ionic_bondvalence - Ionic Bonding Pauling's Rules and the...

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Unformatted text preview: Ionic Bonding Pauling's Rules and the Bond Valence Method Chemistry 754 Solid State Chemistry Dr. Patrick Woodward Lecture #14 Linus Pauling, J. Amer. Chem. Soc. 51, 1010 (1929) 51, 1. The Cation-Anion distance is determined by summing ionic radii. The cation coordination environment is determined by radius ratio. 2. The bond valence sum of each ion should equal its oxidation state. Pauling' Rules for Ionic Structures 3. Crystals tend to avoid shared polyhedral edges and/or faces. This is particularly true for cations with high oxidation state & low coordination number. 4. In a crystal containing different cations those with large valence and small coord. # tend not to share anions. 5. The number of chemically different coordination environments for a given ion tends to be small. 1 Pauling's 1st Rule - Ionic Radii The Cation-Anion distance is determined by sums of ionic radii. The cation coordination environment is determined by radius ratio. The most accurate set of ionic radii are those tabulated initially by initially R.D. Shannon and C.T. Prewitt [Acta Cryst. B 25, 925, (1969)] and [Acta Cryst. later updated by Shannon [Acta Cryst. A 32, 751, (1976)]. [Acta Cryst. There are two sets of radii. "Crystal radii", are based on the best estimate of the size from accurate X-ray crystallography studies. "Traditional ionic radii" are based on the sizes of oxide and fluoride ions as chosen by Pauling. The crystal radii of the cations are 0.14 larger while the crystal radii of the anions are 0.14 smaller. Ionic radii are most accurate for oxides and fluorides, and for smaller, highly charged (hard) cations in regular coordination environments. The sizes of softer ions are more dependent upon their surroundings, and thus are less transferable. Radii are dependent upon both the oxidation state of the ions involved and the coordination number. Ionic Radii Notable Trends There are several trends in ionic radii that should be familiar to you The radius increases as you move down a column. Al+3 = 0.675 , Ga+3 = 0.760 , In+3 = 0.940 , Tl+3 = 1.025 The radius decreases as you move across a period. La+3 = 1.172 , Nd+3 = 1.123 , Gd+3 = 1.078 , Lu+3 = 1.001 The 4d & 5d metals have similar radii due to the lanthanide contraction. Nb+5 = 0.78 , Ta+5 = 0.78 , Pd+4 = 0.755 , Pt+4 = 0.765 The cation radius decreases as you increase the oxidation state. Mn+2 = 0.810 , Mn+3 = 0.785 , Mn+4 = 0.670 The radius increases as the coordination number increases. Sr+2: CN=6 1.32 , CN=10 1.50 CN=8 1.40 , CN=12 1.58 2 Pauling's 1st Rule Radius Ratio The Cation-Anion distance is determined by sums of ionic radii. The cation coordination environment is determined by radius ratio. The radius ratio rule is a geometric argument based on the number of number anions you can get around a cation. It predicts that as the cation size cation decreases its coordination number decreases. 1.00 < R(cation)/R(anion) > 0.732 Cubic (CN=8) 0.732 > R(cation)/R(anion) > 0.414 Octahedral (CN=6) 0.414 > R(cation)/R(anion) > 0.225 Tetrahedral (CN=4) If the cation radius as anion radius remains constant. It will leads to unfavorable overlap of anions in order to maintain optimal cation-anion contact. cation- While this is generally true, the reasons are probably not the simple simple geometrical ones suggested by Pauling. Furthermore, there are so many . Pauling exceptions to this rule that it is of marginal significance. Pauling's 2nd Rule Bond Valence The bond valence sum of each ion should equal its oxidation state. This idea is well known in organic chemistry where each