The Kα line of a hydrogen-like atom has wavelength 0.32 Å. The Kβ line of the same atom has wavelength (x/10 + y/100) Å, where x and y are integers from 0 to 9. Find y − x.
Physics · Mixed Chapter
JEE Mains

Question

The Kα line of a hydrogen-like atom has wavelength 0.32 Å. The Kβ line of the same atom has wavelength (x/10 + y/100) Å, where x and y are integers from 0 to 9. Find y − x.

G1 Physics · Accepted Answer

Step 1 — For a hydrogen-like atom, 1/λ = RZ²[(1/n₁²) − (1/n₂²)]. Step 2 — The Kα line is the 2 → 1 transition. Thus 1/λ₁ = RZ²(1 − 1/4) = (3/4)RZ². Step 3 — The Kβ line is the 3 → 1 transition. Thus 1/λ₂ = RZ²(1 − 1/9) = (8/9)RZ². Step 4 — Divide the equations to eliminate R and Z: λ₂/λ₁ = (3/4)/(8/9) = 27/32. Step 5 — Given λ₁ = 0.32 Å for Kα, find the Kβ wavelength: λ₂ = (27/32) × 0.32 Å = 0.27 Å. Step 6 — The required form is (x/10 + y/100) Å. Hence x = 2 and y = 7. y − x = 7 − 2 = 5. Answer…