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Lecture Comments (7)

3 answers

Last reply by: Joseph Carroll
Mon Apr 4, 2016 1:54 AM

Post by Joseph Carroll on March 30 at 01:08:41 AM

Professor Hovasapian, your lectures never cease to amaze me, especially how, this one in particular, relates so closely to the truth of our very existence in correlation with God.

It reminds me of how our lives are like how a basic equation that computes incorrectly/short when it comes to the reality of what is seen in experimentation. The Father, Son, and Holy Spirit are our correction for our sin nature which was caused from the fall in the Garden of Eden by Adam and Eve who disobeyed God and ate of the apple of the tree of knowledge.

Just like the vib-rot, centrifugal distortion, and anharmonicity corrections are needed to provide the truth of what is actually observed in reality with respect to molecular spectra lines, we need the (1) correction of God the Father, whom sent His Son Jesus Christ of Nazareth to atone/correct(2) for our sin
[a "correction" for something that we could never do fully do since the time God established the 10-commandments covenant through Moses on Mt. Sinai to prove to the Israelites, and to the Gentiles (us), that we could not do follow them, not even King David of Israel (reigned as king sometime between c.1040 - c.970 BCE) who loved God with all his heart and tried his best to do God's will]
by sacrificing himself on a cross(c.33AD) for our disobedient hearts. The final and third correction is His Holy Spirit (3) (sent to baptized, repented believers, {first sent to jesus apostles shortly after His death}) to correct for how we should live a life pleasing to God and ultimately obtain resurrection from the dead just like Jesus Christ did 3 days after his crucifixion near jerusalem. All who confess that Jesus Christ is Lord and believe with all his heart that the Father rose him from the dead will be saved from eternal death and rise to eternal life to worship his creator forever.

2 answers

Last reply by: Professor Hovasapian
Thu Dec 17, 2015 2:27 AM

Post by Van Anh Do on December 14, 2015

I'm not sure why E tilda is E/hc. I thought anything with a tilda is just itself divided by c? Thanks!

Spectroscopic Overview: Which Equation Do I Use & Why

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
  • Spectroscopic Overview: Which Equation Do I Use & Why 1:02
    • Lesson Overview
    • Rotational & Vibrational Spectroscopy
    • Frequency of Absorption/Emission
    • Wavenumbers in Spectroscopy
    • Starting State vs. Excited State
    • Total Energy of a Molecule (Leaving out the Electronic Energy)
    • Energy of Rotation: Rigid Rotor
    • Energy of Vibration: Harmonic Oscillator
    • Equation of the Spectral Lines
  • Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections) 28:37
    • Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)
    • Vibration-Rotation Interaction
    • Centrifugal Distortion
    • Anharmonicity
    • Correcting for All Three Simultaneously
    • Spectroscopic Parameters
  • Summary 47:32
    • Harmonic Oscillator-Rigid Rotor Approximation
    • Vibration-Rotation Interaction
    • Centrifugal Distortion
    • Anharmonicity
    • Correcting for All Three Simultaneously

Transcription: Spectroscopic Overview: Which Equation Do I Use & Why

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

Today, we are going to start our discussion of molecular spectroscopy.0004

I had a little difficulty deciding on how to actually present molecular spectroscopy.0013

What I decided to do was to start with a lesson which gives an overview of primarily the rotational and vibration spectroscopy.0017

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.0026

But I wanted to give a big picture of what it is that is going on0036

because you are going to be sort of swimming in this ocean of equations.0041

And a lot of it is like where is this coming from and when do I use this?0043

I want to let you know why we are choosing the equations we are choosing and 0047

how to actually choose when you are faced with a specific problem.0052

Again, this is the big picture so you have an idea of what is going on with the details,0055

when we get to the details of subsequent lessons.0059

Let us get started.0061

Let me go ahead and write that down and repeat that.0064

This lesson intends to provide a concise and broad overview of rotational and vibrational spectroscopy.0067

There is nothing in this particular lesson that you actually have to like know for sure in terms of an equation because again, 0110

all of this is going to be discussed in detail for the next lesson and the lessons that follow.0116

