WEBVTT chemistry/physical-chemistry/hovasapian
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Hello, welcome back to www.educator.com, welcome back to Physical Chemistry.
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Today, we are going to start our discussion of molecular spectroscopy.
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I had a little difficulty deciding on how to actually present molecular spectroscopy.
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What I decided to do was to start with a lesson which gives an overview of primarily the rotational and vibration spectroscopy.
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The reason I want to do this was the lessons that come after this are actually going to discuss in detail, what it is that I present here.
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But I wanted to give a big picture of what it is that is going on
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because you are going to be sort of swimming in this ocean of equations.
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And a lot of it is like where is this coming from and when do I use this?
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I want to let you know why we are choosing the equations we are choosing and
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how to actually choose when you are faced with a specific problem.
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Again, this is the big picture so you have an idea of what is going on with the details,
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when we get to the details of subsequent lessons.
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Let us get started.
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Let me go ahead and write that down and repeat that.
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This lesson intends to provide a concise and broad overview of rotational and vibrational spectroscopy.
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There is nothing in this particular lesson that you actually have to like know for sure in terms of an equation because again,
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all of this is going to be discussed in detail for the next lesson and the lessons that follow.
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This is just a big picture.
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Just get an idea of what is going on before you get down to the nitty gritty.
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What you see here will be discussed in detail in the next 4 lessons.
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In the lessons that follow and in your book, it can appear that you are swimming in an ocean of equations.
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This lesson here hopes to answer the question which equation and why, which equation do I use and why.
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Very important equation, which is a very important question.
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The discussion that we do for all spectroscopy is only going to concern diatomic molecules.
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This discussion concerns diatomic molecules.
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When a molecule absorbs radiation of a given frequency of transitions to a higher state,
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you know this and you are reasonably familiar with spectroscopy from your work in organic.
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The transitions to a higher rotational, vibrational, or electronic state.
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Microwave radiation tends to affect only the rotational state.
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The infrared tends to affect the vibrational state but with vibration you also get rotational changes.
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The visible ultraviolet region of the electromagnetic spectrum tends to promote electronic transitions.
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Along with electronic transitions, you also get vibrational changes and you also get rotational changes.
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The higher the energy, the more it does.
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The frequency of this absorbed radiation or emission, emission is just the other way, excited state down to lower state.
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There is no real difference but for our purposes emission.
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The frequency is given by, we have is relation that you remember from early on.
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If there is a change in the energy, the energy of the final state - the initial - the energy of the initial state.
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It is energy final - energy initial and that was equal to H × ν.
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We have this equation already.
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Therefore, the frequency of this transition is going to be the final energy - the initial energy
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divided by planks constant, in terms of frequency, in terms of hertz.
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The energy of the final state - the energy of the lower state, or the arrival state to the departure state.
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However you want to say it, how it started and where it ended divided by planks constant.
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That gives you the frequency that we see on the spectrum.
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That is all that is happening.
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Or we can say, we can call it energy upper - energy lower divided by that.
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This is going to be important for us because these frequencies are what we going to see on the spectrum.
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That is what we are reading off is this.
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Any time you want to know what the frequency of the spectral line is,
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take the higher energy - the lower energy divide by planks constant.
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In spectroscopy, we usually work in wave numbers not Hz.
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In other words, inverse cm.
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A wave number is anything with a ̃ symbol on top of it, means it is in inverse cm.
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The definition is very easy to state whether frequency you have divided by the speed of light,
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that will give you the wave number.
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It is also equal to 1/ the wavelength λ.
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In cm then, in inverse cm then the frequency of the spectral line that we see
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is going to be the final energy - the initial energy divided by HZ.
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That was going to be important for us.
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What we are going to try to do, we will try to find equations that explain, that predict the experimental spectra that we see.
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We run an experiment, we get some lines on the spectrum.
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We try to come up with equations that explain those lines, that is all we are doing.
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It is really all we are doing.
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The ν above, the wave number above are the frequencies that we see on the spectra.
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In other words, they represent the differences in energy between the starting state and the final state.
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I’m going to say the starting state and excited state.
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How is that, it is probably a little bit better.
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Usually, we can really be going from ground stage to excited state.
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We are going to be seeking equations for the energy of a given quantum state.
