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Chemical Science - Hydrogen Atom - Lecture 6

Chemical Science/n * Email this page/nVideo Lectures - Lecture 6/nTopics covered: /nThe Hydrogen Atom/nInstructor: /nProf. Sylvia Ceyer/nTranscript - Lecture 6/nAll right. Let's get going. Where were we? We were at the point where we started out the course wondering about the structure of the atom, how the electron and the nucleus hung together. And we saw that we could not explain how that nucleus and electron hung together using classical ideas, classical physics, classical mechanics and classical electromagnetism./nAnd so we put that discussion aside and started to talk about the wave-particle duality of light and matter. And we saw that both light and matter can behave as a wave or it can behave as a particle./nAnd we needed that discussion in order to come back to talk about the structure of the atom. And, in particular, what was so important last time that we met is that we saw the results of an experiment, that Davidson, Germer and George Thompson experiment that demonstrated that mass of particle could exhibit wave-like behavior./nWe saw the interference pattern of electrons reflected from a nickel single crystal. An actual original paper that reports that result is on our website. You're welcome to take a look at it. But that really was the impetus, this observation of the wave-like behavior of matter./nThat was the impetus for this gentleman here, Erwin Schrödinger in 1926-27 to write down an equation of motion for waves. That is he thought maybe the answer here is that if particles can behave like waves then maybe we have to treat the wave-like nature of particles, the wave-like nature of electrons, in particular in the case when the wavelength, the de Broglie wavelength of the particle or the electron is comparable to the size of its environment./nMaybe in those cases we have to use a different kind of equation of motion, a wave equation of motion. And that is what he did. So, he wrote down this equation. We briefly looked at it last time. This equation has some kind of operator called the Hamiltonian operator./nIt has a hat on it, a carrot on the top of it. That tells us it is an operator that operates on this thing, psi. That psi is what's going to represent our particle. That psi is a wave, since we're going to give it a functional form in another day or so we're going to call it a wave function./nSomehow that psi represents our particle. Exactly how it represents our particle is something we're going to talk about again in a few days. But right now the important thing is to realize that this psi represents the presence of a particle, in this case the electron./nAnd when H operates on psi we get back psi. We do this operation and out comes psi again, the original wave function, but it's multiplied by something. That something is the energy. It's the binding energy./nIt's the energy with which the electron is bound to the nucleus. This equation is an equation of motions. This wave equation, Schrödinger's equation is to this new kind of mechanics, called quantum mechanics, like Newton's equation of motion, and I show you just the second law here, are to classical mechanics./nThis equation here tells us how psi changes with position and also with time. It tells us something about where the electron is and also tells us where the electron is as a function of time. It's an equation of motion./nAnd this is what Schrödinger realized, is that maybe when the wavelength of a particle is on the order of the size of its environment you have to treat it with a different equation of motion. You can no longer use F = ma, this classical equation of motion./nYou have to use a different equation of motion. And that's Schrödinger's equation, this wave equation. Can we dim the front lights a little bit? Because the screen is just a little bit hard to see, I think./nWe are going to let your electron be represented by this wave ?. And so psi, since it's going to be representing the electron, is going to be a function of some position coordinates, and also in the broadest sense a function of time./nNow, we, of course, can label psi in Cartesian coordinates giving it an x, y and z. If I gave you x, y and z for this electron in this kind of coordinate system where the nucleus is at the origin of the coordinate system, if I give you x, y and z coordinates, you'd know where the electron was./nBut it turns out that this problem of the hydrogen atom is really impossible to solve if I use a Cartesian coordinate system. So, I am going to use a spherical coordinate system. How many of you are familiar and have used a spherical coordinate system before? A few of you./nNot all of you. Well, it's not hard to understand. And it is important that you understand it. So, instead of giving you an x, y and z to locate this electron, this particle in space, we're going to give you an r, a T and a ?./nAnd the definitions of r, T and ? are the following. If here is my nucleus at the origin and here is the electron, r is the distance of