Chemistry Physical How to Solve: Atomic Structure (Bohr’s Formula, Quantum Numbers, Orbital Shapes, Electronic Configuration) – IIT JEE Guide

How to Solve: Atomic Structure (Bohr’s Formula, Quantum Numbers, Orbital Shapes, Electronic Configuration) – IIT JEE Guide

Introduction

Mastering atomic structure unlocks 10-15% of IIT JEE Chemistry—that’s 30-45 marks in JEE Main and 50+ marks in JEE Advanced. It’s the foundation for periodic trends, chemical bonding, and spectroscopy, and it appears in every single JEE paper. If you can solve these problems fast and accurately, you’re already ahead of 80% of test-takers.

WHAT YOU NEED TO KNOW FIRST

Before diving in, ensure you understand:
1. Rutherford’s nuclear model – Electrons revolve around a dense nucleus.
2. Electromagnetic spectrum basics – Wavelength, frequency, and energy relationships (E = hν).
3. Basic algebra and logarithms – For solving Bohr’s formula and quantum number problems.

If any of these are shaky, stop now and review them first.

KEY TERMS & FORMULAS

1. Bohr’s Model & Energy Formulas

Bohr’s Radius (rₙ)

Formula: rₙ = (0.529 Å) × n² / Z - rₙ = Radius of the nth orbit (in Ångströms) - n = Principal quantum number (1, 2, 3, ...) - Z = Atomic number (number of protons) MEMORISE THIS – You’ll use it for hydrogen-like atoms (H, He⁺, Li²⁺, etc.).

Energy of an Electron in the nth Orbit (Eₙ)

Formula: Eₙ = – (13.6 eV) × Z² / n² - Eₙ = Energy of the electron in the nth orbit (in eV) - Z = Atomic number - n = Principal quantum number MEMORISE THIS – Negative sign means the electron is bound to the nucleus.

Wavelength of Emitted/Absorbed Light (Rydberg Formula)

Formula: 1/λ = R × Z² × (1/n₁² – 1/n₂²) - λ = Wavelength of light (in meters) - R = Rydberg constant = 1.097 × 10⁷ m⁻¹ (given on exam sheet) - n₁ = Lower energy level - n₂ = Higher energy level - Z = Atomic number Given on exam sheet – But you must know how to apply it.

2. Quantum Numbers

MEMORISE THIS TABLE – Examiners love asking for valid quantum number sets.

3. Orbital Shapes & Nodes

MEMORISE THIS – Questions on nodes and shapes appear every year.

4. Electronic Configuration Rules

1. Aufbau Principle – Fill orbitals in order of increasing energy: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f... MEMORISE THE ORDER – Use the (n + l) rule to confirm:
2. Lower (n + l) = Lower energy

3. If same (n + l), lower n = lower energy

4. Pauli Exclusion Principle – No two electrons can have the same four quantum numbers.

5. Each orbital holds max 2 electrons (opposite spins).

6. Hund’s Rule – Electrons fill degenerate orbitals (same energy) singly first, with parallel spins.

MEMORISE THESE RULES – Violating them is an instant wrong answer.

STEP-BY-STEP METHOD

Step 1: Identify the Problem Type

Ask yourself: - Is it about Bohr’s model (energy, radius, wavelength)? - Is it about quantum numbers (valid sets, max electrons)? - Is it about orbital shapes/nodes? - Is it about electronic configuration (ground state, exceptions)?

Step 2: Extract Given Data

• For Bohr’s problems: Note Z, n₁, n₂, λ, or E.
• For quantum numbers: Note n, l, mₗ, or mₛ.
• For electronic config: Note the element or ion.

Step 3: Apply the Correct Formula/Rule

• Bohr’s problems → Use rₙ, Eₙ, or Rydberg formula.
• Quantum numbers → Check if the set is valid (e.g., l must be < n).
• Orbital shapes → Recall the table above.
• Electronic config → Follow Aufbau, Pauli, Hund’s.

Step 4: Solve & Check Units

• Bohr’s problems: Convert eV to Joules if needed (1 eV = 1.6 × 10⁻¹⁹ J).
• Wavelength problems: Ensure λ is in meters (not nm or Å).
• Quantum numbers: Verify mₗ is within –l to +l.

Step 5: Cross-Verify with Rules

• For electronic config: Does it follow Aufbau, Pauli, Hund’s?
• For quantum numbers: Is the set physically possible?

