The diagram shows the transition of an electron in a hydrogen atom from 𝑛 equals four to 𝑛 equals two, emitting a photon as it does so. What is the energy of the photon in electron volts? Give your answer to two decimal places. What is the energy of the photon in joules? Use a value of negative 1.60 times 10 to the negative 19 coulombs for the charge of an electron. Give your answer in scientific notation to two decimal places.
Both parts of this question are asking about the same quantity, just expressed in two different ways. Let’s begin with the first part and calculate the photon’s energy in electron volts. Now, because the electron is transitioning downward, energy must be transferred out of it. And this energy leaves the electron by means of the photon. Recall that the energy difference between the electron’s initial and final levels, which we can call 𝛥𝐸, corresponds to or has the same value as the emitted photon. We can determine 𝛥𝐸, and therefore the energy of the photon, by subtracting the binding energy at level two from the binding energy at level four.
Substituting in these values from the diagram, we have negative 0.85 electron volts minus negative 3.4 electron volts, which comes out to 2.55 electron volts. This is the energy transferred away from the electron by means of the emitted photon. Therefore, to two decimal places, the energy of the photon is 2.55 electron volts.
Now, moving on to the second part of the question, we basically just need to convert the photon’s energy from electron volts to joules. And to do this, we’ve been given the charge of an electron. Let’s talk about the units here. An electron volt is the amount of energy involved in moving one electron across one volt of potential difference. To express this mathematically, let’s multiply the magnitude of the electron’s charge, which is called the elementary charge, and represent it by a lowercase 𝑒 by one volt. Next, we can recall that one volt equals one joule per coulomb. And then, we can cancel out units of coulombs, leaving only units of joules. Thus, we have one electron volt equals 1.60 times 10 to the negative 19 joules.
Okay, now back to the photon, we can multiply its energy, which we found earlier in electron volts, by this conversion factor, which is just equal to one, and then cancel out units of electron volts, leaving only joules. Now, typing this into a calculator, we get 4.08 times 10 to the negative 19 joules. This is already in scientific notation to two decimal places. So we have our answer. The energy of the photon is 4.08 times 10 to the negative 19 joules. Now, although our two answers are equivalent in value, notice how different they are in magnitude. This is a good illustration of how electron volts can be more convenient than joules for use in very small-scale contexts, such as electron energy level transitions.