that electron from the nucleus. It's just the length of this line right here./nThat's one coordinate. A second coordinate is T. T is the angle that this r makes from the z axis. And then the third coordinate is psi. Psi is the following. If I take that electron and I just drop it perpendicular to the xy plane, and I then draw a line in the xy plane from that point of intersection to the origin, the angle that that line makes with the x-axis here is ?./nSo, I am going to give you an r, a T and a ?. R is just the distance of the electron from the nucleus. T and psi tell us something about the angular position. And then, as I said, in the largest sense, there is also time./nBut we'll talk about time a little bit later. Psi represents our electron. Now, what does the Schrödinger equation specifically, for a hydrogen atom, actually look like? This Hpsi times Epsi is kind of generic Schrödinger equation./nAnd now we've got to write a specific one, one specific for the hydrogen atom. We need to know what this is, H. That's our Hamiltonian, our operator. And so the operator here for that hydrogen atom is this./nWhat is it is essentially three second derivatives, one here is with respect to r, a second is a second derivative effectively with respect to T, and another one, the final one is a second derivative with respect to psi./nIn other words, if this whole Hamiltonian is operating on psi, what you're going to do is essentially take the first derivative of psi with respect to r, multiply it by r2. And then take a first derivative with respect to r again and multiple it by 1 / r2./nAnd add to that the first derivative of psi with respect to T multiplied by psi and T, etc. You don't have to know this. I'm just showing this to you so you recognize it later on. This is a differential equation./nIn 18.03, you learned how to solve these differential equations. And then there is another very important term here, so it's all of this plus this, u of r. What is u(r)? Potential energy of interaction./nAnd the potential energy of interaction, of course, is the Coulomb potential energy right here, one over r dependence. We've talked about the Coulomb force. This is the potential energy of interaction that corresponds to the Coulomb force./nSo, that's the specific Schrödinger equation for the hydrogen atom. Now, what we have got to do is we have to solve this equation for the hydrogen atom. And when I say solve this equation, what I mean is we're going to have to find E, these binding energies of the electron to the nucleus./nThat is part of our goal when we say solve this differential equation is knowing what E is, is figuring out what E is. And, actually, this is what we're going to do today, finding those energies. But then the second goal is to find psi./nWe want to find what is the functional form of psi that represents the electron and the hydrogen atom? Therefore, we're going to want to find the wave functions for psi. And, you know what, those wave functions are nothing other than what you already sort of know, and that is orbitals./nYou talked in high school about S orbitals and P orbitals and B orbitals. Those orbitals are nothing other than wave functions. They come from solving Schrödinger's equation for the hydrogen atom./nThat's where they come from. Now, specifically the orbital is something called the spatial part of the wave function as opposed to the spin part. But for all intents and purposes they are the same./nWe're actually going to use these terms interchangeably, orbital-wave function, wave function-orbital. The bottom line is that when you solve Schrödinger's equation for the energy and the wave function it makes predictions for the energies and the wave functions that agree with our observations, as we're going to see today, in particular for the case of the binding energies of the electron to the nucleus./nThis equation predicts having a stable hydrogen atom, a hydrogen atom that seemingly lives forever in contrast to when we use classical equations of motion. When we used classical equations of motion we got a hydrogen atom that lived for all of 10-seconds./nBut here we finally have some way to understand the stability of the hydrogen atom. It makes the Schrödinger equation, it makes predictions that agree with our observations of the world we live in./nAnd, therefore, we believe it to be correct. That is it. It agrees with the observations that we make. Let's start. And we're actually not going to solve the equation, as I said, but you will do so if you take 5.61, which is the quantum course in chemistry, or 8.04, I think it is, which is the quantum course in physics after you take differential equations so that you know how to solve the differential equations./nBut we're going to write down the solution in particular here for E, the binding energies of the electrons to the nucleus. Now we're going to need this. We've got H? = E?. And when we solve that equation, we get the following expression for E, these binding energies./nE = 1 / n2 (me4/8?o2 h2). That is what we get out of it. And there is a minus