TOPICS
Group 18 elements in the solid state
5.13 Lattice energy: estimates from an electrostatic model
The perovskite (CaTiO
3) lattice: a double oxide
Perovskite is an example of a double oxide; it does not, as the formula might imply, contain [TiO3]2ions, but is a mixed Ca(II) and Ti(IV) oxide. Figure 5.23a shows one represen-tation of a unit cell of perovskite (see problem 5.13at the end of the chapter). The cell is cubic, with Ti(IV) centres at the corners of the cube, and O2 ions in the 12 edge sites.
The 12-coordinate Ca2þ ion lies at the centre of the unit cell. Each Ti(IV) centre is 6-coordinate, and this can be appreciated by considering the assembly of adjacent unit cells in the crystal lattice.
Many double oxides or fluorides such as BaTiO3, SrFeO3, NaNbO3, KMgF3 and KZnF3crystallize with a perovskite lattice. Deformations of the lattice may be caused as a consequence of the relative sizes of the ions, e.g. in BaTiO3, the Ba2þ ion is relatively large (rBa2þ¼ 142 pm compared with rCa2þ ¼ 100 pm) and causes a displacement of each Ti(IV) centre such that there is one short TiO contact. This leads to BaTiO3 possessing ferroelectric properties (seeSection 27.6).
The structures of some high-temperature superconductors are also related to that of perovskite. Another mixed oxide lattice is that of spinel, MgAl2O4(seeBox 12.6).
5.12 Crystal structures of
The change in internal energy can be estimated from equation 5.9 by considering the Coulombic attraction between the ions. For an isolated ion-pair:
U¼ jzþj jzje2 4"0r
!
ð5:9Þ where U¼ change in internal energy (unit ¼ joules);
jzþj ¼ modulus† of the positive charge (for Kþ, jzþj ¼ 1;
for Mg2þ, jzþj ¼ 2); jzj ¼ modulus† of the negative charge (for F,jzj ¼ 1; for O2,jzj ¼ 2); e ¼ charge on the electron¼ 1:602 1019C; "0¼ permittivity of a vacuum¼ 8:854 1012F m1; r¼ internuclear distance between the ions (units¼ m).
Coulombic interactions in an ionic lattice
Now consider a salt MX which has an NaCl lattice. A study of the coordination geometry in Figure 5.15 (remembering that the lattice extends indefinitely) shows that each Mzþ ion is surrounded by:
. 6 Xzions, each at a distance r . 12 Mzþions, each at a distance ffiffiffi
p2 r . 8 Xzions, each at a distance ffiffiffi
p3 r . 6 Mzþions, each at a distance ffiffiffi p4
r¼ 2r and so on.
The change in Coulombic energy when an Mzþ ion is brought from infinity to its position in the lattice is given by equation 5.10.
U¼ e2 4"0
6 rjzþj jzj
12 ffiffiffi2 p rjzþj2
þ
8 ffiffiffi3
p rjzþj jzj
6 ffiffiffi4 p rjzþj2
. . .
¼ jzþj jzje2 4"0r
6
12jzþj ffiffiffi2 p jzj
þ
8 ffiffiffi3 p
3jzþj
jzj
. . .
ð5:10Þ
The ratio of the charges on the ions,jzþj
jzj, is constant for a given type of structure (e.g. 1 for NaCl) and so the series in square brackets in equation 5.10 (which slowly converges and may be summed algebraically) is a function only of the crystal geometry. Similar series can be written for other crystal lattices, but for a particular structure type, the series is independent of jzþj, jzj and r. Erwin Madelung first evaluated such series in 1918, and the values appropriate for various lattice types are Madelung constants, A (seeTable 5.4). Equation 5.10 can therefore be written in the more simple form of equation 5.11, in which the lattice energy is estimated in joules per mole of compound.
U¼ LAjzþj jzje2
4"0r ð5:11Þ
where L¼ Avogadro number ¼ 6:022 1023mol1, and A¼ Madelung constant (no units).