bond has a valence of 1. So that C always has a valence sum of 4 (4 bonds), O always forms 2 bonds, H forms 1 bond, etc. Pauling took this concept and extended it to inorganic compounds, including extended ionic lattices. Pauling's proposed that the valence of a bond, sij, could be non-integer, and the sum of the bond valences around each atom, should equal its oxidation state. Vi = The oxidation state of atom i sij = The valence of the bond between atoms i and j. If there are n equivalent bonds around a central atom with valence m, then valence of each bond is equal to: Vi = sij sij = m/n. 3 Bond Valence Example 1 Consider the rutile form of TiO2. What is the valence of the Ti-O bonds? TiFirst lets calculate the valence of the Ti-O bonds. Ti- Vi = sij 4 = 6(sij) 6(s sij = 2/3 Now use this to determine the coordination number of oxygen. Vi = sij 2 = n(2/3) n = 3 Bond Valence Example 2 What is the oxygen coordination in the mineral zircon, ZrSiO4, where Si+4 is tetrahedrally coordinated and Zr+4 is eight coordinate? First lets calculate the valence of the Si-O bonds. SiVi = sij 4 = 4(sij) 4(s sSi-O = 1 SiNow lets calculate the valence of the Zr-O bonds. ZrVi = sij 4 = 8(sij) 8(s sZr-O = 1/2 ZrVi = 2 = n1 = n1sSi-O + Sin1(1) + 1 n2 n2sZr-O Zrn2(1/2) = 2 Now lets determine the coordination number of oxygen. 4 Pauling's 3rd Rule The presence of shared edges and particularly shared faces decreases the stability of a structure. This is particularly true for cations with large valences and small coordination number. When polyhedra share a common edge or face it brings the cations closer together, thereby increasing electrostatic repulsions. Tetrahedra Octahedra Cation-Cation Distance CationCorner Edge Face 2 M-X M2 M-X M1.16 M-X M1.41 M-X M- 0.67 M-X M1.16 M-X M- In a crystal containing different cations those with large valence and small coordination number tend not to share polyhedron elements with each other. The logic behind this rule comes in part from the bonding preferences preferences of the anion. Consider the following example. CaWO4 has the scheelite structure where the W+6 ions are tetrahedrally coordinated. The valence of the W-O bond is: WIf you were to propose a structure where two WO4 tetrahedra shared a corner the shared oxygen would be bound to 2 W+6 ions. Its valence is: Pauling's 4th Rule Vi = sij 6 = 4(sW-O) 4(s sW-O = 1.5 This is clearly too high and a violation of the second rule. This effect is This exacerbated as the cation valence increases its coordination number number decreases, because sij increases. VO = 2(1.5) = 3 VO = 2 sW-O 5 Pauling's 5th Rule The number of chemicaly different coordination environments for a given ion in a crystal tends to be small. (Rule of Parsimony) We have already discussed how ionic forces favor high symmetry and regular coordination environments. This is a generalization of this concept. Quantitative Bond Valence Method A powerful advance on Pauling's bond valence method is to quantiquantitatively link the bond valence to the bond distance (rather than the oxidation state). While a number of scientists have helped in this this endeavor the principle driving force has largely been I.D. Brown from McMaster University.1,2 The bond valence is calculated using the following relationship: Where dij is the distance between atoms i and j, Rij is the empirically determined distance for a given cation-anion pair, and b is also an cationempirical value generally set equal to 0.37. Values of Rij that give bond valence sums near the oxidation state have been tabulated.3,4,5 I.D. Brown, Chem. Soc. Reviews 7, 359-376 (1978). 359I.D. Brown, "The chemical bond in inorganic chemistry: the bond bond valence model" Oxford Univ. Press, New York (2002). 