This is just a big picture.0121

Just get an idea of what is going on before you get down to the nitty gritty.0123

What you see here will be discussed in detail in the next 4 lessons.0128

In the lessons that follow and in your book, it can appear that you are swimming in an ocean of equations.0158

This lesson here hopes to answer the question which equation and why, which equation do I use and why.0200

Very important equation, which is a very important question.0228

The discussion that we do for all spectroscopy is only going to concern diatomic molecules.0237

This discussion concerns diatomic molecules.0246

When a molecule absorbs radiation of a given frequency of transitions to a higher state, 0267

you know this and you are reasonably familiar with spectroscopy from your work in organic.0289

The transitions to a higher rotational, vibrational, or electronic state.0293

Microwave radiation tends to affect only the rotational state.0320

The infrared tends to affect the vibrational state but with vibration you also get rotational changes.0326

The visible ultraviolet region of the electromagnetic spectrum tends to promote electronic transitions.0339

Along with electronic transitions, you also get vibrational changes and you also get rotational changes.0347

The higher the energy, the more it does.0359

The frequency of this absorbed radiation or emission, emission is just the other way, excited state down to lower state.0366

There is no real difference but for our purposes emission.0404

The frequency is given by, we have is relation that you remember from early on.0409

If there is a change in the energy, the energy of the final state - the initial - the energy of the initial state.0416

It is energy final - energy initial and that was equal to H × ν.0425

We have this equation already.0432

Therefore, the frequency of this transition is going to be the final energy - the initial energy0435

divided by planks constant, in terms of frequency, in terms of hertz.0442

The energy of the final state - the energy of the lower state, or the arrival state to the departure state.0449

However you want to say it, how it started and where it ended divided by planks constant.0454

That gives you the frequency that we see on the spectrum.0459

That is all that is happening.0461

Or we can say, we can call it energy upper - energy lower divided by that.0464

This is going to be important for us because these frequencies are what we going to see on the spectrum.0475

That is what we are reading off is this.0480

Any time you want to know what the frequency of the spectral line is, 0483

take the higher energy - the lower energy divide by planks constant.0486

In spectroscopy, we usually work in wave numbers not Hz.0492

In other words, inverse cm.0514

A wave number is anything with a ̃ symbol on top of it, means it is in inverse cm.0519

The definition is very easy to state whether frequency you have divided by the speed of light, 0525

that will give you the wave number.0531

It is also equal to 1/ the wavelength λ.0534

In cm then, in inverse cm then the frequency of the spectral line that we see 0544

is going to be the final energy - the initial energy divided by HZ.0558

That was going to be important for us.0564

What we are going to try to do, we will try to find equations that explain, that predict the experimental spectra that we see.0567

We run an experiment, we get some lines on the spectrum.0589

We try to come up with equations that explain those lines, that is all we are doing.0604

It is really all we are doing.0608

The ν above, the wave number above are the frequencies that we see on the spectra.0612

In other words, they represent the differences in energy between the starting state and the final state.0638

I’m going to say the starting state and excited state.0687

How is that, it is probably a little bit better.0692

Usually, we can really be going from ground stage to excited state.0696

We are going to be seeking equations for the energy of a given quantum state.0702

We, then form energy final - energy initial to give us the frequency that we observe.0721

This is really what we are doing here, for the next 4 or 5 lessons all we are really concerned with, 0737

we want to find an expression for the energy of a given quantum mechanical system.0741

We subject that quantum mechanical system to the radiation, microwave, infrared, visible UV.0747

The rotational, vibrational, electronic transitions that take place taken from one level to another.0755

