Ionic Bonding - Electrostatic Interactions and Polarization

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1 Ionic Bonding - Electrostatic Interactions and Polarization Chemistry 754 Solid State Chemistry Dr. Patrick Woodward Lecture #13 Born-Haber Cycle for NaCl It is energetically unfavorable for Na metal and Cl gas to react to form Na + (g) + Cl - (g). This highly exothermic reaction occurs because of the large lattice energy, that comes largely from ionic bonding. The Born-Haber cycle is used tool for determining lattice energy given standard thermodynamic data How can we calculate lattice energy without experimental data? 1

Electrostatic Interactions The potential energy of two interacting charges is given by the following equation: E = Q 1 Q 2 /(4πε 0 r) E = e 2 Z 1 Z 2 /(4πε 0 r) Q 1,Q 2 = The absolute charges on the

2 Electrostatic Interactions The potential energy of two interacting charges is given by the following equation: E = Q 1 Q 2 /(4πε 0 r) E = e 2 Z 1 Z 2 /(4πε 0 r) Q 1,Q 2 = The absolute charges on the two particles Z 1,Z 2 = The integral charges on the two particles (Q = Ze) e = The charge of an electron = C ε 0 = The permitivity of free space = C/(m-J) r = The distance between particles (meters) Madelung Energy We can calculate the total potential energy holding an ionic crystal together (the total ionic bonding if you will) by summing up all of the electrostatic interactions, both attractive (cation-anion) anion) and repulsive (anion-anion, anion, cation-cation). cation). To do so we need to know the charges on the ions and the geometry of the crystal. This is done below for NaCl, where r is the Na-Cl distance. A = Madelung Constant E = [e 2 Z + Z - /(4πε 0 )] {(6/r)-(12/r (12/r 2)+ 2)+(8/r 3) 3)-(6/ (6/r 4)+ 4)+ } Nearest Neighbor (Na-Cl Cl) Next Nearest Neighbor (Na-Na) Na) E = [e 2 Z + Z - /(4πε 0 r)] AN A Next, Next Nearest Neighbor (Na-Cl Cl) A A = Madelung Constant (depends on structure type) N A = Avogadro s Number r r = Cation-Anion distance 2

3 These are experimental values, determined from a Born-Haber cycle. The lattice energy increases as 1. The ionic charges increase (primary importance) 2. The distance between ions decreases (secondary importance) Electrostatic Considerations Factors that maximize the electrostatic interactions (ionic bonding) holding a crystal together optimize the cation-anion anion distance (dictated largely by ionic radii) and maximize the cation-cation and anion-anion distances. The latter condition is optimal for: High Symmetry Regular Coordination Environments High Coordination Number Structure CN A Structure CN A CsCl 8, , NaCl 6, , Wurtzite 4, ,

4 Limitations of Madelung Energy Calc s 1. Repulsive interactions arising from electron-electron repulsion are neglected. These decrease the lattice energy by 10-15%. 15%. 2. van der Waals interactions and zero point energy are also neglected. The former can be important, particularly when dealing with polarizable anions. 3. The oxidation states are not the true charges. The deviation between oxidation states and true charges increases s as the oxidation states increase and/or as the electronegativity difference decreases. Some estimates for true charges are SiO 2 Si 2+, O 1- CuO Cu 1.6+, O Covalent contributions are not accounted for, yet they are always present. Thus simple calculations tend not to be very accurate when The oxidation state of the cation and/or anion is high The cation and/or anion is not a closed shell ion Comprehensive Lattice Energy A more complete equation for lattice energy is taken from West 1 and Greenwood 2 U = [AN A e 2 Z + Z - /(4πε 0 r)] + Bne -r/ r/ρ - CN A r Nhν max [A] Coulomb energy of [C] van der point charges (assuming Waals forces full ionic charge) [B] Repulsive energy arising from overlap of electronic charge clouds [D] Zero point vibrational energy Substance [A] kj [B] kj [C] kj [D] kj NaCl (U=-766) MgO (U=-3921)

Polarization If an ion is placed in an asymmetric environment its electron cloud can be deformed or polarized by the potential field created by the surrounding ions.

5 Polarization If an ion is placed in an asymmetric environment its electron cloud can be deformed or polarized by the potential field created by the surrounding ions. Anion core Asymmetric environment anion polarized toward cations Anion valence electron cloud Symmetric environment no dipolar polarization The most important type of polarization is generally thought to polarization of the anion electron cloud by smaller, more highly charged cations. Polarization effects increase as the: Cations get smaller and their oxidation state increases (harder) Anions get larger and their oxidation state increases (softer) The asymmetry of the anion environment increases Polarization is one way of introducing covalency into an ionic model. Keep in mind though that covalency is a more general phenomenon. Structures of MX 2 Metal Halides as predicted by lattice energy calculations F (1.15 Å) Cl (1.60 Å) Br (1.74 Å) I (1.95 Å) Mg (0.86 Å) Zn (0.95 Å) Mn (1.01 Å) Ca (1.20 Å) Sr (1.34 Å) Ba (1.56 Å) CdI 2 5

6 Structures of MX 2 Metal Halides observed F (1.15 Å) Cl (1.60 Å) Br (1.74 Å) I (1.95 Å) Mg (0.86 Å) Zn (0.95 Å) Mn (1.01 Å) Ca (1.20 Å) Sr (1.34 Å) PbCl 2 SrI 2 Ba (1.56 Å) PbCl 2 PbCl 2 CdI 2 Polarization and Layered Structures In fluorite, rutile and sphalerite the anion environment is symmetrical and polarization effects are minimal. In the layered CdI 2 and CdCl 2 structures the anion environment is pyramidal and polarization effects shift anion charge cloud toward the cations, thereby minimizing repulsions between halide layers and cation- cation repulsions. Calculated energies for MCl 2 crystals as a function of cation size, (a) fluorite, (b) rutile, (c) layered with polarization, (d) layered w/out polarization. The energies of fluorite and rutile are nearly identical with or without polarization. [Taken from P.A. Madden & M. Wilson, Chem. Soc. Reviews 25(5), 339 (2000). Polarization favors pushing highly polarizable anions into unsymmetrical sites 6

interactions Actual β-cristobalite (SiO 2 ) Space Group = I-42d I (Tetragonal) Si-O-Si =

7 Polarization in SiO 2 Cristobalite Si Idealized β-cristobalite (SiO 2 ) Space Group = Fd3m (Cubic) Si-O-Si = 180 No polarization, sp bonding at O 2- O Si Optimized for electrostatic interactions Actual β-cristobalite (SiO 2 ) Space Group = I-42d I (Tetragonal) Si-O-Si = 147 Some polarization, sp 2 bonding at O 2- Stabilized by polarization/covalency Si O Si 7

Ionic Bonding. Chem

Ionic Bonding. Chem Whereas the term covalent implies sharing of electrons between atoms, the term ionic indicates that electrons are taken from one atom by another. The nature of ionic bonding is very different than that