Lattice energy

In chemistry, the lattice energy is the energy change (released) upon formation of one mole of a crystalline compound from its infinitely separated constituents, which are assumed to initially be in the gaseous state at 0 K. It is a measure of the cohesive forces that bind crystalline 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]

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:

${\displaystyle {\ce {Na^+ (g) + Cl^- (g) -> NaCl (s)}}}$

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 ${\displaystyle \Delta U_{l}}$ and the lattice enthalpy ${\displaystyle \Delta H_{l}}$ at pressure ${\displaystyle P}$ is given by the following equation:

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

where ${\displaystyle \Delta U_{l}}$ is the lattice energy (i.e., the molar internal energy change), ${\displaystyle \Delta H_{l}}$ 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 ${\displaystyle \Delta U_{l}<0}$. 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). Both, lattice energy and lattice enthalpy are identical at 0 K and the difference may be disregarded in practice at normal temperatures^[5].

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 ${\displaystyle \Delta H_{l}}$. 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.

In 1918^[6] 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.^[2] This equation estimates the lattice energy based on electrostatic interactions and a repulsive term characterized by a power-law dependence (using a Born exponent, ${\displaystyle n}$). It was published building on earlier work by Born on ionic lattices.

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

where ${\displaystyle N_{A}}$ is the Avogadro constant, ${\displaystyle M}$ is the Madelung constant, ${\displaystyle z^{+}}$/${\displaystyle z^{-}}$ are the charge numbers of the cations and anions, ${\displaystyle e}$ is the elementary charge (1.6022×10⁻¹⁹ C), ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space (${\displaystyle 4\pi \varepsilon _{0}}$ = 1.112×10⁻¹⁰ C²/(J·m)), ${\displaystyle r_{0}}$ is the distance to the closest ion (nearest neighbour) and ${\displaystyle n}$ is the Born exponent (a number between 5 and 12, determined experimentally by measuring the compressibility of the solid, or derived theoretically).^[7]

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)

The Madelung constant ${\displaystyle M}$ reflects the effect of the geometry of the lattice (relative distribution of ions) on the strength of the net Coulombic interaction. It typically increases with coordination number. This dependence reflects the fact that a large contribution comes from nearest neighbours, and such neighbours are more numerous when the coordination number is large. However, a high coordination number does not necessarily mean that the interactions are stronger because the potential energy also depends on the scale of the lattice. For example, ${\displaystyle r_{0}}$ may be so large in lattices with ions big enough to adopt eightfold-coordination that the separation of the ions reverses the effect of the small increase in the Madelung constant and results in a smaller lattice energy^[8].

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.

In 1932^[9], Born and Joseph E. Mayer refined the Born-Landé equation by replacing the power-law repulsive term with an exponential term ${\displaystyle e^{-r_{0}/\rho }}$ which better accounts for the quantum mechanical repulsion effect between the ions^[10]. This equation improved the accuracy for the description of many ionic compounds:

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

where ${\displaystyle N_{A}}$ is the Avogadro constant, ${\displaystyle M}$ is the Madelung constant, ${\displaystyle z^{+}}$/${\displaystyle z^{-}}$ are the charge numbers of the cations and anions, ${\displaystyle e}$ is the elementary charge (1.6022×10⁻¹⁹ C), ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space (8.854×10⁻¹² C² J⁻¹ m⁻¹), ${\displaystyle r_{0}}$ is the distance to the closest ion and ${\displaystyle \rho }$ 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^[5]. Same as before, it can be seen that large values of ${\displaystyle r_{0}}$ results in low lattice energies, whereas high ionic charges result in high lattice energies.

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^[11]. 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^[2].

${\displaystyle \Delta U_{l}=-{\frac {\kappa Z|z^{+}z^{-}|}{r_{0}}}\left(1-{\frac {\rho }{r_{0}}}\right)}$

where ${\displaystyle \kappa }$ is the Kapustinskii constant (1.202·10⁵ (kJ·Å)/mol), ${\displaystyle Z}$ is the number of ions per formula unit, ${\displaystyle z^{+}}$/${\displaystyle z^{-}}$ are the charge numbers of the cations and anions, ${\displaystyle r_{0}}$ is the distance to the closest ion and ${\displaystyle \rho }$ 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.