One rotational level to another rotational level.0761

One vibrational level to another vibrational level.0764

One electronic state to another electronic state.0767

We can find the energies of those two states.0770

We want to find equations that will give us the energy for those two states.0773

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,0776

finding energies for the different quantum mechanical systems that we are dealing with, 0782

particle in a box rigid rotator harmonic oscillator, whatever it was.0786

If I take the difference between the ground state or the beginning state and the excited state, 0790

what I get are the frequency that I see on the spectra.0795

We want to find equations for those.0799

We find the equations for the energies, that is what is important.0801

And then, we take the difference between the lower and the higher energy level and 0803

that gives us the frequency that we see on the spectra.0807

We just want equations for E and for ν.0810

Let us see, I’m going to leave off the electronic energy for right now.0817

Just know that it is actually there and in the subsequent lessons where we introduce it.0826

But for right now, I just want to talk about vibration and rotation.0831

If you understand those well, everything else after that is very very simple because it is the same thing.0833

I’m just adding one more term for the electronic energy.0838

Leading out the electronic energy of an atom for the moment, 0843

the total energy of a molecule is equal the energy of the rotation of the molecule +0865

the energy of the vibration of the molecule.0892

Let me stop for a second.0895

A molecule has 4 types of energy.0898

A molecule is translational energy, it is moving.0900

It has electronic energy, the energy of the electronic states.0904

It has energy of vibration, it is vibrating.0911

It has the energy of rotation, it is rotating.0913

Notice that I have left off the electronic energy.0916

We have also just automatically left off the energy of translational because0918

the energy of translation is not affected by spectroscopic interaction, by radiation interaction.0922

Basically, we will deal with the translational energy a little bit later when we talk about statistical thermodynamics.0928

But for spectroscopy, we are only concerned with electronic, rotation, and vibration.0935

For right now, I'm leaving off the electronic just to concentrate on rotation and vibration.0938

I think that gives us the best big picture.0942

For our purposes, the total energy comes from the energy of rotation and energy of vibration.0946

Let us deal with the rotational energy first.0952

The energy of rotation, the model for that is our rigid rotator.0956

We pretend that a diatomic molecule is just two bodies stuck together and it is rotating,0968

that gives us a model for this.0975

Now rotator, the energy sub J we said was equal to H ̅² / 2I × J × J + 1.0979

J was equal to 0, 1, 2, and so forth.0991

I was the rotational inertia, it was the reduced mass × the bond length².0996

It had degeneracy as a function of J is equal to 2J + 1.1003

This is energy in Joules.1012

We want to express the energy in wave numbers.1025

Basically, then take any energy in Joules and just divide by HZ and that will give you an energy in wave numbers.1053

In inverse cm, our energy sub J, we are going to express this now in terms of wave numbers.1078

We will get a new symbol F of J and we write it this way BJ × J + 1,1089

where B is equal to planks constant / 8 π² C I.1101

The rotational energy in inverse cm is given by this equation B × J × J + 1.1109

I’m not going to get into great detail here about what each of all the stuff is because 1117

I will discuss it again in the subsequent lesson, in the next lesson.1121

In fact, we are going to start off with vibration and rotation.1124

We will talk about all of this in great detail.1127

Do not worry, I just want to show you again what is happening with the equations,1129

why we are choosing the equations we are choosing.1133

This gives us the equation for the rotational energy of a molecule.1136

For the vibrational energy, for E sub V vibration, this is from the harmonic oscillator.1146

That is our model so we are going to begin with that equation to represent the vibrational energy of the molecule.1155

The energy sub R was equal to ν × R +, it was H μ + ½ values of 0, 1, 2, 3, and so on.1164

Again, these are the vibrational quantum numbers.1180

Here, ν was equal to 1 / 2 π K / μ ^½.1183

In inverse cm, our expression is energy sub R given new symbol G of R is equal to ν ̃ × R + ½.1193

Here, ν~ is equal to ½ π C / μ ^½.1210

Again, do not worry about it this is just big picture stuff.1218

Now, I have the vibrational energy, it is given by this thing.1221

I have the rotation energy given by what you saw.1225

Our total energy is equal to the vibrational energy + the rotational energy.1228

We have that the total energy is equal to, total energy is a function of R, 1241

the vibrational quantum number and J the rotational quantum number.1251

It is equal to G of R + F of J.1255

E sub RJ is equal to this thing ν × R + ½ + B~ × J.1266

Let me make this J a little bit more clear here.1279

J × J + 1.1283

R takes on the values 0, 1, 2, and so on.1287

J takes on the values 0, 1, 2, so on, independently.1291

This, under the rigid rotator harmonic oscillator approximation for the energy of a molecule, 1299

this equation gives me the energy of a molecule who is in vibrational state R, rotational state J.1312

