JEE Chemistry The Solid State Notes Set 01

Types of Solids

A solid is called amorphous when its particles have only short-range order and no repeating pattern over long distances. Glass, rubber and plastics are typical examples. They soften gradually on heating and have no sharp melting point.
A crystalline solid has particles (atoms, ions or molecules) arranged in a regular pattern that repeats throughout the crystal. It has a sharp melting point and is anisotropic, that is, properties such as refractive index and electrical conductivity can differ with direction.

Lattice and Unit Cell

A space lattice is the three-dimensional array of points that shows how the particles of a crystal are repeated. The smallest repeating block, which on stacking in all directions reproduces the whole lattice, is the unit cell. A unit cell is described by three edge lengths a, b, c and three angles \( \alpha, \beta, \gamma \).
A primitive cell has particles only at its corners. Cells with an extra particle at the body centre (body-centred), at the centre of every face (face-centred) or at the centre of one pair of opposite faces (end-centred) are called centred cells.

Seven Crystal Systems and Fourteen Bravais Lattices

The numbers of lattices in the seven systems are 3, 2, 4, 1, 1, 2 and 1, which add up to 14 Bravais lattices.

Particles per Unit Cell (Z)

A particle at a corner is shared by 8 cells, on a face by 2 cells, on an edge by 4 cells, and inside the cell by none. So its contribution to one cell is 1/8, 1/2, 1/4 and 1 respectively.

Packing efficiency = (volume occupied by particles in the cell / volume of the cell) \(\times\) 100. For FCC: \( \dfrac{4 \times \tfrac{4}{3}\pi r^3}{(2\sqrt{2}\,r)^3} = \dfrac{\pi}{3\sqrt{2}} = 0.74 \).

Close Packing

In a single layer, spheres can be arranged in square close packing (each touches 4 neighbours, coordination number 4) or hexagonal close packing (each touches 6, coordination number 6). Hexagonal packing is denser, because every sphere sits in the gaps of the line before it.
In three dimensions, the second layer placed in the depressions of the first can be followed by a third layer in two ways. If the third layer lies exactly above the first, the sequence is ABAB... and the structure is hexagonal close packed (hcp). If the third layer lies over the gaps not used by the second layer, the sequence is ABCABC... and the structure is cubic close packed (ccp), equal to the face-centred cubic lattice. In both, each sphere has 12 neighbours and 74% of the space is filled.

Interstitial Voids

The gaps left between close-packed spheres are of two kinds. A tetrahedral void is surrounded by 4 spheres, and an octahedral void by 6 spheres.
For N close-packed spheres there are N octahedral voids and 2N tetrahedral voids. An FCC cell therefore has 4 octahedral voids (one at the body centre and 12 edge centres each shared by 4 cells, giving 1 + 3) and 8 tetrahedral voids (all wholly inside the cell).
Radius of the sphere that just fits: tetrahedral void 0.225 R and octahedral void 0.414 R, where R is the radius of the host sphere.

Formula of an Ionic Solid from a Unit Cell

Count each kind of ion using the sharing fractions, then write the simplest whole-number ratio. If the ions occupy voids of a close-packed lattice of N ions, use N octahedral and 2N tetrahedral voids and multiply by the fraction of voids filled.

Worked Examples

Illustration 1. A solid has atoms X at the corners of a cube and Y at the face centres. Find the formula.
Answer:
X = 8 \(\times\) 1/8 = 1 and Y = 6 \(\times\) 1/2 = 3.
The formula is XY₃.

Illustration 2. Atoms A form a ccp lattice and atoms B occupy one-third of the tetrahedral voids. Find the formula.
Answer:
For N atoms of A, tetrahedral voids = 2N, so B = 2N/3.
A : B = N : 2N/3 = 3 : 2.
The formula is A₃B₂.

Illustration 3. A metal crystallises in BCC with edge 300 pm. Find the atomic radius.
Answer:
Along the body diagonal \( \sqrt{3}\,a = 4r \).
\( r = \dfrac{1.732 \times 300}{4} \) = 129.9 pm.