2Crystal StructuresAtoms (and later ions) will be viewed as hard spheres. In the case of pure metals, the packing pattern often provides the greatest spatial efficiency (closest packing).Ionic crystals can often be viewed as a close-packed arrangement of the larger ion, with the smaller ion placed in the “holes” of the structure.
3Unit CellsCrystals consist of repeating asymmetric units which may be atoms, ions or molecules. The space lattice is the pattern formed by the points that represent these repeating structural units.
4Unit CellsA unit cell of the crystal is an imaginary parallel-sided region from which the entire crystal can be built up.Usually the smallest unit cell which exhibits the greatest symmetry is chosen. If repeated (translated) in 3 dimensions, the entire crystal is recreated.
5Close PackingSince metal atoms and ions lack directional bonding, they will often pack with greatest efficiency. In close or closest packing, each metal atom has 12 nearest neighbors.The number of nearest neighbors is called the coordination number. Six atoms surround an atom in the same plane, and the central atom is then “capped” by 3 atoms on top, and 3 atoms below it.
6Close PackingIf the bottom “cap” and the top “cap” are directly above each other, in an ABA pattern, the arrangement has a hexagonal unit cell, or is said to be hexagonal close packed.If the bottom and top “caps” are staggered, the unit cell that results is a face-centered cube. This arrangement is called cubic close packing.
8Close PackingEither arrangement utilizes 74% of the available space, producing a dense arrangement of atoms. Small holes make up the other 26% of the unit cell.
9Holes in Close Packed Crystals There are two types of holes created by a close-packed arrangement. Octahedral holes lie within two staggered triangular planes of atoms.
10Holes in Close Packed Crystals The coordination number of an atom occupying an octahedral hole is 6.For n atoms in a close-packed structure, there are n octahedral holes.
11Octahedral HolesThe green atoms are in a cubic close-packed arrangement. The small orange spheres show the position of octahedral holes in the unit cell. Each hole has a coordination number of 6.
12Octahedral HolesThe size of the octahedral hole = .414 rwhere r is the radius of the cubic close-packed atom or ion.
13Holes in Close Packed Crystals Tetrahedral holes are formed by a planar triangle of atoms, with a 4th atom covering the indentation in the center. The resulting hole has a coordination number of 4.
14Tetrahedral HolesThe orange spheres show atoms in a cubic close-packed arrangement. The small white spheres behind each corner indicate the location of the tetrahedral holes.
15Tetrahedral Holes The size of the tetrahedral holes = .225 r For a close-packed crystal of n atoms, there are 2n tetrahedral holes.The size of the tetrahedral holes = .225 rwhere r is the radius of the close-packed atom or ion.
16# of Atoms/Unit Cell Atoms in corners are ⅛ within the cell For atoms in a cubic unit cell:Atoms in corners are ⅛ within the cell
17# of Atoms/Unit Cell Atoms on faces are ½ within the cell For atoms in a cubic unit cell:Atoms on faces are ½ within the cell
18# of Atoms/Unit CellA face-centered cubic unit cell contains a total of 4 atoms: 1 from the corners, and 3 from the faces.
19# of Atoms/Unit Cell For atoms in a cubic unit cell: Atoms in corners are ⅛ within the cellAtoms on faces are ½ within the cellAtoms on edges are ¼ within the cell
20Other Metallic Crystal Structures Body-centered cubic unit cells have an atom in the center of the cube as well as one in each corner. The packing efficiency is 68%, and the coordination number = 8.
21Other Metallic Crystal Structures Simple cubic (or primitive cubic) unit cells are relatively rare. The atoms occupy the corners of a cube. The coordination number is 6, and the packing efficiency is only 52.4%.
22PolymorphismMany metals exhibit different crystal structures with changes in pressure and temperature. Typically, denser forms occur at higher pressures.Higher temperatures often cause close-packed structures to become body-center cubic structures due to atomic vibrations.
23Atomic Radii of MetalsMetallic radii are defined as half the internuclear distance as determined by X-ray crystallography. However, this distance varies with coordination number of the atom; increasing with increasing coordination number.
24Atomic Radii of MetalsGoldschmidt radii correct all metallic radii for a coordination number of 12.Coord # Relative radius
25AlloysAlloys are solid solutions of metals. They are usually prepared by mixing molten components. They may be homogeneous, with a uniform distribution, or occur in a fixed ratio, as in a compound with a specific internal structure.
26Substitutional Alloys Substitutional alloys have a structure in which sites of the solvent metal are occupied by solute metal atoms. An example is brass, an alloy of zinc and copper.
27Substitutional Alloys These alloys may form if:1. The atomic radii of the two metals are within 15% if each other.2. The unit cells of the pure metals are the same.3. The electropositive nature of the metals is similar (to prevent a redox reaction).
28Interstitial AlloysInterstitial alloys are solid solutions in which the solute atoms occupy holes (interstices) within the solvent metal structure. An example is steel, an alloy of iron and carbon.
29Interstitial AlloysThese alloys often have a non-metallic solute that will fit in the small holes of the metal lattice. Carbon and boron are often used as solutes. They can be dissolved in a simple whole number ratio (Fe3C) to form a true compound, or randomly distributed to form solid solutions.
30Intermetallic Compounds Some mixtures of metals form alloys with definite structures that may be unrelated to the structures of each of the individual metals. The metals have similar electronegativities, and molten mixtures are cooled to form compounds such as brass (CuZn), MgZn2, Cu3Au, and Na5Zn2.
31Ionic CompoundsSince anions are often larger than cations, ionic structures are often viewed as a close-packed array of anions with cations added, and sometimes distorting the close-packed arrangement.