sign out in front. Now, what is m? M is the mass of the electron. What is e? E is the charge on the electron. Eo is this permittivity of vacuum that we talked about before./nIt's really just a unit conversation factor here. H is Planck's constant. Here comes Planck's constant again. It is ubiquitous. It's everywhere. And what we do is that we typically take all of these constants and group them together into another constant that we call the Rydberg constant./nAnd we denote it as a RH. All of that is equal to RH, so this is over n2, minus 1. And the value of RH, that Rydberg constant, and this is something you're going to need to use a lot in the next few weeks, is 2.17987 times 10 to the negative 8 joules./nBut you also see, in this expression for the binding energies of the electron to the nucleus, that there is this n here. What's n? N is an integer. When you solve that differential equation, you find that n has only certain allowed values./nN can be as low as 1, 2, 3, and n can go all the way up to infinity. N is what we call the principal quantum number. I am going to explain that a little bit more by looking right now at an energy level diagram./nThat is the expression. But now let's plot it out so that we can understand what is going on here a little more. We are going to be plotting this expression as n goes from 1 to infinity. I have the energy access here./nEnergy is going to be going up in that direction. When n is equal to 1, the binding energy of that electron to the nucleus is effectively the Rydberg constant. Here I rounded it off, minus 2.18 times 10 to the negative 18 joules./nBut our expression here says that there can be another binding energy of the electron to the nucleus. It says that n can be equal to 2. And if n is equal to 2, well, then the binding energy of the electron is one-quarter of the Rydberg constant, because it is the Rydberg constant over 22./nIf n is equal to 3, well, our expression says that the binding energy is minus one-ninth of the Rydberg constant. If n is equal to 4 it is minus a sixteenth of the Rydberg constant, n is equal to 5, minus a twenty-fifth, n equal to 6, minus a thirty-sixth, n equal to 7, minus a forty-ninth, all the way up to n equals infinity./nAnd you know what the value of the binding energy is when n is equal to infinity? Zero. Our equation says that the electron can be bound to the nucleus with this much energy or this much energy or this much energy and so on, but it cannot be bound to the nucleus with this much energy, somewhere in between, or this much energy or that much energy./nIt has to be exactly this, exactly this, exactly this, so on and so forth. That is important. What we see here is that the binding energies of the electron to the nucleus are quantized, that that binding energy can only have specific allowed values./nIt doesn't have a continuum of values for the binding energy. Yes? Those are identically the same size. Because this is an operator, right? I left the hat off here. Remember that we took a second derivative of psi? So, you cannot cancel this./nThis is an operator taking the derivative of psi. You cannot just cancel that. This is to multiply by on this side. This side is. This is E times psi, but not over here. That's really important./nWe have these quantization of the allowed binding energies of the electron to the nucleus. Where did that quantization come from? That quantization came from solving the Schrödinger equation. It drops right out of solving the Schrödinger equation./nHow did that happen? Well, in differential equations, as you will see, when you solve a differential equation, what you have to do to solve it so that it adequately describes your physical situation is you have to often impose boundary conditions onto the problem./nAnd it's that imposition of boundary conditions that gives you that quantization. That is where it comes from mathematically. In other words, remember one of those angles that I showed you, the phi angle? You can see it would run from zero to 360 degrees./nBut you also know, if you go 90 degrees beyond 360 degrees, suppose you to go to 450 degrees, well, that should give you the same result as if you had F = 90. What you have to do is you have to cut off your solution at 360 degrees./nWhen you cut off that solution, well, then that gives you, in these differential equations, these quantization. That is physically where it comes from. Again, this is not something you're responsible for, but when you do differential equations later on in 18.03, you will see how that happens./nLet's talk some more about these loud energy levels. When the electron, or when n = 1, the language we use is that we say that the hydrogen atom, or we say that the electron or the hydrogen atom is in the ground state./nWe call this the ground state because this is the lowest energy state. It has got the most negative energy. It's the lowest energy state. We call n = 1 the ground state of the hydrogen atom or the ground state for the electron./nWe use those terms of the electron or the hydrogen atom