So 
far we have discussed the background that
is required to now enable us to understand
what microwave spectra look like and why.
One thing we have said already is that the
separation has to be equal.
Why does it have to be equal?
Because the selection rule is ?J=ą1.
? J is BJ(J+1).
So of course, ?J+1 would be B(J+1)(J+2).
So if I write ?, ? is the wave number, it
one way of writing the frequency of transition.
For J to J+1, that turns out to be how much,
2B(J+1).
So where will the first line, where do you
expect the first line?
Do you expect the first line a 0?
See, it cannot be 0, isn't it, because if
there is a transition, transition has to take
place ?M is 0, ?J is what matters.
It cannot be 0 because if a line occurs at
0 that means there is no transition.
So first line occurs at put J=0, J=0 is the
smallest value anyway.
first line occurs at 2B. what about the next
one?
Then?
Somehow I managed to invent a new numeral
letter, new alphanumeric character?
Next one?
6B.
So there is no point in working out each and
every one of those.
We understand that they are going to be equispaced
and so on and so forth.
Equispaced for what kind of a rotor?
Don't forget, for a rigid rotor.
Before we finish today, we hope to introduce
expression for energy for a non-rigid rotor
and then I'll leave it to you to figure out
what the spectrum of a non-rigid rotor is
supposed to look like.
So x-axis is defined, now we have to worry
about the y-axis.
What do you have on the y-axis?
Intensity.
What does intensity depend upon?
It depends upon the intrinsic probability
of transition.
If you want to put it in terms of something
that is experimentally observable that would
be your epsilon, molar absorption coefficient.
So let us for the moment assume that molar
absorption coefficient of each transition
is the same.
All ?J=ą transitions are equally probable.
Let us just assume this.
Later on we'll come back to it and see whether
it is absolutely correct or whether it is
correct to an extent that we can live with
it or whether it is an absolutely wrong assumption.
For now, let us go with it.
So then the only thing that is left is population.
If you have a line arising out of a level
with a greater population, then by brute force,
it would be a more intense line.
So now we have to worry about population of
the Jth level.
Population of Jth rotational level.
Now, I believe you are familiar with Boltzmann
distribution.
What is Boltzman distribution?
Well, ratio of population of two levels is
equal to degeneracy multiplied by e-?E/kT.
So I can write like this PJ is proportional
to, what is degeneracy off each J level?
2(J+1), we are all confident about that.
2(J+1)e-?J/kT, isn't it.
So let me write the expression for EJ.
What is the expression for EJ?
BJ(J+1)/kT, is that right?
Well, generally, you don't write kT in centimetre
inverse.
So it is better to convert ?Jto EJ.
So you have to write hc here also.
That is why I left the gap.
So -Bhc.J(J+1)/kT.
Before we go ahead and discuss this, let me
digress a little bit and ask you a different
question which will not tell us anything about
the spectrum but it's a fundamental question
about rotational levels.
What is the energy of the lowest energy level
of rotation?
For J=0, what is the energy?
0 which means the molecule is not rotating.
If it's not rotating then its energy is 0ą0
and no potential energy, only kinetic energy.
So momentum is 0ą0.
So no uncertainty in momentum.
But then if it is fixed then uncertainly in
position, what is uncertainty in position?
Why?
Well, you are right.
What we are saying is uncertainty in position
is infinity that is why uncertainty in momentum
equal to 0 is okay but can you explain to
me why uncertainty in position is infinity?
Position is absolutely uncertain, why?
Well, you don't know what is ? and you don't
know what is f, isn't it?? can be anything
from 0 to 180 degrees.
? can range from 0 to 180 degrees and in this
case, ? can be anything from 0 to 180 degrees.
So it is uncertain completely over the range
of ?. Similarly f can be anything.
That is why it still does not violate uncertainty
principle.
Well, this is the wrong CH107 batch to ask
the question because I think we asked you
this problem in quiz or something like that
and I think also know this.
Please remember.
Because when we talk about vibration we are
going to discuss zero point energy and we
are going to say that for vibration energy
can never be equal to 0 because then uncertainty
principle is violated.
We'll discuss it at that time but for rotation,
it's okay, it can still stop because you have
complete uncertainty about ? as well as f.
