Lattice Energy of an Ionic Crystal

The lattice energy of an ionic crystal is the amount of energy required at absolute zero temperature to convert one mole of crystalline component into constituent ions in a gaseous state at infinite distance. It is composed of the various forms of energies, as shown above. The calculation is in fact somewhat more complex because of the presence of various ions of alternating charges in a regular tridimensional network. 

To approach the complexity of a real tridimensional structure, let us first consider the case of a monodimensional array of alternating positive and negative charges, each at distance r from its first neighbor. We will assume for the sake of simplicity that the dispersive potential is negligible. The total potential is therefore 

The potential at the (positive) central charge in figure 1.11 is that generated by the presence of two opposite charges at distance r, by two homologous charges at distance 2r, by two opposite charges at distance 3r, and so on. The coulombic potential is then represented by a serial expansion of the type 

The term in parentheses in equation 1.61, which in this particular case corresponds to the natural logarithm of 2 (In 2 = 0.69), is known as the Madelung constant (M) and has a characteristic value for each structure. 

Let us now go back to the monodimensional array of figure 1.11. The repulsive potential at the central charge is composed of 

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The ability of an ionic solid to dissolve depends on its lattice energy, as well as the degree to which its ions can become hydrated. The lattice energy of an ionic crystal is a measure of the strength of its three-dimensional network of bonds. If these interactions are weaker than the solute-solvent attractions, the ionic bonds will be easily disrupted by water molecules. 

Adopting the equation of Catti (1981), the lattice energy of an ionic crystal may be expressed by means of the double summations... 

Sodium fluoride, NaF, is a favorable choice for X-ray analysis of the lattice energy of an ionic crystal. Both Na and F are relatively light atoms, and the Na 3s-radial distribution, though diffuse, is not quite as spread out as the Li 2s shell (single-C values are 0.8358 and 0.6396 au-1, respectively see appendix F), and therefore contributes to a larger number of reflections. 

A second major contributor or the lattice energy of an ionic crystal is the repulsive energy. Following Born and Huang (1954), the repulsive energy per mole may be written as... 

The Born-Lande equation for the lattice energy of an ionic crystal was derived. 

The lattice energy of an ionic crystal may be defined as the energy emitted when the correct number of ions emerge from distant locations and station themselves in their appropriate places in the crystal lattice. For a mole of sodium fluoride, for example, one may obtain such a lattice energy by multiplying the potential energy (as given in Equation... 

In so doing, the final expression for the lattice energy of an ionic crystal containing 2N ions was given by M. Born and A. Lande (Bom and Lande, 1918) as ... 

Historically the quantitative theory of ionic crystals was developed between 1918 and 1924 by Born 14), Bom and Lande 19, 20), Made-lung 88) and Haber. In particular Born devised formulae which permit the calculation of the lattice energy of an ionic crystal. The lattice energy of such a crystal may be defined as the increase in internal energy at ab-... 

Quite apart from its theoretical calculation, by the use of one of the expressions developed above, it is possible to relate the lattice energy of an ionic crystal to various measurable thermodynamic quantities by means of a simple Hess s law cycle. This cycle was first proposed and used by Bom 15) and represented in its familiar graphical form by Haber (45). It is now usually referred to as the Born-Haber cycle. The cycle is given below for a uni-univalent salt in terms of enthalpies. 

The lattice energy of an ionic crystal (e.g. Na+Cl ) is the difference between the potential energy of a mol of the crystal at absolute zero and the sum of the potential energies of its ions in the state of an infinitely dilute gas. It was calculated by E. Madelung and others, a simple method being given by Kapustinsky. ... 

Once the Madelung constant is determined from simple structural arguments, the lattice energy of an ionic crystal can be determined easily. These lattice energies can be used in cycles to evaluate energies of difficult-to-determine chemical processes. Such cycles are called Born-Haber cycles. 

The lattice energy of an ionic crystal is derived from a balance of attractive and repulsive forces within the crystal. The strength of attraction and repulsion between ions are represented by potential functions. The simplest of these is based on Coulomb s law which essentially... 

This is the Born-Lande equation Tor the lattice energy of an ionic compound. As we shall see, it is quite successful in predicting accurate values, although it omits certain energy factors to be discussed below. It requires only a knowledge of the crystal structure (in order to choose the correct value for A) and the interionic distance, rc> both of which are readily available from X-ray diffraction studies... 

The total potential energy of an ionic crystal, which is often referred to as the lattice energy, U, per mole, may be represented as the sum of the electrostatic and repulsive energy terms. For a halite stmcture crystal, MX, by summing Equations (2.3) and (2.5), we obtain the lattice energy, (7l. per mole ... 

The lattice enthalpy of an ionic crystal is the enthalpy lowering when 1 mol of crystal is formed from its ions. It consists particularly of the attraction between ions of different signs of charge, according to Equation 6.1. The standard formation enthalpy of crystal is the energy lowering from the pure phases, usually the metal phase for the metal ion and the halogen gas phase at standard conditions. 

The lattice enthalpy of an ionic crystal is the heat energy absorbed (at constant... 

The concept of charge density nvolves the total charge of an ion divided by the space that the ion occupies. Using Table 21.5, determine a trend between charge density of an ion and the lattice energy of similar ionic crystals. Can you justify this trend on physical principles ... 

An important property of an ionic crystal is the energy required to break the crystal apart into individual ions, this is the crystal lattice energy. It can be measured by a thermodynamic cycle, called the Born-Haber cycle. 

Madelung constant A term that accounts for the particular structure of an ionic crystal when the lattice energy is evaluated from the coulombic interactions. The value is different for each crystalline structure. 

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