Video Transcript
The following graph shows the intensity of light observed at different wavelengths for three different objects that may be radiating blackbodies. Which two lines cannot simultaneously represent the radiation emitted by the blackbodies? (A) Line one and line two, (B) line one and line three, (C) line two and line three, (D) no two lines can simultaneously represent the radiation emitted by blackbodies.
Taking a look at our graph, we see these three curves: line one, line two, and line three. The peak of line one lines up, we see, with the peak of line two. In this question, we want to figure out which two of the lines shown cannot simultaneously represent the radiation emitted by blackbodies.
Each of our three lines has a general shape that agrees with the shape of an intensity-versus-wavelength curve of a blackbody. However, the fact that the peak of line one and the peak of line two are shown to occur at the same wavelength is an important clue in helping us figure out whether these two lines can both represent blackbody radiation.
In this question, we’re told that we have three different objects and that these objects may be radiating as blackbodies. Well, let’s say we have an object here that really is a blackbody. As such, if we create a plot of intensity versus wavelength for this object, we would see for different temperatures of our blackbody curves that look like this. This curve in orange represents our blackbody’s radiation at the highest of all these temperatures. Then, if the blackbody cools down a bit, it would radiate, say, with this curve. At a lower temperature still, its curve would look like this green line, and so on.
Notice that among these curves, the peaks where intensity is highest all occur at different wavelengths. As the temperature of our blackbody decreases, that peak wavelength moves to the right. This phenomenon is actually described by a law. It’s called Wien’s law or Wien’s displacement law. This law says that when a blackbody is radiating, the wavelength of its peak intensity is given by a constant value, this value 𝑏, divided by the temperature of the blackbody in kelvin. What this law is saying is that the higher the temperature of a blackbody gets, the shorter the peak wavelength at which that blackbody radiates its maximum intensity becomes. Note that this agrees with our four curves here representing radiation from this blackbody.
This now brings us back to our graph with lines one, two, and three. Because line one and line two have a peak at the same wavelength, according to Wien’s law, these lines should represent blackbody radiation coming from objects with the same temperature. That is, if 𝜆 sub peak is the same for two curves and 𝑏 is the same, which it must be since it’s a constant, that would imply that the two blackbodies must have the same temperature.
But because lines one and two don’t overlap, they can’t represent radiation from two different blackbodies at the same temperature. And that’s where our problem arises. If these two curves really do have the same peak wavelength, and they do, then Wien’s law states that they must represent blackbodies at the same temperature. But they don’t.
So imagining that all three of these lines represent curves generated by blackbodies, we see that lines one and two are incompatible with one another. On the other hand, when we compare line one with line three or line two with line three, either one of those combinations is at least theoretically possible. That’s because line one and line two both have a peak wavelength which is different from the peak wavelength of line three. Specifically, they occur at a shorter wavelength than the peak of line three.
And compared to the temperature of a blackbody that would generate the curve shown as line three, blackbodies that would generate either line one or line two would have a higher temperature than this. So the line pairs of line one and line three are possible and line two and line three. Therefore, our answer choice is not option (D) but instead remains option (A). Line one and line two cannot simultaneously represent radiation emitted by blackbodies.
