[MUSIC] In this lecture I'll be introducing Quantum Numbers. These are numbers that are needed to describe an electron. Remember, an electron is a particle that behaves as a wave, and to really describe it, we need a mathematical equation called a wave function. The wave function contains partial derivatives and Hamiltonian operators, things that are beyond the scope of the mathematics we'll be using in this class. However, we can still learn something from examining the quantum numbers that are used to describe the condition of the electron. The quantum numbers are part of a field of study called quantum mechanics. So you might be asking yourself, what exactly is quantum mechanics? Simply put, and many people have said this, quantum mechanics is the study of matter and radiation at an atomic level. In other words quantum mechanics is needed to describe things, or to study things that are extremely small on the link scale of the atom. We've been studying the Bohr model of the hydrogen atom quite a bit and in that Bohr model, we have something called n. What exactly is n? Well, n as it turns out, is a quantum number. We've seen n before, in Rydberg's equation for the Hydrogen atom's spectral lines, but n is also used in the solution for the wave function of atoms. Now I keep talking about a wave function. But isn't the electron a particle? In fact, we know electrons have mass. I gave you the mass of the electron several weeks ago. Well deBroglie said, what if the electron isn't a particle? What if we treat the electron as if it is a wave. If it's a wave, it needs to look like this. And in fact if you put a wave through a double slit in the famous Double Slit Experiment, you can see interference. Waves have a certain way that they need to behave and if they have boundaries, they start doing things in a certain way with their nodes. If you look at the wave here that's next to deBroglie, you can see that the wave oscillates back and forth, back and forth in a very even way and we could measure the wavelength, right, we could measure from peak to peak, and we could determine the wavelength. What waves don't do, is they never have a flaw like this. So waves have certain shapes that are allowed, and other shapes that are not allowed. So deBroglie's contribution was that the electron, which was thought of as a particle, and there was evidence for it being a particle we can calculate, and measure the charge on it. There were experiments such as the cathode ray tube experiment and the oral drop experiment, where electrons were moved around, indicating that they were particles and they were, they had mass. But the electron also exhibited wave-like properties such as interference, and deBroglie got the Nobel Prize for this discovery. The Wave-Particle Duality of the Electron, just as we have wave-particle duality of light. We're going to describe this particle that behaves like a wave, using a mathematical expression that contains variables that we are going to call quantum numbers. So there's going to be four variables in this mathematical expression, we need all four of them to fully describe the condition of the electron. The two that we'll be talking about in this particular presentation are n, the principle quantum number, that's the same one we've been using all along as n, and l, the orbital angular momentum quantum number. There's two other quantum numbers that we'll do within the next lecture. Ml, which is a magnetic orbital quantum number, and ms, the spin quantum number. All four of these values are necessary to fully describe the condition of the electron, but when we were describing the condition of the electron, there's n element of probability in that, but there's also an element of uncertainty. So we'll talk about that in the next lecture as well. Let's focus for now on n, the principal quantum number. Remember, n is giving us an idea of the approximate distance from the nucleus. So it's designating the level or the shell, which is now viewed as a cloud of electrons. About how large is that cloud? Is that cloud really far from the nucleus or is relatively close? N is a primary indicator of the electron's energy. The closer to the nucleus it is, the lower its energy is. And remember n has natural number of values, it's a positive integer. The lowest value that n is allowed to have is 1, and it can go up from there. The next quantum number that we're going to consider in this lecture, is l. L is the orbital angular momentum quantum number. It designates the sublevel and tells us something about the orbital shape. It's a secondary indicator of the electron's energy, because some shapes have higher energy than others. As it turns out mathematically, l has to be less than n, so, l can have a value from 0 up to n-1 depending on what n is. Here's some examples. If l equals 0, we have another notation that we use. We use a letter to describe the shape that l is designating. So if l is 0, the shape is a sphere, and we give that the notation s. If L is 1 the shape is a little bit different, and we give that the notation p. We say that's a p orbital. I'll show you some pictures of those in a minute. If L is 2, then the shape is different yet again, and it's a d orbital, and if L is 3 we have an f orbital. So l's a little bit unique because not only does l have numerical designations, but we also have given letters, to each of those shapes. Let's look at the shapes, as examples. Let's look first at s and p, because those are the two most common orbital shapes that we see in the lower atomic number elements. So for example we can have orbitals that are designated as 1s, 2s or 3s. You may remember these designations from a previous chemistry class that you had. This number 1, 2 or 3 is in. So n is 1, or n is 2, or n is 3. S is an indication of the orbital angular momentum quantum number, and here, if it's an s orbital, that means l is 0. An s orbital implies that the cloud has a spherical shape. The higher the value is for n, the larger that, that spherical shape is. And we think about it as being a shell that's kind of hollow in the middle, where the electrons travelling in this cloud that's around the perimeter of where the sphere is. Three is bigger even still. So I have these concentric shells, around a nucleus if that particular