Chapter 1: The Solid State – Detailed Notes for NEET/JEE Mains

1. Introduction to Solids

• Solids: Characterized by definite shape, volume, and rigidity. They have strong intermolecular forces and fixed positions for their constituent particles (atoms, molecules, or ions), allowing only vibrational motion. This strong inter-particle attraction gives solids their characteristic mechanical strength and low compressibility.
• Types of Solids:
  • Crystalline Solids:
    • Definition: Have a regular, repeating, three-dimensional arrangement of constituent particles (atoms, ions, or molecules) that extends throughout the entire crystal. This highly ordered arrangement is known as a crystal lattice.
    • Long-range order: The pattern of arrangement of particles is repeated periodically over long distances.
    • Sharp melting points: They melt abruptly at a specific, characteristic temperature because all the bonds/intermolecular forces are of similar strength and require the same amount of energy to break.
    • Anisotropic: Their physical properties (like electrical conductivity, thermal conductivity, refractive index, mechanical strength) show different values when measured along different directions within the crystal. This is due to the ordered arrangement of particles, meaning the environment of a particle differs depending on the direction of measurement.
    • True solids: They possess a definite heat of fusion.
    • Cleavage: When cut with a sharp-edged tool, they break into two pieces with smooth, flat surfaces.
    • Examples: NaCl, Quartz, Ice, Diamond, all metals.
  • Amorphous Solids:
    • Definition: Have an irregular, random arrangement of constituent particles. They do not possess a well-defined geometric shape.
    • Short-range order: They may possess some order over very short distances, but this order does not repeat periodically throughout the material.
    • Melt over a range of temperatures: They gradually soften over a range of temperatures because their particles are arranged randomly, leading to bonds/intermolecular forces of varying strengths.
    • Isotropic: Their physical properties are the same in all directions. This is because the random arrangement of particles makes the overall environment statistically uniform in all directions.
    • Supercooled liquids or pseudo solids: They have a tendency to flow, though very slowly. For example, old glass panes are slightly thicker at the bottom due to slow flow over centuries. They do not have a definite heat of fusion.
    • Irregular cleavage: When cut with a sharp-edged tool, they break into two pieces with irregular surfaces.
    • Examples: Glass, Rubber, Plastics, Tar.

2. Classification of Crystalline Solids

Crystalline solids are classified into four types based on the nature of intermolecular forces or chemical bonds holding the constituent particles together:

