In this video, I'll be discussing how the atomic emissions spectra we observed in the gas discharge tube demonstration, provided evidence for the quantized nature of the electrons in atoms. That helped lead Niels Bohr to his model of the hydrogen atom in 1913. And we will discuss this model for atoms, with the smallest element as a starting point for later lectures on heavier element electronic structures. Just to remind you, here are some atomic emission spectra. The top is a continuous spectrum of visible light. Below that are some representations of the spectral lines observed for hydrogen, neon, and iron respectively. In hydrogen we can see red, blue, and purple lines. Blue and purple here are a little bit difficult to see because they're dark. Neon has more lines than hydrogen. Mostly in the red yellow and orange region. And Iron has a very large number of light emission lines. Just based upon these three atoms what general trend are you observing about number of emission lines for different elements? Thank you for your submission. A general trend we can observe is that as the atoms become heavier, the light wave length they emit after excitation with electricity become more numerous. As recall that demonstration, let's zero in and more carefully consider the visible line spectrum observed from the hydrogen gas discharge tube. There are four lines which can be observed, which means there are four distinct and discrete light photon energies which are emitted. That's really important. We can calculate the energies associated with each of these types of photons, because we know the wavelength associated with those photons, and we know some key equations. Do you remember those equations? One of them I hope you remember is C equal lambda nu. And another one I hope you remember is E equals h nu and of course you can substitute one into the other using some simple algebra. We also know that the overall energy change of the universe must be zero. So that leads us to two conclusions. First, the energy emitted from the photon, which is equal to H new. Remember, H is plus constant. Must come from an energy change within the atom. So, we can write that as an equation, the change in the energy in the atom has to equal the energy of the photon we observed the atom emit. A second conclusion is since the energy of the emitted photon is quantized, quantized meaning some things are allowed and some things are not allowed. Just like steps allowing us to be only certain distances from the ground. The energy of the atom also must be quantized. In other words there are some energies within the atom that are allowed and there are other energies within the atom that are not allowed. If something is quantized that means that some values are allowed and some values are not allowed. Swif mathematician Johann Balmer was the first to fit all of the observed lines into an equation that contained only a wavelength. A constant h which is now called the Balmer constant. This value, which you can look up if you like, is 3.6456 times 10 to the minus 7 meters. And it's called H, but don't confuse it with Planck's constant, which is also given the symbol H. They're different values. And the integers here in his equation are M and N. So m and n are integers. Of course this is the wave length. If we were going to afind, define these variables, h is Balmer's constant. Which is always just some number if we can look up. M is an integer and it turns out m is equal to for the visible light lines that Bahmer was describing. Three, four, five or six and n is also an integer and in this equation n is always equal to two. A few years after Balmer developed this equation, Swedish physicist Johannes Rydberg was able to fit all of those know wavelength of light emitted for hydrogen. Including those in the ultraviolet and infrared regions. With a similar, more general, but still simple equation containing a new constant R. In this case, I've given the equation in wave number units, but it can also be given in frequency units. In this particular equation, the Rydberg constant RH. Is expressed as 1.097373 times 10 to the seven reciprocal meters. And again we have wavelength in the equation, don't we, just like we did in the balmer equation. And we have integers n. And n here doesn't necessarily have to be two, three, four, five, or six. n could also be one, in the Rydberg equation. And this can apply to visible light, which is included in the Balmer equation. So the Balm equation was for the visible lines only. And the Rydberg equation was for the UVIR and visible lines. Here, the reason I've labeled nlow and nhigh, which is a little bit different than you'll see sometimes in other books. You might see it labeled as n1 and n2. Is that I put the high here to help remember that the larger value of n is always the second one. So n could be one and two. Or n could be one and three. Or n could be two and four. We always put the higher number here, and in each case, n is squared. So we square the low number and we square the high number. With this equation, we can predict the wavelengths of light that are emitted from the hydrogen atom. These equations mathematically fit the data quite well. But a physical model for the hydrogen atom was not able to explain these equations, until Niels Bohr presented his model for the hydrogen atom in 1913. His model was a refinement of Rutherford's model of the atom which was developed in 