The Born-Haber Cycle

The net reaction resulting from the series of steps in Table 8.2 is 

TheiMeiira Suncs of Sleps th the I nrmaliun of NaCl from Its Consiitueut Hlcments 

Sample Problem 8.3 shows how to use the Boin-Haber cycle to calculate the lattice energy. 

1 Will the lattice energy of KF be larger or smaller than that of LiF, larger or smaller than that of KCl. and larger or smaller than that of KI  

For the sake of simplicity, the shared pair of electrons can be represented by a dash, rather than by two dots H—H 

3 Lattice energies are graphed for three series of compounds in which the ion charges are +2, -2 +2, -1 and +1, -1. The ions in each series of compounds are separated by different distances. Identify the series. 

The tabulated value of Mi° for Cl(g) gives the amount of energy needed to convert i mole of Cl2(g) to 1 mole of Cl(g)  

Lattice energies cannot be measured experimentally since they represent hypothetical processes  

However, the following reaction sequence, relating the heat of formation, A//f of a crystal 

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From the Born-Haber cycle it follows that... 

Lattice energies may be derived from the Born-Haber cycle or calculated using the Kapustinskii equation. ... 

J 15 Calculate a lattice enthalpy by using the Born-Haber cycle (Example 6.13). 

The ligand field stabilization is expressed in the lattice energies of the halides MX2. The values obtained by the Born-Haber cycle from experimental data are plotted v.v. the d electron configuration in Fig. 9.5. The ligand field stabilization energy contribution is no more than 200 kJ mol-1, which is less than 8% of the total lattice energy. The ionic radii also show a similar dependence (Fig. 9.6 Table 6.4, p. 50). 

The enthalpy of formation of a compound is a so-called thermodynamic state function, which means that the value depends only on the initial and final states of the system. When the formation of crystalline NaCl from the elements is considered, it is possible to consider the process as if it occurred in a series of steps that can be summarized in a thermochemical cycle known as a Born-Haber cycle. In this cycle, the overall heat change is the same regardless of the pathway that is followed between the initial and final states. Although the rate of a reaction depends on the pathway, the enthalpy change is a function of initial and final states only, not the pathway between them. The Born-Haber cycle for the formation of sodium chloride is shown as follows ... 

In fact, it may be impossible to measure the heat associated with an atom gaining two electrons, so the only way to obtain a value for the second electron affinity is to calculate it. As a result, the Born-Haber cycle is often used in this way, and this application of a Born-Haber cycle will be illustrated later in this chapter. In fact, electron affinities for some atoms are available only as values calculated by this procedure, and they have not been determined experimentally. 

Before considering further the thermodynamic nature of particular macrocyclic effects, it is necessary to consider the various components of a typical complexation reaction. These are best illustrated by the Born-Haber cycle illustrated in Figure 6.1. Each of the steps 1-5 has a AG, AH and a A5 term associated with it and the overall values of these parameters reflect the respective sums of the individual components. To understand fully the nature of a particular macrocyclic effect, it is necessary to have data available for each of these steps for both the macrocyclic system and open-chain (reference) system. Although some progress has been made... 

The next term in Equation (5) is the halide affinity of the metal halide, Ax (MtXJ. This is a very important term for the final value of AH4e and a term which, were it generally available, would facilitate worthwhile conclusions as to which metal halide would be the best anionogen. However, only two halide affinities for the metals of interest are known to us, viz. the fluoride affinities of BF3 and WF5 [26, 27]. In Scheme 3 the role of the halide affinity in the Born-Haber cycle for the BIE [Equation (iv)] is emphasised and an alternative route for the same energy contribution to AH4B is given. The individual BDE D(XBMt-X) is, like the halide affinity, known only for a very few examples of interest,... 

The situation presented above is, of course, a simplification. It is well documented that metal halides can take part in a self-ionisation reaction [34] involving only the metal halide. The Born-Haber cycle for such a BIE is given in Scheme 4. 

The errors involved in the above analysis of the Born-Haber cycle for the organic halide + metal halide are, of course, so large (of the order of 100 kj mol 1) that to convert the derived energy change into an equilibrium constant would be meaningless, but the analysis is still worthwhile since, as has been demonstrated, it provides the basis for a comparison between systems it permits predictions as to the effectiveness of untried systems and it provides explanations for some of the established experimental results. 

It is quite remarkable that electrostatic calculations based on a simple model of integral point charges at the nuclear positions of ionic crystals have produced good agreement with values of the cohesive energy as determined experimentally with use of the Born-Haber cycle. The point-charge model is a purely electrostatic model, which expresses the energy of a crystal relative to the assembly of isolated ions in terms of the Coulombic interactions between the ions. 

FIGURE 1.56 The Born-Haber cycle for a metal chloride (MCI). 

TABLE 1.18 Values of the Born-Haber cycle terms for NaCb and MgCb/kJ moE ... 

Use the Born-Haber cycle in Figure 1.59 and the data in Table 1.19 to calculate the lattice energy of solid calcium chloride, CaCl2 ... 

In principle, we can use the Born-Haber cycle to predict whether a particular ionic compound should be thermodynamically stable, on the basis of calculated values of U, and so proceed to explain all of the chemistry of ionic solids. The relevant quantity is actually the free energy of formation, AGf, and this is calculable if an entropy cycle is set up to complement the Born-Haber enthalpy cycle. However, in practice AHf dominates the energetics of formation of ionic compounds. 

Solubility equilibria can be treated in terms of the free energy of solution, as outlined in Section 2.3, and the temperature dependences can be related to the enthalpy and entropy of solution. The Born-Haber cycle for solutions, in terms of enthalpies, can be written as follows ... 

E. A. Hylleraas, Z. Physik 63, 771 (1930). The calculated value of the crystal energy is 219 kcal/mole, and the Born-Haber cycle value is 218 kcal/mole, using for the electron affinity of hydrogen the reliable quantum-mechanical value 16.480 kcal/mole (see Introduction to Quantum Mechanics, Sec. 29c). The calculated value for the lattice constant, 4.42 A, is less reliable than the value... 

FIGURE 6.31 The Born-Haber cycle used to determine the lattice enthalpy of potassium chloride (see Example 6.13). 

SOLUTION The Born-Haber cycle for KCl is shown in Fig. 6.31. The sum of the enthalpy changes for the complete cycle is 0, so we can write... 

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