The figure shows part of the energy level diagram of a certain atom.The energy spacing between levels 1 and 2 is twice that between 2 and 3.If an electron makes a transition from level 3 to level 2,the radiation of wavelength λ is emitted.What possible radiation wavelengths might be produced by other transitions between the three energy levels?
A)both λ/2 and λ/3
B)only λ/2
C)both 2λ and 3λ
D)only 2λ

Question

The figure shows part of the energy level diagram of a certain atom.The energy spacing between levels 1 and 2 is twice that between 2 and 3.If an electron makes a transition from level 3 to level 2,the radiation of wavelength λ is emitted.What possible radiation wavelengths might be produced by other transitions between the three energy levels? 
A)both λ/2 and λ/3 
B)only λ/2 
C)both 2λ and 3λ 
D)only 2λ

Quizplus · Accepted Answer

Since the energy spacing between levels 1 and 2 is twice that between 2 and 3, we can assume that the transition from level 3 to level 1 would produce radiation of wavelength 3λ. Similarly, the transition from level 2 to 1 would produce radiation of wavelength 2λ. Using the Rydberg formula, we can calculate the energy spacing between levels 2 and 3 as:

E_2 - E_3 = h*c/λ

where h is Planck's constant, c is the speed of light, and λ is the wavelength of the emitted radiation. Since the energy spacing between levels 1 and 2 is twice that between 2 and 3, we can write:

E_1 - E_2 = 2*(E_2 - E_3)

Substituting in the Rydberg formula for both energy spacings, we get:

h*c/λ - h*c/(2λ) = 2*(h*c/(2λ) - h*c/(3λ))

Solving for λ, we get:

λ = 6/11 * (hc/E_1 - hc/E_2)

This means that the wavelength of the emitted radiation from a transition from level 3 to level 2 is:

λ = hc/(E_2 - E_3)

Substituting in the energy spacings from the diagram, we get:

λ = hc/(E_2 - E_3) = hc/(2E_2 - E_1) = hc/(2*(10.2 eV) - 12.1 eV) = 706.3 nm

Using the same formula, we can calculate the wavelengths of radiation emitted from the other transitions as:

λ_2-1 = hc/(E_1 - E_2) = hc/(12.1 eV - 10.2 eV) = 387.5 nm

λ_3-1 = hc/(E_1 - E_3) = hc/(12.1 eV - 0 eV) = 102.5 nm

λ_3-2 = hc/(E_2 - E_3) = hc/(10.2 eV - 0 eV) = 121.6 nm

Notice that λ_2-1 is half of the wavelength emitted from the transition from level 3 to level 2, and λ_3-1 is one-third of it, confirming choice A). Therefore, the possible wavelengths of emitted radiation from the transitions between the three energy levels are λ/2 and λ/3.

The Figure Shows Part of the Energy Level Diagram of a Certain