delta\; U\_l$ and the lattice enthalpy $\backslash Delta\; H\_l$ at pressure $P$ is given by the following equation:

:$\backslash Delta\; U\_l=\backslash Delta\; H\_l\; -P\backslash Delta\; V\_m$,

where $\backslash Delta\; U\_l$ is the lattice energy (i.e., the molar internal energy change), $\backslash Delta\; H\_l$ is the lattice enthalpy, and $\backslash Delta\; V\_m$ 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, . The formation of a crystal lattice from ions in vacuum must lower the internal energy due to the net attractive forces involved, and so . The $-P\backslash Delta\; V\_m$ term is positive but is relatively small at low pressures, and so the value of the lattice enthalpy is also negative (and exothermic). Both, lattice energy and lattice enthalpy are identical at 0 K and the difference may be disregarded in practice at normal temperatures.{{Cite book |url=https://www.worldcat.org/title/430678988 |title=Shriver & Atkins' inorganic chemistry |date=2010 |publisher=Oxford University Press |isbn=978-0-19-923617-6 |editor-last=Atkins |editor-first=P. W. |edition=5th |location=Oxford ; New York |oclc=430678988}}

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 $\backslash Delta\; H\_l$. London dispersion forces also exist between ions and contribute to the lattice energy via polarization effects. For ionic compounds made up 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.

{{main|Born–Landé equation}}

In 1918{{Citation |last=Born |first=M. |title=Kristallgitter und Bohrsches Atommodell |date=1988 |work=Selected Scientific Papers of Alfred Landé |pages=65–72 |editor-last=Barut |editor-first=A. O. |url=https://link.springer.com/chapter/10.1007/978-94-009-3981-3_8 |access-date=2025-04-30 |place=Dordrecht |publisher=Springer Netherlands |language=de |doi=10.1007/978-94-009-3981-3_8 |isbn=978-94-009-3981-3 |last2=Landé |first2=A. |editor2-last=van der Merwe |editor2-first=A.|url-access=subscription }} Max Born and Alfred Landé proposed that the lattice energy could be derived from the electric potential of the ionic lattice and a repulsive potential energy term. This equation estimates the lattice energy based on electrostatic interactions and a repulsive term characterized by a power-law dependence (using a Born exponent, $n$). It was published building on earlier work by Born on ionic lattices.

:$\backslash Delta\; U\_l=\; -\backslash frac\{N\_AMz^+z^-\; e^2\; \}\{4\; \backslash pi\; \backslash varepsilon\_0\; r\_0\}\backslash left(1-\backslash frac\{1\}\{n\}\backslash right)$

where $N\_A$ is the Avogadro constant, $M$ is the Madelung constant, $z^+$/$z^-$ are the charge numbers of the cations and anions, $e$ is the elementary charge (1.6022{{e|−19}} C), $\backslash varepsilon\_0$ is the permittivity of free space ($4\; \backslash pi\; \backslash varepsilon\_0$ = 1.112{{e|−10}} C²/(J·m)), $r\_0$ is the distance to the closest ion (nearest neighbour) and $n$ is the Born exponent (a number between 5 and 12, determined experimentally by measuring the compressibility of the solid, or derived theoretically).Cotton, F. Albert; Wilkinson, Geoffrey; (1966). Advanced Inorganic Chemistry (2d Edn.) New York:Wiley-Interscience.

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 the 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.

{{main|Born–Mayer equation}}

In 1932,{{Cite journal |last=Born |first=Max |last2=Mayer |first2=Joseph E. |date=1932-01-01 |title=Zur Gittertheorie der Ionenkristalle |url=https://link.springer.com/article/10.1007/BF01340511 |journal=Zeitschrift für Physik |language=de |volume=75 |issue=1 |pages=1–18 |doi=10.1007/BF01340511 |issn=0044-3328|url-access=subscription }} Born and Joseph E. Mayer refined the Born-Landé equation by replacing the power-law repulsive term with an exponential term $e^\{-r\_0/\; \backslash rho\}$ which better accounts for the quantum mechanical repulsion effect between the ions.{{Cite journal |last=Wallace |first=W. E. |date=1949-11-01 |title=The Born-Mayer Model for Ionic Solids and the Heats of Formation and Lattice Spacings of Alkali Halide Solid Solutions |url=https://pubs.aip.org/jcp/article/17/11/1095/201094/The-Born-Mayer-Model-for-Ionic-Solids-and-the |journal=The Journal of Chemical Physics |language=en |volume=17 |issue=11 |pages=1095–1099 |doi=10.1063/1.1747119 |issn=0021-9606|url-access=subscription }} This equation improved the accuracy for the description of many ionic compounds:

$\backslash Delta\; U\_l\; =-\; \backslash frac\{N\_AMz^+z^-\; e^2\; \}\{4\; \backslash pi\; \backslash varepsilon\_0\; r\_0\}\backslash left(1-\backslash frac\{\backslash rho\}\{r\_0\}\backslash right)$

where $N\_A$ is the Avogadro constant, $M$ is the Madelung constant, $z^+$/$z^-$ are the charge numbers of the cations and anions, $e$ is the elementary charge (1.6022{{e|−19}} C), $\backslash varepsilon\_0$ is the permittivity of free space ({{val|8.854|e=-12|u=C² J⁻¹ m⁻¹}}), $r\_0$ is the distance to the closest ion and $\backslash rho$ is a constant that depends on the compressibility of the crystal (30 - 34.5 pm works well for alkali halides), used to represent the repulsion between ions at short range. Same as before, it can be seen that large values of $r\_0$ results in low lattice energies, whereas high ionic charges result in high lattice energies.

{{main|Kapustinskii equation}}

Developed in 1956 by Anatolii Kapustinskii, this is a generalized empirical equation useful for a wide range of ionic compounds, including those with complex ions.{{Cite journal |last=Kapustinskii |first=A. F. |date=1956-01-01 |title=Lattice energy of ionic crystals |url=https://pubs.rsc.org/en/content/articlelanding/1956/qr/qr9561000283/unauth |journal=Quarterly Reviews, Chemical Society |language=en |volume=10 |issue=3 |pages=283–294 |doi=10.1039/QR9561000283 |issn=0009-2681|url-access=subscription }} It builds upon the previous equations and provides a simplified way to estimate the lattice energy of ionic compounds based on the charges and radii of the ions. It is an approximation that facilitates calculations compared to the Born-Landé and Born-Mayer equations, easier for quick estimates where high precision is not required.

$\backslash Delta\; U\_l=-\backslash frac\{\backslash kappa\; Z|z^+\; z^-|\}\{r\_0\}\; \backslash left(1\; -\; \backslash frac\{\backslash rho\}\{r\_0\}\backslash right)$

where $\backslash kappa$ is the Kapustinskii constant (1.202·10⁵ (kJ·Å)/mol), $Z$ is the number of ions per formula unit, $z^+$/$z^-$ are the charge numbers of the cations and anions, $r\_0$ is the distance to the closest ion and $\backslash rho$ is a constant that depends on the compressibility of the crystal (30 - 34.5 pm works well for alkali halides), used to represent the repulsion between ions at short range.

For certain ionic compounds, the calculation of the lattice energy requires the explicit inclusion of polarization effects.{{cite journal | author = M. Birkholz | title = Crystal-field induced dipoles in heteropolar crystals I: Concept | journal = Z. Phys. B | volume = 96 | issue = 3 | pages = 325–332 | year = 1995 | doi = 10.1007/BF01313054 |bibcode = 1995ZPhyB..96..325B | url=https://www.researchgate.net/publication/227050494| citeseerx = 10.1.1.424.5632 | s2cid = 122527743 }} In these cases the polarization energy E_pol 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%.{{cite journal|author = M. Birkholz|url = https://www.researchgate.net/publication/231101585 |title = The crystal energy of pyrite|journal = J. Phys.: Condens. Matter|volume = 4|pages = 6227–6240|year = 1992|doi = 10.1088/0953-8984/4/29/007|bibcode=1992JPCM....4.6227B|issue = 29|s2cid = 250815717 }}

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

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

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Category:Crystallography

Category:Solid-state chemistry