An electron in an unknown energy level of a hydrogen atom transitions to the n=2 level and emits a photon with wavelength 410 nm in the process. What was the initial energy level?

The initial energy level = 6

Explanation:

Photon wavelength is proportional to energy. The wavelength of emitted photons is related to the energy levels of the atom as given by the Rydberg formula:

ₕ₁₂

(1/λ) = Rₕ [(1/n₂²) − (1/n₁²)]

where n₂ = final energy level = 2

n₁ = initial energy level = ?

Rₕ = Rydberg's constant = 1.097 × 10⁷ m⁻¹

λ = wavelength = 410 nm = 410 × 10⁻⁹ m

1/(410 × 10⁻⁹) = (1.097 × 10⁷) [(1/2²) − (1/n₁²)]

0.223 = [(1/4) − (1/n₁²)]

(1/n₁²) = 0.02778

n₁² = 1/0.02778 = 36

n₁ = 6.

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