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We, then form energy final - energy initial to give us the frequency that we observe.
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This is really what we are doing here, for the next 4 or 5 lessons all we are really concerned with,
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we want to find an expression for the energy of a given quantum mechanical system.
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We subject that quantum mechanical system to the radiation, microwave, infrared, visible UV.
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The rotational, vibrational, electronic transitions that take place taken from one level to another.
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One rotational level to another rotational level.
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One vibrational level to another vibrational level.
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One electronic state to another electronic state.
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We can find the energies of those two states.
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We want to find equations that will give us the energy for those two states.
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We actually have them, that is what we did and what we have been doing for the last 30 or 40 lessons in quantum mechanics,
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finding energies for the different quantum mechanical systems that we are dealing with,
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particle in a box rigid rotator harmonic oscillator, whatever it was.
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If I take the difference between the ground state or the beginning state and the excited state,
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what I get are the frequency that I see on the spectra.
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We want to find equations for those.
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We find the equations for the energies, that is what is important.
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And then, we take the difference between the lower and the higher energy level and
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that gives us the frequency that we see on the spectra.
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We just want equations for E and for ν.
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Let us see, I’m going to leave off the electronic energy for right now.
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Just know that it is actually there and in the subsequent lessons where we introduce it.
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But for right now, I just want to talk about vibration and rotation.
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If you understand those well, everything else after that is very very simple because it is the same thing.
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I’m just adding one more term for the electronic energy.
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Leading out the electronic energy of an atom for the moment,
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the total energy of a molecule is equal the energy of the rotation of the molecule +
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the energy of the vibration of the molecule.
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Let me stop for a second.
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A molecule has 4 types of energy.
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A molecule is translational energy, it is moving.
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It has electronic energy, the energy of the electronic states.
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It has energy of vibration, it is vibrating.
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It has the energy of rotation, it is rotating.
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Notice that I have left off the electronic energy.
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We have also just automatically left off the energy of translational because
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the energy of translation is not affected by spectroscopic interaction, by radiation interaction.
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Basically, we will deal with the translational energy a little bit later when we talk about statistical thermodynamics.
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But for spectroscopy, we are only concerned with electronic, rotation, and vibration.
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For right now, I'm leaving off the electronic just to concentrate on rotation and vibration.
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I think that gives us the best big picture.
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For our purposes, the total energy comes from the energy of rotation and energy of vibration.
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Let us deal with the rotational energy first.
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The energy of rotation, the model for that is our rigid rotator.
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We pretend that a diatomic molecule is just two bodies stuck together and it is rotating,
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that gives us a model for this.
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Now rotator, the energy sub J we said was equal to H ̅² / 2I × J × J + 1.
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J was equal to 0, 1, 2, and so forth.
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I was the rotational inertia, it was the reduced mass × the bond length².
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It had degeneracy as a function of J is equal to 2J + 1.
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This is energy in Joules.
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We want to express the energy in wave numbers.
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Basically, then take any energy in Joules and just divide by HZ and that will give you an energy in wave numbers.
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In inverse cm, our energy sub J, we are going to express this now in terms of wave numbers.
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We will get a new symbol F of J and we write it this way BJ × J + 1,
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where B is equal to planks constant / 8 π² C I.
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The rotational energy in inverse cm is given by this equation B × J × J + 1.
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I’m not going to get into great detail here about what each of all the stuff is because
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I will discuss it again in the subsequent lesson, in the next lesson.
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In fact, we are going to start off with vibration and rotation.
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We will talk about all of this in great detail.
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Do not worry, I just want to show you again what is happening with the equations,
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why we are choosing the equations we are choosing.
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This gives us the equation for the rotational energy of a molecule.
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For the vibrational energy, for E sub V vibration, this is from the harmonic oscillator.
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That is our model so we are going to begin with that equation to represent the vibrational energy of the molecule.
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The energy sub R was equal to ν × R +, it was H μ + ½ values of 0, 1, 2, 3, and so on.
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Again, these are the vibrational quantum numbers.
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Here, ν was equal to 1 / 2 π K / μ ^½.
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In inverse cm, our expression is energy sub R given new symbol G of R is equal to ν ̃ × R + ½.
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Here, ν~ is equal to ½ π C / μ ^½.
