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The energy level of Hydrogen atom are determned by the formula
`E = -E_0/n^2` , where n is the principal quantum number and `E_0 = 13.6 eV` .
The energy of the photon emitted is the difference of the energy levels of the atom in different states:
`E_ph=E_0(1/n_1^2 - 1/n_2^2)`
Here `n_2 = 4` , the principal quantum number for the final state of the atom.
The frequency and the energy of the photon are related by the formula
`E_ph= hF` , where h is the Planck constant `h= 4.14*10^-15 eV*s`
So the energy of the photon with given frequency is
`E_(ph) = 4.14*10^-15*1.41*10^14 = 0.58 eV`
`1/n_1^2 - 1/n_2^2=E_(ph)/E_0 = 0.58/13.6=0.043`
Plugging in `n_2 = 4` , we get
`1/n_1^2 = 0.043+1/n_2^2 =0.043+1/16 = 0.1055`
From here `n_1 = 3` .
This means the electron the Hydrogen atom originally had principal quantum number n = 3.
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