1. Molecular Solids:
   • Constituent particles: Individual molecules (which can be non-polar, polar, or capable of hydrogen bonding).
   • Forces: Held together by weak intermolecular forces.
     • Non-polar molecular solids: Held by weak London dispersion forces (van der Waals forces). E.g., H2​, Cl2​, I2​, CO2​ (dry ice), CH4​, Argon. They are generally soft and have very low melting points, often existing as gases or liquids at room temperature.
     • Polar molecular solids: Held by stronger dipole-dipole interactions in addition to London forces. E.g., HCl, SO2​, solid NH3​. They are relatively soft and have slightly higher melting points than non-polar molecular solids.
     • Hydrogen-bonded molecular solids: Molecules are held together by strong hydrogen bonds. E.g., Ice (H2​O), solid HF. They are generally harder than other molecular solids and have comparatively higher melting points.
   • General Properties: Soft, low melting points (often volatile), poor electrical conductors (insulators) because electrons are localized within molecules and ions are absent.
2. Ionic Solids:
   • Constituent particles: Cations (positively charged ions) and anions (negatively charged ions).
   • Forces: Held together by strong electrostatic forces of attraction (ionic bonds) between oppositely charged ions. These forces are non-directional.
   • Properties: Hard and brittle (due to strong, fixed electrostatic forces which cause repulsion upon displacement), high melting points (a large amount of energy is required to overcome strong ionic bonds).
   • Electrical Conductivity: Good conductors in molten state or aqueous solution (due to the mobility of ions), but are insulators in the solid state (ions are fixed in the lattice and cannot move).
   • Examples: NaCl, KCl, MgO, CaF2​, ZnS.
3. Metallic Solids:
   • Constituent particles: Positive metal ions (called ‘kernels’ or ‘cores’ – the atom minus its valence electrons) immersed in a “sea” of delocalized electrons. The valence electrons are not bound to individual atoms but are spread out over the entire crystal.
   • Forces: Metallic bonding, which is the attraction between the positive metal ions and the mobile sea of electrons.
   • Properties: Hard (variable), malleable (can be hammered into sheets), ductile (can be drawn into wires) because the delocalized electron cloud allows layers of atoms to slide past each other without breaking the metallic bond significantly. They have high melting points (variable, e.g., Hg is liquid at room temp).
   • Electrical and Thermal Conductivity: Excellent conductors of electricity and heat due to the highly mobile and delocalized electrons.
   • Lustre: Exhibit characteristic metallic lustre (shininess) because the free electrons absorb and re-emit light.
   • Examples: Fe, Cu, Ag, Mg, Au, Al.
4. Covalent or Network Solids:
   • Constituent particles: Atoms.
   • Forces: Held together by strong covalent bonds forming a continuous, three-dimensional network throughout the entire crystal. These bonds are directional.
   • Properties: Very hard and rigid, very high melting points (often decompose before melting) due to the extensive network of strong covalent bonds.
   • Electrical Conductivity: Usually insulators because electrons are localized in covalent bonds and are not free to move.
   • Exceptions: Graphite is a notable exception. It consists of layers of carbon atoms covalently bonded within the layer, but these layers are held by weak van der Waals forces. This layered structure allows layers to slide over each other (making it soft and a good lubricant) and provides free electrons within layers, making it a good electrical conductor.
   • Examples: Diamond (3D network of sp3 hybridized carbon atoms), Graphite (2D layers of sp2 hybridized carbon atoms), Quartz (SiO2​), Silicon Carbide (SiC).

3. Crystal Lattice and Unit Cell

• Crystal Lattice: A regular three-dimensional arrangement of points in space that represents the positions of constituent particles (atoms, molecules, or ions) in a crystalline solid. Each point in a crystal lattice is identical to every other point in its surroundings.
• Unit Cell: The smallest repeating three-dimensional portion of a crystal lattice which, when repeated over and over again in all three dimensions (translationally), generates the entire crystal lattice. It is the basic building block of a crystal.
  • Parameters of a Unit Cell: A unit cell is characterized by six parameters:
    • Edge lengths: a,b,c (representing the dimensions along the x, y, z axes respectively).
    • Axial angles: α (angle between edges b and c), β (angle between edges a and c), γ (angle between edges a and b).
• Types of Unit Cells:
  • Primitive Unit Cell (P) / Simple Unit Cell (SC): Constituent particles are present only at the corners of the unit cell.
  • Centered Unit Cells: Particles are present at corners as well as at other positions within the unit cell.
    • Body-Centered Cubic (BCC): Particles are at all 8 corners plus one particle at the exact center of the body of the unit cell.
    • Face-Centered Cubic (FCC): Particles are at all 8 corners plus one particle at the center of each of the 6 faces.
    • End-Centered (E) / Base-Centered: Particles are at all 8 corners plus one particle at the center of two opposite faces (e.g., top and bottom faces).
• Contribution of Particles per Unit Cell:
  • A particle at a corner is shared by 8 unit cells, so its contribution to one unit cell is 1/8.
  • A particle at a face center is shared by 2 unit cells, so its contribution to one unit cell is 1/2.
  • A particle at a body center belongs entirely to one unit cell, so its contribution is 1.
  • A particle at an edge center is shared by 4 unit cells, so its contribution to one unit cell is 1/4.
• Seven Crystal Systems (Bravais Lattices): Based on the geometry of the unit cell (axial lengths and angles), Bravais showed that there are only 7 unique crystal systems possible, which lead to 14 possible three-dimensional lattices (Bravais Lattices).
  1. Cubic: a=b=c, α=β=γ=90∘. (Possible Lattices: Primitive (P), Body-Centered (BCC), Face-Centered (FCC)). Example: NaCl, Cu, Fe.
  2. Tetragonal: a=b=c, α=β=γ=90∘. (Possible Lattices: Primitive (P), Body-Centered (BCC)). Example: TiO2​ (Rutile), SnO2​.
  3. Orthorhombic: a=b=c, α=β=γ=90∘. (Possible Lattices: Primitive (P), Body-Centered (BCC), Face-Centered (FCC), End-Centered (E)). Example: Rhombic sulphur, KNO3​.
  4. Hexagonal: a=b=c, α=β=90∘, γ=120∘. (Possible Lattice: Primitive (P)). Example: Graphite, ZnO, CdS.
  5. Rhombohedral (Trigonal): a=b=c, α=β=γ=90∘. (Possible Lattice: Primitive (P)). Example: Calcite (CaCO3​), Cinnabar (HgS).
  6. Monoclinic: a=b=c, α=γ=90∘, β=90∘. (Possible Lattices: Primitive (P), End-Centered (E)). Example: Monoclinic sulphur.
  7. Triclinic: a=b=c, α=β=γ=90∘. (Possible Lattice: Primitive (P)). Example: K2​Cr2​O7​, CuSO4​⋅5H2​O.
  • Mnemonic for 7 Crystal Systems: “Cubic Teachers Often Have Many Right Triangles.” (Cubic, Tetragonal, Orthorhombic, Hexagonal, Monoclinic, Rhombohedral/Trigonal, Triclinic).

