Video Transcript
What is the difference in energy between a 2.00 times 10 to the 14 hertz photon and a 5.00 times 10 to the 15 hertz photon? Use a value of 6.63 times 10 to the negative 34 joule-seconds for the Planck constant. Give your answer in scientific notation to two decimal places.
We are looking for a difference in energy between two photons. And what we know about these photons is that one is a 2.00 times 10 to the 14 hertz photon and one is a 5.00 times 10 to the 15 hertz photon. Now, remember, hertz is a unit of frequency, so what we know about these photons is their frequency. So we know the frequency of two photons, and we need to figure out the energy difference between these photons.
The relationship that we need between the energy of photons and their frequency is that 𝐸 equals ℎ𝑓, where 𝐸 is the energy of the photon, ℎ is the Planck constant, and 𝑓 is the photon’s frequency. Since we have a value for the Planck constant, we could use this formula to calculate the energy of each photon and then subtract. Instead though, let’s do a little bit of algebra so we only have to substitute values into a formula once.
If we call the two photons photon one and photon two, then we can write 𝐸 one, the energy of the first photon, is equal to ℎ times 𝑓 one, where 𝑓 one is the frequency of the first photon, and 𝐸 two, the energy of the second photon, is equal to ℎ times 𝑓 two, the frequency of the second photon. We then express the difference in energy 𝐸 two minus 𝐸 one as ℎ𝑓 two minus ℎ𝑓 one. Factoring the ℎ out of both terms, we have that the difference in energy is equal to the Planck constant times the difference in frequencies. If we call the difference in energy Δ𝐸 and the difference in frequency Δ𝑓, then we can rewrite this relationship as Δ𝐸 equals ℎ times Δ𝑓.
Note the similarity between this formula and our original formula for the energy of a single photon. In fact, these are the same; we’ve just replaced energy and frequency with difference in energy and difference in frequency. The reason this works is because energy and frequency are directly proportional.
Anyway, we are now ready to substitute values and calculate the difference in energy. We have 6.63 times 10 to the negative 34 joule-seconds times the quantity 5.00 times 10 to the 15 hertz minus 2.00 times 10 to the 14 hertz.
Before we actually evaluate this expression, let’s pay special attention to the difference in frequencies expressed here. The important thing to be careful of is that 10 to the 15th hertz is a different order of magnitude than 10 to the 14th hertz. So we can’t simply subtract two from five. We have to take into account this different order of magnitude. To make both of these quantities have the same order of magnitude, we note that five times 10 to the 15th is the same thing as 50 times 10 to the 14th.
50 times 10 to the 14th minus two times 10 to the 14th is 48 times 10 to the 14th, or 4.8 times 10 to the 15th. Note that 4.8 is actually quite close to five. And the reason for this is again because the two frequencies initially have different orders of magnitude. Now, of course, a calculator would smoothly handle these numbers anyway. The reason we’ve gone into so much detail is to drive home that it is very important to be careful not only of the leading number in a quantity but also the exponent.
Using a calculator, we find that this quantity is equal to 3.1824 times 10 to the negative 18 joule-seconds hertz. We now recall that hertz are defined as inverse seconds. So seconds times hertz is the same thing as seconds per second, which is just one. So the final units of this quantity are joules, which is exactly what we need because we are looking for an energy.
Lastly, we need to round this answer to two decimal places. So rounding 3.1824 to two decimal places gives us a difference in energy between these two photons of 3.18 times 10 to the negative 18 joules.
