How do I solve this ionization problem?

Question

Consider an electron for a hydrogen atom in an excited state. The maximum wavelength of electromagnetic radiation that can completely remove (ionize) the electron from the H atom is 1460 nm. What is the initial excited state for the electron (n = ?)?

Answer 1

The electron starts with energy

 

E_n = -13.60 eV / n²

 

To remove it from the H atom means to increase its energy to zero by absorbing a photon of energy hc/λ.

 

hc/λ = -E_n = 13.60 eV / n²

 

n² = (λ / hc)(13.6 eV)

 

n² = {(1460 x 10^-9 m) / [(4.136 x 10^-15 eV·s)(3.00 x 10⁸ m/s)]} (13.60 eV)

 

n² ≅ 16.003 ≅ 16  [To get an integer you need really precise values of the constants.]

 

n = 4