Chapter 1: Solid State Mock Test
Time: 1 Hour | Maximum Marks: 25
- All questions are compulsory.
- Section A contains Q1 (Multiple Choice) and Q2 (Very Short Answer).
- Section B contains Short Answer Type I questions (2 marks each). Attempt any 4.
- Section C contains Short Answer Type II questions (3 marks each). Attempt any 2.
- Section D contains Long Answer questions (4 marks each). Attempt any 1.
- Use of logarithmic tables is allowed. Calculators are not permitted.
SECTION A
Q1. Select and write the most appropriate answer from the given alternatives: [4 Marks]
-
The coordination number of a sphere in a cubic close-packed (ccp) structure is:
(A) 6(B) 8(C) 12(D) 4
-
Which of the following defects decreases the density of the crystal?
(A) Frenkel defect(B) Schottky defect(C) Interstitial defect(D) F-centers
-
The packing efficiency of a simple cubic unit cell is:
(A) 52.4%(B) 68%(C) 74%(D) 100%
-
In an n-type semiconductor, the majority charge carriers are:
(A) Protons(B) Holes(C) Positive ions(D) Electrons
Q2. Answer the following questions in one sentence: [3 Marks]
- Define: Unit cell.
- What is meant by an isotropic substance?
- Write the formula to calculate the density of a unit cell.
SECTION B
Attempt any FOUR of the following: [8 Marks]
- Distinguish between crystalline solids and amorphous solids. (Any 2 points).
- A metal crystallizes in a face-centered cubic (fcc) lattice. The edge length of the unit cell is 408 pm. Calculate the radius of the metal atom.
- Explain ferromagnetism with a suitable example.
- State the characteristics of ionic solids.
- What are F-centers? How do they impart color to a crystal?
SECTION C
Attempt any TWO of the following: [6 Marks]
- Derive the relationship between the density ($\rho$) of a unit cell and its edge length ($a$).
- Calculate the packing efficiency of a Body-Centered Cubic (BCC) unit cell.
- An element has a bcc structure with an edge length of 288 pm. The density of the element is 7.2 g/cm³. Calculate the molar mass of the element. ($N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$).
SECTION D
Attempt any ONE of the following: [4 Marks]
- (a) Distinguish between Schottky and Frenkel defects. [3 Marks]
(b) What is polymorphism? Give one example. [1 Mark] - (a) Calculate the number of atoms present in a unit cell of a simple cubic and a face-centered cubic (fcc) lattice. [2 Marks]
(b) Explain p-type semiconductors. [2 Marks]
Solutions & Marking Scheme
SECTION A [7 Marks]
Q1. Multiple Choice Answers:
1. (C) 12 [1 Mark for correct option]
2. (B) Schottky defect [1 Mark for correct option]
3. (A) 52.4% [1 Mark for correct option]
4. (D) Electrons [1 Mark for correct option]
Q2. Very Short Answers:
1. Unit cell:
A unit cell is the smallest repeating fundamental structural portion of a crystal lattice which, when repeated in three-dimensional space, generates the entire crystal lattice. [1 Mark for correct definition]
2. Isotropic substance:
A substance whose physical properties (like refractive index, electrical conductivity) are identical in all directions is called an isotropic substance (e.g., amorphous solids). [1 Mark for correct definition]
3. Density formula:
$\rho = \frac{z \cdot M}{a^3 \cdot N_A}$ [1 Mark for correct formula]
SECTION B [8 Marks]
Q3. Crystalline vs Amorphous solids:
| Crystalline Solids | Amorphous Solids |
|---|---|
| Regular, long-range order of particles. | Irregular, short-range order of particles. |
| Have a sharp and characteristic melting point. | Soften over a range of temperatures. |
[1 Mark for each correct point of distinction. Total 2 Marks]
Q4. Radius of fcc unit cell:
For an fcc lattice, the relationship between edge length ($a$) and radius ($r$) is: $r = \frac{a}{2\sqrt{2}}$ [1/2 Mark]
Given $a = 408 \text{ pm}$.