4. Number of Particles in a Unit Cell (Z)

This represents the effective number of atoms, molecules, or ions belonging to one unit cell.

• Primitive/Simple Cubic (SC): Atoms are only at the 8 corners. Z=8 (corners)×(1/8) (contribution per corner atom)=1 atom.
• Body-Centered Cubic (BCC): Atoms at 8 corners + 1 at the body center. Z=[8×(1/8)]+[1×1]=1+1=2 atoms.
• Face-Centered Cubic (FCC) / Cubic Close Packed (CCP): Atoms at 8 corners + 6 at face centers. Z=[8×(1/8)]+[6×(1/2)]=1+3=4 atoms.

5. Close Packing in Solids

Close packing refers to the most efficient arrangement of spheres (representing constituent particles) in a crystal lattice to minimize empty space, maximizing the packing efficiency.

• Coordination Number (CN): The number of nearest neighbors with which a given constituent particle is in direct contact in a crystal lattice.
• One-Dimensional Close Packing:
  • Arrangement: Spheres arranged in a single row, touching each other.
  • CN = 2 (each sphere touches two neighbors).
• Two-Dimensional Close Packing:
  • Square Close Packing (SCP): Rows are stacked vertically and horizontally, such that the spheres in adjacent rows are directly above each other.
    • CN = 4 (each sphere touches four neighbors, forming a square).
    • Less efficient packing.
  • Hexagonal Close Packing (HCP) in 2D: Rows are stacked in a staggered manner, where the spheres of the second row fit into the depressions of the first row.
    • CN = 6 (each sphere touches six neighbors, forming a hexagon).
    • More efficient packing in 2D.
• Three-Dimensional Close Packing: Formed by stacking two-dimensional layers.
  • Hexagonal Close Packing (HCP): Formed by stacking 2D hexagonal close-packed layers in an ABAB… pattern. The spheres in the third layer are directly aligned with the spheres in the first layer.
    • CN = 12 (each atom is surrounded by 6 atoms in its own layer, 3 in the layer above, and 3 in the layer below).
    • Packing efficiency = 74%.
    • Examples: Mg, Zn, Cd, Ti.
  • Cubic Close Packing (CCP) or Face-Centered Cubic (FCC): Formed by stacking 2D hexagonal close-packed layers in an ABCABC… pattern. The spheres in the fourth layer are directly aligned with the spheres in the first layer. This structure is identical to the FCC unit cell.
    • CN = 12.
    • Packing efficiency = 74%.
    • Examples: Cu, Ag, Au, Al, Ni, Pt.
  • Body-Centered Cubic (BCC): Not a close-packed structure, but common. The central atom touches 8 corner atoms.
    • CN = 8.
    • Packing efficiency = 68%.
    • Examples: Fe, Cr, W, Na, K.
  • Simple Cubic (SC): Least efficient packing. Each corner atom touches 6 other corner atoms.
    • CN = 6.
    • Packing efficiency = 52.4%.
    • Example: Polonium (only true example).

