Q.34 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 34 Solution

Topic: Dual Nature of Radiation and Matter · Official NTA Booklet Code 45

Question

De-Broglie wavelength of an electron orbiting in the n = 2 state of hydrogen atom is close to (Given Bohr radius = 0.052 nm)

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Verified Against NTA Final Answer Key

Option (2) : 0.67 nm

📚 Dual Nature of Radiation and Matter Easy

📋 Given Parameters & Problem Setup

Principal Quantum Number (n) 2

Bohr Radius (a₀) 0.052 nm

Target Variable de Broglie wavelength λ of electron

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The 30-Second Exam Hall Shortcut: ⚡ Bohr-de Broglie Standing Wave Shortcut

⏱️ Save ~40 seconds

2π r_n = n λ ⟹ For n = 2: 2π r₂ = 2λ ⟹ λ = π r₂ = π (a₀ n²) = π (0.052 × 4) = 3.14 × 0.208 ≈ 0.67 nm!

1 Recall de Broglie’s standing wave condition for stationary Bohr orbits: Circumference = n wavelengths ⟹ 2π r_n = n λ.

2 For the n = 2 state: 2π r₂ = 2 λ ⟹ λ = π r₂.

3 Calculate radius of n = 2 orbit: r₂ = a₀ · n² = 0.052 nm × 2² = 0.052 × 4 = 0.208 nm.

4 Calculate wavelength: λ = π × 0.208 nm ≈ 3.1416 × 0.208 ≈ 0.653 nm ≈ 0.67 nm (using standard a₀ = 0.0529 nm gives exactly 0.665 nm ≈ 0.67 nm).

5 Match Option (2) in under 20 seconds!

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Why You Can 100% Trust This Shortcut: de Broglie’s Standing Wave Quantization (NCERT Physics · Atoms: 2πr = nλ).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Bohr-de Broglie Condition for Stationary Orbits

Louis de Broglie explained Bohr’s angular momentum quantization by proposing that an electron in an orbit behaves as a standing matter wave. Constructive interference requires the orbital circumference to contain an integral number of de Broglie wavelengths:

2\pi r_n = n \lambda \implies \lambda = \frac{2\pi r_n}{n}

• For n = 2: λ = 2π r₂ / 2 = π r₂.

Step 2

Calculate Radius of n = 2 Orbit

The radius of the n-th Bohr orbit scales with the square of the principal quantum number:

r_n = a_0 n^2 \implies r_2 = 0.052\text{ nm} \times (2)^2 = 0.052 \times 4 = 0.208\text{ nm}

• r₂ = 0.208 nm (or 0.212 nm with standard a₀ = 0.0529 nm).

Step 3

Calculate de Broglie Wavelength

Substitute r₂ into the wavelength relation:

\lambda = \pi r_2 = 3.1416 \times 0.208\text{ nm} \approx 0.653\text{ nm} \approx 0.67\text{ nm}

• Matches Option (2) 0.67 nm.

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360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 A

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Off by a factor of 10 (decimal place error).

⚠️

Student Fallacy (Why candidates pick this trap): Order of magnitude calculation slip.

2 B

✓ 100% Correct

Verdict & Scientific Reason:

100% correct. λ = π r₂ = π (4 × 0.052 nm) ≈ 0.67 nm.

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Why this is airtight: None; textbook matter wave standing wave relation.

3 C

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Exceeds the circumference of the orbit.

⚠️

Student Fallacy (Why candidates pick this trap): Multiplying by extraneous factors.

4 D

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Corresponds to n = 4 calculation error.

⚠️

Student Fallacy (Why candidates pick this trap): Using n = 4 or incorrect orbit radius.

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NTA Trap Alert: Calculating via Velocity Instead of 2πr = nλ

Students often try to compute electron velocity v = e² / (2ε₀ n h), then momentum p = mv, then λ = h/p. This lengthy method introduces high chances of numerical arithmetic mistakes. Using 2πr = nλ gives the answer in one line!

🛡️ How to avoid the -1 mark: Whenever de Broglie wavelength in a Bohr orbit is asked, always use 2πr = nλ.

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Core NEET Syllabus Standard

NEET Physics Core Syllabus · Dual Nature of Radiation and Matter Standard Benchmark

Core Scientific Principle Tested:

"Circumference contains integer wavelengths: 2π r_n = n λ."

Syllabus Focus: de Broglie Wavelength & Bohr Orbitals

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

n = 2 ⟹ λ = π r₂ = π (4 × 0.052 nm) ≈ 0.67 nm!

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NEET 2025 Physics Question 34 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Bohr-de Broglie Standing Wave Shortcut

Step-by-Step Rigorous Workflow & Mathematical Proof

Bohr-de Broglie Condition for Stationary Orbits

Calculate Radius of n = 2 Orbit

Calculate de Broglie Wavelength

360° Option Elimination Engine Key Feature

NTA Trap Alert: Calculating via Velocity Instead of 2πr = nλ

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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