Energetics

The lattice energy of a solid is the difference in potential energy of the ions packed together in a solid and widely separated as a gas. The lattice energy is always positive; a high lattice energy indicates that the ions interact strongly with one another to give a tightly bonded solid. The lattice enthalpy, ∆HL, is the change in standard molar enthalpy for the process , MX(s)→M⁺(g)+X⁻(g) , and its equivalent for other charge types and stoichiometries. The lattice enthalpy is equal to the lattice energy at T=0; at normal temperatures they differ by only a few kilojoules per mole, and the difference is normally neglected. Each ion in a solid experiences electrostatic attractions from all the other oppositely charged ions and repulsions from all the other like-charged ions. The total Coulombic potential energy is the sum of all the electrostatic contributions. Each cation is sur rounded by anions, and there is a large negative contribution from the attraction of the opposite charges. Beyond those nearest neighbours, there are cations that con tribute a positive term to the total potential energy of the central cation. There is also a negative contribution from the anions beyond those cations, a positive contribution from the cations beyond them, and so on to the edge of the solid. These repulsions and attractions become progressively weaker as the distance from the central ion increases, but the net outcome of all these contributions is a lowering of energy. First, consider a simple one-dimensional model of a solid consisting of a long line of uniformly spaced alternating cations and anions, with d the distance between their centres, the sum of the ionic radii (Fig. 20.40). If the charge numbers of the ions have the same absolute value (+1 and −1, or +2 and −2, for instance), then z1=+z, z2=−z, andz1z2=−z2. The potential energy of the central ion is calculated by summing all the terms, with negative terms representing attractions to oppositely charged ions and positive terms representing repulsions from like-charged ions. For the interaction with ions extending in a line to the right of the central ion, the lattice energy is

We have used the relation 1 – 1/2 + 1/3− 1/4+···=ln 2. Finally, we multiply E_P by 2 to obtain the total energy arising from interactions on each side of the ion and then multiply by Avogadro’s constant, NA, to obtain an expression for the lattice energy per mole of ions. The outcome is

With d= r_cation + r_anion. This energy is negative, corresponding to a net attraction. The calculation we have just performed can be extended to three-dimensional arrays of ions with different charges:

The factor A is a positive numerical constant called the Madelung constant; its value depends on how the ions are arranged about one another. For ions arranged in the same way as in sodium chloride, A = 1.748. Table 20.4 lists Madelung constants for other common structures. There are also repulsions arising from the overlap of the atomic orbitals of the ions and the role of the Pauli principle. These repulsions are taken into account by sup posing that, because wavefunctions decay exponentially with distance at large distances from the nucleus, and repulsive interactions depend on the overlap of orbitals, the repulsive contribution to the potential energy has the form , E* P = N_AC′e^{−d/d ,} with C′ and d* constants; the latter is commonly taken to be 34.5 pm. The total potential energy is the sum of EP and E* P, and passes through a minimum when d(EP + E* P)/dd = 0 (Fig. 20.41). A short calculation leads to the following expression for the minimum total potential energy (see Exercise 20.21a):

This expression is called the Born–Mayer equation. Provided we ignore zero-point contributions to the energy, we can identify the negative of this potential energy with the lattice energy. We see that large lattice energies are expected when the ions are highly charged (so |zAzB| is large) and small (so d is small). Experimental values of the lattice enthalpy (the enthalpy, rather than the energy) are obtained by using a Born–Haber cycle, a closed path of transformations starting and ending at the same point, one step of which is the formation of the solid com pound from a gas of widely separated ions. A typical cycle, for potassium chloride, is shown in Fig. 20.42. It consists of the following steps (for convenience, starting at the elements):

1. Sublimation of K(s) +89 [dissociation enthalpy of K(s)]

2. Dissociation of 1/2 Cl2(g) +122 [ ×dissociation enthalpy of Cl2(g)]

3. Ionization of K(g) +418 [ionization enthalpy of K(g)]

4. Electron attachment to Cl(g) −349 [electron gain enthalpy of Cl(g)]

5. Formation of solid from gas −∆HL/ (kJ mol⁻¹)

6. Decomposition of compound +437 [negative of enthalpy of formation of KCl(s)]

Because the sum of these enthalpy changes is equal to zero, we can infer from

89 +122+418−349−∆H_L/ (kJ mol⁻¹) +437=0

that ∆HL =+717 kJ mol−1. Some lattice enthalpies obtained in this way are listed in Table 20.5. As can be seen from the data, the trends in values are in general accord with the predictions of the Born–Mayer equation. Agreement is typically taken to imply that the ionic model of bonding is valid for the substance; disagreement implies that there is a covalent contribution to the bonding.

Fig. 20.40A line of alternating cations and ions used in the calculation of the Madelung constant in one dimension.

Fig. 20.41 The contributions to the total potential energy of an ionic crystal.

Fig. 20.42 The Born–Haber cycle for KCl at 298 K. Enthalpies changes are in kilojoules per mole.

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مواضيع ذات صلة

Structure

Close packing

Structure refinement

The phase problem

Bragg’s law

X-ray diffraction

Lattices and unit cells

Surface pressure

Membrane formation

Micelle formation

The electrical double layer

Classification and preparation