6. Voids (Interstitial Sites)

When constituent particles (spheres) are packed closely, some empty spaces or holes, called voids or interstitial sites, are left in between them.

• Types of Voids:
  • Tetrahedral Voids: Formed by four spheres, with the center of the void forming a tetrahedron with the centers of the four spheres.
    • Number of tetrahedral voids = 2N (where N is the number of spheres in the close-packed structure, i.e., the effective number of atoms per unit cell, Z).
    • Location in FCC: 8 tetrahedral voids per unit cell (2 along each body diagonal).
  • Octahedral Voids: Formed by six spheres, with the center of the void forming an octahedron with the centers of the six spheres.
    • Number of octahedral voids = N (where N is the effective number of atoms per unit cell, Z).
    • Location in FCC: 4 octahedral voids per unit cell (1 at the body center, 12 at edge centers, each shared by 4 unit cells, so 1+12×1/4=4).
  • Cubic Void: In a simple cubic array, there is one cubic void at the body center of the unit cell.
• Relationship between Radius of Void (rvoid​) and Radius of Sphere (Rsphere​): This is the radius ratio (r/R).
  • Tetrahedral void: rvoid​=0.225Rsphere​
  • Octahedral void: rvoid​=0.414Rsphere​
  • Cubic void (in simple cubic): rvoid​=0.732Rsphere​
  • Triangular void (in 2D): rvoid​=0.155Rsphere​

7. Density of Unit Cell

The density (ρ) of the unit cell is a crucial property, as it directly relates the macroscopic density of a crystalline solid to its atomic properties and unit cell dimensions.

The formula for the density of a unit cell is: ρ=NA​×a3Z×M​

Where:

• Z = Number of atoms/molecules/ions per unit cell (effective number, calculated as discussed in section 4).
  • For SC: Z=1
  • For BCC: Z=2
  • For FCC/CCP: Z=4
• M = Molar mass of the substance (in g/mol). If it’s an element, it’s the atomic mass.
• NA​ = Avogadro’s number (6.022×1023 mol−1).
• a = Edge length of the unit cell.
  • Units are critical: If density is required in g/cm3, then a must be in cm. If a is given in pm (1 pm=10−12 m=10−10 cm), convert it before calculation.

Relationship between edge length (a) and atomic radius (r):

• Simple Cubic (SC): Atoms touch along the edge. a=2r.
• Body-Centered Cubic (BCC): Atoms touch along the body diagonal. Body diagonal = 3​a=4r. So, a=3​4r​.
• Face-Centered Cubic (FCC): Atoms touch along the face diagonal. Face diagonal = 2​a=4r. So, a=2​4r​=22​r.

8. Imperfections in Solids (Defects)

Any deviation from the perfectly ordered arrangement of constituent particles in a crystal is called a defect or imperfection. These defects significantly influence the properties of solids.