ERJ is equal to ν~ R + ½ + B × J × J + 1 under the harmonic oscillator rigid rotator approximation,1327

because molecules are not rigid and they are not harmonic.1354

The first approximation, I’m going to make one correction for this.1358

Under the harmonic oscillator rigid rotator approximation, this equation gives the energy.1361

Very important.1375

The frequency of absorption because the energy of the molecule in vibrational state R and rotational state J.1376

That is very straightforward.1399

To find the equation of the spectral line, find the equations of the spectral lines.1402

In other words, the transitions from one energy level to another.1428

To find the equations of the spectral lines, we take the energy R upper J upper - the energy R lower J lower.1437

The upper energy - the lower energy, whatever those happen to be.1460

For example, if I would want the equation for the observed, for the ν of the spectral line1465

for the 0, 2 to 1, 3 transition, this is R lower and this is the J lower.1500

This is the R upper, this is the J upper.1511

I would form the Δ E.1519

In other words, I would form the energy of the 1, 3 - the energy to 0, 2.1523

That will give me an equation for the spectral line.1534

The equation that predicts where it should be.1538

Running the experiment tells me where actually is.1541

The extent to which is a good match depends on my equation.1544

This is an approximation that is going to be off when I start making corrections to 1548

this harmonic oscillator rigid rotator approximation, that gives me a better and better predictions until it is almost exact.1552

That is what is happening here.1559

It is the energy equation that is important.1570

The spectral line equation, the observed frequency or the predicted frequency, 1582

that I can derive just by taking the energy / - the energy lower.1586

It involves a lot of algebra but it is doable.1590

It is the energy equation that is important.1594

The equations for the absorption emission frequencies can be derived with algebra, energy upper - energy lower.1601

A lot of the mess that you see, as far as all these equations, all the derivations that you see1646

in the spectroscopy, that is the stuff right here.1651

We are taking upper energy - lower energy.1659

We are coming up with different equations.1662

The thing is we have the harmonic oscillator rigid rotator approximation.1664

You will see in a minute that would give us one set of equations.1668

When I start making corrections to that, for different phenomenon that I observe1672

to make my equations match more of what the real spectra look like give me the different equations.1676

It is not really 150 equations that you have to know, you have to know just one.1683

The corrections to that one can where everything else comes from.1690

That is what I'm trying to do with this lesson.1693

I’m trying to show you which one or two equations are important.1695

And then from there, depending on what corrections you make, you can derive everything else.1699

That is what you are seeing is the derivations.1703

Do not get lost in the ocean.1706

That is why this is probably the most important lesson of spectroscopy, the overview to the big picture,1708

the forest before we get into the trees.1713

Let me go back to black here.1721

The harmonic oscillator rigid rotator approximation is precisely that, just an approximation.1726

Approximation is just that.1733

It is an approximation.1746

We will now make corrections to the vibrational term, rotational term, 1750

to make the equations more closely match what we see in experiment.1774

To make the equations better match and predict what we see in reality.1780

We are going to correct for three things.1805

We will correct for three things.1815

The lessons, each one, we will talk about a different correction.1818

We will correct for three things, we will talk about what they are.1821

This is all big picture stuff.1825

It is actually really important.1829

I wish that more people would spend more time on the big picture stuff because it will make all the details1831

and you will know exactly what is going on because I can see the big picture.1835

It is better to see from the outside in than it is to be the inside trying to look out, that is the idea.1842

Any time you find yourself lost in science or math or whatever it is,1851

99% of the time it is going to be because you are inside trying to look out.1856

Try to find someone or some book or some other way to get yourself on the outside looking in the big picture.1862

