Graphite and diamond are allotropes of carbon in which atoms are held by extended covalent bonding (network solids). Hence both are covalent crystals.
Corners: 8 corners × 1/8 = 1 A per unit cell. Faces: 6 faces × 1/2 = 3 B per unit cell. Formula = A1B3 = AB3.
In close packing (fcc/ccp) number of tetrahedral holes = 2N for N close-packed atoms. So ratio atoms : tetrahedral holes = 1 : 2.
Solid CO2 (dry ice) consists of discrete CO2 molecules held by weak van der Waals forces — a molecular solid.
Monoclinic sulphur is an example of the monoclinic crystal system. The monoclinic unit cell has unequal edges (a ≠ b ≠ c) with two angles 90° and one different (α = γ = 90°, β ≠ 90°). The reason correctly explains the assertion.
In fluorite (CaF2) Ca2+ is surrounded by 8 F- (cubic coordination) and each F- is tetrahedrally coordinated to 4 Ca2+. So CN(Ca2+) = 8, CN(F-) = 4.
Moles of X = 8/40 = 0.2 mol ⇒ number of atoms = 0.2 N_A. In bcc each unit cell has 2 atoms, so unit cells = (0.2 N_A)/2 = 0.1 N_A = 0.1×6.023×10^23 = 6.023×10^22.
In fcc (ccp) there are 4 M atoms per unit cell and 8 tetrahedral holes. If 1/2 of tetrahedral holes are occupied, N = 8×1/2 = 4. Thus M:N = 4:4 ⇒ formula MN. (Question OCR showed fraction; interpreted as 1/2.)
Radius ratio r+/r- = 0.98/1.81 ≈ 0.541. A radius ratio in the range 0.414–0.732 corresponds to octahedral coordination (CN = 6).
In CsCl structure the distance between Cs and Cl (corner to body centre) = (√3/2) a. So interatomic distance = (√3/2)×400 pm ≈ 0.866×400 = 346.4 pm. (Option d corresponds to the factor √3/2).
In NaCl (rocksalt) the anions form fcc: a = 2√2 r_–. Also along edge a/2 = r_+ + r_–. Combining: √2 r_– = r_+ + r_– ⇒ r_+ = (√2 − 1)r_– ⇒ r_– = r_+/(√2 − 1). For r_+ = 100 pm, r_– ≈ 100/0.4142 ≈ 241.4 pm (option corresponds to 100/0.414).
Packing efficiency of bcc = 68.0% occupied, so vacant space = 100% − 68% = 32%.
For fcc, a = 2√2 r. Thus a = 2√2×300 pm = 600×1.4142 ≈ 848.5 pm.
In simple cubic r = a/2. Fraction occupied = (volume of one atom)/(cell volume) = (4/3)πr^3 / a^3 = (4/3)π (a/2)^3 / a^3 = π/6.
Colouring in NaCl (yellow) arises from F-centres: electrons trapped in anion vacancies absorb visible light by electronic excitation.
r_sc = a/2; r_bcc = √3 a/4; r_fcc = √2 a/4. Hence r_sc : r_bcc : r_fcc = 1/2 : √3/4 : √2/4 = 1 : √3/2 : √2/2 (or multiplying by 4/a gives 2 : √3 : √2).
Corner at (0,0,0) and body centre at (a/2,a/2,a/2). Distance = √[(a/2)^2+(a/2)^2+(a/2)^2] = (√3/2) a.
Nearest neighbour distance d = 4.52×10^-10 m; for bcc a = 2d/√3 = 5.219×10^-10 m. Volume per cell = a^3 ≈ 1.421×10^-28 m^3. Mass per cell = 2×39/ N_A ≈ 1.295×10^-25 kg. Density = mass/volume ≈ 1.295×10^-25 /1.421×10^-28 ≈ 9.12×10^2 kg m^-3 ≈ 915 kg m^-3.
Schottky defect: vacancies of both cations and anions (equal numbers) are missing from lattice sites, preserving electrical neutrality.
Frenkel defect: a cation (usually smaller) moves from its lattice site to an interstitial site, creating a vacancy–interstitial pair.
Frenkel defect: a cation is displaced from its lattice site to an interstitial site; no ion leaves the crystal and stoichiometry is unchanged. Because no ions are lost, density does not decrease appreciably. Thus both the assertion and the reason are false.
FeO exhibits metal deficiency (non-stoichiometry) due to cation vacancies and variable oxidation states (Fe2+/Fe3+), giving a formula like Fe1–xO. NaCl and KCl are stoichiometric; ZnO commonly shows metal excess (oxygen vacancies or interstitial electrons).
Counting atoms in the repeating unit cell gives 1 X and 2 Y (corner/sharing contributions accounted). Hence simplest formula = XY2.