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Again, do not worry about it this is just big picture stuff.
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Now, I have the vibrational energy, it is given by this thing.
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I have the rotation energy given by what you saw.
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Our total energy is equal to the vibrational energy + the rotational energy.
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We have that the total energy is equal to, total energy is a function of R,
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the vibrational quantum number and J the rotational quantum number.
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It is equal to G of R + F of J.
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E sub RJ is equal to this thing ν × R + ½ + B~ × J.
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Let me make this J a little bit more clear here.
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J × J + 1.
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R takes on the values 0, 1, 2, and so on.
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J takes on the values 0, 1, 2, so on, independently.
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This, under the rigid rotator harmonic oscillator approximation for the energy of a molecule,
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this equation gives me the energy of a molecule who is in vibrational state R, rotational state J.
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ERJ is equal to ν~ R + ½ + B × J × J + 1 under the harmonic oscillator rigid rotator approximation,
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because molecules are not rigid and they are not harmonic.
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The first approximation, I’m going to make one correction for this.
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Under the harmonic oscillator rigid rotator approximation, this equation gives the energy.
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Very important.
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The frequency of absorption because the energy of the molecule in vibrational state R and rotational state J.
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That is very straightforward.
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To find the equation of the spectral line, find the equations of the spectral lines.
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In other words, the transitions from one energy level to another.
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To find the equations of the spectral lines, we take the energy R upper J upper - the energy R lower J lower.
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The upper energy - the lower energy, whatever those happen to be.
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For example, if I would want the equation for the observed, for the ν of the spectral line
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for the 0, 2 to 1, 3 transition, this is R lower and this is the J lower.
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This is the R upper, this is the J upper.
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I would form the Δ E.
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In other words, I would form the energy of the 1, 3 - the energy to 0, 2.
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That will give me an equation for the spectral line.
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The equation that predicts where it should be.
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Running the experiment tells me where actually is.
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The extent to which is a good match depends on my equation.
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This is an approximation that is going to be off when I start making corrections to
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this harmonic oscillator rigid rotator approximation, that gives me a better and better predictions until it is almost exact.
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That is what is happening here.
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It is the energy equation that is important.
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The spectral line equation, the observed frequency or the predicted frequency,
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that I can derive just by taking the energy / - the energy lower.
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It involves a lot of algebra but it is doable.
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It is the energy equation that is important.
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The equations for the absorption emission frequencies can be derived with algebra, energy upper - energy lower.
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A lot of the mess that you see, as far as all these equations, all the derivations that you see
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in the spectroscopy, that is the stuff right here.
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We are taking upper energy - lower energy.
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We are coming up with different equations.
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The thing is we have the harmonic oscillator rigid rotator approximation.
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You will see in a minute that would give us one set of equations.
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When I start making corrections to that, for different phenomenon that I observe
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to make my equations match more of what the real spectra look like give me the different equations.
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It is not really 150 equations that you have to know, you have to know just one.
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The corrections to that one can where everything else comes from.
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That is what I'm trying to do with this lesson.
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I’m trying to show you which one or two equations are important.
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And then from there, depending on what corrections you make, you can derive everything else.
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That is what you are seeing is the derivations.
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Do not get lost in the ocean.
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That is why this is probably the most important lesson of spectroscopy, the overview to the big picture,
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the forest before we get into the trees.
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Let me go back to black here.
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The harmonic oscillator rigid rotator approximation is precisely that, just an approximation.
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Approximation is just that.
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It is an approximation.
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We will now make corrections to the vibrational term, rotational term,
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to make the equations more closely match what we see in experiment.
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To make the equations better match and predict what we see in reality.
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We are going to correct for three things.
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We will correct for three things.
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The lessons, each one, we will talk about a different correction.
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We will correct for three things, we will talk about what they are.
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This is all big picture stuff.
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It is actually really important.
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I wish that more people would spend more time on the big picture stuff because it will make all the details
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and you will know exactly what is going on because I can see the big picture.
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It is better to see from the outside in than it is to be the inside trying to look out, that is the idea.
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Any time you find yourself lost in science or math or whatever it is,
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99% of the time it is going to be because you are inside trying to look out.