interchangeably. Now, what's the significance of this binding energy? And this is important. The significance is that the binding energy is minus the ionization energy for the hydrogen atom, because if I put this energy in from here to there into the system then I will be ripping off the electron and will have a free electron./nSo, the ionization energy is minus the value of this binding energy. The ionization energy is always positive. The binding energy, the way we're going to treat this, is going to be negative because the electron is bound./nAnd then the separated limit, the electron far away from the nucleus, well, that energy is zero. So, the binding energy is minus the ionization energy or conversely the ionization is minus the binding energy./nThat is the physical significance of these binding energies. And when we talk about an ionization energy for an atom, we are typically talking about the ionization energy when the atom is in the ground state./nThis is the ionization energy we're talking about. But we also said that the binding energy of the electron can be this much meaning it's in the n = 2 state. That can be possible also. Not at the same time as it's in the n = 1 state, but you can have a hydrogen atom in a state, which is the n = 2 state./nWhat that means is that the electron is bound by less energy. When that is the case, we talk about the hydrogen atom being in the first excited state. This is the ground, this is the first excited state, but n is equal to 2./nIn that case, the electron is not as strongly bound because it is going to require less energy to rip that electron off. The binding energy is n = 2 is minus the ionization energy if you have a hydrogen atom in the first excited state./nMake sense to you? Yeah. OK, so we can have atoms in this state, too. Then the ionization energy is less. It takes less energy to pull the electron off. Yes? In everything that we are going to deal with, we are going to have binding energies that are negative./nLet's do that. You can, of course, have a binding energy that is positive, but the problem is that isn't a stable situation. OK. Good. Other questions? Yes. When we're dealing with a solid, we talk about a work function as opposed to an ionization energy./nWhen we're dealing with an atom or a molecule, we talk about an ionization energy as opposed to the work function. It's really the same thing. Historically there is a reason for calling the ionization energy off of a solid the work function./nOh, one other thing. I just wanted to point out again right here is that when n is equal to infinity the binding energy is zero. That is the ionization limit. That is when the electron is no longer bound to the nucleus./nNow, one other point here is that this solution to the Schrödinger equation for the hydrogen atom works. It predicts the allowed energy levels for any one electron atom. What do I mean by one electron atom? Well, helium plus is one electron atom./nBecause helium usually has two electrons, but if you take one away you have only one electron left. And so this helium plus ion, that's a one electron atom, or if you want to say it more preciously one electron ion./nOr, lithium double plus, that's a one electron atom or a one electron ion. Because lithium usually has three electrons, but if you take two away and you only have one left that's a one electron atom./nUranium plus 91 is a one electron atom. Because you took 92 of them away, one is left, that's a one electron atom or an ion. And the bottom line is that this expression for the energy levels predicts all of the binding energies for one electron atoms as long as you remember to put in the Z2 up here./nFor a hydrogen atom that is, of course, Z = 1, so we just have -RH / n2. But for these other one electron atoms you have to have the Z in there, the charge on the nucleus. Why? Well, because that Z comes from the potential energy of interaction./nThe Coulomb potential energy of interaction is the charge on the electron times Z times E, the charge on the nucleus. That is where the Z comes from. That is important. How do we know that the Schrödinger equation is making predictions that agree with our observations? Well, we've got to do an experiment./nAnd the experiment we're going to do is we're going to take a glass tube like this. We're going to pump it out and we're going to fill it with hydrogen, H2. And then in this glass tube there are two electrodes, a positive electrode and a negative electrode./nAnd what I am going to do here is I am going to crank up the potential difference between these positive and negative electrodes, higher, higher, higher until at a point we're going to have the gas break down, a discharge is going to be ignited, just like I am going to do over here./nDid I ignite a discharge? Yes. There it is. And the gas is going to glow. We are going to have a plasma formed here. Oh, and what happens in this plasma is that the H2 is broken down into hydrogen atoms./nAnd these hydrogen atoms are going to emit radiation. That is some of the radiation that you're seeing here in this