All the details are not irrelevant but are secondary.1868

If you have the big picture, you really understand, then science becomes a beautiful thing that really is.1873

We will correct for three things.1883

The first thing we are going to correct for, we would be correcting for something called vibration rotation interaction.1884

We are also going to correct for something called centrifugal distortion.1900

We are going to correct for anharmonicity.1910

Each correction will modify our basic equation, our basic energy equation E sub RJ is equal to ν ̃ R + ½ + B~ × J × J + 1.1916

This is our basic equation, it is a rigid rotator harmonic oscillator approximation to the energy of a molecule.1959

Each correction we make is going to modify this equation and give us a new energy equation.1968

I can do one correction.1973

I can correct for 1, I can correct for 2 of them, any two.1975

Or I can correct for all three, depending the equation becomes more and more complicated.1977

That is all that is happening.1983

The three corrections above R will be discussed in the lessons that follow.1995

Let us talk about our first one which is vibration rotation.2019

Let us talk about the vibration rotation correction.2032

We are making a correction for something called vibration rotation interaction.2049

Let us talk about our basic equation E sub RJ is equal to ν~ R + ½ + that × J × J + 1.2058

B itself, the correction that we are going to make B actually depends on R.2079

We symbolize that with a B sub R.2096

B sub E - Α sub E × R + ½.2099

We take this expression and we have to put into here for the correction.2105

When we do that, we get the following modified basic equation.2112

We get E sub RJ is equal to ν × R + ½ + B sub R - Α sub E × R + ½ × J × J + 1.2118

We now get a slightly more complicated, slightly more complicated equation for the energy of a molecule.2138

This equation accounts for something called the vibration rotation interaction.2149

When we use this equation, when we take the difference of the upper and lower 2153

to get a new equation for the observed frequency, what we get is closer to what we see.2157

It gives us a little bit closer.2161

This is the harmonic oscillator rigid rotator corrected for vibration rotation interaction.2164

That is one of the corrections that we are going to make.2183

Again, I'm going to do each one of these corrections one at a time.2189

I took the basic equation, I corrected for vibration rotation interaction.2194

Now, I’m going to go back to the basic equation correct for centrifugal distortion.2197

I’m going to take the basic equation, I’m going to correct for anharmonicity.2201

We can put them all together, 2 at a time, 1 at a time, 3 at a time.2206

That is what we do.2210

Let us talk about centrifugal distortion basic equation.2211

We have E sub RJ is equal to ν × R + ½ + B × J × J + 1.2227

Real quickly, centrifugal distortion what is it is when a molecule rotates, the rigid rotator assumes the bond is rigid.2249

A diatomic molecule speeds faster and faster, it is not rigid, the things actually pull apart.2258

Notice from your experience, if you spin something, it starts to pull apart.2264

Because of the bond actually stretches, we have to make an adjustment for that.2267

The adjustment that we make gives us the following equation E sub RJ is equal to ν × R + ½ + B × J × J + 1.2274

That is our basic equation, the correction is DJ² J + 1².2291

This is the new energy equation under a correction for centrifugal distortion.2301

Let us go ahead and do our third correction.2309

We are going to correct for anharmonicity.2311

Once again, we have our basic equation E sub RJ is equal to ν ̃ R + ½ + B~ × J × J + 1.2318

As you move to higher and higher vibrational states, as it starts to vibrate more and more violently, 2344

it starts to deviate from harmonic behavior.2349

It does not become harmonic.2352

We have to adjust for that.2355

The equation that we end up with is E sub RJ is equal to ν sub E × R + ½. 2357

It is not ν~, it is ν sub E ̃, different set of numbers.2367

X sub E ν sub E × R + ½² + B × J × J + 1.2373

This is the new equation for the energy of a molecule after we have made a correction for the anharmonic behavior.2386

Notice each time the basic equation was corrected for one phenomenon, we can correct for 1, 2, or all 3 phenomena simultaneously.2396

Correcting for all three simultaneously gives us the best agreement for what we actually see in the spectra. 2454

