Lattice energy the Born-Haber cycle

g) = Enthalpy change associated with the attaehment of an electron Ajif°(MX, s) = Standard enthalpy of formation Lattice enthalpy change (see text) 

In NaCl, the contributions to the total lattice energy (—766kJmor ) made by electrostatic attractions, electrostatic and Bom repulsions, dispersion energy and zero point energy are —860, - -99, —12 and - -7kJmol respectively. The error introduced by neglecting the last two terms (which always tend to compensate each other) is very small. 

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 thermochentical cycle caUed the Born-Haber cycle. If the anion in the salt is a hahde, then aU 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 6.16. 

Rearranging this expression and introducing the approximation that the lattice energy AC/(0K) Aiauice/T(298K) gives equatirai 6.20. AU the quantities on the right-hand side of the equation are obtained from standard tables of data. (Enthalpies of atomizatiOTi see Appendix 10 irmization energies see Appendix 8 electron affinities see Appendix 9.) 

Given that the standard enthalpy of formation at 298 K of CaF2 is -1228kJmor, determine the lattice energy for CaF2 using appropriate data from the Appendices. 

By assuming an electrostatic model, estimate the lattice energy of MgO (NaCI lattice) values of rj are listed in Appendix 6. /4 v. —3926 kJ mol  

Lattice energies obtained from the Born-Lande equation are approximate, and for more accurate evaluations of their values, several improvements to the equation can be made. 

The most important of these arises by replacing the term  

The constant p has a value of 35 pm for all alkali metal halides. Note that ro appears in the Born repulsive term (compare equations 5.16 and 5.17). 

Further refinements in lattice energy calculations include the introduction of terms for the dispersion energy and the zero-point energy (see Section 2.9). Dispersion forces arise from momentary fluctuations in electron density which produce temporary dipole moments that, in turn, induce dipole moments in neighbouring species. Dispersion forces are also referred to as induced-dipole-induced-dipole interactions. They are non-directional and give rise to a dispersion energy that is related to the internuclear separation, r, and the polarizability, a, of the atom (or molecule) according to equation 5.18. 

Lattice energies derived using the electrostatic model are often referred to as calculated values to distinguish them from values obtained using thermochemical cycles. It should, however, be appreciated that values of rQ obtained from X-ray diffraction studies are experimental quantities and may conceal departures from ideal ionic behaviour. In 

Worked example 6.6 Application of the Born-Haber cycle 

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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. 

Chapters Lattice energy the Born-Haber cycle 155... 

CALCULATION OF LATTICE ENERGIES THE BORN-HABER CYCLE... 

Calculation of Lattice Energies The Born-Haber Cycle... 

Calculation of Lattice Energies The Born-Haber Cycle 304 Oxidation Numbers, Formal Charges, and Actual Partial Charges 319... 

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

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). 

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 ... 

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... 

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. 

The lattice energies calculated using this equation are compared with those obtained from the Born-Haber cycle in Table 4.2.4. 

In cases where the lattice energy is known from the Born-Haber cycle, the Kapustinskii equation can be used to derive the ionic radii of complex anions such as S042- and P043-. The values determined in this way are known as thermochemical radii some values are shown in Table 4.2.6. 

The Born-haber cycle It is used to estimate the lattice energy of ionic solids. It makes use of Hess s law... 

The number of electrons available for empirical evaluation of metal-metal bonding has been taken as the Pauling metallic valence less the number of H ions per metal. In this connection the valence numbers of Borelius (6) give somewhat better correlations—e.g., in differentiating Pd from Ag (valences 7 and 1, respectively). Heats of formation calculated from the lattice energies by the Born-Haber cycle are not yet sufficiently accurate to be useful numerically, but they provide an interesting rationalization of the formation of many hydrides. This is the principal reason for considering such a naive model. 

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... 

In the sulphides, selenides, tellurides and arsenides, all types of bond, ionic, covalent and metallic occur. The compounds of the alkali metals with sulphur, selenium and tellurium form an ionic lattice with an anti-fluorite structure and the sulphides of the alkaline earth metals form ionic lattices with a sodium chloride structure. If in MgS, GaS, SrS and BaS, the bond is assumed to be entirely ionic, the lattice energies may be calculated from equation 13.18 and from these values the affinity of sulphur for two electrons obtained by the Born-Haber cycle. The values obtained vary from —- 71 to — 80 kcals and if van der Waal s forces are considered, from 83 to -- 102 kcals. 

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. 

The values of Huggins are probably, the most accurate lattice energies obtainable and agree with the Born-Haber cycle values to within the experimental accuracy of the cycle terms. The values given by the Kapustinskii equation will be seen to be rather low. The Bom cycle values are obtained from the values of AH/ M+(g) and aH/ MX(s) given by the U. S. Bureau of Standards, circular 500, and the values of AH/ X (g) decided upon by Pritchard 108), as a result of a review of all the experimental data. 

Ionic lattice energies are measured experimentally by means of a thermodynamic cycle developed by Max Born and Fritz Haber. The Born-Haber cycle is an application of Hess s law (the first law of thermodynamics). It is illustrated by a determination of the lattice energy of sodium chloride, which is A for the reaction... 

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. 

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