Kapustinskii observed that^[5], if the Madelung constants for a number of structures are divided by the number of ions per formula unit, ${\displaystyle Z}$, then approximately the same value is obtained for them all. He also noted that the value so obtained increases with the coordination number. Therefore, because ionic radius also increases with coordination number, the variation in ${\displaystyle M/Z}$ from one structure to another can be expected to be fairly small. This observation led Kapustinskii to propose that there exists a hypothetical rock-salt (NaCl) structure that is energetically equivalent to the true structure of any ionic solid and therefore that the lattice enthalpy can be calculated by using the rock-salt Madelung constant and the appropriate ionic radii for (6,6)-coordination.

For certain ionic compounds, the calculation of the lattice energy requires the explicit inclusion of polarization effects.^[12] 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%.^[13]

The lattice energy of a molecular compound corresponds to the energy of sublimation at 0 K. This energy cannot be measured directly, but it is equal to the enthalpy of sublimation at a temperature ${\displaystyle T}$ plus the thermal energy needed to warm the sample from 0 K to this temperature, minus ${\displaystyle RT}$. Here ${\displaystyle RT}$ is the amount of energy required to expand one mole of a gas at a temperature ${\displaystyle T}$ to an infinitely small pressure. These amounts of energy, in principle, can be measured and therefore the lattice energy can be determined experimentally in this case. However, the measurement is not simple and is subject to various uncertainties^[14].

Overall, the lattice energy ${\displaystyle \Delta U_{l}}$ can be calculated in an approximate fashion^[15] by summing the contributions from the London dispersion forces ${\displaystyle E_{D}}$ (always attractive), the repulsion ${\displaystyle E_{A}}$ due to the interpenetration of electron shells of close atoms, the electrostatic interaction ${\displaystyle E_{C}}$ of molecules with polar bonds (dipoles or multipoles) and the zero-point energy ${\displaystyle E_{0}}$ (always present even at 0 K) from a Debye solid:

${\displaystyle \Delta U_{l}=N_{A}\sum _{i}E_{i}=N_{A}(E_{D}+E_{A}+E_{C}+E_{0})=N_{A}\sum _{i,j}\left[-C_{ij}r_{ij}^{-6}+B_{ij}\exp \left(-\alpha _{ij}r_{ij}\right)+{\frac {q_{i}q_{j}e^{2}}{4\pi \epsilon _{0}r_{ij}}}+{\frac {9}{8}}h\nu _{\max }\right]}$

The atoms of one molecule are counted with the index ${\displaystyle i}$, while all atoms of all other molecules in the crystal are counted with the index ${\displaystyle j}$. In this way the interaction energy of one molecule with all other molecules is calculated. The distance between the atoms ${\displaystyle i}$ and ${\displaystyle j}$ is ${\displaystyle r_{ij}}$, while ${\displaystyle q_{i}}$ and ${\displaystyle q_{j}}$ are their partial charges in units of the electric unit charge. ${\displaystyle N_{A}}$ is the Avogadro constant and ${\displaystyle \nu _{\max }}$ is the frequency of the highest occupied vibrational state in the Debye crystal. ${\displaystyle B_{ij}}$, ${\displaystyle \alpha _{ij}}$ and ${\displaystyle C_{ij}}$ are parameters that have to be determined experimentally; they are optimized to reproduce the measured sublimation enthalpies at 300 K correctly. As the contributions of the terms in this equation decrease with growing distances ${\displaystyle r_{ij}}$, sufficient accuracy can be obtained by considering only atoms up to some upper limit for ${\displaystyle r_{ij}}$^[15].

The zero point energy makes a considerable contribution for molecules with a very small mass and those being held together via hydrogen bonds^[16]. For H₂ and He it even amounts to the predominant part of the lattice energy. For H₂O it contributes about 30 %, and for N₂, O₂ and CO about 10 %. For larger molecules the contribution of the zero point energy is marginal.

In molecules with low polarity like hydrocarbons, electrostatic forces have only a minor influence. Although the forces exerted by a multipole (polar molecules) only have appreciable influence on close-lying molecules, they can contribute significantly to the lattice energy. Values for the partial charges of atoms can be derived from quantum mechanical calculations, from the molecular dipole moments and from rotation–vibration spectra. However, often they are not well known. If the contribution of the Coulomb energy cannot be calculated precisely, no reliable lattice energy calculations are possible^[14].

Generally, increasing molecular size, heavier atoms and more polar bonds contribute to an increased lattice energy of a molecular crystal. Typical values are: argon 7.7 kJ/mol ; krypton 11.1 kJ/mol; organic compounds 50 - 150 kJ/mol^[17].

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