CHEMCA
EXAM MASTER FORMULA SHEET
The Solid State
High-Yield Revision for JEE Main, Advanced & NEET
1. Cubic Unit Cell Analysis
| Property | Simple Cubic (SCC) | Body Centered (BCC) | Face Centered (FCC/CCP) | HCP (Hexagonal) |
|---|---|---|---|---|
| Effective Atoms ($Z$) | 1 | 2 | 4 | 6 |
| Relation ($a$ & $r$) | $a = 2r$ | $a = \frac{4r}{\sqrt{3}}$ | $a = 2\sqrt{2}r$ | $a=2r, c=4r\sqrt{2/3}$ |
| Nearest Distance ($d$) | $d = a$ | $d = \frac{\sqrt{3}a}{2}$ | $d = \frac{a}{\sqrt{2}}$ | $d = a$ |
| Coordination No. | 6 | 8 | 12 | 12 |
| Packing Efficiency | 52.4% | 68.0% | 74.0% | 74.0% |
2. Important Ionic Crystal Structures
Rock Salt (NaCl) Type
- • $\ce{Cl^-}$ forms FCC lattice ($Z=4$).
- • $\ce{Na^+}$ occupies all Octahedral Voids ($Z=4$).
- • Coordination Number: 6 : 6.
- • Edge length: $a = 2(r_+ + r_-)$.
- Examples: Halides of Li, Na, K, Rb; $\ce{AgF, AgCl, AgBr}$.
Cesium Chloride (CsCl) Type
- • $\ce{Cl^-}$ at corners (Simple Cubic).
- • $\ce{Cs^+}$ occupies Cubic Void (body center).
- • Coordination Number: 8 : 8.
- • Body diagonal: $\sqrt{3}a = 2(r_+ + r_-)$.
- Examples: $\ce{CsBr, CsI, TlCl}$.
Fluorite (CaF₂) Type
- • $\ce{Ca^{2+}}$ forms FCC lattice ($Z=4$).
- • $\ce{F^-}$ occupies all Tetrahedral Voids ($Z=8$).
- • Coordination Number: 8 : 4 (Cation : Anion).
- Examples: $\ce{BaF2, SrF2, CdF2}$.
Anti-Fluorite (Na₂O) Type
- • $\ce{O^{2-}}$ forms FCC lattice ($Z=4$).
- • $\ce{Na^+}$ occupies all Tetrahedral Voids ($Z=8$).
- • Coordination Number: 4 : 8 (Cation : Anion).
- Examples: $\ce{Li2O, K2O, Rb2O}$.
Zinc Blende (Sphalerite - ZnS) Type
- • $\ce{S^{2-}}$ forms FCC lattice ($Z=4$).
- • $\ce{Zn^{2+}}$ occupies alternate (half of) Tetrahedral Voids ($Z=4$).
- • Coordination Number: 4 : 4.
- Examples: $\ce{CuCl, CuBr, CuI, AgI}$. (Note: Wurtzite is the HCP variant of ZnS).
3. Density Calculation & Bragg's Law
Density of Unit Cell ($\rho$):
\[ \rho = \frac{Z \times M}{a^3 \times N_A} \]
Crucial Units for Calculation:
If $a$ is in pm, use $a^3 \times 10^{-30}$ for volume in $cm^3$.
$\rho$ will be in $\text{g/cm}^3$. $M$ = Molar mass in g/mol.
Bragg's Equation (X-ray Diffraction):
\[ n\lambda = 2d \sin \theta \]
$n$ = Order of reflection (1, 2, 3...)
$\lambda$ = Wavelength of X-rays
$d$ = Interplanar spacing. For cubic crystal with Miller indices $(h, k, l)$:
\[ d = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \]
4. Voids & Radius Ratio Rule
Tetrahedral Voids (TV)
- • Formed by 4 spheres.
- • Quantity: $2N$ (2 per atom in lattice).
- • Location in FCC: 2 voids on each body diagonal.
- • Distance from corner: $\frac{\sqrt{3}a}{4}$
Octahedral Voids (OV)
- • Formed by 6 spheres.
- • Quantity: $N$ (1 per atom in lattice).
- • Location in FCC: At the Body Center (1) and Edge Centers ($12 \times \frac{1}{4} = 3$).
- • Distance from corner: $\frac{a}{2}$
Radius Ratio Rule ($r_+ / r_-$):
| Ratio Range | Coord. No. | Geometry | Example |
|---|---|---|---|
| 0.155 – 0.225 | 3 | Trigonal Planar | $\ce{B2O3}$ |
| 0.225 – 0.414 | 4 | Tetrahedral | $\ce{ZnS}$ |
| 0.414 – 0.732 | 6 | Octahedral | $\ce{NaCl}$ |
| 0.732 – 1.000 | 8 | Cubic | $\ce{CsCl}$ |
5. Crystal Defects
Schottky Defect
Equal number of cations and anions are completely missing from their lattice sites.
- • Effect: Density Decreases.
- • Condition: High Coordination No., similar sizes of cation/anion.
- • Examples: $\ce{NaCl, KCl, CsCl, AgBr}$.
Frenkel Defect
An ion (usually the smaller cation) leaves its normal site and occupies an interstitial site.
- • Effect: Density Remains Constant.
- • Condition: Low Coordination No., large difference in ion sizes.
- • Examples: $\ce{ZnS, AgCl, AgBr, AgI}$.
AgBr exhibits BOTH Schottky and Frenkel defects!
Non-Stoichiometric Defects:
-
Metal Excess Defect (F-Centers): Anion vacancy occupied by an unpaired electron. Imparts Color and Paramagnetism.
E.g., Yellow $\ce{NaCl}$, Pink $\ce{LiCl}$, Lilac $\ce{KCl}$. - Metal Excess (Interstitial Cations): Extra cation in interstitial site with electron in another site. E.g., $\ce{ZnO}$ turns yellow on heating.
- Metal Deficiency Defect: Cation vacancy compensated by higher oxidation state of nearby metal. E.g., $\ce{FeO}$, $\ce{FeS}$, $\ce{NiO}$ (Transition metals).
Impurity Defects:
Addition of impurities of ions with different valency.
Example: Addition of $\ce{SrCl2}$ to $\ce{NaCl}$. Each $\ce{Sr^{2+}}$ replaces two $\ce{Na+}$ ions. It occupies one site and leaves the other site vacant.
Number of Cation Vacancies = Number of $\ce{Sr^{2+}}$ ions added.
6. Magnetic & Electrical Properties
Magnetic Alignments (Domains)
Ferromagnetic: Domains align parallel $\uparrow\uparrow\uparrow\uparrow$. Strong attraction, permanent magnetism. (e.g., $\ce{Fe, Co, Ni, CrO2}$).
Antiferromagnetic: Domains align anti-parallel in equal numbers $\uparrow\downarrow\uparrow\downarrow$, cancelling out. Net $\mu=0$. (e.g., $\ce{MnO}$).
Ferrimagnetic: Domains align anti-parallel in unequal numbers $\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow$. Weak net attraction. (e.g., $\ce{Fe3O4, MgFe2O4}$).
Semiconductors (Doping)
n-type (Electron rich): Doping Group 14 ($\ce{Si, Ge}$) with Group 15 elements ($\ce{P, As, Sb}$). Extra electron conducts electricity.
p-type (Electron deficient): Doping Group 14 ($\ce{Si, Ge}$) with Group 13 elements ($\ce{B, Al, Ga}$). Creates electron "holes" which conduct electricity.
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