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.
Lattice energy vs lattice enthalpy
The lattice energy 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. The lattice enthalpy is reported as a positive value.
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)
Some textbooks and the commonly used CRC Handbook of Chemistry and Physics define lattice energy with the opposite sign, i.e. 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.
The relationship between the molar lattice energy and the molar lattice enthalpy is given by the following equation:
where is the molar lattice energy, the molar lattice enthalpy and 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 .
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.
- NA 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;
- qe 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.
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.
The Kapustinskii equation can be used as a simpler way of deriving lattice energies where high precision is not required.
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. 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.
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