1911. In Rutherford's model, there was a dense nucleus containing the protons and neutrons, surrounded by the electrons in a planetary like orbit. But in Rutherford's model the electrons were allowed to be any distance from the nucleus they would like. Bohr refined this model to say that the electrons were only allowed to occupy certain discreet distances from the nucleus. Those are shown here as concentric circles. So the electron can't be between the pink circle and the red circle. It can only be at the distance that is. Circumscribed by the pink circle or by the red circle. In other words some values are allowed and some values are not allowed. Remember before when we were learning about electro-negativity and ionization energy. I showed you that sometimes chemists, instead of drawing this whole picture of the atom. Just draw a little cross section of the atom, like this, and then use little horizontal lines to show positions where electrons can be. That's what I'm going to do for the Bohr Model of the atom right now. So here's that picture of Bohr's Model of the hydrogen atom. Remember that the nucleus is down here, in this case it is a single positive charge from one proton, because it's hydrogen. And in fact, I've show down at the bottom of the screen, but it's much farther down than that. Because the distance between the proton and the electron is quite large. In Bohr's model, the electron is only allowed to be certain discreet distances from the nucleus. It can be on this distance which we could measure,. Or it can be at a distance that's greater than that. All right, or it can be any of these spots that are shown with these little horizontal line. Which remember again, is a cross section of those concentric circles, which are supposed to be representing spheres that I showed you on the previous slide. Now these distances from the nucleus are associated with a certain energy. Here I've shown the energy times Plunk's constant and Rydberg's constant and that comes from this equation. These you see these lines are labeled with numbers and those numbers are n. Remember we talked about n being integer before. So n can be 1 or 2 or 3 or 4, et cetera. The energy associated with the allowed level, given the number N is equal to minus, and I'll define these variables in a minute,Z squared hR divided by n squared. That's a pretty simple equation. What do these variables stand for? E of course is energy. Let's call it the electron energy, just to be more specific. Z is the nuclear charge, which for hydrogen is plus one. And in fact, the bore model only works for. Atoms that have a nuclear charge of one. H is Planck's constant, no Wilmer's constant that we saw on the previous slide. R is Rydberg's constant. But is Rydberg's constant in a unit, of frequency which is a little bit difference in the unit and the equation we showed on the, on the previous slide. And N is an integer. But it's a positive integer, so n can be one, two, three, four, et cetera. [SOUND] Okay. So R here is still Rydberg's constant, Rydberg's constant can have different units remember? Because we have those equations E equals h nu and C equals lambda nu. And it kind of depends on the form that Rydberg's equation is given in, what the units are for Rydberg's constant. In this particular case, I want Rydberg's constant to have units of frequency. >> So 120 I'm going to say R is approximately equal to 3.290 times 10 to the 15th.2/1 Cycles per second. Which makes h times r equal to 2.180 times 10 to the minus 18 jewels. So this equation is very useful. One thing I want you to notice is that the equation gives us all negative values. It allows us to calculate the allowed energies of the electrons in a hydrogen atom. And if we know what the allowed energies are, we can calculate what the allowed energy differences are. As the electron moves around in the atom. Let me show you what I mean. What if we take the hydrogen atoms and we stimulate them somehow? What kinds of reactions to that electrical energy do you think can occur? One thing that can happen is that the negatively charged electron can absorb the energy by moving farther away from the positively charged nucleus. This process is not spontaneous. Remember the opposite charges are attracted to each other, so pulling them apart requires energy. But the electrons can't just go to any random distance away from the nucleus. It has to have enough energy to jump to a higher allowed energy level. For example, here is the electron. Remember the proton is down here, here is the electron in the closest allowed energy level, n=1. And that's called the Ground State. If we put some energy into the system. And that could be electrical energy or we could bombard the atoms with light. The atom is going to absorb the energy of E equals h nu. And that will allow the electron from the ground state to move to one of the higher energy allowed levels. So now after the absorption of the energy, the electron is moved up here. This is called the "excited state". The excited state is a short lived state. Once the electron is in the excited state which is a higher allowed level than the minimum allowed level where it was allowed to be, it will spontaneously. Move back to lower energy. We call this process relaxation so it started at a higher energy level and it's moving to a lower allowed energy level. When that elect, relaxation process happens light will be emitted. So the electron