The Bohr model of the hydrogen atom allows us to find that the transition with the shortest wavelength up to the level n = 8, has a response is:
λ = 92.4 nm
The spectrum of the hydrogen atom is explained by the Borhr model, where the energy of the atom is given by
[tex]E_n = - \frac{13.606}{n^2} \ [eV][/tex]
Where Eₙ is the energy, n is an integer, the constant is the energy of the base state of hydrogen expressed in electron-volts.
It's using Planck's equation and that the light speed is related to the frequency, we can find an expression for the transition between two levels.
E = h f
c = λ f
E = [tex]\frac{h c}{\lambda }[/tex]
ΔE = [tex]E_{n' } - E_n[/tex] n ’> n
Where h is the Plank constnat, c the light speed and λ the wavelength
It's combining the equations we find
[tex]\frac{1}{\lambda} = R_H ( \ \frac{1}{n_{i}^2 } - \frac{1}{n_f^2} \ )[/tex]
where λ is the transition wavelength, [tex]R_H[/tex] is the Rydberg constant
([tex]R_H[/tex] = 1.0997 10⁷ m⁻¹), n_f and n_i are the final and initial levels of the electron.
In this case, the transition is requested up to the level n = 8, the sublevel p does not influence the value of the transition for atoms with a single electron such as hydrogen.
To find which is the initial nevel we use that they ask for the shortest wavelength, for which the electron must leave the level; n_i = 1
[tex]\frac{1}{\lambda}[/tex] = 1.0997 10⁷ ( [tex]\frac{1}{1^2 } - \frac{1}{8^2 }[/tex] )
[tex]\frac{1}{\lambda}[/tex] = 1.082517 10⁷
λ = 0.923772 10⁻⁷ m
λ = 9.23772 10⁻⁸ m
It is very common to give this wavelength in nanometers let's reduce the magnitude and use the criterion of three significant figures indicated
λ = 9.23772 10⁻⁸ m ([tex]\frac{10^9 nm}{1m}[/tex])
λ = 92.4 nm
In conclusion, using the Bohr model of the hydrogen atom, we can find that the transition with the shortest wavelength for the novel n = 8, has a response of 92.4 nm
Learn more about Borh's model here:
https://brainly.com/question/21600357