• Point Defects: Irregularities or deviations from the ideal arrangement around a point or an atom in a crystalline substance.
  • Stoichiometric Defects (Intrinsic or Thermodynamic Defects): These defects do not disturb the stoichiometry (the ratio of cations to anions) of the solid. They are further classified:
    • Vacancy Defect:
      • Description: Some lattice sites are vacant. This defect arises when a particle is missing from its regular lattice site.
      • Effect: Leads to a decrease in the density of the substance.
      • Occurrence: Typically found in non-ionic solids (e.g., metals). This defect is created when a substance is heated.
    • Interstitial Defect:
      • Description: Some constituent particles (atoms or molecules) occupy interstitial sites (empty spaces) between the regular lattice sites.
      • Effect: Leads to an increase in the density of the substance.
      • Occurrence: Typically found in non-ionic solids.
    • Schottky Defect:
      • Description: In ionic solids, it’s a pair of vacant sites (one cation vacancy and one anion vacancy) so that the electrical neutrality of the crystal is maintained.
      • Effect: Decreases the density of the crystal because mass is lost while volume remains constant.
      • Occurrence: Found in highly ionic compounds with high coordination number and where the sizes of cations and anions are nearly similar.
      • Examples: NaCl, KCl, CsCl, AgBr.
    • Frenkel Defect (Dislocation Defect):
      • Description: An ion (usually the smaller cation due to its ability to fit into interstitial sites) leaves its original lattice site and occupies an interstitial site within the crystal. This creates a vacancy at the original lattice site and an interstitial defect at the new site.
      • Effect: Does not change the density of the solid, as no ions are missing from the crystal; they are just displaced. Electrical neutrality is maintained.
      • Occurrence: Found in ionic compounds with a large difference in size between cations and anions and low coordination number.
      • Examples: AgCl, AgBr, AgI, ZnS. (Note: AgBr shows both Schottky and Frenkel defects due to the similar size of Ag+ and Br− but also the small size of Ag+ allowing it to fit into interstitial sites).
  • Non-Stoichiometric Defects: These defects disturb the stoichiometry of the solid, meaning the ratio of cations to anions becomes different from the ideal chemical formula.
    • Metal Excess Defects: Occur due to the presence of excess metal ions.
      • Anion Vacancies: Anions are missing from their lattice sites, and the void is occupied by an electron to maintain electrical neutrality. These electron-filled anionic vacancies are called F-centers (from German Farbenzentrum, meaning color center) and are responsible for the color of the crystal.
        • Example: NaCl heated in sodium vapor appears yellow (due to F-centers absorbing light). KCl appears violet, LiCl appears pink.
      • Presence of Interstitial Cations: Extra cations occupy interstitial sites, and electrons are also present in other interstitial sites to maintain electrical neutrality.
        • Example: Heating ZnO turns it yellow. Zn2+ ions and electrons occupy interstitial sites.
    • Metal Deficiency Defects: Occur due to the presence of less metal ions than required by the ideal stoichiometry, typically found in compounds where the metal can show variable valency.
      • Cation Vacancies: Some cations are missing from their lattice sites, and the charge is balanced by other metal ions having a higher oxidation state.
        • Example: FeO (ferrous oxide) typically exists as Fe0.95​O. Some Fe2+ ions are missing, and to maintain electrical neutrality, an equal number of Fe3+ ions (with higher charge) are present.
• Impurity Defects: Introduced when foreign atoms are added to the host crystal.
  • Example: Doping NaCl with SrCl2​. When SrCl2​ is added to molten NaCl, Sr2+ ions (charge +2) replace Na+ ions (charge +1). To maintain electrical neutrality, for every Sr2+ ion introduced, one Na+ vacancy is created. These cation vacancies can conduct electricity.

9. Electrical Properties

Solids exhibit a wide range of electrical conductivities, which are explained by the Band Theory of Solids.