$r = \frac{408}{2 \times 1.414} = \frac{408}{2.828}$ [1 Mark for substitution]
$r = 144.27 \text{ pm}$ [1/2 Mark for correct answer with units]
Q5. Ferromagnetism:
Substances that are strongly attracted by a magnetic field and can be permanently magnetized are called ferromagnetic substances. [1 Mark]
In these solids, the metal ions group together into small regions called 'domains'. In the presence of a magnetic field, all domains align in the direction of the field. Example: Iron (Fe), Cobalt (Co), Nickel (Ni). [1 Mark for explanation/example]
Q6. Characteristics of Ionic Solids:
- The constituent particles are positively charged cations and negatively charged anions. [1/2 Mark]
- The particles are held together by strong electrostatic forces of attraction (Coulombic forces). [1/2 Mark]
- They have high melting and boiling points due to strong bonding. [1/2 Mark]
- They are electrical insulators in the solid state but are good conductors in the molten state or aqueous solution. [1/2 Mark]
Q7. F-Centers:
When an alkali metal halide (like NaCl) is heated in the atmosphere of alkali metal vapor, anions (halides) leave their lattice sites, leaving holes. These holes are occupied by electrons. The electron-trapped anionic vacancies are called F-centers (Farbenzenter). [1 Mark]
They impart color to the crystal because the trapped electrons absorb energy from visible light and get excited to higher energy levels, radiating the complementary color (e.g., yellow for NaCl). [1 Mark]
SECTION C [6 Marks]
Q8. Density Derivation:
Consider a cubic unit cell of edge length $= a$.
Volume of the unit cell ($V$) $= a^3$ [1/2 Mark]
If there are $z$ atoms per unit cell and mass of one atom is $m$, mass of unit cell $= z \times m$ [1/2 Mark]
Mass of a single atom $m = \frac{\text{Molar mass } (M)}{\text{Avogadro's number } (N_A)}$ [1 Mark]
$\text{Density } (\rho) = \frac{\text{Mass}}{\text{Volume}} = \frac{z \cdot (M / N_A)}{a^3}$ [1/2 Mark]
$\rho = \frac{z \cdot M}{a^3 \cdot N_A}$ [1/2 Mark]
Q9. Packing Efficiency of BCC:
For BCC, atoms touch along the body diagonal. Body diagonal $= \sqrt{3}a = 4r \implies a = \frac{4r}{\sqrt{3}}$ [1 Mark]
Volume of unit cell $= a^3 = \frac{64r^3}{3\sqrt{3}}$
Number of atoms in BCC ($z$) = 2. Volume of 2 atoms $= 2 \times \frac{4}{3}\pi r^3 = \frac{8}{3}\pi r^3$ [1 Mark]
$\text{P.E.} = \frac{\text{Vol. of atoms}}{\text{Vol. of unit cell}} \times 100 = \frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}} \times 100 = \frac{\sqrt{3}\pi}{8} \times 100 \approx 68\%$ [1 Mark]
Q10. Numerical on Density:
Given: BCC structure $\implies z = 2$, $a = 288 \text{ pm} = 2.88 \times 10^{-8} \text{ cm}$, $\rho = 7.2 \text{ g/cm}^3$, $N_A = 6.022 \times 10^{23}$. [1/2 Mark for given/formula]
$a^3 = (2.88 \times 10^{-8})^3 = 23.88 \times 10^{-24} \text{ cm}^3$
Formula: $M = \frac{\rho \cdot a^3 \cdot N_A}{z}$
$M = \frac{7.2 \times 23.88 \times 10^{-24} \times 6.022 \times 10^{23}}{2}$ [1.5 Marks for substitution & calculation]
$M = \frac{1035.5}{2} = 51.77 \text{ g/mol}$ [1 Mark for correct answer with units]
SECTION D [4 Marks]
Q11. (a) Schottky vs Frenkel [3 Marks] (b) Polymorphism [1 Mark]
| Schottky Defect | Frenkel Defect |
|---|---|
| Equal number of cations and anions are missing from lattice sites. | An ion (usually cation) leaves its site and occupies an interstitial site. |
| Decreases the density of the solid. | Density of the solid remains unchanged. |
| Shown by compounds with high coordination number (e.g., NaCl). | Shown by compounds with low coordination number (e.g., ZnS). |
[1 Mark for each point of distinction = 3 Marks]
(b) Polymorphism: The phenomenon where a single substance exists in two or more different crystalline structures under different conditions. Example: Carbon (Diamond and Graphite). [1 Mark]
Q12. (a) Atoms in Unit Cells [2 Marks] (b) p-type semiconductors [2 Marks]
(a) Number of atoms:
- Simple Cubic: Atoms only at 8 corners. Total $= 8 \times \frac{1}{8} = 1$ atom. [1 Mark]
- FCC: Atoms at 8 corners and 6 face centers. Total $= (8 \times \frac{1}{8}) + (6 \times \frac{1}{2}) = 1 + 3 = 4$ atoms. [1 Mark]
(b) p-type semiconductors:
Formed by doping a Group 14 element (Si/Ge) with a Group 13 element (like Boron or Gallium). Group 13 elements have 3 valence electrons. When doped into the Si lattice, they form 3 covalent bonds, leaving a vacancy or "hole" where the fourth bond should be. This hole can accept an electron, making the hole appear to move through the crystal. Since electrical conductivity is due to the movement of these positive holes, it is called a p-type semiconductor. [2 Marks for explanation]
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