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The Non-Rigid Rotator

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • The Non-Rigid Rotator 0:09
    • Pure Rotational Spectrum
    • The Selection Rules for Rotation
    • Spacing in the Spectrum
    • Centrifugal Distortion Constant
    • Fundamental Vibration Frequency
    • Observed Frequencies of Absorption
    • Difference between the Rigid Rotator & the Adjusted Rigid Rotator
    • Adjusted Rigid Rotator
    • Observed Frequencies of Absorption

Transcription: The Non-Rigid Rotator

Hello, welcome back to www.educator.com, welcome back to Physical Chemistry.0000

We are going to continue our discussion of molecular spectroscopy, let us jump right on in.0005

We started our discussion of molecular spectroscopy by talking about vibration rotation transitions.0011

When molecules absorb infrared, they are going to make it vibration transition.0018

With those vibration transitions, you get accompanying rotational transitions.0028

If you stay in the microwave range, we actually cannot just stay in the pure rotational state.0033

That is what we are going to investigate today, pure rotational spectra0039

not the vibration rotation spectra that we did in the previous two lessons.0043

I think I will do it in black.0048

If we stay in the microwave region, we could get a pure rotational spectrum.0054

Again, remember what we said for a pure rotational spectrum, there are some selection rules.0082

The δ J = + or -1, but what is more important about it is this existence of a permanent dipole0086

If the molecule does not have a permanent dipole like N2 or O2, something like that,0094

we are not going to see any spectrum for it.0096

There needs to be a permanent dipole in order for us to actually see some microwave activity.0103

If we stay in the microwave region, we can get a pure rotational spectrum.0112

It is always good to review the equations over and over and over again.0116

We had that the energy is equal to, we said H ̅²/ 2I × J × J + 1, or J takes on the values 0, 1, 2, and so fourth.0121

We said that I was equal to the rotational inertia, the moment of inertia MR².0138

We said that the degeneracy of these rotational states is 2J + 1.0148

In terms of inverse cm which is really what we have been dealing with.0154

In terms of the inverse cm wave numbers, we had this rotational term F of J was equal to this B~ × J × J + 1.0158

J × J + 1 where B ~ we said was planks constant divided by 8 π² C I.0175

Like we said before, the selection rules for rotation.0192

Remember, we are just talking about pure rotation now.0201

We have to have δ J = + or -1, one quantum stay at a time and a permanent dipole in the molecule.0204

The frequencies that we observe in the rotational spectrum R, that we should observe are the upper – lower.0220

It is going to be F of J + 1 - F of J.0230

F of J + 1, put it in here.0238

J + 1 to J + 2, put it in here.0241

I will go ahead and do the algebra, it is not a problem.0244

It is going to be B~ × J + 1 × J + 1 + 1 – B ~ × J × J + 1.0248

After a little bit of algebra, what we end up with is a ν observed equal to 2B × J + 1.0264

We know this already, it is not a problem.0277

J = 0, 1, 2.0280

J is the initial rotational state.0283

What we get is the following.0307

As we make 0 to 1, these are J values, the 0 to 1 transition.0310

Our ν observed is equal to 2B.0314

Our R2 transition, our ν observed which is putting the J values in the equation that we just got.0322

It is equal to 4B.0328

Our 2 to 3 transition, our ν observed is going to be 6B, and so on.0332

The spacing between the transition.0345

In other words, we see a line at 2B, we see a line at 4B, we see a line at 6B.0346

2B, 4B, 6B.0353

The spacing between these lines 2 to 4 is 2B, 4 to 6 is 2B, 6 to 8B is 2B.0356

The spacing between these lines, the spacing between these transition lines representing the transitions is 2B.0370

In other words, evenly spaced.0388

Again, we are just talking about pure rotation spectrum not the same as the vibration rotation spectrum.0392

We expect to see the following.0399

We expect to see this.0409

Notice the arrows go down, this is emission.0417

Absorption is just the other way around, from lower to higher, higher to lower.0422

The energy difference is the same.0426

It actually does really matter.0427

What we are seeing in this particular case is an emission spectrum.0428

Notice the 0 to 1 transition is 2B, we see a line at 2B.0433

The transition from 1 to 2 is 4B, we see a line at 4B.0441

The 2 to 3, the 2 state and the 3 state is, the difference between the energy is 6B.0451

