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In two Bohr orbits of a hydrogen-like species, the de Broglie wavelengths of the electron are in the ratio 3: 5. If the ratio of kinetic energies of the electron in these two orbits is 25: x, find the value of x.
- 9
- 16
- 25
- 36
Correct answer: 9
Solution
In Bohr model, angular momentum mvr = n * h / (2 * pi) and r = n² * a0 / Z. So mv = n * h / (2 * pi * r) = Z * h / (2 * pi * n * a0). Thus lambda = h / (mv) = 2 * pi * n * a0 / Z, meaning lambda is proportional to n. Given lambda₁ / lambda₂ = 3 / 5, we get n1 / n2 = 3 / 5. Kinetic energy KE = m * e⁴ * Z² / (8 * epsilon0² * h² * n²) is proportional to 1 / n². So KE₁ / KE₂ = n2² / n1² = 25 / 9. Therefore x = 9.
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