A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. Upto which energy level the hydrogen atoms would be excited?

Energy of the electron in the nth state of an atom = -13.6z²/n² eV
Here, z is the atomic number of the atom.
For the hydrogen atom, z is equal to 1.
Energy required to excite an atom from the initial state (ni) to the final state (nf) = -13.6/nf² + 13.6/ni² eV
This energy must be equal to or less than the energy of the incident electron beam.
∴ -13.6/nf² + 13.6/ni² eV = 12.5
(Energy of the electron in the ground state = -13.6/1² = -13.6 eV)
-13.6/nf² + 13.6 eV = 12.5
→ 13.6 - 12.5 = 13.6/nf²
→ 1.1 = 13.6/nf²
→ nf² = 13.6/1.1 = 12.36
→ nf = √12.36 = 3.51
Since nf can't be a fraction
∴ nf = 3
Hence hydrogen atom would be excited upto 3rd energy level
First member of lyman: 1/λH = RH(1)²[1/1² - 1/2²]
R = 1.097 × 10⁷ m⁻¹
λ = 1215 Å
First member of Balmer: 1/λ23 = RH(1)²[1/2² - 1/3²]
λ = 6563 Å