The diagram shows the binding energy of each energy level of a hydrogen atom. If an electron is in the ground state, what frequency of photon must it absorb in order to move to the energy level 𝑛 equals three? Use a value of 4.14 times 10 to the negative 15 electron volt seconds for the Planck constant. Give your answer in scientific notation to two decimal places.
Here we have a diagram that shows us several energy levels available to an electron in a hydrogen atom. And we’ve been told that the electron is initially at the ground state, or energy level 𝑛 equals one. In order to move to a higher energy level, 𝑛 equals three in this case, the electron needs to gain energy, which it can do by absorbing a photon. Recall that we can relate the energy 𝐸 of a photon to its frequency 𝑓 using the formula 𝐸 equals ℎ𝑓, where ℎ is the Planck constant. And to solve the formula for frequency, let’s copy it over here and divide both sides by ℎ, so we can cancel that term from the right-hand side. Now flipping it the other way and writing it a bit more neatly, the formula reads photon frequency equals photon energy divided by the Planck constant.
Now we already know the value of the Planck constant. So let’s focus on finding the required photon energy. This photon has to account for the difference in energy between the electron’s initial and final levels. We’ll call this amount Δ𝐸, which equals the energy at 𝑛 equals three minus the energy at the ground state. From the graph, we can substitute in their numeric values, and we have Δ𝐸 equals negative 1.51 electron volts minus negative 13.6 electron volts. This comes out to 12.09 electron volts, which is the required photon energy 𝐸.
Now let’s substitute 𝐸 along with ℎ into our formula for frequency, which equals energy divided by the Planck constant. Notice that we can cancel out units of electron volts from the numerator and denominator, leaving only units of per seconds, or hertz, which is a good sign because hertz is the SI unit for frequency. Now calculating, this comes out to about 2.920 times 10 to the 15 hertz. And finally, rounding our answer to two decimal places, we have found that in order for the electron to move from the ground state to the third energy level, it must absorb a photon with a frequency of 2.92 times 10 to the 15 hertz.