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Try to find someone or some book or some other way to get yourself on the outside looking in the big picture.
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All the details are not irrelevant but are secondary.
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If you have the big picture, you really understand, then science becomes a beautiful thing that really is.
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We will correct for three things.
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The first thing we are going to correct for, we would be correcting for something called vibration rotation interaction.
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We are also going to correct for something called centrifugal distortion.
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We are going to correct for anharmonicity.
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Each correction will modify our basic equation, our basic energy equation E sub RJ is equal to ν ̃ R + ½ + B~ × J × J + 1.
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This is our basic equation, it is a rigid rotator harmonic oscillator approximation to the energy of a molecule.
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Each correction we make is going to modify this equation and give us a new energy equation.
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I can do one correction.
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I can correct for 1, I can correct for 2 of them, any two.
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Or I can correct for all three, depending the equation becomes more and more complicated.
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That is all that is happening.
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The three corrections above R will be discussed in the lessons that follow.
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Let us talk about our first one which is vibration rotation.
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Let us talk about the vibration rotation correction.
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We are making a correction for something called vibration rotation interaction.
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Let us talk about our basic equation E sub RJ is equal to ν~ R + ½ + that × J × J + 1.
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B itself, the correction that we are going to make B actually depends on R.
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We symbolize that with a B sub R.
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B sub E - Α sub E × R + ½.
00:35:05.100 --> 00:35:12.300
We take this expression and we have to put into here for the correction.
00:35:12.300 --> 00:35:18.200
When we do that, we get the following modified basic equation.
00:35:18.200 --> 00:35:38.600
We get E sub RJ is equal to ν × R + ½ + B sub R - Α sub E × R + ½ × J × J + 1.
00:35:38.600 --> 00:35:49.500
We now get a slightly more complicated, slightly more complicated equation for the energy of a molecule.
00:35:49.500 --> 00:35:53.300
This equation accounts for something called the vibration rotation interaction.
00:35:53.300 --> 00:35:57.500
When we use this equation, when we take the difference of the upper and lower
00:35:57.500 --> 00:36:01.100
to get a new equation for the observed frequency, what we get is closer to what we see.
00:36:01.100 --> 00:36:03.700
It gives us a little bit closer.
00:36:03.700 --> 00:36:23.300
This is the harmonic oscillator rigid rotator corrected for vibration rotation interaction.
00:36:23.300 --> 00:36:28.800
That is one of the corrections that we are going to make.
00:36:28.800 --> 00:36:33.800
Again, I'm going to do each one of these corrections one at a time.
00:36:33.800 --> 00:36:37.400
I took the basic equation, I corrected for vibration rotation interaction.
00:36:37.400 --> 00:36:41.400
Now, I’m going to go back to the basic equation correct for centrifugal distortion.
00:36:41.400 --> 00:36:46.400
I’m going to take the basic equation, I’m going to correct for anharmonicity.
00:36:46.400 --> 00:36:50.300
We can put them all together, 2 at a time, 1 at a time, 3 at a time.
00:36:50.300 --> 00:36:51.600
That is what we do.
00:36:51.600 --> 00:37:06.700
Let us talk about centrifugal distortion basic equation.
00:37:06.700 --> 00:37:29.400
We have E sub RJ is equal to ν × R + ½ + B × J × J + 1.
00:37:29.400 --> 00:37:38.400
Real quickly, centrifugal distortion what is it is when a molecule rotates, the rigid rotator assumes the bond is rigid.
00:37:38.400 --> 00:37:44.500
A diatomic molecule speeds faster and faster, it is not rigid, the things actually pull apart.
00:37:44.500 --> 00:37:47.600
Notice from your experience, if you spin something, it starts to pull apart.
00:37:47.600 --> 00:37:54.000
Because of the bond actually stretches, we have to make an adjustment for that.
00:37:54.000 --> 00:38:11.600
The adjustment that we make gives us the following equation E sub RJ is equal to ν × R + ½ + B × J × J + 1.
00:38:11.600 --> 00:38:21.400
That is our basic equation, the correction is DJ² J + 1².
00:38:21.400 --> 00:38:28.700
This is the new energy equation under a correction for centrifugal distortion.
00:38:28.700 --> 00:38:31.200
Let us go ahead and do our third correction.
