The first four energy levels for the hydrogen atom are shown in the diagram below. The lowest energy state is labeled E₁, and the higher states are labeled E₂, E₃, and E₄, respectively. When a hydrogen atom undergoes a transition from E₃ to E₁, it emits a photon with λ = 102.6 nm. Similarly, if the atom undergoes a transition from E₃ to E₂, it emits a photon with λ = 656.3 nm. Find the wavelength of light emitted by an atom making a transition from E₂ to E₁.

Strategy The energy-level diagram shows us the relationship between the various energy levels involved. The vertical direction in the diagram is energy. So, if we represent each transition by an arrow, as shown, then the length of that arrow will be proportional to the energy lost by the atom in that transition. From the diagram, we can see that the lengths of the arrows for the E₃ to E₂ transition and for the E₂ to E₁ transition must add up to the length of the arrow for the E₃ to E₁ transition. (Another way of thinking about this is that an atom going from the E₃ state to the E₁ state must lose the same amount of energy whether or not it stops at E₂ along the way.) Because we know the wavelengths for two of the transitions, we can find the corresponding photon energies. That will let us find the energy for the third transition, and then finally we can convert that back to wavelength.

Transcribed Image Text:

E E4- E3 E₂- E₁- A = 656.3 nm λ = 102.6 nm