Question

A parallel beam of light of wavelength 100 nm passes through a sample of atomic hydrogen gas in ground 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 same direction as the incident photon. Neglecting the light emitted by the excited hydrogen atoms in the direction of the incident beam, what wavelengths may be absorbed in the transmitted beam? (b) A radiation detector is placed near the gas to detect radiation coming perpendicular to the incident beam. Find the wavelengths of radiation that may be detected by the detector.

Answer 1

$\lambda =100nm$

$E=\frac{hc}{\lambda }=\frac{1242}{100}=12.42eV$

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

$E_1$ to $E_2$, energy absorbed = 10.2 eV.

Energy left = 12.42 - 10.2 = 2.22 eV.

2.22 eV = $\frac{hc}{\lambda }=\frac{1242}{\lambda }$

or $\lambda =559.45=560nm$

$E_1$ to $E_3$, Energy absorbed = 12.1 eV

Energy left = 12.42 - 12.1 = 0.32 eV

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

or $\lambda =\frac{1242}{0.32}=3881.2=3881nm$

$E_3$ to $E_4$, Energy absorbed = 0.65

Energy left = 12.42-0.65 = 11.77eV

$11.77=\frac{hc}{\lambda }=\frac{1242}{\lambda }$

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 $\lambda =\frac{1242}{10.2}=121.76nm$

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

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