• Conductors: Substances that allow the passage of electricity easily.
  • Conductivity range: 104 to 107 ohm−1 m−1.
  • Examples: All metals (Cu, Ag, Au), Graphite.
• Insulators: Substances that do not allow the passage of electricity.
  • Conductivity range: 10−20 to 10−10 ohm−1 m−1.
  • Examples: Wood, Glass, Plastics, Rubber, Diamond, Sulphur, Phosphorus.
• Semiconductors: Substances with intermediate conductivity between conductors and insulators.
  • Conductivity range: 10−6 to 104 ohm−1 m−1.
  • Examples: Silicon (Si), Germanium (Ge).
• Band Theory of Solids:
  • Valence Band (VB): This is the band of energy levels occupied by valence electrons. It can be partially or completely filled with electrons.
  • Conduction Band (CB): This is a higher energy band that is usually empty or partially filled. Electrons in the conduction band are free to move and conduct electricity.
  • Energy Gap (Band Gap, Eg​): The energy difference between the top of the valence band and the bottom of the conduction band. Electrons must gain energy equal to or greater than the band gap to move from the valence band to the conduction band.
  • Conductors:
    • The valence band and conduction band either overlap significantly or have a very small energy gap.
    • Electrons can easily move from the valence band to the conduction band (or even within a single, continuous band), allowing for excellent electrical conductivity.
    • Conductivity generally decreases with increasing temperature (due to increased vibrations of atoms hindering electron flow).
  • Insulators:
    • Have a very large energy gap (>3 eV) between the valence band and the conduction band.
    • Electrons require a very large amount of energy to jump from the valence band to the conduction band, hence they cannot conduct electricity.
  • Semiconductors:
    • Have a small energy gap (<3 eV) between the valence band and the conduction band.
    • At absolute zero, they behave as insulators. However, at room temperature, some electrons gain enough thermal energy to jump from the valence band to the conduction band.
    • The empty spaces left in the valence band are called holes, which act as positive charge carriers. Both electrons (in CB) and holes (in VB) contribute to conductivity.
    • Conductivity increases significantly with increasing temperature (more electrons jump to CB, more holes are created).
• Doping of Semiconductors (Extrinsic Semiconductors): The conductivity of semiconductors can be increased by adding a small amount of suitable impurity (doping).
  • n-type Semiconductors:
    • Process: Doping a pure semiconductor (e.g., Si or Ge, Group 14) with a Group 15 element (e.g., P, As, Sb). Group 15 elements have 5 valence electrons.
    • Mechanism: Four of the impurity’s valence electrons form covalent bonds with the host atoms. The fifth electron is extra and becomes delocalized, contributing to conductivity.
    • Charge Carriers: Electrons are the majority charge carriers, and holes are minority charge carriers. “n” stands for negative charge carriers (electrons).
  • p-type Semiconductors:
    • Process: Doping a pure semiconductor (e.g., Si or Ge, Group 14) with a Group 13 element (e.g., B, Al, Ga, In). Group 13 elements have 3 valence electrons.
    • Mechanism: The impurity atom forms three covalent bonds with host atoms. It has one electron less than required for forming four covalent bonds, thus creating an electron vacancy or “hole.” This hole can accept an electron from a neighboring atom, effectively creating movement of positive charge.
    • Charge Carriers: Holes are the majority charge carriers, and electrons are minority charge carriers. “p” stands for positive charge carriers (holes).

10. Magnetic Properties

The magnetic properties of solids arise from the magnetic moments associated with the electrons (due to their orbital motion and spin).

• Diamagnetic Substances:
  • Property: Weakly repelled by an external magnetic field.
  • Electron Configuration: All electrons are paired (no unpaired electrons). The magnetic moments of paired electrons cancel each other out.
  • Examples: NaCl, H2​O, C6​H6​ (benzene), TiO2​, Zn.
• Paramagnetic Substances:
  • Property: Weakly attracted by an external magnetic field.
  • Electron Configuration: Possess one or more unpaired electrons. The magnetic moments of these unpaired electrons align weakly in the direction of the applied field.
  • Behavior: Lose their magnetism in the absence of a magnetic field.
  • Examples: O2​, Cu2+, Fe3+, Cr3+.
• Ferromagnetic Substances:
  • Property: Strongly attracted by an external magnetic field. Can be permanently magnetized (even after the removal of the external field).
  • Electron Configuration: Possess unpaired electrons. In the solid state, the metal ions are grouped together into small regions called domains. Within each domain, the magnetic moments of the electrons are aligned in the same direction.
  • Behavior: In the absence of an external field, domains are randomly oriented. When an external field is applied, domains orient themselves in the direction of the field, leading to strong magnetism.
  • Examples: Fe, Co, Ni, Gadolinium (Gd), CrO2​.
• Antiferromagnetic Substances:
  • Property: Show paramagnetism or ferromagnetism at high temperatures, but their magnetic moments align in opposite directions and cancel each other out at low temperatures.
  • Behavior: The domains are aligned in anti-parallel directions, resulting in a net magnetic moment of zero.
  • Examples: MnO, Mn2​O3​, FeO.
• Ferrimagnetic Substances:
  • Property: Weakly attracted by an external magnetic field compared to ferromagnetic substances.
  • Behavior: The magnetic moments of the domains are aligned in parallel and anti-parallel directions in unequal numbers, resulting in a net magnetic moment that is not zero but is smaller than in ferromagnetism.
  • Examples: Ferrites like Fe3​O4​ (magnetite), MgFe2​O4​, ZnFe2​O4​.

