Now that we understand something about the energy of radiation, we can use that to analyze the energies of electrons and atoms. Remember in the previous lecture I mentioned something about spectroscopy, which is the study of matter by its interaction with radiation. In particular, we can shine light on matter, atoms or molecules, and see what frequencies of light are absorbed by them, by the atom or molecule. Or, we can energize atoms and molecules, and see what frequencies of light that they emit. We're going to focus on this particular lecture on one substance, most notably, hydrogen, the simplest of all of the elements. We're going to analyze something called the Hydrogen Atom Spectrum. Here is the concept. We're going to take a sample of hydrogen. We're going to put it in the electric arc, that's going to energize the hydrogen in there. And as a consequence of that, the hydrogen's actually going to emit a number of frequencies of radiation. To see each of those frequencies, we're going to pass them through a simple prism, that will separate the frequencies of light spatially so that we can see them either with our own eyes or with instruments. Here, we've illustrated what is called the visible spectrum of Hydrogen. It's four different frequencies of light which are emitted, or four different wavelengths of light which are emitted. But these aren't the only ones, these are just the ones that our eyes will respond to. In fact there are a huge number, of frequencies of light which are omitted when hydrogen is excited in this way. And, many of these have to be detected by instruments. Here is a complete set well, a nearly complete set of the set of frequencies which, and wavelengths. Which are omitted by excited hydrogen atoms. And you can see, here, the relationship that we, talked about before. Which is the inverse relationship between the wavelength and the frequency. The shorter the wavelength, the higher the frequency and vice versa. In this case, we've also identified the region of the spectrum particularly there are those for frequencies that we saw in the previous slide for the visible spectrum of hydrogen. We look at the pattern of the data here, let's say we just sort of look down these frequencies and say, anything there that looks like a pattern that we can observe. and the answer turns out to be, not particularly, there just seem to be a set of number there. We've put them in sequence. But they don't necessarily seem to correspond to anything. But it turns out there actually is a remarkably beautiful relationship amongst these numbers. Won't go into the history of how that relationship was developed, but it goes under the name of the Rydberg equation. It turns out the Rydberg equation can predict every one of the frequencies which is in this chart. And it is this simple formula that we've illustrated back over here. And what is the idea here? The idea is a pretty simple one. Pick a particular integer n. And predict, say, one or two. We're taking another integer m. Make it two or three. Don't make it the same number, and calculate a frequency of radiation based upon that value of n and m. And it turns out for every choice of n and m you want to make, there is a frequency over here that will show up in the table. Alternative for every frequency that's in that table there exists a choice of N and M, which can predict it from the Rydberg equation. Not it's not obvious why the Rydberg equation looks the way it does. Why should there be a relationship between the frequencies and two different integers, and what do those integers mean? None of that is told to us immediately by this experimental data. Because the experimental data is simply collected and then analyzed to determine to have this particular form. Let's see how we can now interpret the fact there are only a select set of frequencies which were emitted by the hydrogen atom. We're going to walk through a line of reasoning here, that tells us that individual atoms are, of hydrogen, are only emitting very specific frequencies of radiation. That's the experimental observation. Now we're going to combine that information with what we learned in the previous Concept of Elements study. Which is that the frequency of radiation is related to the energy of the photons which are being emitted since only certain frequencies are emitted. Then only certain photons with certain energies are emitted. And as a consequence, what we can say is that the atom itself is only capable of losing rather specific energies. That is, if I'm emitting, if I'm an atom, and I emit a photon. I lose the energy associated with that photon. If only certain energies of photons are permitted then I can only lose certain energies. But I can only lose certain energies, it must be true, that I can only have certain kinds of energy transitions that can take place. In particular, we think back over here, let's imagine first. I've got an atom and it is emitting radiation, and that radiation corresponds to an energy loss of H nu. And that must correspond to the energy loss of the individual electrons in the atoms and as a consequence, if I try to plot What the energies must for the electrons. They can't just be anywhere. Because if they could be anywhere I could lose any amount of energy. Since I can only lose specific amounts of energy, then there must only be specific energy levels that can exist for the electrons. Such that I could lose that amount of energy, or I could lose that amount of energy. But I couldn't lose just some sort of random amount of energy out here corresponding to, say that energy loss. Because that wouldn't correspond to the transition between any two energy levels here. I might have that one. But I could not have just any random number energy loss. Therefore since only certain energy transitions are possible, I can conclude that only certain energies are possible within the atom. There are only certain energies that the electron can have in the Hydrogen atom. As a consequence, looking at the spectrum of hydrogen atom, we can clearly conclude, that the hydrogen atom electrons must be in one, of a number of quantized energy levels. Let's see if we can figure out what those energy levels are, from the Rydberg equation. Let's back our way up, here's the Rydberg equation again. Remember, this is the frequency of light which is being emitted over here in the diagram. What that corresponds to, a certain photon frequency, h nu, that's being emitted. But that photon energy must correspond to a certain amount of energy lost by the electron. So in the next line in the equation here, what we have specified Is that the energy of the photon is now the negatives of the energy change of the electrons. So I've inserted a minus sign in there. So the amount of the energy that the electron can lose is minus h times the frequency of the photon emitted and the photon emitted must fit the rydberg equation. Well if that's the case then we can also just algabraically rewrite the change in the energy of the electrons. All we've really done here is to divide the previous equation into two pieces corresponding to the n squared and the m squared, so here's the n squared piece. And here's the n squared piece. But if I now analyze those, what I clearly see, is that this looks like the difference, which between two different terms, that look very similar. Both of those terms have a minus hR, divided by an integer squared And I subtract by a minus HR divided by a different integer square. So the energy difference is the difference between two very similar terms that is strongly suggestive that the energy the electron is simply given by a very simple formula. Minus H a proportionality constant called Plank's Constant times R. Proportionality constant called Rydberg's constant, divided by n squared, where n is just some integer. One, this actually tells us what the energies are, associated with an electron in an hydrogen atom. And this formula exactly predicts the spectrum from the Rydberg equation because we derived it from the Rydberg equation. But it also tells us something funny here, it says that the energy of a hydrogen electron depends upon an integer, n, and that integer is a quantum number. A quantum number n that has just shown up rather naturally, by examining the experimental data. So we have our first observation, then, of quantized energy levels corresponding to individual quantum numbers. Now that's all true for Hydrogen, what about for other atoms? Turns out each atom has it's own characteristic spectrum, you can look these up on the internet, on any place that you want to. I actually recommend a particular site from the University of Oregon. Which I have pulled up here for you. That actually allows you to see the frequencies of light which are emitted, by whatever element you might be interested in. Lets click on hydrogen here. Here are those four frequencies of light that we've seen before. Lets click on Helium. There are quite a few frequencies of light here. Notice again that these are only the visible spectrum that we are looking at. There are a number of frequencies which are outside the visible range for each of these elements. Pick on your favorite element. How about phosphorus? We can click on this. There are a significant number of different frequencies in the visible range. The important point of this conversation is that each Atom from each element has it's own characteristic set of frequencies. And since it has it's own characteristic set of frequencies, they don't apply the Rydberg equation doesn't apply. So this formula that we created does not apply to other atoms only to the Hydrogen atom. However, we can conclude that since each atom has its own characteristic set of frequencies, then each atom must have its own characteristic set of energy levels. And those energy levels can actually be determined experimentally and measured simply by measuring the spectrum that is omitted, the spectrum of frequencies that is omitted. By each individual atom. Now it's a different question to ask why can you only have certain energy levels. Or for that matter, why are those energy levels characterized by a quantum number. We're going to pick that up by digging into quantum mechanics in the next lecture.