We see a line at 6B.0459

The energy differences increase 2, 4, 6, 8, 10, 12, between the energy levels.0462

On the spectrum that we see, we see that the actual energy values, the difference in energies,0467

we see the 2, 4, 6, 8, 10, 12.0471

The difference in between them, that is the 2B.0474

The difference in the spacing of the lines of the spectrum is not the same as the difference0480

in the spacing of the energy levels.0484

Be very clear about that, work slowly.0486

This is what we expect to see.0491

We expect to see evenly spaced lines in the rotational spectrum.0493

However, when we investigate this more closely, in other words, when we look at real spectra,0497

we see that the lines are not equally spaced.0504

We expect to see based on what we just did.0512

We expect to see equally spaced lines in the spectrum, we do not.0519

In a real spectra, the lines are not equally spaced.0541

The rigid rotator approximation gave us that the energy is equal to B × J × J + 1.0556

That is the rigid rotator approximation.0577

In other words, we are presuming that molecule of the bond is actually rigid.0579

However, bond is not rigid.0584

As a molecule moves to a higher and higher rotational states, in other words as it starts to spin faster and faster,0586

because of centrifugal forces, it actually starts to stretch.0593

Again, when the bond goes up, the bond length increases, the value of the B rotational constant decreases.0598

The rigid rotator approximation gave us this.0617

The bond in a diatomic molecule is not rigid.0621

As the molecule rotates faster and faster, increase in quantum state J, the bond stretches due to centrifugal forces.0641

It make sense, you will know this from your own experiences in physics.0671

What we are going to do is we are going to introduce an adjustment to the rigid rotator approximation.0679

We are going to make a correction for the fact that the bond is not rigid.0684

We introduced the adjusted rotational energy.0695

I will go to the next page.0706

F of J is equal to B J × J + 1, it is what you expect – term.0712

The actual energy is going to be slightly less than what the rigid rotator approximation tells us it will be.0724

- D ~ J² J + 1².0734

We are not going to go the mathematics of where this actually comes.0744

It comes from the branch of mathematics called perturbation theory but the details are not important for us.0747

This D ~ is called the centrifugal distortion constant.0752

And again, it is one of the constants that is already tabulated for multiple molecules.0765

Or sometimes we use the spectroscopic data to find it ourselves, it is called the centrifugal distortion constant.0770

D is related to the fundamental vibration frequency ν.0787

The fundamental vibration frequency, my D is equal to 4 × B³/ ν sub 0².0803

In other words, if I have a fundamental vibration frequency has to do with a force constant, the reduced mass of the molecule.0826

If I had a really stiff molecule, it is not going to stretch very much even though it is spinning faster.0833

If I have a very small rotational constant K, it is a very loose bond.0838

The faster it spins, it is going to stretch easy.0844

It actually does depend on the vibrational frequency.0848

The observed frequencies of absorption, in other words the spectral frequencies0855

that we see are as always the difference in energy between the two levels.0871

Ν observed = F of J + 1 - F of J.0883

This is our new F of J.0890

When I put F of J + 1 subtract F of J from it, I end up with, I will just go ahead and algebra here.0895

I will go through the algebra and you will go through if you want to.0904

What we end up with is an observed frequency is equal to 2B J + 1 – 4 D × J + 1³.0908

Before, the observe frequency we had was 2B J + 1.0926

Again, the observed frequency is slightly less than the frequency we observed.0931

This is the adjustment, the adjustment based on the fact that the bond actually stretches as it goes to higher and higher states.0937

This is an important equation.0945

As J increases, the magnitude of the second term increases.0949

The deviation from the rigid rotator approximation increases.0976

As it goes to higher and higher rotational states, the differences between energies starts to increase.0996

The difference between the rigid rotator and the adjusted rigid rotator can look like this.1017

This is the energy, this is the energy axis.1043

Over here, we have the rigid rotator which is going to be B × J × J + 1.1047

The energy of non rigid rotator, the adjusted rigid rotator is going to be B × J × J + 1 -4 D - D × J² × J + 1²².1054