A unit cell is the smallest repeating structural unit of a crystal lattice that, by translation in three dimensions, reproduces the entire crystal. It is defined by the edge lengths a, b, c and interaxial angles α, β, γ and contains the arrangement of atoms for the crystal.
Additional features: composed of alternate cations and anions, generally soluble in polar solvents, crystalline and have high lattice energies.
1. High melting and boiling points due to strong electrostatic forces. 2. Hard but brittle (cleave along planes). 3. Conduct electricity only in molten state or in solution (ions mobile); poor conductors as solids.
Crystalline: atoms/molecules arranged periodically; Amorphous: only short-range order and random arrangement over long distances (e.g., glass, plastics).
Crystalline solids: long-range order, definite geometric shape, sharp melting points, anisotropic. Amorphous solids: no long-range order, no definite shape, show glass transition rather than sharp melting, isotropic.
Molecular solids: held by van der Waals (P4, I2). Metallic: delocalised electrons (brass). Covalent network: strong covalent bonds throughout (diamond). Ionic: electrostatic attraction between ions (NaCl).
a. P4 — molecular solid; b. Brass — metallic (alloy) solid; c. Diamond — covalent (network) solid; d. NaCl — ionic solid; e. Iodine (I2) — molecular solid.
Brief descriptions: - Cubic: a=b=c, α=β=γ=90°. (e.g. NaCl, fcc, bcc) - Tetragonal: a=b≠c, α=β=γ=90°. - Orthorhombic: a≠b≠c, α=β=γ=90°. - Monoclinic: a≠b≠c, α=γ=90°, β≠90°. - Triclinic: a≠b≠c, α≠β≠γ≠90° (no right angles). - Hexagonal: a=b≠c, α=β=90°, γ=120° (e.g. graphite, hcp metals). - Trigonal/rhombohedral: a=b=c, α=β=γ≠90° (a rhombohedron).
The seven crystal systems (types of unit cells): cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, trigonal (rhombohedral).
Other differences: symmetry — hcp is hexagonal lattice, ccp is cubic (face-centered). Examples: Mg (hcp), Cu (fcc).
hcp stacking sequence = ABAB...; ccp (fcc) stacking sequence = ABCABC.... Both have packing efficiency 74% and coordination number 12, but unit cells differ: hcp has 6 atoms per conventional cell, ccp (fcc) has 4 atoms per unit cell.
Locations: tetrahedral voids lie between a triangular face and opposite atom; octahedral voids lie at edge centers and body centers of close-packed arrays.
Tetrahedral void: formed by four atoms surrounding a small sphere; coordination number 4; number of tetrahedral voids = 2 per atom in close packing; radius ratio r/R ≈ 0.225 for fit. Octahedral void: formed by six atoms; coordination number 6; number of octahedral voids = 1 per atom; radius ratio r/R ≈ 0.414.
Point defects are localized imperfections involving one or a few atomic or ionic sites in a crystal (zero-dimensional defects). Types include:
- Vacancies (missing atoms/ions, e.g., Schottky defect)
- Interstitials (extra atoms/ions in interstitial sites, e.g., Frenkel defect)
- Substitutional impurities (foreign atoms replacing host atoms)
- Interstitial impurities (foreign atoms in interstitial sites)
They affect properties such as density, electrical conductivity and diffusion.
Schottky defect: paired vacancies of both cations and anions in an ionic solid such that stoichiometry is maintained (e.g., NaCl, KCl). It lowers the density of the solid and increases ionic mobility; its concentration depends on temperature and lattice energy.
In metal excess two mechanisms: (i) metal ions in interstitial sites or (ii) anion vacancies with trapped electrons (F-centres). Metal deficiency often arises from cation vacancies compensated by higher oxidation states.
Metal excess: occurs when there are extra metal atoms or electrons (e.g. Na excess in NaCl gives F-centres — electrons trapped in anion vacancies causing colour). Metal deficiency: occurs when metal sites are vacant or metals have mixed valency causing vacancies (e.g. FeO is metal deficient: some Fe2+ oxidises to Fe3+ and vacancies form).
Contribution: 8 corner atoms × 1/8 = 1; 6 face-centred atoms × 1/2 = 3; total = 1 + 3 = 4 atoms.
4 atoms per fcc unit cell.
Stacking sequences: - AAAA: every layer identical (no offset). - ABAB: second layer fits into holes of first, third layer repeats first. - ABCABC: third layer occupies a different set of holes so that three distinct layers repeat. (All represent arrangements of close-packed spheres with packing efficiency 74%.)
AAAA: each layer lies directly above previous (simple hexagonal stacking). ABAB: alternate layers repeat (hexagonal close packing, hcp). ABCABC: three-layer repeat (cubic close packing, ccp or fcc).
Thus rigidity gives hardness but directional shift under stress results in fracture rather than plastic deformation.
