How to Use This Page
Welcome to the exercise solutions for Chapter 2 — Quantum Mechanical Model of Atom. Here is how to make the best use of this resource:
1. Attempt each question independently before checking the solution. 2. Compare your answer with the provided solution and understand the reasoning. 3. Focus on the Key Concepts tables for a quick revision before exams. 4. Practise writing electronic configurations — this is a frequently tested skill. 5. Use the Exam Checklist to verify you have covered all important topics.
Key Concepts (Quick Recap)
Quantum Numbers
| Quantum Number | Symbol | Values | Determines |
|---|
| Principal | n | 1, 2, 3, ... | Energy level / shell |
| Azimuthal | l | 0 to n-1 | Subshell shape (s, p, d, f) |
| Magnetic | m_l | -l to +l | Orbital orientation |
| Spin | m_s | +1/2 or -1/2 | Spin direction of electron |
Subshell Notation and Capacity
| l value | Subshell | Number of orbitals | Max electrons |
|---|
| 0 | s | 1 | 2 |
| 1 | p | 3 | 6 |
| 2 | d | 5 | 10 |
| 3 | f | 7 | 14 |
Maximum Electrons per Shell
| Shell (n) | Max electrons = 2n² |
|---|
| K (n=1) | 2 |
| L (n=2) | 8 |
| M (n=3) | 18 |
| N (n=4) | 32 |
Rules for Electronic Configuration
| Rule | Statement |
|---|
| Aufbau principle | Electrons fill orbitals in order of increasing energy |
| Pauli exclusion principle | No two electrons in an atom can have the same set of four quantum numbers; max 2 electrons per orbital with opposite spins |
| Hund's rule | In degenerate orbitals, electrons fill singly with parallel spins before pairing |
Energy Ordering of Orbitals
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p
Bohr's Model — Key Points
- Electrons revolve in fixed circular orbits (stationary orbits) without radiating energy.
- Energy is absorbed or emitted when electrons jump between orbits.
- Angular momentum is quantised: mvr = nh/2π, where n = 1, 2, 3, ...
- Radius of nth orbit: r_n = 0.529 × n² A (for hydrogen)
I. Choose the Best Answer (MCQs)
1. The maximum number of electrons that can be accommodated in the L shell is
Answer
(a) 2 (b) 8 (c) 18 (d) 32
Answer(b) 8
ExplanationThe L shell corresponds to n = 2. Maximum electrons = 2n² = 2(2)² = 8.
2. Which of the following sets of quantum numbers is NOT possible?
Answer
(a) n = 1, l = 0, m_l = 0, m_s = +1/2 (b) n = 2, l = 1, m_l = -1, m_s = -1/2 (c) n = 3, l = 3, m_l = -3, m_s = +1/2 (d) n = 4, l = 2, m_l = 0, m_s = +1/2
Answer(c) n = 3, l = 3, m_l = -3, m_s = +1/2
Explanationl can range from 0 to (n-1). For n = 3, l can be 0, 1, or 2 — but NOT 3. So this set is impossible.
3. The shape of p-orbitals is
Answer
(a) Spherical (b) Dumb-bell (c) Double dumb-bell (d) Complex
Answer(b) Dumb-bell
Explanations-orbitals are spherical, p-orbitals are dumb-bell shaped, d-orbitals are double dumb-bell, and f-orbitals have complex shapes.
4. According to Hund's rule, the electronic configuration of carbon (Z=6) is
Answer
(a) 1s² 2s² 2p² (both electrons in same orbital) (b) 1s² 2s² 2p_x¹ 2p_y¹ (c) 1s² 2s¹ 2p³ (d) 1s² 2s² 2p_x²
Answer(b) 1s² 2s² 2p_x¹ 2p_y¹
ExplanationHund's rule states that electrons occupy degenerate orbitals singly with parallel spins before pairing. In carbon, the two 2p electrons occupy separate p orbitals (p_x and p_y) rather than pairing in the same orbital.
5. The number of nodal planes in a d-orbital is
Answer
(a) 0 (b) 1 (c) 2 (d) 3
Answer(c) 2
Explanationd-orbitals have 2 nodal planes (except d_z² which has a different shape but still conforms to the angular momentum rules). The general number of angular nodes equals the azimuthal quantum number l, so for d-orbitals l = 2 → 2 nodal planes.
6. The energy of an electron in the nth orbit of hydrogen atom is given by
Answer
(a) E_n = -13.6/n eV (b) E_n = -13.6/n² eV (c) E_n = 13.6/n² eV (d) E_n = -13.6 × n² eV
Answer(b) E_n = -13.6/n² eV
ExplanationThe energy of an electron in the nth orbit of a hydrogen atom is E_n = -13.6/n² eV. The negative sign indicates the electron is bound to the nucleus.
