Measuring Lattice Energies Experimentally
M+(g) + X(g) MX(s) H ~ Elat
This is a difficult measurement - hard to get enough gas phase ions to achieve big enough heats
Thermochemical cycles: Born-Haber cycles
The sum of the known reactions gives the desired reaction; summing the energies gives the lattice energy. Born-Haber cycles must be set up for each compound to account for the correct states of matter, stoichiometry, etc.
|Na(s) Na(g)||Hsublimation = +107.8 kJ/mole (M atomization)|
|½Br2(l) ½Br2(g)||½Hvaporization = ½(+53.4) = + 26.7 kJ/mole|
|½Br2(g) Br(g)||½Hdissociation = ½(+190.2) = +95.1 kJ/mole|
|(Xatomization = ½Hvaporization + ½Hdissociation)|
|Na(g) Na+(g) + e||IP = +495.4 kJ/mole|
|Br(g) + e Br(g)||EA = 324.6 kJ/mole|
|Na+(g) + Br(g) NaBr(s)||Elat|
|NaBr(s) Na(s) + ½Br2(l)||Hf = (361.4) = 361.4 kJ/mole|
Summing all of these reactions gives nothing so that summing the energies must add to zero:
Hsublimation + ½Hvaporization + ½Hdissociation + IP EA Elat Hf = 0
Elat = +107.8 + 26.7 + 95.1 + 495.4 324.6 + 361.4 = 761.8 kJ/mole
Compares to 712.4 kJ/mole from Born-Lande equation
With such large lattice energies, why do ionic compounds dissolve in water?
Consider another Born-Haber cycle:
|Na+(aq) + Cl(aq)NaCl(s)||Hsolvation|
|NaCl(s) Na+(g) + Cl(g)||Elat|
So that Hsolvation + Elat + Esolv+ + Esolv = 0
Hsolvation is usually pretty small - a few 10s of kJ/mol; clearly much different from Elat
This says that Esolv+ + Esolv- ~ Elat but with opposite signs
Find Esolv in a manner similar to finding Elat.
r = ionic radius
Z = ionic charge
= solvent dielectric constant
+ or refers to either cation or anion Z or r
1. Cations are smaller than anions so that r is much smaller for cations than for anions. This means that Esolv for cations is much larger than for anions, i.e., solvation is caused primarily by the cations and not the anions.
2. Solvation energy increases with the square of ionic charge.
3. Solvents with high dielectric constants solvate ionic compounds better;
a dielectric constant near 1 will give near zero solvation energy (water
has ~ 87 at room T)
Deviations from ionic character:
Consider AgCl: has NaCl structure, r0 = 2.77 Å (compared to 2.81 Å for NaCl) yet AgCl is only sparingly soluble in water, mp is ~350 oC lower than NaCl (801 vs 455 oC), Lattice energy is Elat = 907.4 kJ/mol (compared to 786.8 kJ/mol for NaCl).
Why so different?
Polarization effects (covalency): AgCl has more covalent character than NaCl so has less classical ionic character. Consider the limits of bonding: the difference between polarized ionic and polar covalent is merely one of degree. Polarization of "ionic" compounds occurs when the cation is very small and the anion is very large and polarizable. This also can occur in transition metal ions where the d shell electrons are more polarizable than s or p electrons. Transition metals are also more polarizing to anions because of a larger Z*.
Since ions are basically spherical objects, the structures of ionic compounds arise from packing of spheres. Generally the most effective way to do this is to put objects as close to each other as possible (closest packed structures). There are two common closest packed structures.
Hexagonally closest packed (hcp): an ABAB sequence of spheres
solid circles are the 1st, 3rd, 5th, etc., layers open circles are the 2nd, 4th, 6th, etc. layers
There are two types of vacancies or holes: those with four neighbors (Td holes) and those with six neighbors (Oh holes). ~74% of the volume is filled; each sphere has twelve nearest neighbors adjacent to it
Cubic closest packed: ABCABC sequence of spheres
ccp = face centered cubic, i.e. draw a cube and put spheres at the corners and the center of each of the faces
Ionic compounds nearly always form closest packed structures if the ions are simple and spherical (nonspherical ions change things). Normally, the largest ion (typically the anion) forms the lattice and the smaller ion fills the holes - either Td or Oh holes.
When closest packed structures are not formed (and even when they are), the usual way of thinking about solid state crystalline structures is to find the smallest unit that repeats itself via translational symmetry over the whole macroscopic structure. This smallest repeat unit is called the unit cell. There are 14 types of unit cells called Bravais Lattices. The atoms or ions in the crystal do not determine the unit cell; it is determined by the translational symmetry!
To determine the Bravais lattice, 6 parameters are specified, 3 distances and 3 angles, and the occupation of the cell
Ionic Radii - the effective radius of an ion in a crystal
We assume that ionic radius is constant from compound to compound
How are these found?
Crystal structures only give internuclear distances r0
Assume that ro = r+ + r (r+ = cation radius r = anion radius)
More information is still required to find the ionic radii.
Landé assumed that in LiI the iodides "touched" and that the Li+ fit completely in the holes
d = r + r
ro = r+ + r
So now both radii can be determined.
Note: ionic radii are coordination number (CN) dependent.
Thus, radii are reported with CN, since this is important as well.
ro = r+ + r and where Z±* is the effective nuclear charge found from Slater's rules
Thus, so that the two necessary conditions are given to solve for the radii
This approach works well for univalent (Z=1) ions but breaks down for higher ionic charges so Pauling introduced a correction:
where Z is the ion charge, n is the Born exponent, and r is the radius
These corrections lead to a self consistent set of radii, fairly similar to the Landé approach
In real life lattices are not perfect; defects always exist, even at 0 K (this is thermodynamically required: enthalpy prefers perfection but entropy prefers randomness; get a balance with mostly perfect but a few defects).
Schottky defects: equal numbers of cations and anions are missing
from the lattice
Frenkel defects: small ion (usually the cation) moves to interstitial
site creating a hole that doesn't belong or pushing larger ion off the lattice site
Nonstoichiometric Compounds: many compounds do not follow the freshman chemistry idea that stoichiometry of compounds is in the ratio of small whole numbers.
FeO is more typically is found as FexO where x~0.95-0.99
Charge neutrality must still be maintained so a better description is (Fe2+)3x-2(Fe3+)2-2xO
F centers: e become part of the lattice
NaCl + Na Na1+(Cl)(e)