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Question

What is the wavelength of a photon emitted when an electron drops from \( n = 4 \) to \( n = 3 \) in a
hydrogen atom? (Use \( h = 6.6 \times 10^{-34} \, \text{J·s} \), \( c = 3 \times 10^8 \, \text{m/s} \),
1 eV = \( 1.6 \times 10^{-19} \, \text{J} \))

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Explanation

**Thomson's plum pudding model** positive charge uniformly distributed in sphere with electrons embedded, positive charge spread, fails to explain large angle scattering observed. Bohr's model introduces stationary orbits with quantized angular momentum L = n h/2π, physical basis de Broglie standing wave condition 2πr = n λ, circumference fits n wavelengths, explains line spectrum. Δ E = 0.66 eV = 1.056 × 10⁻¹⁹ J . λ = (hc/Δ E) = (6.6 × 10⁻³⁴ × 3 × 10⁸/1.056 × 10⁻¹⁹) ≈ 1.875 × 10⁻⁶ m . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm

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