Question

Photons are emitted by the electron making downward transitions that could eventually carry it from the n = 4 state to the n = 1 state. find the wavelengths of all such photons (in nm).

Answer 1

The wavelength of the photons emitted during the transition of an electron from the n = 4 state to the n = 1 state is approximately 91 {nm}.

Step-by-step explanation:

The wavelengths of photons emitted during electron transitions between different energy levels in a hydrogen atom can be calculated using the Rydberg formula:

{1}\{λ} = R ({1}\{n₁²} -{1}\{n₂²})

Where:

- λ is the wavelength of the emitted photon.

- R is the Rydberg constant (1.097 × 10⁷ {m}⁻¹).

- n₁ is the initial energy level.

- n₂ is the final energy level.

Given that the electron transitions from n = 4 to n = 1, let's calculate the wavelengths for these transitions:

For n₁ = 4 and n₂ = 1:

{1}\{λ} = R ({1}\{1²} - {1}\{4²})

{1}\{λ} = R (1 - {1}\{16})

{1}\{λ} = R ({15}\{16})

λ= {1}\{R} . {16}\{15}

Now, plug in the value for the Rydberg constant:

λ = {1}\{1.097 × 10⁷ {m}⁻¹} . {16}\{15}

λ ≈ 91 nm

Therefore, the wavelength of the photons emitted during the transition of an electron from the n = 4 state to the n = 1 state is approximately 91 {nm}.