end up back at and equals one so it's back at the ground state so it's traveling from the excited state back to the ground state. That happens spontaneously and that emits light. We can measure the distance between these two energy levels, right? And that tells us the energy of the photon. That photon has the amount of energy that is the difference between those two energy levels. In this case, the light from this emission is the electron is traveling from N equals 2 to N equals 1 is actually ultraviolet light. We can't detect that light with our eyes but we have instruments that can detect it. The four lines of light that we are seeing with our eyes, are not from the transitions back to the ground state where N equals 1. The lines of light that we could see were intermediate transitions. So if for example these electrons were given lots of electrical energy. And they could jump up not just inequal two, but also to inequal three or inequal four or inequal five. They have enough energy to jump up to other allowed energy levels. On the way back down they can relax all the way from where they are, all the way back to the ground state. Or they can stop and rest at any of the intermediate allowed levels. Just like if I was going down a flight of stairs, I could do the little kid method and I could jump from the top stair all the way down to the bottom of the stairs. And I have to tell you, I haven't tried to do that since I was about ten years old. Or they can take steps on their way back down. For example if it, if the electron went up to energy level 3. It could relax on its way back down by going from 3 to 2. Which would give off red light. And then it can continue back to the ground state by going from 2 to 1. Which would give off ultra-violet light. The ones that the emissions that we see are in the visible region are called the Balmer Series and these other emissions also have names but it's the Balmer Series that we were observing in the demonstration. Wow so we've done a bunch of math and we've talked about a bunch of new words. So before we go any further I want to make sure you are getting the main idea of this model and not getting too bogged down by an potential levels of detail. So let's look at this simple picture. This is a picture that Bohr drew of the hydrogen atom. Which of these two electrons, electron A or electron B, is at higher energy? So here's two spots where the electron could be, either A or B. Which of those two electrons is higher in energy? Thank you for answering that question. Electron B has higher energy because it is a greater away from the positively charged nucleus. Because the electron is negative and the nucleus is positive and we know opposites attract which is governed by Hulum's law. The electron is at lower energy if it is closer to the nucleus. One analogy that sometimes helps students remember this is an analogy about running around a track, like at a track meet. Let's suppose that we had to run around the track, a 400 meter track, in one minute. Which is something I've never been able to do, but some of you can probably run that fast. Now if it was exactly 400 meters, right, that would be like the inside lane, start to finish, correct? But what if we said, no, you're not, if we're going to start here, we're going to start from phase five. We're going to start at the same spot, but electron b, you have to start from where. You are and get all the way around the track in one minute back to where you were. Would electron b have to be moving faster than electron a, to travel this circle in the same amount of time that electron a could travel that circle? Of course it would have to be. So electron b has a higher energy than electron a. [SOUND] Here is another problem for you to try from Ractice. . This time let's look at a single electron moving. The initial state of the electron is out here. It, what I would call n equals three, right? We can label these one, two, three. And the final state of that electron is going to be n equals two. What is the sign of delta e, the change in energy for this process. Is that positive? Or a negative. Well in order to answer that question, we need to remember the equation for delta e. Remember, delta e is the energy of the final state minus the energy of the initial state. Here the energy of the final state is a lower value or a smaller number. Right? And the energy of the initial state is a higher energy or a larger number. So the sign for delta E here is going to be a small number minus a big number, which is a negative number. [SOUND] All right, here's another question about this particular scenario. Is this process spontaneous. Yes. The process is spontaneous. So going from higher energy state to a lower energy state is spontaneous. Finally, is this process going to be an absorption or an emission of light? What do you think? Thank you for answering. Because we're going from a higher energy state to a lower energy state some energy has to be given off by the atom. And then in this case the atom does that by emitting light. This concludes the first part of the introduction to the bore model for the hydrogen atom. This model gave scientists their first insights into the quantized nature of atomic structure. In this lecture we talked about energy absorption and emission from the hydrogen atom. The difference between the ground state and the excited state. And we reviewed the energetics of these states.