00:38:31.200 --> 00:38:38.500
We are going to correct for anharmonicity.
00:38:38.500 --> 00:39:04.500
Once again, we have our basic equation E sub RJ is equal to ν ̃ R + ½ + B~ × J × J + 1.
00:39:04.500 --> 00:39:09.300
As you move to higher and higher vibrational states, as it starts to vibrate more and more violently,
00:39:09.300 --> 00:39:12.000
it starts to deviate from harmonic behavior.
00:39:12.000 --> 00:39:14.800
It does not become harmonic.
00:39:14.800 --> 00:39:17.000
We have to adjust for that.
00:39:17.000 --> 00:39:26.800
The equation that we end up with is E sub RJ is equal to ν sub E × R + ½.
00:39:26.800 --> 00:39:33.000
It is not ν~, it is ν sub E ̃, different set of numbers.
00:39:33.000 --> 00:39:46.400
X sub E ν sub E × R + ½² + B × J × J + 1.
00:39:46.400 --> 00:39:56.500
This is the new equation for the energy of a molecule after we have made a correction for the anharmonic behavior.
00:39:56.500 --> 00:40:53.700
Notice each time the basic equation was corrected for one phenomenon, we can correct for 1, 2, or all 3 phenomena simultaneously.
00:40:53.700 --> 00:41:06.800
Correcting for all three simultaneously gives us the best agreement for what we actually see in the spectra.
00:41:06.800 --> 00:41:44.900
Correcting for all three simultaneously gives us the best agreement for the energies and
00:41:44.900 --> 00:41:57.200
absorption emission frequencies we see in the actual experiments.
00:41:57.200 --> 00:42:00.600
Correcting for all three, the equation becomes this.
00:42:00.600 --> 00:42:29.700
E sub RJ is equal to ν sub E × R + ½ - X sub E ν sub E × R + ½² + B sub E - Α sub E × R + ½ ×,
00:42:29.700 --> 00:42:38.600
I will put it on the second line here because I want to see that it is actually three different corrections.
00:42:38.600 --> 00:43:02.700
+ B sub E - Α sub E × R + ½ × J × J + 1 - DJ² J + 1².
00:43:02.700 --> 00:43:13.600
The correction for anharmonicity correction for vibration rotation interaction, correction for centrifugal distortion,
00:43:13.600 --> 00:43:23.700
this equation gives me the total energy of the molecule that comes from rotation and vibration.
00:43:23.700 --> 00:43:28.400
Adjustments made to account for more of the reality of what is going on.
00:43:28.400 --> 00:43:31.800
In fact that the molecule is not a rigid rotator.
00:43:31.800 --> 00:43:38.800
The fact that the vibration of the molecule does not follow a harmonic model.
00:43:38.800 --> 00:43:51.900
The predicted frequency of absorption and emission, I will call it predicted if you want, equation predicted.
00:43:51.900 --> 00:43:57.000
Let us write predicted or calculated.
00:43:57.000 --> 00:44:06.500
The equation is going to be the ERJ for the upper state - the ERJ for the lower state.
00:44:06.500 --> 00:44:11.100
In other words, one of these whole equations for the upper state and one of these whole equations
00:44:11.100 --> 00:44:14.200
for the lower state depending on what RJR.
00:44:14.200 --> 00:44:17.300
You put those values in here and you get this equation.
00:44:17.300 --> 00:44:21.100
You will do a whole bunch of algebra to reduce it down as an equation
00:44:21.100 --> 00:44:28.100
that will predict what the frequency of the spectral line is.
00:44:28.100 --> 00:44:35.100
Now, the symbols, do not worry about these symbols.
00:44:35.100 --> 00:44:46.000
The symbols ν sub E X sub E ν sub E is actually a single symbol by the way, we will get to that.
00:44:46.000 --> 00:45:10.100
B sub ED, Α sub E are called spectroscopic parameters.
00:45:10.100 --> 00:45:20.800
I have it tabulated for you for a bunch of diatomic molecules.
00:45:20.800 --> 00:45:27.200
In the problems you come across here in your book, on your exams,
00:45:27.200 --> 00:45:30.600
you will be given the spectroscopic parameters and you have to find information.