Correcting for all three simultaneously gives us the best agreement for the energies and2467

absorption emission frequencies we see in the actual experiments.2505

Correcting for all three, the equation becomes this.2517

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 ×, 2520

I will put it on the second line here because I want to see that it is actually three different corrections.2550

+ B sub E - Α sub E × R + ½ × J × J + 1 - DJ² J + 1².2558

The correction for anharmonicity correction for vibration rotation interaction, correction for centrifugal distortion,2583

this equation gives me the total energy of the molecule that comes from rotation and vibration.2593

Adjustments made to account for more of the reality of what is going on.2604

In fact that the molecule is not a rigid rotator.2608

The fact that the vibration of the molecule does not follow a harmonic model.2612

The predicted frequency of absorption and emission, I will call it predicted if you want, equation predicted.2619

Let us write predicted or calculated.2632

The equation is going to be the ERJ for the upper state - the ERJ for the lower state.2637

In other words, one of these whole equations for the upper state and one of these whole equations 2646

for the lower state depending on what RJR.2651

You put those values in here and you get this equation.2654

You will do a whole bunch of algebra to reduce it down as an equation 2657

that will predict what the frequency of the spectral line is.2661

Now, the symbols, do not worry about these symbols.2668

The symbols ν sub E X sub E ν sub E is actually a single symbol by the way, we will get to that.2675

B sub ED, Α sub E are called spectroscopic parameters.2686

I have it tabulated for you for a bunch of diatomic molecules.2710

In the problems you come across here in your book, on your exams, 2721

you will be given the spectroscopic parameters and you have to find information.2727

Or you will be given certain spectroscopic information and you have to find the spectroscopic parameters.2730

It pretty much all comes down to.2735

Not always easy but that is what it comes down to.2737

That is the big picture.2740

When you do your homework problems, you will be told which corrections, if any you need to make to the basic equation.2745

You will be told which corrections to account for.2768

The particular corrections that they are asking for, that is going to decide which equation you use.2785

These will decide which equation you use.2798

You might have a problem that says under the harmonic oscillator rigid rotator approximation, 2813

you know to use the basic equation.2817

You might say under the anharmonic oscillator also accounted for centrifugal distortion, you use the appropriate equation.2819

That is what is happening.2831

Again, the ocean of the equations that you see in your book, 2835

that is all the various energy upper - energy lower equations that they are driving for you.2838

Let us go ahead and turn to our summary.2847

Here is our summary, very important.2854

For harmonic oscillator rigid rotator approximation, this is the basic equation 2856

under the harmonic oscillator rigid rotator approximation.2861

The vibration rotation interaction, this is a correction to the rotational term.2865

This is the correction.2870

The centrifugal distortion, this is also a correction to the rotational term.2873

That is this one, it is - this thing.2877

The anharmonicity, this is a correction to the vibrational term.2881

It is this thing we are subtracting from this.2884

This is the vibrational term.2886

This is the rotational term.2893

For the vibration rotation interaction, use and take this thing into here.2895

If you are accounting for centrifugal distortion, you subtract it here.2902

The vibrational term, you take this thing and you subtract it here.2910

The vibrational term and the rotational term.2917

These two correct for the rotation and this one correct for the vibration.2918

Correcting for all three simultaneously, this is the equation that makes a correction for all three simultaneously.2926

Here is a correction for vibration rotation.2937

Here is the correction for centrifugal distortion.2940

Here is the correction for anharmonicity.2944

This gives you the energy of a molecule.2947

It is in a rotational state given by J and a vibrational state given by R.2951

If that molecule is excited to another vibration and or rotational level, the frequency that we see on the absorption or emission spectrum 2960

is given by the energy of the upper state - the energy of the lower state.2972

All that is happening in the next 5 lessons.2980

The difficulty, rather the tedium and the mess, come from working out the algebra for this.2983

I hope that helped to see the forest, now we can get into the trees.2994

Thank you so much for joining us here at www.educator.com.3000

We will see you next time, bye.3001