32Common Crystal Types 1. The Rock Salt (NaCl) structure- Can be viewed as a face-centered cubic array of the anions, with the cations in all of the octahedral holes, or
33Common Crystal Types 1. The Rock Salt (NaCl) structure- A face-centered cubic array of the cations with anions in all of the octahedral holes.
34Common Crystal Types 1. The Rock Salt (NaCl) structure- The coordination number is 6 for both ions.
35Common Crystal Types 2. The CsCl structure- Chloride ions occupy the corners of a cube, with a cesium ion in the center (called a cubic hole) or vice versa. Both ions have a coordination number of 8, with the two ions fairly similar in size.
36Common Crystal Types 3. The Zinc-blende or Sphalerite structure- Anions (S2-) ions are in a face-centered cubic arrangement, with cations (Zn2+) in half of the tetrahedral holes.
37Common Crystal Types4. The Fluorite (CaF2) and Antifluorite structuresA face-centered cubic arrangement of Ca2+ ions with F- ions in all of the tetrahedral holes.
38Common Crystal Types4. The Fluorite (CaF2) and Antifluorite structuresThe antifluorite structure reverses the positions of the cations and anions. An example is K2O.
39Ionic RadiiIonic radii are difficult to determine, as x-ray data only shows the position of the nuclei, and not the electrons.Most systems assign a radius to the oxide ion (often 1.26Å), and the radius of the cation is determined relative to this assigned value.
40Ionic RadiiLike metallic radii, ionic radii seem to vary with coordination number. As the coordination number increases, the apparent ionic radius increases.
41Ionic Radii 1. Ionic radii increase as you go down a group. 2. Radii of ions of similar charge decrease across a period.3. If an ion can be found in many environments, its radius increases with higher coordination number.4. For cations, the greater the charge, the smaller the ion (assuming the same coordination #).5. For atoms near each other on the periodic table, cations are generally smaller than anions.
42Predicting Crystal Structures General “rules” have been developed, based on unit cell geometry, to predict crystal structures using ionic radii.Radius ratios, usually expressed as the (radius of the cation)/(radius of the anion) are used.
43Predicting Crystal Structures General “rules” have been developed, based on unit cell geometry, to predict crystal structures using ionic radii.Radius ratios, usually expressed as the (radius of the cation)/(radius of the anion) are used. This assumes that the cation is smaller than the anion.
45Energetics of Ionic Bonds The lattice energy is a measure of the strength of ionic bonds within a specific crystal structure. It is usually defined as the energy change when a mole of a crystalline solid is formed from its gaseous ions.M+(g) + X-(g) MX(s)
46M+(g) + X-(g) MX(s) ∆E = Lattice Energy Lattice energies cannot be measured directly, so they are obtained using Hess’ Law. They will vary greatly with ionic charge, and, to a lesser degree, with ionic size.
471/2 bond energy of Cl2Electron Affinity of ClIonization energy of KLattice Energy of KCl∆Hsub of K}∆Hf of KCl
48Ionic charge has a huge effect on lattice energy.
49Lattice EnergyAttempts to predict lattice energies are generally based on coulomb’s law:VAB = (Zae)(Zbe)4πεorABZa and Zb = charge on cation and anione= charge of an electron (1.602 x 10-19C)4πεo=permittivity of vacuum ( x 10-10J-1C2m-1)rAB = distance between nuclei
50Lattice EnergySince ionic crystals involve more than 2 ions, the attractive and repulsive forces between neighboring ions, next nearest neighbors, etc., must be considered.
51The Madelung ConstantThe Madelung constant is derived for each type of ionic crystal structure. It is the sum of a series of numbers representing the number of nearest neighbors and their relative distance from a given ion.The constant is specific to the crystal type (unit cell), but independent of interionic distances or ionic charges.
52Madelung Constants Crystal Structure Madelung Constant Cesium chlorideFluoriteRock salt (NaCl)SphaleriteWurtzite
53Estimating Lattice Energy Ec = NM(Z+)( Z-) e24πεorwhere N is Avogadro’s number, andM is the Madelung constant (sometimes represented by A)This estimate is based on coulombic forces, and assumes 100% ionic bonding.
54Estimating Lattice Energy A further modification, the Born-Mayer equation corrects for complex repulsion within the crystal.Ec = NM(Z+)( Z-) e2 (1-ρ/r)4πεorfor simple compounds, ρ=30pm
55Solubility of Ionic Crystals The dissolving of ionic compounds in water may be viewed in terms of lattice energy and the solvation of the gaseous ions. MX(s) M+(g) + X-(g) Lattice energy M+(g) + H2O(l) M+(aq) Solvation X-(g) + H2O(l) X-(aq) Solvation MX(s) ) + H2O(l) M+(aq) + X-(aq) ΔHsoln
56Solubility of Ionic Crystals Factors such as ionic size and charge, hardness or softness of the ions, crystal structure and electron configuration of the ions all play a role in the solubility of ionic solids. The entropy of solvation will also play a role in solubility.
57Ionic SizeSmaller ions have a stronger coulombic attraction for each other and also for water. They also have less room to accommodate the waters of hydration. Larger ions have weaker electrostatic attraction for each other and also for water. They also have accommodate more waters of hydration.
58Ionic SizeThe overall result of these factors result in low solubility of salts containing two large ions (soft-soft) or two small ions (hard-hard). For salts containing two small ions, especially with the same magnitude of charge, the greater lattice energy dominates, and cannot be easily overcome by the hydration energy of the ions.