Chemistry 401

The Solid State

In solids, atoms or molecules are near each other to the point of 'touching' (high density)

In the simplest case, a solid can be thought of as an array of spheres put together in some arrangement (spherical symmetry certainly is a good assumption for metals, perhaps poorer for molecular solids)

The packing of spheres can fit into one of two extremes:

Crystalline Solids : x-rays diffract sharply which implies regular order on the scale of nm

spheres pack into a regular, repeatable fashion called the crystal lattice

the smallest repeat unit in the lattice is called the unit cell

the unit cells fit together snugly like bricks in a wall-this is translational symmetry

Amorphous Solids: x-rays scatter broadly and diffusively which implies no long range order

spheres pack together closely but there is no translational symmetry

no regular order to the unit cells (there still may be short range order)

like the pile of bricks before being laid into a wall

Closest packed structures

ABAB (hexagonally closest packed, hcp)

Coordination Number = CN = the number of nearest neighbors

T_d = tetrahedral holes (4 nearest neighbors, CN = 4)

O_h = octahedral holes (6 nearest neighbors, CN = 6)

ABCABC (cubic closest packed, ccp or face centered cubic, fcc)

fcc still has CN = 12, still 26% void volume, the unit cell has 4 spheres in it.

Closest packed structures are the most efficient way to pack spheres but are not required or even the most common.

Other common structures:

body centered cubic (bcc):

spheres only 'touch' along the body diagonal of the cubic unit cell

CN = 8, 2 spheres per unit cell, ~32% void volume

 

primitive cubic:

spheres only touch along cell edges, CN = 6, void volume ~48%

1 sphere per unit cell

The structure of metals often changes as a function of temperature - phase transitions occur - this is called polymorphism or polytypism

 

Alloys are form by mixing different types of metals together:

Substitutional alloys: one metal replaces another metal at some lattice sites. This requires that both types of metals be roughly the same size.

Interstitial alloys: the new atoms are introduced into the hole sites (called interstices) of the host lattice. The new atoms can occupy either T_d or O_h holes.

Ionic solids

Ionic materials form crystalline lattices like metals but the lattice structure is determined by one ion (the larger, usually the anion) and the other ion fits into the holes. Since the smaller ion may not just right, lattice expansions can occur.

 

What holds an ionic lattice together?

A simple model: assume each ion is represented by a point charge, then sum up all the Coulomb interactions.

E_Coul = N_AAq₊q_–                4πε_od

q₊, q_– are charges on the ion, d the distance between ions, ε₀ = permittivity of free space, N_A is Avogadro's number, A is the Madelung constant

The Madelung constant is a purely geometric factor that has been tabulated for many lattice types. It represents the potential energy associated with different relative geometries. d is the distance between ions so accounts for the different energy associated with different compounds with the same lattice type (hence, the same Madelung constant).

E_Coul is negative, thus attractive, but maximizes at d = 0, i.e. when anions and cation occupy the the same point in space. Need a repulsive term (to account for nuclear-nuclear repulsion when the ions are close enough) which is done in an ad hoc fashion.

Born-Mayer: E_rep = Ce^d*/d where C and d* are constants

Born-Landé: E_rep = B/dⁿ where B and n are constants

Then, E_lat = E_Coul + E_rep graphically:

The total lattice energy has a minimum which is the equilibrium value for d (i.e., the observed distance between ions in a lattice). E_rep is larger than E_Coul at small distances (for large n or d*) but the reverse is true at short distances. We evaluate the constants by using experimental data at equilibrium and compressibility (this gives the slope up the low d part of the graph)

E_lat = N_AAe²Z⁺Z^–                   4πε_od_o (1 – 1/n) Born-Landé

 

E_lat = N_AAe²Z⁺Z^–                   4πε_od_o (1 – d*/d_o) Born-Mayer

e = electronic charge, Z = ion charge in units of e (i.e., +1, +2, –1, –2, etc.)

Note: the lattice energy is only a slight modification of a simple dipole energy! (n ~ 5-10)

 

For NaCl

d_o = 2.814 Å = 2.814×10^–10 m

A = 1.748

n = 7.66

N_A = 6.022×10²³ mole^–1

e = 1.602×10^–19 C

4πε_o = 1.113×10^–10 J^–1 C² m^–1

E_lat = (6.022×10²³ mole^–1) (1.748)(1.602×10^–19 C)²(+1)(–1)                                                                                           (1.113×10^–10 J^–1 C² m^–1) (2.814×10^–10 m)(1 – 1/(7.66)) = –750.0 kJ/mol

experimental value = 787 kJ/mol

note the sign difference: E_lat is usually written without the negative sign (thermodynamically wrong)