7. Which of the following has the electronic configuration [Ar] 3d⁵ 4s¹?
Answer
(a) Cr (b) Fe (c) Mn (d) Cu
Answer(a) Cr
ExplanationChromium (Z=24) has the expected configuration [Ar] 3d⁴ 4s², but due to the extra stability of half-filled d-subshell, it is [Ar] 3d⁵ 4s¹. This is an exception to the Aufbau principle.
II. Fill in the Blanks
1. The principal quantum number n determines the energy level or shell of an electron.
2. The azimuthal quantum number l for an f-orbital is 3.
3. The maximum number of electrons in the third shell (M shell) is 18.
4. According to Pauli exclusion principle, no two electrons in an atom can have the same set of four quantum numbers.
5. The shape of an s-orbital is spherical.
6. The magnetic quantum number m_l can have values from -l to +l.
7. The ground state electronic configuration of sodium (Z=11) is 1s² 2s² 2p⁶ 3s¹.
8. The number of orbitals in the d-subshell is 5.
9. Bohr's model is applicable only to hydrogen and hydrogen-like species (single electron systems).
10. The energy difference between two orbits is called quantum of energy or photon energy.
11. The spin quantum number m_s can have only two values: +1/2 and -1/2.
12. The number of radial nodes for an orbital with quantum numbers n and l is given by n - l - 1.
13. The electronic configuration of Cu (Z=29) is [Ar] 3d¹⁰ 4s¹ (an exception due to full d-subshell stability).
14. The total number of nodes in an orbital equals n - 1.
15. De Broglie's equation relating wavelength and momentum is λ = h/mv.
III. True or False (with Corrections)
1. Statement: The principal quantum number n can have zero as a value.
AnswerFalse. Correction — The principal quantum number n can take only positive integer values: 1, 2, 3, ... It cannot be zero because n = 0 would mean no orbit exists.
2. Statement: In a given subshell, all orbitals have the same energy.
AnswerTrue. Orbitals within the same subshell (e.g., p_x, p_y, p_z) are degenerate — they have the same energy in the absence of an external field.
3. Statement: An s-orbital has one nodal plane.
AnswerFalse. Correction — An s-orbital is spherically symmetric and has zero nodal planes. The number of angular nodes (nodal planes) equals l. For s-orbitals, l = 0, so there are 0 nodal planes.
4. Statement: The 4s orbital is filled before the 3d orbital according to the Aufbau principle.
AnswerTrue. The (n+l) rule states that the orbital with a lower (n+l) value is filled first. 4s: n+l = 4+0 = 4; 3d: n+l = 3+2 = 5. Since 4 < 5, 4s fills before 3d.
5. Statement: The magnetic quantum number determines the energy of an orbital.
AnswerFalse. Correction — The magnetic quantum number (m_l) determines the orientation of an orbital in space, not its energy. The principal quantum number (n) primarily determines the energy of an orbital.
IV. Match the Following
| Column A | Column B |
|---|
| 1. Principal quantum number | (a) Shape of orbital |
| 2. Azimuthal quantum number | (b) Orientation of orbital |
| 3. Magnetic quantum number | (c) Energy level / shell |
| 4. Spin quantum number | (d) +1/2 or -1/2 |
| 5. Hund's rule | (e) Maximum multiplicity in degenerate orbitals |
| 6. Pauli exclusion principle | (f) No two electrons with same four quantum numbers |
Answer
1 — (c): Principal quantum number determines the energy level or shell 2 — (a): Azimuthal quantum number determines the shape of the orbital 3 — (b): Magnetic quantum number determines the orientation of the orbital 4 — (d): Spin quantum number takes values +1/2 or -1/2 5 — (e): Hund's rule states maximum multiplicity in degenerate orbitals 6 — (f): Pauli exclusion principle — no two electrons with identical four quantum numbers
V. Answer Briefly (2-mark)
1. What are quantum numbers? Name them.
Answer
Quantum numbers are a set of four numbers that completely describe the position, energy, and spin of an electron in an atom. The four quantum numbers are:
1. Principal quantum number (n) — determines the energy level or shell 2. Azimuthal quantum number (l) — determines the subshell and shape of the orbital 3. Magnetic quantum number (m_l) — determines the orientation of the orbital in space 4. Spin quantum number (m_s) — determines the spin of the electron (+1/2 or -1/2)
2. State Pauli exclusion principle.
Answer
Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. This means that a single orbital can accommodate a maximum of two electrons, and they must have opposite spins (m_s = +1/2 and m_s = -1/2).