First level, first level, second level slightly lower.1076

Because it is not be BJ × 1 - a little something, and a little something is based on J.1089

You will go a little higher, you deviate a little bit more.1096

The rigid rotator approximation tells us that these are the energies.1107

The just agreed rotator tells us that these are the energies.1112

The energies are going to be slightly less than what we expect from the rigid rotator approximation.1115

Note, very important, the Y axis is energy not absorption energy, not observed frequency absorption or emission.1125

I will say not observed spectral frequency.1146

The same thing is going to happen, we expect the ones to be equally spaced but because of this adjustment,1161

as J increases the lines in the rotational spectrum 2B 2B 2B, they are not equally spaced.1166

They are actually get smaller.1172

It is not observed spectral frequency, not this.1181

But the observed spectral frequency will look the same time1190

Because our ν observed is equal to 2B J + 1 is going to be - 4 D + 1 Q.1201

It has an adjustment to it.1219

What we see is there is going to be in increments of 2B.1221

There are going to be increments of 2B – something.1224

And as we get higher and higher with J, that - is going to be more.1227

We said that this centrifugal distortion constant is equal to 4B³/ the fundamental vibration frequency².1238

Let me make sure that it is clear t3, where ν 0 is the fundamental separation frequency.1258

We said that is equal to 1/2 π C × the force constant divided by the reduced mass ^½, the square root of that.1271

In the previous lesson, let me come over here.1284

In the previous lesson, we saw that this rotational constant actually depends on the vibrational quantum number.1291

A B depends on R, the vibrational rotation quantum number.1312

The vibration quantum number, we said that this dependence is expressed as B sub R is equal to B sub E - Α sub E × R + ½.1325

We can, if we want to introduce this B sub R into the equation that we just got for the adjusted rigid rotator approximation.1340

I say if we want, because we will not usually.1351

We can if we want, include this dependence in the adjusted rigid rotator.1356

We said that F of J is equal to B J × J + 1 - this centrifugal distortion constant × J² × J + 1².1383

We also said that B sub R is equal to B sub E - Α sub E × R + ½.1399

We can put this B sub R into there to get F of J = B sub E - Α sub E × R + ½ × J × J + 1 - D × J² × J + 1².1411

What we will we end up with is F of J is equal to B sub E × J × J + 1 - Α sub E × R + ½ × J × J + 1 - D J² × J + 1².1450

The rotational energy actually depends on both J, which is the rotation quantum number1488

and R the vibration quantum number.1526

If we want, we can use this value.1541

The extra term here is, notice this is the one where we just take the adjusted rotational energy1548

without worrying about the dependence of this rotational constant on R, the vibration quantum number.1558

If I want to include that, this is my thing, my equation for the energy.1565

This is the only extra term which is not altogether necessary.1570

If I want to lead off, it is totally up to you if you want to leave it off.1574

Just to finish this off, do I have another page here?1578

Yes, I do.1584

When we find the observed frequency of absorption, we will get, like always the observed frequency of rotation is F of J + 1 - F of J1586

in upper state - the lower state, with a whole bunch of algebra based on the equation that I just got with those three terms,1609

I end up with the observed frequency of rotation equal to 2 B sub E × J + 1 -2 Α sub E × R + ½ - 4 D × J + 1³.1617

This right here, this term, this is the only addition if I want to include the dependents of B on R.1647

If I do not, in general, what we had before was sufficient.1665

Our ν observed, the frequencies that we observe in the rotational spectrum are going to be,1686

what we would normally expect + this adjustment.1693

And again, these are the frequencies that I observed in the spectra, that I expect to observe in the spectra.1704

These are not the energies.1709

The energies are different.1711

Be very clear about which is which.1728

Energy is this expression, the F of J + 1.1732

In other words, δ F of J is what gives me this observed frequency, what it is that I actually see in the spectrum.1737

If I were to take the δ ν observed, what that gives me is the spacing in between the lines of the spectrum.1746

Thank you so much for joining us here at www.educator.com.1758

We will see you next time for more discussion of molecular spectroscopy.1760

Take care, bye.1763