00:45:30.600 --> 00:45:34.700
Or you will be given certain spectroscopic information and you have to find the spectroscopic parameters.
00:45:34.700 --> 00:45:37.500
It pretty much all comes down to.
00:45:37.500 --> 00:45:40.300
Not always easy but that is what it comes down to.
00:45:40.300 --> 00:45:45.600
That is the big picture.
00:45:45.600 --> 00:46:08.300
When you do your homework problems, you will be told which corrections, if any you need to make to the basic equation.
00:46:08.300 --> 00:46:25.100
You will be told which corrections to account for.
00:46:25.100 --> 00:46:38.500
The particular corrections that they are asking for, that is going to decide which equation you use.
00:46:38.500 --> 00:46:53.000
These will decide which equation you use.
00:46:53.000 --> 00:46:57.300
You might have a problem that says under the harmonic oscillator rigid rotator approximation,
00:46:57.300 --> 00:46:59.500
you know to use the basic equation.
00:46:59.500 --> 00:47:10.800
You might say under the anharmonic oscillator also accounted for centrifugal distortion, you use the appropriate equation.
00:47:10.800 --> 00:47:15.600
That is what is happening.
00:47:15.600 --> 00:47:18.400
Again, the ocean of the equations that you see in your book,
00:47:18.400 --> 00:47:27.300
that is all the various energy upper - energy lower equations that they are driving for you.
00:47:27.300 --> 00:47:33.800
Let us go ahead and turn to our summary.
00:47:33.800 --> 00:47:36.300
Here is our summary, very important.
00:47:36.300 --> 00:47:41.300
For harmonic oscillator rigid rotator approximation, this is the basic equation
00:47:41.300 --> 00:47:45.000
under the harmonic oscillator rigid rotator approximation.
00:47:45.000 --> 00:47:49.800
The vibration rotation interaction, this is a correction to the rotational term.
00:47:49.800 --> 00:47:53.100
This is the correction.
00:47:53.100 --> 00:47:57.300
The centrifugal distortion, this is also a correction to the rotational term.
00:47:57.300 --> 00:48:00.700
That is this one, it is - this thing.
00:48:00.700 --> 00:48:04.000
The anharmonicity, this is a correction to the vibrational term.
00:48:04.000 --> 00:48:06.100
It is this thing we are subtracting from this.
00:48:06.100 --> 00:48:13.100
This is the vibrational term.
00:48:13.100 --> 00:48:15.100
This is the rotational term.
00:48:15.100 --> 00:48:21.900
For the vibration rotation interaction, use and take this thing into here.
00:48:21.900 --> 00:48:30.200
If you are accounting for centrifugal distortion, you subtract it here.
00:48:30.200 --> 00:48:36.700
The vibrational term, you take this thing and you subtract it here.
00:48:36.700 --> 00:48:38.400
The vibrational term and the rotational term.
00:48:38.400 --> 00:48:46.600
These two correct for the rotation and this one correct for the vibration.
00:48:46.600 --> 00:48:56.800
Correcting for all three simultaneously, this is the equation that makes a correction for all three simultaneously.
00:48:56.800 --> 00:49:00.400
Here is a correction for vibration rotation.
00:49:00.400 --> 00:49:03.800
Here is the correction for centrifugal distortion.
00:49:03.800 --> 00:49:07.100
Here is the correction for anharmonicity.
00:49:07.100 --> 00:49:11.600
This gives you the energy of a molecule.
00:49:11.600 --> 00:49:19.700
It is in a rotational state given by J and a vibrational state given by R.
00:49:19.700 --> 00:49:32.500
If that molecule is excited to another vibration and or rotational level, the frequency that we see on the absorption or emission spectrum
00:49:32.500 --> 00:49:40.400
is given by the energy of the upper state - the energy of the lower state.
00:49:40.400 --> 00:49:43.500
All that is happening in the next 5 lessons.
00:49:43.500 --> 00:49:53.800
The difficulty, rather the tedium and the mess, come from working out the algebra for this.
00:49:53.800 --> 00:49:59.700
I hope that helped to see the forest, now we can get into the trees.
00:49:59.700 --> 00:50:01.200
Thank you so much for joining us here at www.educator.com.
00:50:01.200 --> 00:50:02.000
We will see you next time, bye.