particular discharge lamp. We are going to take that radiation and we're going to disburse it./nThat is we're going to send the light to a diffraction grading. This is kind of like the two-slit experiment. And when you look at it you're going to see constructive and destructive interference./nBut when you look at the bright spots of constructive interference you're going to find that those bright spots now are broken down into different colors, purple, blue, green, etc. And that is because the different colors of light have different wavelengths./nAnd if they have different wavelengths, well, then the points in space of maximum constructive interference are going to be a little different. And so we're going to literally separate the light out in space depending on their colors./nAnd we're going to see what colors come out of this. And so now, in order to help you do that, we've got some diffraction grading glasses for you. You should put them on and look into this light. And you will be able to see off to your left and to the right some very distinct lines./nAnd if you look into the lights above you can see all different colors from the white light. All right. Do you see the hydrogen lamp? I know that the white lights above the room are more interesting because there is a whole rainbow there./nI am going to turn the lamp a little bit since not all of you, if you are way on the side, can see it. I am going to start over here and I am going to turn the lamp a bit. Can you see that now? You should see a bunch of lines here to your left and some to your right./nAnd then, of course, you will see some up here. But they will probably dispersed best to your left and to your right. Pardon? Can we dim the bay lights? Can we dim those big lights over there? Probably not./nI am going to turn it over here. Can you see it? The spectrum that you should see is what I am showing on the center board there. You see it? Pardon? You have to look at the light. Oh, thank you./nThank you very much. Can you see that better? I will turn it back there. Do you see the emission spectrum now? It's a little better. Let's see if we can try to understand this emission spectrum that you're seeing./nWhat you should see the brightest is a purple line. No? Well, let's see. The purple line is actually rather weak, I have to say. If you come really close you can see it. And you're invited to come up a little bit closer./nThe purple line is kind of weak. What did I do? [LAUGHTER] Oh, I see. Yes. Interference phenomena work. Hey, look at that. [LAUGHTER] Fantastic. All right. The purple line is kind of weak, but the blue line is really strong./nAnd then there is a green line, which is also a little bit weak. And I can see because I'm really close, well, I'm not going to tell you that. There is a green line there. And then there is this red line./nLet's see if we can understand where these lines are coming from. What is happening is that this discharge, not only does it pull the H2 apart, break bonds, make hydrogen atoms, but it puts some of those hydrogen atoms into these excited states./nAnd so a hydrogen atom might be in this excited state. This initial excited state, high energy state. And, of course, that's a high energy state. It is unstable. The system wants to relax. It wants to relax to a lower energy state./nAnd when it does so, because it's going to lower energy state, it has to emit radiation. And that radiation is going to come out as a photon whose energy is exactly the energy difference between these two states./nThat's the quantum nature here of the hydrogen atom. The photon that comes out has to have an energy psi e which is exactly the energy of the initial state minus that of the final state. And, therefore, the frequency of that radiation is going to have one value given by this energy difference divided by Planck's constant H./nThat is what's happening in the discharge. What we've got is some hydrogen atoms excited to say, for example, this B state, which is a lower energy state, and so when it relaxes there is a small energy difference between here and this bottom state./nTherefore, you are going to have a low frequency of radiation. If you have some other hydrogen atoms in the discharge that are excited to this state up here, well, this is a big energy difference. And so psi e is going to be large./nAnd, therefore, you're going to have some radiation emitted that's at a high frequency because psi e is large. If it's at a high frequency, it's going to have a short wavelength. These hydrogen atoms are going to have a low frequency emission./nIt's going to be a long wavelength. So, we've got a mixture of atoms in this state or in this state or in any other state in this discharge. Now, let's try to understand this spectrum. And to do that I have drawn an energy level diagram for the hydrogen atom./nHere is n = 1 state, n = 2, n = 3, n = 4, all the way up to n = 0 here on the top. They get closer and closer together as we go up. This purple line, it turns out, or the purple color comes from a transition made from a hydrogen atom