Calculate the wavelength (in nanometers) of a photon emitted when an electron undergoes a transition from n = 4 to n = 2 in a hydrogen atom.
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To calculate the wavelength of the photon emitted, we can use the Rydberg formula:
1/λ = R_H * (1/n_2^2 - 1/n_1^2)
where:
- λ is the wavelength of the photon,
- R_H is the Rydberg constant for hydrogen (approximately 1.097 x 10^7 m^-1),
- n_2 is the principal quantum number of the final state (n = 2),
- n_1 is the principal quantum number of the initial state (n = 4).
Plugging in the values:
1/λ = (1.097 x 10^7 m^-1) * (1/2^2 - 1/4^2)
1/λ = (1.097 x 10^7 m^-1) * (1/4 - 1/16)
1/λ = (1.097 x 10^7 m^-1) * (3/16)
1/λ ≈ 0.205 x 10^7 m^-1
Now, we need to convert this into nanometers by dividing by 10^9:
1/λ ≈ 0.205 x 10^-2 nm^-1
Taking the reciprocal:
λ ≈ 48.78 nm
Therefore, the wavelength of the photon emitted during the transition is approximately 48.78 nanometers.
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