WORKED EXAMPLES

Example 1 – Basic (Bohr’s Radius)

Question: Calculate the radius of the 2nd orbit of He⁺ (Z = 2).

Step 1: Identify problem type → Bohr’s radius. Step 2: Given: n = 2, Z = 2. Step 3: Use rₙ = (0.529 Å) × n² / Z. Step 4: Plug in values: r₂ = (0.529 Å) × (2)² / 2 = (0.529 × 4) / 2 = 1.058 Å. Step 5: Check units → Å is correct.

Answer: 1.058 Å

What we did and why: We used Bohr’s radius formula because the question asked for the size of an orbit in a hydrogen-like ion (He⁺). The formula directly relates n and Z to the radius.

Example 2 – Medium (Rydberg Formula)

Question: Find the wavelength of light emitted when an electron in Li²⁺ (Z = 3) jumps from n = 3 to n = 2.

Step 1: Identify problem type → Wavelength using Rydberg formula. Step 2: Given: Z = 3, n₁ = 2, n₂ = 3. Step 3: Use 1/λ = R × Z² × (1/n₁² – 1/n₂²). Step 4: Plug in values: 1/λ = (1.097 × 10⁷ m⁻¹) × 3² × (1/2² – 1/3²) = 1.097 × 10⁷ × 9 × (1/4 – 1/9) = 1.097 × 10⁷ × 9 × (5/36) = 1.097 × 10⁷ × 1.25 = 1.371 × 10⁷ m⁻¹ Step 5: Take reciprocal: λ = 1 / (1.371 × 10⁷) ≈ 7.29 × 10⁻⁸ m = 72.9 nm.

Answer: 72.9 nm

What we did and why: We used the Rydberg formula because the question involved an electron transition in a hydrogen-like ion (Li²⁺). The formula connects energy levels to wavelength, and we converted the final answer to nm for convenience.

Example 3 – Exam-Style (Quantum Numbers & Electronic Config)

Question: Which of the following is not a valid set of quantum numbers for an electron in a 3d orbital? (A) n = 3, l = 2, mₗ = –1, mₛ = +½ (B) n = 3, l = 2, mₗ = 3, mₛ = –½ (C) n = 3, l = 2, mₗ = 0, mₛ = +½ (D) n = 3, l = 2, mₗ = –2, mₛ = –½

Step 1: Identify problem type → Quantum number validity. Step 2: Recall rules: - l must be 0 to (n–1) → For n = 3, l can be 0, 1, 2. - mₗ must be –l to +l → For l = 2, mₗ can be –2, –1, 0, +1, +2. - mₛ can only be +½ or –½. Step 3: Check each option: - (A) Valid (mₗ = –1 is allowed). - (B) Invalid (mₗ = 3 is not allowed for l = 2). - (C) Valid (mₗ = 0 is allowed). - (D) Valid (mₗ = –2 is allowed). Step 4: Confirm answer → (B) is invalid.

Answer: (B)

What we did and why: We systematically checked each quantum number against the rules. The trap was in option (B), where mₗ exceeded the allowed range for l = 2. Always verify mₗ first—it’s the most common mistake.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

Listen up—this is your last-minute cheat sheet.

1. Bohr’s formulas:
2. Radius: rₙ = 0.529 × n² / Z (Å).
3. Energy: Eₙ = –13.6 × Z² / n² (eV).

4. Wavelength: 1/λ = R × Z² × (1/n₁² – 1/n₂²).

5. Quantum numbers:

6. n = 1, 2, 3...
7. l = 0 to (n–1).
8. mₗ = –l to +l.

9. mₛ = ±½.

10. Orbitals:

11. s = sphere, p = dumbbell, d = cloverleaf.

12. Nodes: n–1 total (n–l–1 radial, l angular).

13. Electronic config:

14. Fill in order: 1s 2s 2p 3s 3p 4s 3d 4p...

15. Exceptions: Cr (4s¹ 3d⁵), Cu (4s¹ 3d¹⁰).

16. Exam traps:

17. Hydrogen-like ions → Use Z > 1.
18. Quantum numbers → Check mₗ range.
19. Exceptions → Cr, Cu, Mo, Ag.

Final tip: If stuck, draw the orbital diagram—it’s faster than memorising rules. Now go crush that exam!