in the n = 6 state to the n = 2 state./nThe final n here is 2. The blue line comes from a hydrogen atom that has made a transition from n = 5 also to n = 2. The green line is from a hydrogen atom that makes a transition from n = 4 to n =2 and then the red line from n = 3 to n =2./nOf course, the transition from n = 3 to n = 2 is the smallest energy. Therefore, it is going to be the longest wavelength. n = 6 to n = 2 largest energy. Therefore, it is going to have the smallest wavelength./nNow, how do we know that these frequencies agree with what Schrödinger predicted they should be? Well, to know that, what we're going to do is we're going to write down this equation here, which is just telling us what the frequency of the radiation should be psi e over H./nBut we're going to use the predictions from the Schrödinger equation and plug them into here to calculate what the frequency should be. We were told here that the energy, say, of the initial state given by the Schrödinger equation is -RH / the initial quantum number squared./nWe're going to plug that into there. The final state, well, that's also the expression for the energy, we're going to plug that into there. We're then going to rearrange that equation so we get the frequency is the Rydberg constant over H times this quantity 1 / nf2 - 1 / ni2./nAnd since I told you here that all of these lines -- The final quantum number is 2. We can plug that in. And then we can just go in and put in 3, 4, 5, 6 and get predictions for what nu is for the frequency./nAnd what you would find is that predictions that this makes, the Schrödinger equation makes agrees with the observations of the frequencies of these lines to one part and 108. There is really just remarkable agreement between the energies or the frequencies predicted by Schrödinger equation and what we actual observe for the hydrogen atom./nHere is another diagram of the energies of the hydrogen atom n = 1, n = 2, n = 3. And the four lines that we were looking at where shown right here. These are the four lines. Here is n = 6 to n = 2, n = 5 to n = 2./nThese lines are actually called the Balmer series. I want you to know that there is also a transition from n = 6 to n = 1. It is over here. But you can see that that transition is a very high energy transition./nThat transition occurs in the ultraviolet range of the electromagnetic spectrum. And, therefore, you cannot see it, but it is there. Actually, what you can see is that there are transitions from these higher energy states to the ground state, transitions from all of them to the ground state, but they're all in the ultraviolet range of the electromagnetic spectrum./nThat is why you cannot see that right now. But those lines are called the Lyman series. And then there are transitions here to the n = 3 state. These transitions from the larger quantum number to n = 3 are called the Paschen series./nThey occur in the near infrared. Brackett series in the infrared. Pfund series in the far infrared. I got that backwards. And these different series are all labeled by the final state. And they're labeled by the names of the discoverers./nAnd the reason there are so many different discoverers is because in order to see the different kinds of radiation, you have to have a different kind of detector. And, depending on what kind of a detector an experimentalist had, well, that will dictate then what he actually can see, what kind of radiation, which one of these transitions he can view./nNow, we looked at emission. But it is also possible for there to be absorption between these allowed states of a hydrogen atom. That is we can have a hydrogen atom here in a low energy state, the initial state Ei./nAnd if there is a photon around whose energy matches the energy difference between these two states, well, then at photon can be absorbed by the hydrogen atom. Again, the energy of that photon has to be exactly the difference in energy between those two states./nIt cannot be a little larger. If it is a little larger that photon is not going to be absorbed. That's important. That's the quantum nature again of the hydrogen atom. They are specific energies that are allowed and nothing in between./nAnd then from knowing the energy of the photon you can get that frequency. And then in the case of absorption, the frequencies of the radiation that can be absorbed by a hydrogen atom are given by this expression./nThis expression differs from the frequencies for emission only in that I've reversed these two terms. This is 1 / ni2. This is 1 / nf2. I have reversed them so that you come out with a frequency that is a positive number./nFrequencies do have to be positive. So, we've got two different expressions here for the frequency depending on whether we're absorbing a photon or we're emitting a photon. Questions? I cannot see anybody./nThere. Epsilon knot is a conversion factor for electrostatic units. That is all you need at the moment. In 8.02 maybe you will go through the unit conversation there to get you to SI units.

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