Question

Problem 3. Consider two identical boxes filled with hydrogen atoms. The atoms in Box A are illuminated by a beam of ultraviolet light, while the atoms in Box B collide with a beam of electrons. The light has a wavelength of 97.3 nm in vacuum. Each electron in the electron beam has the same energy as each photon in the beam of light.

a) Calculate the energy of the electron beam

b) How many spectral absorption lines do you expect to see from the atoms in each box?

c) How many spectral absorption lines do you expect to see from the atoms in each box?

d) Calculate the energy levels for the photons in the beam, exiting Box A.

e) Calculate the energy levels for the electrons in the beam exiting Box B.

Answer 1

a) Each electron beam has the same energy as each photon in the beam of light. The energy of the photon of wavelength 97.3 nm is

Thus the energy of the electron beam is also 12.733 eV.

Conversion: 1eV = 1.6*10^-19J

Therefore the energy of the electron beam is 12.733 * 1.6*10^-19 J = 20.3728 *10^-19 J

b) The energy level of hyrdogen atoms are discrete in nature given by the relation

where n is the principle quantum number. Thus for both the box, we expect one absorption line each for an energy corresponding to =97.3 nm because each electron beam also has the same energy as each photon in the beam of light . One wavelength corresponds to one energy and each energy are absorbed at once in a discrete level.

d) Box A:

All the wavelengths in the Lyman series are in the ultraviolet region.

.for lyman series n_f = 1.

Therefore = 97.2 nm in vacuum, n_i = 4. For the light with =97.2 nm ,one absorption lines from n =1 to n= 4 is expected to see from atoms in Box A. Hence the energy levels for electrons exiting Box A will be n=4

e) Box B : When the electron beam collides with an atom, it transfers all its energy to the hydrogen atom.The excitation energy required to raise the electron from the ground state to the nth state is given by

Since E = 12.733 eV,  

.  

hence the energy levels for electrons exiting Box B will be n=4