Radius Ratio Rule

In an ionic structure each cation tends to surround itself with anions; the number that can be grouped around it will depend on the relative size of the cations and anions. The Coordination Number (CN) is defined as the number of anions that can fit around a cation. This number increases as the radius ratio increases. The number of anions that can ‘fit’ around a cation is related to the relative size difference between the ions, and this size difference can be described using the radius ratio, which is given by: r cation /r anion

When this number is small, then only a few anions can fit around a cation. When this number is large, then more anions can fit around a cation. When CN is 4, it is known as tetrahedral coordination; when it is 6, it is octahedral; and when it is 8, it is known as cubic coordination. See the following table.

3 4 6 8 12 Corners of equilateral triangle 0.15 - 0.22 Coordination Number (CN) Arrange of Anions around the cation Radius Ratio 3 Corners of equilateral triangle 0.15 - 0.22 4 Corners of a tetrahedron 0.22 - 0.41 6 Corners of an octahedron 0.41 – 2.42 8 corners of a cube 0.73 – 1.37 12 Midpoints of cube edges 1

Holes in which positive ions pack Radius ratio Coordinate number Holes in which positive ions pack 0.225 – 0.414 4 Tetrahedral holes ZnS,CuCl 0.414 – 2.42 6 Octahedral holes NaCl,MgO 1 12 Dodecahedral 0.732 – 1.37 8 Cubic holes CsCl,NH4Br N.B. the C.N. = 12 not found in simple ionic crystals but in the complicated metal oxides

IONIC CRYSTAL TYPES Ionic crystal type Co-ordination number A X Structure type AX AX2 AX3 6 6 8 8 6 3 8 4 6 2 NaCl CsCl Rutile(TiO2) Fluorite (CaF2) ReO3

Example ZnS in which rZn2+ / rS2- = 74 / 184 = 0.4 By using the calculations of radius ratio we can predict the coordination number . Example ZnS in which rZn2+ / rS2- = 74 / 184 = 0.4 From this ratio we can expect that Zn2+ ions prefer to fill the tetrahedral holes in the crystal lattice of S2- ions

By the same method we can predict that Na+ ions will prefer the octahedral holes in the close pack crystal of Cl- ions (rNa+ / rCl-) = 95 / 181 = 0.52 The crystal structure of NaCl is of C.N.=6

Holes in which positive ions pack Radius ratio Coordinate number Holes in which positive ions pack 0.225 – 0.414 4 Tetrahedral holes ZnS,CuCl 0.414 – 2.42 6 Octahedral holes NaCl,MgO 1 12 Dodecahedral 0.732 – 1.37 8 Cubic holes CsCl,NH4Br

IONIC CRYSTAL TYPES Ionic crystal type Co-ordination number A X Structure type AX AX2 AX3 6 6 8 8 6 3 8 4 6 2 NaCl CsCl Rutile(TiO2) Fluorite (CaF2) ReO3

rcation / ranion ranion / rcation In this case there is two types of calculations : rcation / ranion ranion / rcation 1 e.g. CaF2 C.N.=8 ranion / rcation = 0.133 / 0.1= 1.33 2 C.N.=8 But the no. of F- ions must be twice of Ca2+ so the C.N. of Ca2+ ions twice that of F- ions so C.N. of Ca2+ = 8 and C.N. of F- = 4

The structure of SnO2 is the same as the structure of Rutile (TiO2) Example : SnO2 rSn2+ / rO2- = 69 / 140 = 0.49 C.N.=6 ro2- / rSn2+ = 140 / 69 = 2.03 C.N.=6 The structure of SnO2 is the same as the structure of Rutile (TiO2) But the no. of O2- ions must be twice of Sn2+ so the C.N. of Sn2+ ions twice that of O2- ions so C.N. of Sn2+ = 6 and C.N. of O2- = 3

Example : Rb2O C.N.=8 C.N.=8 rRb+ / rO2- = 148 / 140 = 1.06 ro2- / rRb+ = 140 / 148 = 0.95 C.N.=8 The structure of this oxide is inverse that of fluorite which means that C.N. of O2- = 8 and C.N. of Rb+ = 4

Exceptions

This radius ratio rule allows to predict the correct coordination no This radius ratio rule allows to predict the correct coordination no. of ions in the crystal lattice. BUT 1.This rule must be used under precautions , specially in covalent bonding , e.g. CdS , we can find the radius ratio = 0.53 which leads to C.N.=6 Although the structure of this compound is like the structure of ZnS in which C.N.=4 . This means that the covalent bonding between Cd and S prefer C.N.=4

2. LiI, LiBr and LiCl have radius ratio less than 0 2. LiI, LiBr and LiCl have radius ratio less than 0.414 although all of them have the NaCl structure . Covalent Bonding Not pure Ionic Bonding less than 100% ionic bonding 3. The alkali metals of covalent bonding .

As the size of positive ion decreases as the negativity of lattice energy increases which mean more stability in crystal lattice . Ex. ZnS which has the zinc Blende structure in which ZnS has lattice energy more than that if take any crystal structure else

Predicting of Structure of complex of ionic compounds

This part is dealing with the structure of complex ionic compounds Here we are dealing with two types of complex ionic compounds : 1. Compounds of general formula A2BO4 , since A is the metal and B is another metal of higher valence than A or could be nonmetal . Spinel Structure

2. Perovskite structure AB2O3