Crystal Structures of Ionic Compounds

The structure of ionic compounds depends on the size of the ions (radius ratio rule) and their stoichiometry. The larger ions (usually anions) form the close-packed lattice, and the smaller ions (cations) occupy the voids.

• Rock Salt Structure (NaCl Type):
  • Lattice: FCC arrangement of Cl− ions.
  • Voids Occupied: All octahedral voids are occupied by Na+ ions.
  • Coordination Number: Na+ has CN 6 (surrounded by 6 Cl−), and Cl− has CN 6 (surrounded by 6 Na+). Ratio 6:6.
  • Z for NaCl: 4 Na+ and 4 Cl− ions per unit cell.
  • Examples: NaCl, KCl, MgO, LiCl.
• Cesium Chloride Structure (CsCl Type):
  • Lattice: Simple cubic arrangement of Cl− ions.
  • Voids Occupied: Cs+ ion occupies the cubic void at the body center.
  • Coordination Number: Cs+ has CN 8 (surrounded by 8 Cl−), and Cl− has CN 8 (surrounded by 8 Cs+). Ratio 8:8.
  • Z for CsCl: 1 Cs+ and 1 Cl− ion per unit cell.
  • Examples: CsCl, CsBr, CsI.
• Zinc Blende Structure (ZnS Type):
  • Lattice: FCC arrangement of S2− ions.
  • Voids Occupied: Zn2+ ions occupy half of the tetrahedral voids.
  • Coordination Number: Zn2+ has CN 4 (surrounded by 4 S2−), and S2− has CN 4 (surrounded by 4 Zn2+). Ratio 4:4.
  • Z for ZnS: 4 Zn2+ and 4 S2− ions per unit cell.
  • Examples: ZnS (zinc blende form), CuCl, AgI.
• Fluorite Structure (CaF$_2$ Type):
  • Lattice: FCC arrangement of Ca2+ ions.
  • Voids Occupied: F− ions occupy all tetrahedral voids.
  • Coordination Number: Ca2+ has CN 8 (surrounded by 8 F−), and F− has CN 4 (surrounded by 4 Ca2+). Ratio 8:4.
  • Z for CaF$_2$: 4 Ca2+ and 8 F− ions per unit cell.
  • Examples: CaF2​, BaCl2​.
  • Antifluorite Structure (Na2​O Type): Reverse of fluorite structure. Anions form FCC lattice, cations occupy all tetrahedral voids. CN ratio 4:8.

Multiple Choice Questions (MCQs) and Explanations

Here are some MCQs to test your understanding, along with detailed explanations.

Question 1: Which of the following is an example of an amorphous solid? (A) Diamond (B) Graphite (C) Quartz glass (D) Sodium chloride

Explanation 1:

• Correct Answer: (C) Quartz glass.
• Reasoning: Amorphous solids lack a long-range ordered structure and melt over a range of temperatures. Diamond, graphite, and sodium chloride are all crystalline solids with distinct, repeating arrangements of atoms/ions and sharp melting points. Quartz glass (silica glass) is an amorphous form of silicon dioxide, which lacks the regular crystalline structure of quartz.

Question 2: In a face-centered cubic (FCC) unit cell, the number of atoms per unit cell is: (A) 1 (B) 2 (C) 4 (D) 6

Explanation 2:

• Correct Answer: (C) 4.
• Reasoning:
  • In an FCC unit cell, atoms are present at all 8 corners and at the center of each of the 6 faces.
  • Contribution from corners: 8×(1/8)=1 atom.
  • Contribution from face centers: 6×(1/2)=3 atoms.
  • Total number of atoms (Z) = 1+3=4 atoms.

