Lattice enthalpy Born-Haber cycle

FIGURE 6.32 In a Born-Haber cycle, we select a sequence of steps that starts and ends at the same point (the elements, for instance). The lattice enthalpy is the enthalpy change accompanying the reverse of the step in which the solid is formed from a gas of ions. The sum of enthalpy changes around the complete cycle is 0 because enthalpy is a state function. 

The lattice enthalpy of a solid cannot be measured directly. However, we can obtain it indirectly by combining other measurements in an application of Hess s law. This approach takes advantage of the first law of thermodynamics and, in particular, the fact that enthalpy is a state function. The procedure uses a Born-Haber cycle, a closed path of steps, one of which is the formation of a solid lattice from the gaseous ions. The enthalpy change for this step is the negative of the lattice enthalpy. Table 6.6 lists some lattice enthalpies found in this way. 

In a Born-Haber cycle, we imagine that we break apart the bulk elements into atoms, ionize the atoms, combine the gaseous ions to form the ionic solid, then form the elements again from the ionic solid (Fig. 6.32). Only the lattice enthalpy, the enthalpy of the step in which the ionic solid is formed from the gaseous ions, is unknown. The sum of the enthalpy changes for a complete Born-Haber cycle is zero, because the enthalpy of the system must be the same at the start and finish. 

Devise and use a Born-Haber cycle to calculate the lattice enthalpy of potassium chloride. 

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

But what is the magnitude of the lattice enthalpy We cannot measure it directly experimentally, so we measure it indirectly, with a Hess s law energy cycle. The first scientists to determine lattice enthalpies this way were the German scientists Bom and Haber we construct a Born-Haber cycle, which is a form of Hess s-law cycle. 

Figure 3.8 Born-Haber cycle constructed to obtain the lattice enthalpy A//(E, lce) of sodium chloride. All arrows pointing up represent endothermic processes and arrows pointing down represent exothermic processes (the figure is not drawn to scale)...

Fig. 1. Born-Haber cycle for the formation of solvated ions from an ionic crystal [M+X ]w. U lattice energy, Affsoiv. enthalpy of ion solvation...

Born-Haber cycle A closed series of reactions used to express the enthalpy of formation of an ionic solid in terms of contributions that include the lattice enthalpy. 

The lattice enthalpy U at 298.20 K is obtainable by use of the Born—Haber cycle or from theoretical calculations, and q is generally known from experiment. Data used for the derivation of the heat of hydration of pairs of alkali and halide ions using the Born—Haber procedure to obtain lattice enthalpies are shown in Table 3. The various thermochemical values at 298.2° K [standard heat of formation of the crystalline alkali halides AHf°, heat of atomization of halogens D, heat of atomization of alkali metals L, enthalpies of solution (infinite dilution) of the crystalline alkali halides q] were taken from the compilations of Rossini et al. (28) and of Pitzer and Brewer (29), with the exception of values of AHf° for LiF and NaF and q for LiF (31, 32, 33). The ionization potentials of the alkali metal atoms I were taken from Moore (34) and the electron affinities of the halogen atoms E are the results of Berry and Reimann (35)4. 

D is the dissociation enthalpy of Cl2,1 is the ionization potential of Na, E is the electron addition enthalpy of Cl (which is the negative of the electron affinity), and U is the lattice energy. The Born-Haber cycle shows that the lattice energy corresponds to the energy required to separate a mole of crystal into the gaseous ions, and forming the crystal from the ions represents -U. 

Although the differences between the first and second ionization enthalpies, especially for beryllium, might suggest the possibility of a stable +1 state, there is no evidence to support this. Calculations using Born-Haber cycles show that owing to the much greater lattice energies of MX2 compounds, MX compounds would be unstable and disproportionate ... 

The enthalpy of formation of an ionic compound can be calculated with an accuracy of a few percent by means of the Born-Land equation (Eq. 4.13) and the Born-Haber cycle. Consider NaCI. for example. Wc have seen that by using the predicted internuclear distance of 283 pm (or the experimental value of 281.4 pm), the Madelung constant of 1.748, the Born exponent, n, and various constants, a value of —755kJmor could be calculated for the lattice energy. The heat capacity correction is 2.1 kJ mol", which yields = —757 kJ moP. The Bom-Haber summation is then... 

Use the Born-Haber cycle to calculate the enthalpy of formation of MgO, which crystallizes in the mtile lattice. Use these data in the calculation O2 bond energy = 247 kJ/mol AHj ji,(Mg) = 37 kJ/mol. Second ionization energy of Mg = 1451 kJ/mol second electron affinity of O = —744 kJ/inol. 

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. 

Most of the enthalpies associated with steps in the cycle can be estimated, to a greater or less accuracy, by experimental methods. The lattice energy, however, is almost always obtained theoretically rather than from experimental measurement. It might be supposed that the enthalpy of dissociation of a lattice could be measured in the same way as the enthalpy of atomization of the metal and nonmctal, that is, by heating the crystal and determining how much energy is necessary to dissociate it into ions. Unfortunately, this is experimentally very difTicull. When a crystal sublimes (AHj), the result is not isolated gaseous ions but ion pairs and other clusters. For this reason it is necessary to use Eq. 4.13 or some more accurate version of it. We can then use the Born-Haber cycle to check the accuracy of our predictions if we can obtain accurate data on every other step in the cycle. Values computed from the Bom-Haber cycle are compared with those predicted by Eq. 4.13 and its modifications in Table 4.3. 

Lattice energies may be estimated from a thermodynamic cycle known as a Born-Haber cycle, which makes use of Hess Law (see Topic B3Y Strictly speaking, the quantities involved are enthalpy rather than energy changes and one should write HL for the lattice enthalpy. From Fig. 1. 

Fig. 1. Born-Haber cycle for determining the lattice enthalpy ofNaCl.

The Born-Mayer equation is an alternative (and possibly more accurate) form based on the assumption of an exponential form for the repulsive energy. Both equations predict lattice energies for compounds such as alkali halides that are in reasonably close agreement with the experimental values from the Born-Haber cycle. Some examples are shown in Table 1. A strict comparison requires some corrections. Born-Haber values are generally enthalpies, not total energies, and are estimated from data normally measured at 298 K not absolute zero further corrections can be made, for example, including van der Waals forces between ions. 

By considering the definition of lattice energy, it is easy to see why these quantities are not measured directly. However, an associated lattice enthalpy of a salt can be related to several other quantities by a thermochemical cycle called the Born-Haber cycle. If the anion in the salt is a haUde, then all the other quantities in the cycle have been determined independently the reason for this statement will become clearer when we look at applications of lattice energies in Section 5.16. 

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