Q.42 Physics ⭐ 360° Option Elimination Active

NEET 2023 Physics Question 42 Solution

Topic: Atoms · Official NTA Booklet Code G5

Question

The radius of inner most orbit of hydrogen atom is 5.3×10⁻¹¹ m. What is the radius of third allowed orbit of hydrogen atom?

✓

Verified Against NTA Final Answer Key

Option (1) : 4.77 Å

📚 Atoms Easy

📋 Given Parameters & Problem Setup

First Bohr orbit radius (r₁) 5.3 × 10⁻¹¹ m = 0.53 Å

Target principal quantum number (n) 3 (third allowed orbit)

Atomic number (Z) 1 (Hydrogen atom)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Bohr Radius Quantum Scaling: r_n = r₁ × n²

⏱️ Save ~35 seconds

Radius scales as n²: r₃ = r₁ × 3² = 9 × 0.53 Å = 4.77 Å in 5 seconds!

1 Recall Bohr’s quantization condition: orbital radius r_n ∝ n² / Z.

2 For hydrogen atom, Z = 1, so r_n = r₁ × n².

3 First orbit radius r₁ = 5.3 × 10⁻¹¹ m = 0.53 Å.

4 For the third orbit (n = 3), the scaling factor is n² = 3² = 9.

5 Calculate radius: r₃ = 0.53 Å × 9 = 4.77 Å.

6 Lock Option (1) 4.77 Å instantly!

🛡️

Why You Can 100% Trust This Shortcut: Modern Physics · Bohr Model Postulates (Orbital radius in hydrogen-like systems varies strictly with the square of the principal quantum number: r_n = a₀ n² / Z).

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Bohr’s Formula for Stationary Orbit Radii

According to Bohr’s model of the hydrogen atom, by equating the electrostatic Coulomb attraction to the centripetal force and applying angular momentum quantization L = mvr = nh / 2π, the radius of the n-th stationary orbit is derived as:

r_n = \frac{\varepsilon_0 n^2 h^2}{\pi m e^2 Z} = r_1 \cdot \frac{n^2}{Z}

Step 2

Identification of Initial Radius and Units

For hydrogen (Z = 1), the radius of the innermost (ground state, n = 1) orbit is given as:

r_1 = 5.3 \times 10^{-11}\text{ m} = 0.53 \times 10^{-10}\text{ m} = 0.53\text{ \AA}

Step 3

Application to the Third Allowed Orbit (n = 3)

Setting principal quantum number n = 3:

r_3 = r_1 \cdot n^2 = (0.53\text{ \AA}) \times (3)^2 = 0.53 \times 9\text{ \AA}

Step 4

Numerical Evaluation

Multiplying the values:

r_3 = 4.77\text{ \AA}

⭐

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 (1)

✓ 100% Correct

Verdict & Scientific Reason:

Exact radius of third orbit: 9 × 0.53 Å = 4.77 Å.

💡

Why this is airtight: None. Candidate correctly applied the n² scaling.

2 (2)

✗ Incorrect

Verdict & Scientific Reason:

Radius of the first orbit (n = 1), not the third orbit.

⚠️

Student Fallacy (Why candidates pick this trap): Candidate selected ground state radius without scaling.

3 (3)

✗ Incorrect

Verdict & Scientific Reason:

Linear scaling by 2 (0.53 × 2 = 1.06 Å).

⚠️

Student Fallacy (Why candidates pick this trap): Candidate assumed linear dependence on n (r ∝ n).

4 (4)

✗ Incorrect

Verdict & Scientific Reason:

Linear scaling by 3 (0.53 × 3 = 1.59 Å).

⚠️

Student Fallacy (Why candidates pick this trap): Candidate multiplied by n = 3 instead of n² = 9.

⚠️

NTA Trap Alert: Multiplying by n (Linear Scaling) Instead of n²

Many students hastily calculate r₃ = 3 × 0.53 = 1.59 Å (Option 4), confusing the linear scaling of angular momentum (L ∝ n) with the quadratic scaling of orbital radius (r ∝ n²).

🛡️ How to avoid the -1 mark: Remember: Radius scales as n², Velocity scales as 1/n, Energy scales as 1/n².

📖

Core NEET Syllabus Standard

National Eligibility cum Entrance Test (NEET UG) · Atoms Standard Benchmark

Core Scientific Principle Tested:

"The radius of the n-th Bohr orbit is proportional to n²: r_n = r₁ n²."

Syllabus Focus: Bohr Model and Radii of Stationary Orbits

Preserved in Rationalized Curriculum Syllabus & Academic Standards ↗

Ready to attempt under real exam pressure?

Test your speed and accuracy in the authentic 200-minute timed NTA simulator.

Launch NEET 2023 CBT Test →

← Previous (Q.41) All Physics Questions Grid Next (Q.43) →

Spotted a mistake in this solution? Report an error ([email protected])

⚖️ Legal & Fair Dealing Notice: Sourced from publicly released NEET 2023 question papers and NEET answer keys for private study, research, and non-commercial educational practice under Fair Dealing (Section 52(1)(a) of the Indian Copyright Act, 1957). Not affiliated with or endorsed by NTA or NMC.

NEET 2023 Physics Question 42 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Bohr Radius Quantum Scaling: r_n = r₁ × n²

Step-by-Step Rigorous Workflow & Mathematical Proof

Bohr’s Formula for Stationary Orbit Radii

Identification of Initial Radius and Units

Application to the Third Allowed Orbit (n = 3)

Numerical Evaluation

360° Option Elimination Engine Key Feature

NTA Trap Alert: Multiplying by n (Linear Scaling) Instead of n²

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

🔗 Related Questions — Atoms

Quantized Magnetic Flux in Magnetic Field & Electron Magnetic Moment

Bohr Quantization with Constant Inward Central Force

Wavelengths of Balmer Spectral Lines in Hydrogen

Electrical Neutrality, Stability, and Characteristic Emission Spectra of Atoms

Solve 4 NEET Questions a Day — One per Subject