Although we have derived this expression by considering the ions that surround Mzþ, the same equation results by starting from a central Xzion.
Born forces
Coulombic interactions are not the only forces operating in a real ionic lattice. The ions have finite size, and electron–
electron and nucleus–nucleus repulsions also arise; these are Born forces. Equation 5.12 gives the simplest expression for the increase in repulsive energy upon assembling the lattice from gaseous ions.
U¼LB
rn ð5:12Þ
where B¼ repulsion coefficient, and n ¼ Born exponent.
Values of the Born exponent (Table 5.3) can be evaluated from compressibility data and depend on the electronic configurations of the ions involved; effectively, this says that n shows a dependence on the sizes of the ions.
Worked example 5.3 Born exponents
Using the values given in Table 5.3, determine an appropriate Born exponent for BaO.
Ba2þis isoelectronic with Xe, and so n¼ 12 O2is isoelectronic with Ne, and n¼ 7 The value of n for BaO¼12þ 7
2 ¼ 9:5
†The modulus of a real number is its positive value, e.g.jzþj and jzj are both positive.
Table 5.3 Values of the Born exponent, n, given for an ionic compound MX in terms of the electronic configuration of the ions [Mþ][X]. The value of n for an ionic compound is determined by averaging the component values, e.g. for MgO, n¼ 7; for LiCl, n ¼5þ 9
2 ¼ 7.
Electronic configuration of the ions in an ionic compound MX
Examples of ions n
(no units)
[He][He] H, Liþ 5
[Ne][Ne] F, O2, Naþ, Mg2þ 7 [Ar][Ar], or [3d10][Ar] Cl, S2, Kþ, Ca2þ, Cuþ 9 [Kr][Kr] or [4d10][Kr] Br, Rbþ, Sr2þ, Agþ 10 [Xe][Xe] or [5d10][Xe] I, Csþ, Ba2þ, Auþ 12 Chapter 5 . Lattice energy: estimates from an electrostatic model 153
Self-study exercises Use data in Table 5.3.
1. Calculate an appropriate Born exponent for NaF. [Ans. 7 ] 2. Calculate an appropriate Born exponent for AgF.
[Ans. 8.5 ] 3. What is the change in the Born exponent in going from BaO to
SrO? [Ans.1 ]
The Born–Lande´ equation
In order to write an expression for the lattice energy that takes into account both the Coulombic and Born inter-actions in an ionic lattice, we combine equations 5.11 and 5.12 to give equation 5.13.
Uð0 KÞ ¼ LAjzþj jzje2 4"0r þLB
rn ð5:13Þ
We evaluate B in terms of the other components of the equation by making use of the fact that at the equilibrium separation where r¼ r0, the differentialdU
dr ¼ 0. Differen-tiating with respect to r gives equation 5.14, and rearrange-ment gives an expression for B (equation 5.15).
0¼ LAjzþj jzje2 4"0r02 nLB
r0nþ 1 ð5:14Þ
B¼ Ajzþj jzje2r0n 1
4"0n ð5:15Þ
Combining equations 5.13 and 5.15 gives an expression for the lattice energy that is based on an electrostatic model and takes into account Coulombic attractions, Coulombic repulsions and Born repulsions between ions in the crystal lattice. Equation 5.16 is the Born–Lande´ equation.
Uð0 KÞ ¼ LAjzþj jzje2 4"0r0
11
n
ð5:16Þ Because of its simplicity, the Born–Lande´ expression is the one that chemists tend to use; many chemical problems involve the use of estimated lattice energies, e.g. for hypothe-tical compounds. Often lattice energies are incorporated into thermochemical cycles, and so an associated enthalpy change is needed (seeSection 5.14).
Madelung constants
Values of Madelung constants for selected lattices are given in Table 5.4. Remembering that these values are derived by considering the coordination environments (near and far neighbours) of ions in the crystal lattice, it may seem surprising that, for example, the values for the NaCl and CsCl lattices (Figures 5.15 and 5.16) are similar. This is simply a consequence of the infinite nature of the structures:
although the first (attractive) term in the algebraic series for
Ais greater by a factor of86for the CsCl lattice, the second (repulsive) term is also greater, and so on.
