A particular hydrogen-like ion emits radiation of frequency 3 × 10¹⁵ Hz when it makes transition from n = 2 to n = 1. The frequency of radiation emitted in transition from n = 3 to n = 1 is (x/9) × 10¹⁵ Hz, where x = ______.
Physics · Atomic Structure and X-rays · Atomic Structure & Bohr's Atomic Model
JEE Mains 2024 01 Feb Evening Shift

Question

A particular hydrogen-like ion emits radiation of frequency 3 × 10¹⁵ Hz when it makes transition from n = 2 to n = 1. The frequency of radiation emitted in transition from n = 3 to n = 1 is (x/9) × 10¹⁵ Hz, where x = ______.

Physics · Atomic Structure and X-rays · Atomic Structure & Bohr's Atomic Model

JEE Mains 2024 01 Feb Evening Shift

G1 Physics · Accepted Answer

For a hydrogen-like ion: ν ∝ (1/n(f)² - 1/n(i)²) For transition 2 → 1: ν₁ ∝ 1 - 1/4 = 3 4 \frac{3}{4} 4 3 ​ For transition 3 → 1: ν₂ ∝ 1 - 1/9 = 8/9 Therefore: ν₂/ν₁ = (8/9)/( 3 4 \frac{3}{4} 4 3 ​ ) = 32/27 Given: ν₁ = 3 × 10¹⁵ Hz So: ν₂ = (32/27)(3 × 10¹⁵) = (32/9) × 10¹⁵ Hz Hence: x = 32