Which of the following is a reasonable frequency for a photon of red light? A) 4.67 times 10 to the 14 hertz, B) 5.45 times 10 to the 14 hertz, C) 6.65 times 10 to the 14 hertz, D) 9.89 times 10 to the 14 hertz, or E) 3.43 times 10 to the 14 hertz.
Light is an electromagnetic wave. That wave has peaks and troughs. The frequency of that light is the number of full waves passing a point per second. But, when we talk about the color of visible light, we tend to talk about its wavelength far more than we talk about frequency. The distance between one point on the wave and its next equivalent point is the wavelength of the wave. These two points on the wave are equivalent because they have the same value and because they have the same direction of change. So, how do we convert from the frequency of the wave, which has the symbol 𝑓, and the wavelength of the wave, which has the symbol 𝜆?
We can use the equation that relates the frequency of the wave, its wavelength, and the speed of light. It makes sense that the speed of light is the same as the distance per wavelength multiplied by the waves traveled per second. The speed of light is most commonly given in meters per second. The speed of light in a vacuum is equal to 299792458 meters per second. But for our calculations, we can approximate it to 2.9979 times 10 to the nine meters per second. That’s two significant figures more than any of our statements have. So, that should give us the precision we need. We’re now going to work out the equivalent wavelength for each of the frequencies.
The easiest thing to do is keep all the units SI units and then convert to other wavelength units at the end. We calculate our wavelength by dividing the speed of light by the frequency. So for our first value, we take 2.9979 times 10 to the nine meters per second and divide it by 4.67 times 10 to the 14 hertz. The unit hertz is equivalent to per second. So, we can insert that into our equation. This gives us 6.42 times 10 to the minus seven meters. We can replicate this for the other four statements. The equivalent wavelength for a photon of light with a frequency of 5.45 times 10 to the 14 hertz is 5.50 times 10 to the minus seven meters.
And for C, we get a wavelength of 4.51 times 10 to the minus seven meters. For D, we get 3.03 times 10 to the minus seven meters. And for E, we get 8.74 times 10 to the minus seven meters. The next thing is to convert these units of length into ones you’re more familiar with. For instance, nanometers. 10 to the power of nine or a billion nanometers is equivalent to one meter. So, if we multiply our wavelength in meters by 10 to the nine nanometers per meter, we’ll get the wavelength in nanometers. 6.42 times 10 to the seven multiplied by 10 to the nine is 642. So, our wavelength is 642 nanometers. For B, we get 550 nanometers. For C, we get 451. For D, we get 303. And for E, we get 874 nanometers.
Now, all that remains is to compare these wavelengths with those we associate with red light. Red light is typically associated with wavelengths of about 635 to 700 nanometers. This puts A as our top candidate, with a wavelength of 642 nanometers.
If you aren’t sure about which wavelength would be appropriate for red light, you can think about the boundaries of visible light in the electromagnetic spectrum. Typically, human beings can see light with wavelengths as short as 400 nanometers or as long as 700 nanometers. Human beings see wavelengths of light close to 700 nanometers as red and those close to 400 nanometers as blue. Once we go beyond 700 nanometers, we get into the infrared range which human beings can’t see. Likewise, if we go below 400 nanometers, we get into the ultraviolet range.
The wavelength for option A sits about here. And, these are the positions for the wavelengths of B, C, D, and E. D and E are not in the visible range, so they cannot be seen by the human eye and could never be described as red. While C and B are much closer to the blue end of the spectrum than the red end, leaving A as our only reasonable candidate. So, of the five frequencies given, the one most reasonable for a photon of red light is 4.67 times 10 to the 14 hertz.