atom had electrons all the way up to the 3s sublevel. Now if n equals 2, then there's a choice. So if n equals 1, let's just go back for a minute. If N equals 1, l has to equal 0. Because l must be less than n. But if n equals 2, now I have a choice. If n equals 2, l can still be equals 0. Remember, that's an S orbital. But l could also be 1. If l is 1, remember the latter designation we gave to that was a p orbital. P orbitals are not spherical, they have this dumbbell-like shape. And because we no longer have spherical symmetry, we now have p orbitals that have different orientations based on the Cartesian coordinate. So here, a Cartesian coordinate has arbitrarily drawn x, y and z in certain directions. So it can have a p orbit that lays along the, the x axis, the way it's been written there, or I can have one that's coming out towards you, in this case, that's the direction of the y axis. Or it can have one that's going up and down, that's the direction of the z axis. Now, I chose x, y and z arbitrarily here. Right, sometimes people have y going up and z coming out at you, it doesn't matter, it's arbitrary choice. But the point is, that there are three different directions that the p orbital can have. The p orbital has directionality to it. The designation p is telling us the shape and not the direction. Now, I can have 2p orbital, but if n equals anything greater than 2, I can also have a p orbital of that sublevel. So if n equals 3, then l can be 0 as it is up here for the sphere, or l can be 1. And in fact, l can be 2, which is a d shape. But here l can be 1, that gives us the p orbital. The p orbital at the 3p level is slightly larger than the p orbital we saw at the 2p level. Okay. So let's practice determining the n and l quantum numbers for particular sublevels. We talked about 3s, 3p, the next one would be 3d. What are the Quantum numbers for the 3d sublevel? Well n, the principal Quantum number would be 3, and what would l be? L which is orbital angular momentum quantum number, would have to be 2. It does meet the criteria that l is less than n here. Remember, sometimes students have trouble remembering this. When l equals 0, that's the s orbital. Sometimes students want to line up s with 1. But l has to be 0 to have an s orbital, and then we can fill in the rest of this table. If we know the values for n and l, and these can be experimentally determined, then we can determine the electron's primary energy. They're the primary indicators of the electron's energy. So let's suppose we know the values of n and l for a couple of different electrons. Those electrons are in different sublevels. Which of these two sublevels is higher in energy, the 3p sublevel or the 3d sublevel? Well we can actually add up the values of n plus l, to get a primary indication of that sublevel's energy. So for the 3p orbital n equals 3. And what's l again? For the p orbital? That's right, l equals 1. And for the 3d orbital, n equals 3. It's the same value for n. But the shape is different, this time l equals 2. We're going to take a look at some of those d orbitals in the next presentation. If we add up n plus l, we see that for the 3p orbital n equal 3, l equals 1 and that gives us 4. And for the 3d orbital, if we add up n plus l, we get 5 because the n+1 value is larger for the 3d orbital, the 3d orbital is the one that's at higher energy. Okay, here's a question to consider. So we know that the 3d orbital is higher in energy. If an electron makes a transition from the 3d sublevel, so that's were it's starting, to the 3p sublevel, so it's starting at 3d and it's moving to 3p. WIll that electron need to absorb energy, or will that electron emit energy? Go ahead and try and answer that now. Thank you for working that problem. Because where the electron began was higher energy than where the electron ended up, that particular process would emit energy, probably in the form of light. Let's do another practice problem like this. Which of these two sublevels is higher in energy, the 5f or the 7s? So now we're getting down towards the part of the Periodic Table where there's some heavier elements. Remember what we need to do to determine the primary indication of the electron sublevel's energy, is to add up n plus l. So for the 5f, what's n? That's right, n equals 5 for the 5f. What is l for the 5f sublevel? Correct. L equals 3. This is really just a case of kind of rote memorization, I'm sorry to say. But you'll see as we go through the periodic table later and we do electron configurations that theres a method to the madness here, and hopefully it'll all start to come together once we trace it through the periodic table. How about 7s? Well, in there is 7, that's always the number. And what is l for the s orbital again? That's right, l equals 0. So let's add up these n plus l values, and we see that for the 5s orbital we get a total of 8. And for the 7s orbital, we get a total of 7. So the 5f orbital here has higher energy, even though it's n value is lower. So, here's a question to consider. Now, an electron is going to start in the 7s sublevel, and is going to finish in the 5f sublevel. In order for that transition to occur, will the electron need to absorb energy, or will the electron need to emit energy? Thank you for answering. In this case, it's starting at the lower energy level, the 7s and it's going to the higher energy level, the 5f. So that electron needs to absorb energy. Let's end this lecture by reviewing the two quantum numbers that we've talked about. N is the principle quantum number, that's the quantum number that gives us the approximation of the orbital size. Remember it's restricted to the natural number of the positive energers. L is the orbital angular momentum quantum number, and remember it's restricted by n. It must be less than n, and it's a whole number. It can be as low as 0. It indicates the electron's sublevel. It's a secondary indicator of the electron's energy, and it's what describes the shape of the electron's probability region or cloud. In the next lecture, we'll review n and l one more time, and we will also talk about the other two quantum numbers. Remember, we need four quantum numbers to fully describe the condition of an electron. So, in the next lecture, we'll talk about ml and ms. [SOUND]