Question 3: Which of the following defects is also known as dislocation defect? (A) Schottky defect (B) Frenkel defect (C) Vacancy defect (D) Interstitial defect

Explanation 3:

• Correct Answer: (B) Frenkel defect.
• Reasoning: A Frenkel defect involves an ion leaving its lattice site (creating a vacancy) and occupying an interstitial site. This displacement of an ion from its normal position to an interstitial position is essentially a type of dislocation, hence it’s also called a dislocation defect. Schottky defect involves missing ions from the crystal lattice, while vacancy and interstitial defects are more general terms for missing or extra particles in non-ionic solids.

Question 4: A metal crystallizes with a face-centered cubic lattice. The edge length of the unit cell is 408 pm. The diameter of the metal atom is: (A) 144 pm (B) 204 pm (C) 288 pm (D) 408 pm

Explanation 4:

• Correct Answer: (C) 288 pm.
• Reasoning:
  • In an FCC structure, atoms are in contact along the face diagonal.
  • Let ‘a’ be the edge length and ‘r’ be the atomic radius.
  • The face diagonal (df​) is equal to 4r.
  • Using Pythagoras theorem for the face diagonal: df2​=a2+a2=2a2.
  • So, df​=2​a.
  • Therefore, 4r=2​a.
  • r=42​a​=22​a​.
  • Given a=408 pm.
  • r=2×1.414408​=2.828408​≈144.2 pm.
  • The diameter of the atom is 2r=2×144.2 pm=288.4 pm.
  • Rounding to the nearest whole number gives 288 pm.

Question 5: Which of the following statements is incorrect regarding Frenkel defect? (A) It is shown by ionic solids. (B) It leads to a decrease in the density of the crystal. (C) It is a dislocation defect. (D) It is found in AgBr.

Explanation 5:

• Correct Answer: (B) It leads to a decrease in the density of the crystal.
• Reasoning:
  • (A) Frenkel defect is indeed shown by ionic solids where there’s a large difference in ionic sizes.
  • (B) This statement is incorrect. In a Frenkel defect, an ion simply moves from a lattice site to an interstitial site within the same crystal. No ions are missing from the crystal, so the overall mass and volume remain essentially the same, leading to no change in density.
  • (C) As explained in Question 3, it is a dislocation defect.
  • (D) AgBr is a common example that shows both Schottky and Frenkel defects.

Solved Numerical Problem

Problem: An element has a body-centered cubic (BCC) structure with a cell edge of 288 pm. The density of the element is 7.2 g/cm3. Calculate the number of atoms present in 208 g of the element.

Solution:

1. Identify the given information and what needs to be calculated:
   • Structure: BCC
   • Edge length (a) = 288 pm
   • Density (ρ) = 7.2 g/cm3
   • Mass of element = 208 g
   • To find: Number of atoms in 208 g
2. Convert edge length to cm: a=288 pm=288×10−12 m=288×10−10 cm.
3. For a BCC structure, find Z (number of atoms per unit cell): Z=2 atoms/unit cell (from notes section 4).
4. Use the density formula to find the molar mass (M) of the element: ρ=NA​×a3Z×M​ Rearranging for M: M=Zρ×NA​×a3​
   Substitute the values: M=27.2 g/cm3×(6.022×1023 mol−1)×(288×10−10 cm)3​ M=27.2×6.022×1023×(288)3×(10−10)3​ M=27.2×6.022×1023×23887872×10−30​ M≈27.2×6.022×23.88×10−7​ M≈51.8 g/mol (The molar mass of the element is approximately 51.8 g/mol, which corresponds to Chromium).
5. Calculate the number of atoms in 208 g of the element: Number of moles = Molar massGiven mass​=51.8 g/mol208 g​≈4 mol.Number of atoms = Number of moles ×NA​ Number of atoms = 4 mol×6.022×1023 atoms/mol Number of atoms = 24.088×1023 atoms Number of atoms ≈2.409×1024 atoms.

Answer: The number of atoms present in 208 g of the element is approximately 2.409×1024 atoms.