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Question Video: Identifying an Electron Energy Level Transition Given the Wavelength of an Absorbed Photon Physics • 9th Grade

The diagram shows the binding energy of each energy level of a hydrogen atom. If an electron is in the ground state, what energy level would it transition to if it absorbed a photon with a wavelength of 97.4 nm? Use a value of 4.14 × 10⁻¹⁵ eV.s for the value of the Planck constant.

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Video Transcript

The diagram shows the binding energy of each energy level of a hydrogen atom. If an electron is in the ground state, what energy level would it transition to if it absorbed a photon with a wavelength of 97.4 nanometers? Use a value of 4.14 times 10 to the negative 15 electron volt seconds for the value of the Planck constant.

Here, we see several energy levels available to an electron. And we know the electron begins in the ground state or energy level one. Say it absorbs a photon. This transfers the energy of the photon to the electron. And that amount of energy must match the energy difference between the electron’s initial level and some other level, causing the electron to transition to that other level. In other words, when this electron transitions, the difference in binding energy between the electron’s final and initial levels must be accounted for by the energy of the absorbed photon. We call this energy difference Δ𝐸, and it equals 𝐸 final minus 𝐸 initial.

Now, we already know 𝐸 initial from the diagram. So, if we also know the energy of the photon which accounts for Δ𝐸, we can calculate the binding energy of the level that the electron will transition to. Then, we can match it to one of these levels, and we’ll have our answer. Put mathematically, we can solve this formula for 𝐸 final by adding 𝐸 initial to both sides. So 𝐸 initial cancels out of the right-hand side. And writing the formula a bit more neatly, we have that Δ𝐸 plus 𝐸 initial equals 𝐸 final.

Now, we already know 𝐸 initial. And it turns out we also have all the info we need to calculate the energy of the photon, which, remember, acts as Δ𝐸. The photon has a wavelength of 97.4 nanometers. And recall that we can relate the wavelength, 𝜆, of a photon to its energy, 𝐸, using the formula 𝐸 equals ℎ𝑐 over 𝜆, where ℎ is the Planck constant, whose value was given to us in the question statement, and 𝑐 is the speed of light, 3.0 times 10 to the eight meters per second.

Now, substituting in the values on the right-hand side, we have that the photon’s energy equals the Planck constant times the speed of light divided by the photon’s wavelength. But notice that the wavelength is currently expressed in nanometers. And to calculate, it should be written in plain meters. So recall that one nanometer equals 10 to the negative nine meters. And we can make a substitution to write 𝜆 as 97.4 times 10 to the negative nine meters.

Now, calculating, we found that the energy of the photon is 12.75 electron volts, which will act as our Δ𝐸 value. So, now that we know Δ𝐸 and 𝐸 initial, we can substitute them into this formula and find the binding energy at the electron’s final level. Thus, we have 12.75 electron volts plus negative 13.6 electron volts, which equals negative 0.85 electron volts. According to the diagram, this value matches the binding energy at energy level four. Thus, if the electron in the ground state absorbed a photon with a wavelength of 97.4 nanometers, it would transition to energy level four.

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