Hardness: strong electrostatic (ionic) bonds hold ions rigidly in place. Brittleness: when a stress shifts ionic planes, like-charged ions can be brought adjacent causing strong repulsive forces and the crystal cleaves along planes.
Derivation: bcc has 2 atoms/cell. Let atomic radius = R. Cell edge a = 4R/√3. Volume occupied by atoms = 2 × (4/3)πR^3. Volume of cell = a^3 = (4R/√3)^3. Efficiency = [2(4/3)πR^3]/(4R/√3)^3 = (π√3)/8 ≈ 68.02%.
Packing efficiency (bcc) = (π√3)/8 ≈ 0.6802 = 68.02%.
In a square close-packed (square lattice) layer each atom/molecule has four nearest neighbours (one at each side), so 2D coordination number = 4.
4
In bcc a central atom touches eight corner atoms (cube corners), so CN = 8.
Coordination number: the number of nearest neighbouring atoms/ions surrounding a given atom/ion in a structure. In bcc the coordination number = 8.
Given: a = 288 pm = 2.88 × 10^-8 cm, a^3 = (2.88 × 10^-8)^3 = 2.3888 × 10^-23 cm^3. Density ρ = 7.2 g cm^-3. Mass of one unit cell = ρ·a^3 = 7.2 × 2.3888×10^-23 = 1.7199×10^-22 g. For bcc, n = 2 atoms/cell so mass per atom = 1.7199×10^-22 / 2 = 8.5995×10^-23 g. Number of atoms in 208 g = 208 / 8.5995×10^-23 ≈ 2.417×10^24 ≈ 2.42×10^24 atoms.
2.42 × 10^24 atoms (approximately).
In fcc (ccp) the face diagonal = 4r and also = a√2. Hence a = 4r/√2 = 2√2·r. With r = 125 pm,
\(a = 2\sqrt{2}\times125\,\text{pm} = 250\times1.4142\,\text{pm} \approx 353.6\,\text{pm}.\)
a = 2√2 r = 353.6 pm
10^-2 mol% = 10^-2% = 0.01% = 10^-4 (fraction). Doping with SrCl2 introduces Sr^{2+} ions; each Sr^{2+} in a Na^+ site requires one Na^+ vacancy for charge neutrality. Thus vacancy concentration = mole fraction of SrCl2 = 10^-4 (i.e. 0.01%).
1.0×10^-4 (fraction) = 0.01% vacancies
For NaCl type (fcc) unit cell: 4 formula units per cell. M(KF)=39.10+19.00=58.10 g·mol^{-1}. Density ρ=2.48 g·cm^{-3}.
Cell volume V = 4M/(N_A·ρ).
V = 4×58.10/(6.022×10^{23}×2.48) ≈ 1.557×10^{-22} cm^{3}.
a = V^{1/3} ≈ 5.37×10^{-8} cm = 537 pm.
Nearest‑neighbour (K^+–F^-) distance in NaCl structure = a/2 ≈ 537/2 ≈ 268.5 pm ≈ 269 pm.
K–F distance ≈ 269 pm
a = 100 pm = 100×10^{-12} m = 1×10^{-8} cm. Volume of unit cell V = a^3 = (1×10^{-8} cm)^3 = 1×10^{-24} cm^3.
Mass of one unit cell = ρV = 10 g·cm^{-3}×1×10^{-24} cm^3 = 1×10^{-23} g.
In fcc, atoms per cell = 4, so mass per atom = 1×10^{-23} g /4 = 2.5×10^{-24} g.
Number of atoms in 1 g = 1 g / (2.5×10^{-24} g) = 4.0×10^{23} atoms.
4.0×10^{23} atoms
In bcc: 8 corner atoms ×1/8 = 1 atom X per cell; 1 centre atom Y per cell. Formula per cell = X_1Y_1 → XY.
XY
Interpreting edge length a = 4.3×10^{-8} cm (typical value). In bcc the body diagonal = a√3 = 4r. Thus r = (√3/4)·a.
r = (1.732/4)×4.3×10^{-8} cm ≈ 0.433×4.3×10^{-8} cm ≈ 1.86×10^{-8} cm = 1.86×10^{-10} m = 186 pm.
r ≈ 1.86×10^{-8} cm = 186 pm
A Frenkel defect occurs when a smaller ion (usually a cation) leaves its lattice site and occupies an interstitial site, creating a vacancy–interstitial pair. It does not change overall stoichiometry or the number of ions, so mass is unchanged; density is nearly unchanged (small local volume change). Common in ionic crystals with large size difference between ions (e.g. AgCl, ZnS). Frenkel defects increase ionic conductivity (mobile interstitial ions) and create lattice disorder.
Frenkel defect: cation vacancy–interstitial pair; stoichiometry unchanged.