Question

A parallel beam of light of wavelength 100 nm passes through a sample of atomic hydrogen gas in fground state.

(a) Assume that when a photon supplies some of its energy to a hydrogen atom, the rest of the energy appears as another photon moving in the direction of the incident photon. Neglecting the light emitted by the excited hydrogen atoms in the direction of the incident beam,, what wavelengths may be observed in the transmitted beam ?

(b) A radition detector is placed near the gas to detect radition coming perpendicular to the incident beam. Find the wavelengths of raditions that may be detected by the detector.

Answer 1

Here $\lambda$ = 100 nm

E = $\frac{hc}{\lambda }=\frac{1212}{100}$

= 12.42 e V

(a) The possible transitions may be $E_1$ to $E_2$

Energy absorbed in $E_{1}to{E}_2$

= 10.2 e

Energy left = 12.42 - 10.2 = 2.22 e V

2.22 e V = $\frac{hc}{\lambda }=\frac{1242}{\lambda }$

or $\lambda =559.45=560nm$

Energy absorbed in $E_{1}to{E}_3$

= 12.1 e V

Energy left = 12.42 - 12.1 = 0.32 e V

0.32 = $\frac{hc}{\lambda }=\frac{1242}{\lambda }$

$\lambda =\frac{1242}{0.32}$

= 3881.2 = 3881 nm

Energy absorved in $E_{3}to{E}_4=0.65$

Energy left = 12.42 -0.65 = 11.77

11.77 = $\frac{hc}{\lambda }$

or, $\lambda =\frac{1242}{11.77}=105.52$

(b) The energy absorbed by the 'H' atom is now radiated perpendicular to the incident beam.

$\to 10.2=\frac{hc}{\lambda }$

or $\lambda =\frac{1242}{102}=121.76$

$\to 12.1=\frac{hc}{\lambda }$

or $\lambda =\frac{1242}{12.1}=102.64nm$

$\to 0.65=\frac{hc}{\lambda }$

or $\lambda =\frac{1242}{0.65}=1910.76nm$