What is the wavelength of a photon that has an energy of 2.97 times 10 to the negative 17 joules. Use 6.63 times 10 to the negative 34 joule-seconds for the value of the Planck constant and 3.00 times 10 to the eighth meters per second for the value of the speed of light in free space. Give your answer in meters in scientific notation to two decimal places.
The question asks us to find the wavelength of a photon given the energy of that photon, a value for the Planck constant, and a value for the speed of light in free space. A good place to start is a formula that relates these four quantities. 𝐸 is equal to ℎ𝑐 divided by 𝜆, where 𝐸 is the energy of the photon, ℎ is the Planck constant, 𝑐 is the speed of light in free space, and 𝜆 is the wavelength of the photon.
Since we are given 𝐸, ℎ, and 𝑐 and we are looking for 𝜆, we need to rearrange this formula by multiplying both sides by 𝜆 divided by 𝐸. On the left-hand side, 𝐸 divided by 𝐸 is one and we are just left with 𝜆, which is what we want. And on the right-hand side, 𝜆 divided by 𝜆 is one and we are left with ℎ𝑐 divided by 𝐸. This leaves us with 𝜆 equals ℎ𝑐 divided by 𝐸. That is, we have expressed the wavelength of a photon, what we are looking for, in terms of the three quantities we are given in the statement of the question.
All that’s left now is to substitute values. We have 6.63 times 10 to the negative 34 joule-seconds times 3.00 times 10 to the eighth meters per second divided by 2.97 times 10 to the negative 17 joules. Looking at the units, joules divided by joules is one, and seconds per second is also one. So the overall units of this quantity are meters. This is good because first of all 𝜆 is a wavelength, which means that appropriate units are units of length like meters. And we are asked to give our final answer with units of meters. So all we’re left to do is calculate the numerical value of this quotient.
Working out the numbers gives us 6.696969 et cetera times 10 to the negative nine meters. Rounding to two decimal places, we find that the wavelength of our photon is 6.70 times 10 to the negative nine meters, which is part of the X-ray portion of the electromagnetic spectrum.