3. What is the (n + l) rule? Give an example.
Answer
The (n + l) rule, or Aufbau rule, states that orbitals are filled in order of increasing (n + l) value. When two orbitals have the same (n + l) value, the orbital with the lower n value is filled first.
Example: For 4s, n + l = 4 + 0 = 4; for 3d, n + l = 3 + 2 = 5. Since 4 < 5, the 4s orbital fills before 3d.
4. Differentiate between orbit and orbital.
Answer
| Feature | Orbit | Orbital |
|---|
| Definition | Fixed circular path of electron (Bohr model) | 3D region of probability of finding an electron |
| Nature | Well-defined path | Probability distribution |
| Shape | Circular | Spherical (s), dumb-bell (p), complex (d, f) |
| Max electrons | 2n² | 2 |
| Model | Bohr's model | Quantum mechanical model |
VI. Answer in Detail (5-mark)
1. Explain Bohr's model of the atom and its limitations.
Answer
Bohr's model (1913) was proposed to explain the stability of the hydrogen atom and the line spectrum of hydrogen.
Postulates of Bohr's model:
1. Stationary orbits: Electrons revolve in certain fixed circular paths called stationary orbits without radiating energy. 2. Quantised energy: Each orbit has a fixed energy. The energy of an electron in the nth orbit is:
E_n = -13.6 / n² eV (for hydrogen)
3. Quantised angular momentum: The angular momentum of an electron is quantised:
mvr = nh / 2π, where n = 1, 2, 3, ...
4. Radiation on transition: When an electron jumps from a higher orbit (n₂) to a lower orbit (n₁), energy is emitted as a photon:
ΔE = E_n₂ - E_n₁ = hν = hc/λ
5. Radius of nth orbit (hydrogen):
r_n = 0.529 × n² Å = 0.529 n² × 10⁻¹⁰ m
Limitations of Bohr's model:
1. It could not explain the spectra of multi-electron atoms (only works for hydrogen and He⁺, Li²⁺, etc.). 2. It could not explain the fine structure of spectral lines (splitting due to spin-orbit coupling). 3. It could not explain the Zeeman effect (splitting of spectral lines in a magnetic field). 4. It could not explain the Stark effect (splitting in an electric field). 5. It violates Heisenberg's uncertainty principle by assuming exact position and momentum simultaneously. 6. It could not explain the shapes of molecules or chemical bonding.
2. Explain the four quantum numbers with examples and write the electronic configurations of elements with Z = 1 to 30.
Answer
Four Quantum Numbers in Detail:
1. Principal Quantum Number (n):
Answer
- Determines the main energy level or shell.
- Values: n = 1, 2, 3, 4, ... (K, L, M, N shells)
- Maximum electrons = 2n²
2. Azimuthal Quantum Number (l):
Answer
- Determines the subshell and shape of the orbital.
- Values: l = 0 to (n-1)
- l = 0 → s (spherical), l = 1 → p (dumb-bell), l = 2 → d (double dumb-bell), l = 3 → f (complex)
3. Magnetic Quantum Number (m_l):
Answer
- Determines the orientation of the orbital in space.
- Values: m_l = -l to +l (total 2l+1 values)
- For p orbital (l=1): m_l = -1, 0, +1 (3 orbitals)
4. Spin Quantum Number (m_s):
Answer
- Determines the spin of the electron.
- Values: +1/2 (spin up) or -1/2 (spin down)
Electronic Configurations (Z = 1 to 30):
H (1) : 1s¹
He (2) : 1s²
Li (3) : 1s² 2s¹
Be (4) : 1s² 2s²
B (5) : 1s² 2s² 2p¹
C (6) : 1s² 2s² 2p²
N (7) : 1s² 2s² 2p³
O (8) : 1s² 2s² 2p⁴
F (9) : 1s² 2s² 2p⁵
Ne (10) : 1s² 2s² 2p⁶
Na (11) : [Ne] 3s¹
Mg (12) : [Ne] 3s²
Al (13) : [Ne] 3s² 3p¹
Si (14) : [Ne] 3s² 3p²
P (15) : [Ne] 3s² 3p³
S (16) : [Ne] 3s² 3p⁴
Cl (17) : [Ne] 3s² 3p⁵
Ar (18) : [Ne] 3s² 3p⁶ = [Ar]
K (19) : [Ar] 4s¹
Ca (20) : [Ar] 4s²
Sc (21) : [Ar] 3d¹ 4s²
Ti (22) : [Ar] 3d² 4s²
V (23) : [Ar] 3d³ 4s²
Cr (24) : [Ar] 3d⁵ 4s¹ (exception — half-filled d)
Mn (25) : [Ar] 3d⁵ 4s²
Fe (26) : [Ar] 3d⁶ 4s²
Co (27) : [Ar] 3d⁷ 4s²
Ni (28) : [Ar] 3d⁸ 4s²
Cu (29) : [Ar] 3d¹⁰ 4s¹ (exception — full d)
Zn (30) : [Ar] 3d¹⁰ 4s²
Exceptions: Cr (Z=24) and Cu (Z=29) deviate from the Aufbau principle due to the extra stability of half-filled (d⁵) and fully-filled (d¹⁰) d-subshells respectively.
