Energy of the electron used for bombardment of gaseous hydrogen at room temperature = 12.5 eV.

Energy of gaseous hydrogen at ground state = -13.6 eV

When gaseous hydrogen is bombarded with electron beam, then energy of hydrogen will be =

(-13.6 + 12.5) eV= -1.1 eV

Orbital energy is given by the relation E= -13.6/n²

For n = 3, orbital energy will be E= -13.6/3²

E = - 1.5eV

Since the energy is equal to the energy of gaseous hydrogen. Therefore, the electron has made it transition from n= 1 to n= 3

During deexcitation of the hydrogen gas, the electron will return back from n =3 to n= 1 and forms a line of Lyman series of the hydrogen spectrum.

The wave number of the lyman series is given by the relation:

1/λ = R_y (1/1² – 1/n²)

Where R_y is the Rydberg’s constant and is equals to 1.097 × 10⁷m^-1

For n =3, substituting the values, we get

1/λ = 1.097 × 10⁷ (m^-1)[(1 – 1/3²)]

On calculating, we get

λ = 102.55 nm

When electron makes transition from n = 2 to n =1 level

1/λ = 1.097 × 10⁷ (m^-1)[(1 – 1/2²)]

On calculating, we get

λ = 121.54 nm

When the electron makes transition from n=3 to n = 2

1/λ = 1.097 × 10⁷ (m^-1)[(1/4 – 1/3²)]

On calculating, we get

λ = 656.33 nm

∴ In Lyman series, two wavelengths 102.55 nm and 121.54 nm were emitted, and in Balmer series, one wavelength 656.33 nm is emitted.