Table 5.4 shows that Madelung constants for MX2 structures are50% higher than those for MX lattices. We return to this difference in Section 5.16.
Worked example 5.4 Use of the Born–Lande´ equation
Sodium fluoride adopts the NaCl type lattice. Estimate the lat-tice energy of NaF using an electrostatic model.
Data required:
L¼ 6:022 1023mol1A¼ 1:7476 e ¼ 1:602 1019C e0¼ 8:854 1012F m1 Born exponent for NaF¼ 7 Internuclear NaF distance ¼ 231 pm
The change in internal energy (the lattice energy) is given by the Born–Lande´ equation:
Uð0 KÞ ¼ LAjzþj jzje2 4"0r0
11
n
rmust be in m: 231 pm¼ 2:31 1010m
U0¼
6:022 1023 1:7476 1
1 ð1:602 1019Þ2 4 3:142 8:854 1012 2:31 1010 0
B@
1 CA
11
7
¼ 900 624 J mol1
901 kJ mol1
Self-study exercises
1. Show that the worked example above is dimensionally correct given that C, F and J in SI base units are: C¼ A s;
F¼ m2kg1s4A2; J¼ kg m2s2.
2. Estimate the lattice energy of KF (NaCl lattice) using an elec-trostatic model; the KF internuclear separation is 266 pm.
[Ans.798 kJ mol1] Table 5.4 Madelung constants, A, for selected lattice types.
Values of A are numerical and have no units.
Lattice type A
Sodium chloride (NaCl) 1.7476
Caesium chloride (CsCl) 1.7627
Wurtzite (a-ZnS) 1.6413
Zinc blende (b-ZnS) 1.6381
Fluorite (CaF2) 2.5194
Rutile (TiO2) 2.408a
Cadmium iodide (CdI2) 2.355a
aFor these structures, the value depends slightly on the lattice parameters for the unit cell.
3. By assuming an electrostatic model, estimate the lattice energy of MgO (NaCl lattice); values of rion are listed in
Appendix 6. [Ans.3926 kJ mol1]
Refinements to the Born–Lande´ equation
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 1 rnterm in equation 5.12 by er, a change reflecting the fact that wavefunctions show an exponential dependence on r; is a constant that can be expressed in terms of the compressibility of the crystal. This refinement results in the lattice energy being given by the Born–Mayer equation (equation 5.17).
Uð0 KÞ ¼ LAjzþj jzje2 4"0r0
1
r0
ð5:17Þ The constant has a value of 35 pm for all alkali metal halides. Note that r0 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(seeSection 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 inter-actions. They are non-directional and give rise to a dispersion energy that is related to the internuclear separation, r, and the polarizability, , of the atom (or molecule) according to equation 5.18.
Dispersion energy/
r6 ð5:18Þ
The polarizability of a species is a measure of the degree to which it may be distorted, e.g. by the electric field due to an adjacent atom or ion. In the hard sphere model of ions in lattices, we assume that there is no polarization of the ions. This is a gross approximation. The polarizability increases rapidly with an increase in atomic size, and large ions (or atoms or molecules) give rise to relatively large induced dipoles and, thus, significant dispersion forces.
Values of can be obtained from measurements of the relative permittivity (dielectric constant, seeSection 8.2) or the refractive index of the substance in question.
In NaCl, the contributions to the total lattice energy (766 kJ mol1) made by electrostatic attractions, electro-static and Born repulsions, dispersion energy and zero-point energy are860, þ99, 12 and þ7 kJ mol1respectively. In fact, the error introduced by neglecting the last two terms (which always tend to compensate each other) is very small.
Overview
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 r0 obtained from X-ray diffraction studies are experimental quantities and may conceal departures from ideal ionic behaviour. In addition, the actual charges on ions may well be less than their formal charges. Nevertheless, the concept of lattice energy is of immense importance in inorganic chemistry.