Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom undergoes the transition from the energy level n = 4 to the level n = 2.

The Rydberg formula was formulated by Johannes Rydberg for the calculation of the wavelength of an atomic element when an excited electron moves from the quantum shells. The formula is

1/wavelength=R(1/n1^2 - 1/n2^2), where n2 and n1 are the principal quantum numbers and R is the Rydberg constant which is equal to 1.097 x 107 m−1.

As mentioned, the electron moves from an excited state n2, to the ground-level state n1. Therefore, n2 is always greater than n1. Substituting the values to the equation:

1/wavelength=1.097 x 10^7 (1/2^2 - 1/4^2)
wavelength = 4.8617 meters or 48.6 micrometers

johanrusli johanrusli · Answer 2 · 2020-02-01T08:44:52+01:00

The wavelength of the spectral lline produced is about 4.87 × 10⁻⁷ m

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Further explanation

The term of package of electromagnetic wave radiation energy was first introduced by Max Planck. He termed it with photons with the magnitude is :

[tex]\large {\boxed {E = h \times f}}[/tex]

E = Energi of A Photon ( Joule )

h = Planck's Constant ( 6.63 × 10⁻³⁴ Js )

f = Frequency of Eletromagnetic Wave ( Hz )

Let us now tackle the problem !

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Given:

initial shell = n₁ = 4

final shell = n₂ = 2

Asked:

λ = ?

Solution:

Firstly, we will use this following formula to calculate the change in energy of the electron:

[tex]\Delta E = R (\frac{1}{(n_2)^2} - \frac{1}{(n_1)^2})[/tex]

[tex]\Delta E = 2.18 \times 10^{-18} \times ( \frac{1}{2^2} - \frac{1}{4^2})[/tex]

[tex]\Delta E = 2.18 \times 10^{-18} \times ( \frac{1}{4} - \frac{1}{16} )[/tex]

[tex]\Delta E = 2.18 \times 10^{-18} \times \frac{3}{16}[/tex]

[tex]\boxed{\Delta E \approx 4.0875 \times 10^{-19} \texttt{ J}}[/tex]

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Next, we will calculate the wavelength of the light:

[tex]\Delta E = h \frac{c}{\lambda}[/tex]

[tex]4.0875 \times 10^{-19} = 6.63 \times 10^{-34} \times \frac{3 \times 10^8}{\lambda}[/tex]

[tex]\boxed{\lambda \approx 4.87 \times 10^{-7} \texttt{ m}}[/tex]

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Learn more

Photoelectric Effect : https://brainly.com/question/1408276
Statements about the Photoelectric Effect : https://brainly.com/question/9260704
Rutherford model and Photoelecric Effect : https://brainly.com/question/1458544

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Answer details

Grade: College

Subject: Physics

Chapter: Quantum Physics