“1, 2, 4”
Atoms per unit cell
Simple cubic 1, body centred 2, face centred 4. Almost every solid state numerical starts by picking the right Z.
Chemistry, Class 11 and 12 · Gujarat Board, CBSE, GUJCET, JEE and NEET
The Solid State formulas
Solid state is geometry with chemistry attached. Unit cells, packing and density make up nearly all of it, and every numerical uses the same density formula, so it is one of the fastest chapters to secure.
18 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Unit cells
| What | Formula | Symbols and notes |
|---|---|---|
| Simple cubic | atoms per cell Z = 1 | corners only, 8 × ⅛ |
| Body centred cubic | Z = 2 | corners plus one at the centre |
| Face centred cubic | Z = 4 | corners plus six face centres, each shared by two |
| Edge and radius, simple cubic | a = 2r | atoms touch along the edge |
| Body centred cubic | a√3 = 4r | atoms touch along the body diagonal |
| Face centred cubic | a√2 = 4r | atoms touch along the face diagonal |
Packing and density
| What | Formula | Symbols and notes |
|---|---|---|
| Packing efficiency, simple cubic | 52.4% | the most wasteful arrangement |
| Packing efficiency, bcc | 68% | better, and common in alkali metals |
| Packing efficiency, fcc and hcp | 74% | the closest packing possible for equal spheres |
| Coordination number | simple cubic 6, bcc 8, fcc and hcp 12 | nearest neighbours touching an atom |
| Density of a unit cell | ρ = ZM/(a³NA) | a in metres gives density in kg m−3 |
| Voids in close packing | 2 tetrahedral and 1 octahedral per atom | for N atoms, 2N and N respectively |
| Radius ratio | r+/r− decides the geometry | 0.225 to 0.414 tetrahedral, 0.414 to 0.732 octahedral |
Defects
| What | Formula | Symbols and notes |
|---|---|---|
| Schottky defect | a cation and an anion both missing | density falls; seen in NaCl, KCl |
| Frenkel defect | an ion moves to an interstitial site | density unchanged; seen in AgCl, ZnS |
| Metal excess defect | gives colour, as F-centres | for example NaCl heated in sodium vapour turns yellow |
| n-type semiconductor | doped with a group 15 element | extra electrons carry the current |
| p-type semiconductor | doped with a group 13 element | holes carry the current |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
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Memory hooks
How to remember this chapter.
“Edge, face, body: 2, √2, √3”
Relating a and r
Simple cubic touches along the edge, fcc along the face diagonal, bcc along the body diagonal. a = 2r, a√2 = 4r, a√3 = 4r.
“Schottky slims, Frenkel shuffles”
The two defects
Schottky removes a pair of ions, so density falls. Frenkel just moves an ion inside, so density is unchanged.
Where marks go
The mistakes that cost the most.
- Using Z = 1 for a face centred cell. Corners give 1, but the six faces add 3 more.
- Forgetting to convert the edge length from picometres to metres before finding density.
- Mixing up which defect changes the density. Only Schottky does.
- Assuming fcc and hcp differ in packing efficiency. Both are 74%.
- Using the radius ratio rule without checking that the cation is the smaller ion.
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