3. Explain Aufbau principle, Hund's rule, and Pauli exclusion principle with examples.
Answer
Aufbau Principle:
The word Aufbau means "building up." This principle states that electrons fill orbitals in order of increasing energy. The order of filling is determined by the (n + l) rule:
- Orbitals with lower (n + l) values are filled first.
- If two orbitals have the same (n + l) value, the one with lower n is filled first.
Order: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p
Example: Potassium (Z=19) — the 19th electron goes to 4s, not 3d, because 4s has lower energy (n+l = 4) than 3d (n+l = 5).
Hund's Rule of Maximum Multiplicity:
Hund's rule states that when filling degenerate orbitals (orbitals of the same energy), electrons occupy separate orbitals with parallel spins before pairing up. This minimises electron-electron repulsion and maximises stability.
Example — Nitrogen (Z=7): 1s² 2s² 2p³
2p orbitals: ↑ ↑ ↑
p_x p_y p_z
Each of the three p orbitals gets one electron with parallel spins. Only after all three are singly occupied does pairing begin (as in oxygen: 2p⁴).
Pauli Exclusion Principle:
No two electrons in an atom can have the same set of four quantum numbers. This means:
- Each orbital can hold a maximum of 2 electrons.
- The two electrons in the same orbital must have opposite spins (m_s = +1/2 and -1/2).
Example — Helium (Z=2): 1s²
1s orbital: ↑↓
Both electrons share n=1, l=0, m_l=0
But one has m_s = +1/2 and the other m_s = -1/2
No two electrons have identical quantum numbers.
Comparison Table:
| Principle | Governs | Key Statement |
|---|
| Aufbau | Filling order | Lowest energy orbitals fill first |
| Hund's rule | Degenerate orbitals | Singly occupied before pairing |
| Pauli | Same orbital | Max 2 electrons with opposite spins |
Common Mistakes to Avoid
1. Confusing orbit and orbital: An orbit is a fixed path (Bohr model); an orbital is a 3D probability region (quantum model). They are fundamentally different concepts.
2. Writing incorrect quantum number sets: l must be less than n; m_l must range from -l to +l. For example, (n=2, l=2) is impossible since l can only be 0 or 1 when n=2.
3. Forgetting exceptions in electronic configuration: Cr (Z=24) is [Ar] 3d⁵ 4s¹, not [Ar] 3d⁴ 4s². Cu (Z=29) is [Ar] 3d¹⁰ 4s¹, not [Ar] 3d⁹ 4s². Half-filled and fully-filled d-subshells are extra stable.
4. Incorrectly applying Hund's rule: Electrons fill degenerate orbitals singly with parallel spins FIRST, then pair up. Do not put two electrons in the same orbital while other degenerate orbitals are empty.
5. Confusing nodal planes with total nodes: Number of angular nodes = l. Number of radial nodes = n - l - 1. Total nodes = n - 1.
6. Applying Bohr's model to multi-electron atoms: Bohr's model only works for hydrogen and hydrogen-like species (He⁺, Li²⁺). It fails for atoms with more than one electron.
7. Misapplying the (n+l) rule: Remember that when (n+l) values are equal, the orbital with the lower n fills first. Example: 3d (n+l=5) fills after 4s (n+l=4), but 4p (n+l=5) fills after 3d because 3 < 4.
8. Forgetting spin quantum number in counting: Each orbital can hold only 2 electrons (Pauli exclusion). A subshell with l = 1 (p) has 3 orbitals × 2 = 6 electrons maximum, not 3.
Exam Checklist
- [ ] Can define and list the four quantum numbers with their allowed values
- [ ] Can write valid sets of quantum numbers for any electron
- [ ] Can draw the shapes of s, p, and d orbitals
- [ ] Can write ground state electronic configurations for Z = 1 to 30
- [ ] Can identify exceptions (Cr, Cu) and explain why they occur
- [ ] Can state and apply Aufbau principle, Hund's rule, and Pauli exclusion principle
- [ ] Can calculate the maximum number of electrons in a shell (2n²) and subshell
- [ ] Can explain Bohr's model postulates and calculate energy, radius for hydrogen
- [ ] Can distinguish between orbit and orbital
- [ ] Can calculate number of nodes (angular, radial, total) for given quantum numbers