3. Brown & Altermatt, Acta Cryst. B 41, 244-247 (1985). Cryst. 41, 244Altermatt, 4. Brese & O'Keeffe, Acta Cryst. B 47, 192-197 (1991). Cryst. 47, 192O'Keeffe, 5. O'Keeffe, Acta Cryst. A 46, 138-142 (1990). Cryst. 46, 138O'Keeffe, 1. 2. sij = exp [(Rij dij)/b] 6 Uses of the Bond Valence Method 1. To examine experimental structures for accuracy, determine oxidation states, or identify bonding instabilities. 2. To locate light atoms (i.e. H or Li) that are hard to find by X-ray Xdiffraction methods by examining the valences of the surrounding atoms. 3. To predict bond distances, as an alternative to ionic radii. This This can be done by inverting the equation: dij = Rij b{ln(sij)} Advantages over ionic radii include Independent of coordination number Can handle unsymmetrical coordination environments Bond Valences & Structural Analysis FeTiO3 (Ilmenite) AlOOH (Diaspore) Bond Distances Fe-O = 32.07, 32.20 Fe3 Ti-O = 31.88, 32.09 Ti3 Bond Valence Sums Fe = 30.40 + 30.28 = 2.04 3 3 Fe = 30.84 + 30.48 = 3.96 3 3 O = 0.40 + 0.28 + 0.84 + 0.48 = 2.00 Bond Distances O1-Al = 1.85, 1.85, 1.86 O1O2-Al = 1.97, 1.97, 1.98 O2Bond Valence Sums Al = 2.75 O1 = 1.60 OH group O2 = 1.16 7 Bond Valences Predictively CrO3 Two types of Oxide ions Bridging - IIO Terminal - IO Pauling's 2nd rule gives the expected bond valences Cr-IIO sij = 1.0 CrCr-IO sij = 2.0 CrInverting the bond valence calculation we can estimate distances Cr-IIO 1.79 CrCr-IO 1.54 CrThe observed bond distances are Cr-IIO 1.75 CrCr-IO 1.57 Cr- Bond Graph O O O Cr Space Group = P-1 Sp. Group = P21/n PChains of edge-sharing octahedra Corner sharing octahedra edgeMo-O Bond distances = Mo-O Bond distances = MoMo1.67, 1.73, 1.95, 1.95, 2.25, 2.33 Mo(1) 1.65, 1.79, 1.81, 2.09, 2.12, 2.20 Mo(2) 1.80, 1.87, 1.91, 1.92, 2.01, 2.23 -MoO3 -MoO3 8 The Mo6+ ions shift away from the shared edges. -MoO3 A closer look d=1.948 s = 0.90 IIIO d=1.734 s = 1.60 IO IIO d=2.332 s = 0.32 d=1.671 s = 1.90 d=2.251 s = 0.40 Bond Valence Sums Mo = 20.90 + 1.90 + 0.32 + 1.60 + 0.40 = 6.02 2 IO = 1.90 IIO = 1.60 + 0.40 = 2.00 IIIO = 0.90 + 0.90 + 0.32 = 2.12 Distorted Coordination Environments SrTiO3 CaTiO3 Ca-O Distances Ca1 2.36 2 2.37 1 2.46 2 2.62 2 2.65 4 > 3.0 How far should the tilting go? Can you tell from the ionic radii of Ca2+ and O2-? O2- (CN = 2) r = 1.21 Ca2+ (CN=12) r = 1.48 2+ (CN=8) r = 1.26 O2- (CN = 6) r = 1.26 Ca Ca2+ (CN=6) r = 1.14 9 Distorted Coordination Environments Ca-O Distances Ca1 2.36 2 2.37 1 2.46 2 2.62 2 2.65 Ca-O Valences Ca1 0.345 2 0.336 1 0.264 2 0.171 2 0.157 Vca = 1.94 Structure Prediction Using Bond Valences SPuDS is a software program developed for predicting crystal structures of cubic and distorted AMX3 & A2MM'X6 perovskites.It optimizes the structure perovskites. based on the idea of rigid octahedra and the bond valence method through the following steps. 1. Vary the M-X bond length, d, in order Mto optimize the valence of the octahedral cation(s). 2. Vary the octahedral tilting in order to optimize the valence of the A-site Acation(s). 25.0 SPuDS Literature Tilt Angle (/degrees) 20.0 15.0 10.0 5.0 0.0 0.86 0.88 0.90 0.92 0.94 Tolerance Factor 0.96 0.98 1.00 0.60 SPuDS X Literature X A Cation Distance () from High Symmetry Location in X and Z 0.50 SPuDS Z Literature Z 0.40 0.30 0.20 0.10 0.00 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 Tolerance Factor M. Lufaso & P.M. Woodward, Acta Cryst. B 57, 725-738 (2001) 10 ...
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This note was uploaded on 06/11/2011 for the course CHEM 101 taught by Professor Stegemiller during the Spring '07 term at Ohio State.

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