Okay, so now we come back to what we were
actually discussing, population of the Jth
level is proportional to 2(J+1)e-BhcJ(J+1)/kt.
If I want to find out what is the level with
maximum population, how do I do it?
Even before that, are the higher levels at
all occupied?
That is an important question.
What is a typical energy gap between, say,
J=0 and J=1?
Well, that was on microwave spectroscopy.
So what kind of energy are we talking about
here?
Say, five centimetre inverse, 10 centimetre
inverse, no more than that.
So for this kind of energy gaps, higher levels
actually get populated at room temperature.
So it makes sense to talk about a population
of the higher J levels as well.
So this is what it is.
So dPJ/dJ has to be equal to 0 for J=Jmax.Jmax
means the J level with maximum population.
What is dPJ/dJ from there?
What is dPJ/dJ?
One thing I know is that this will definitely
come out e-BhcJ(J+1)/kt.
This comes out if I differentiate the first
time, then I get 2.
If I differentiate the second term, then I
get -Bhc/kT multiplied by J2+J. So it is 2J+1.
Is this right?
Is this dPJ/dJ?
Just check if I made any mistake.
Sushnatu?
Of course, you have to work it out yourself
if you have to tell me whether I made a mistake
or not.
Let's do it quickly.
Easy differentiation.
e-BhcJ(J+1)/kt that comes out anyway.
Yes sir.
Why?
Yes.
So what will it be?
Will it be (2J+1)2?
Yes.
See, Sushnatu, you have to work it out otherwise
I can make a mistake and I can teach you a
wrong expression and you live with it for
the rest of your life and that's not so good.
So when I work out something you also work
it out.
Don't think that I'll always do it -- more
often than not I make mistakes.
So is this okay now?
So when J=Jmax then what happens?
This should be equal to 0.
If this is equal to 0 then what happens?
This I don't have to worry about.
What I write is 2-(Bhc/kT)(2Jmax+1)2=0.
Is it right?
So please do it yourself.
Please do it yourself independent of what
I am doing so that you can tell me whether
I am right or wrong.
So (2Jmax+1)2=2kt/Bhc and then we have 2Jmax=(2kt/Bhc)1/2-1.
So Jmax=(kT/2Bhc)1/2-1/2.
Is that right?
Question?
Okay, why the 0th level is not maximum population?
Because first of all, if I don't do all this,
if I just look at this population, what do
you see (2J+1)e-Bhc whatever, right?
So this exponential term is a decay.
What about this?
It is a linear increase with respect to J.
So you multiply a rising straight line with
an exponential decay, what kind of thought
do you expect?
It goes from maximum.
So that is the answer that I get from simple
algebra or geometry or whatever it is.
I am drawing pictures, so maybe it's geometry
or it's algebra.
Otherwise if you think, if you think of the
same thing, Boltzmann distribution, if I write
PJ/P0, what will it be?
So you just subtract and then here instead
of E, you have to write ?E, that is all.
And what is ?E, what is the energy off J=0?
You just told me it is 0.
So same expression.
So what we see is that the ratio actually
goes to a maximum.
So since the energy gap is small, if I think
physically now, the room temperature is enough
to cause a promotion.
Suppose, do a thought experiment.
Let us think that there is no rotation.
We start with a situation like that somehow
and then we expose it to the room temperature.
The temperature will be enough to cause a
redistribution of the population so that the
upper levels also get populated.
So as you go higher the levels, the factor
that favours population of higher levels,
of course, is degeneracy.
There are more level at a given energy.
The factor that disfavours population of the
higher levels is the energy gap.
So the temperature will not be enough after
a while.
That is why it will eventually fall down.
Are you answered?
So this is your expression for Jmax.
What is the problem with this approach?
Okay, Shivam, tell us.
Okay, I'll side, perhaps that is what you
are trying to say anyway.
see, when do we do differentiation?
When you have continuous crows.
Do we actually have a continuous PJ versus
J curve?
We don't. we have PJ values for discrete J
values.
So we are kind of extending the scope a little
bit.
So the answer we that we get, do you think
this answer will always be integral?
No.
So you'll get something like 3.2 or 10.5 or
something like that.
But actually Jmax has to have an integral
value.
So from this approach you are only going to
get an approximate value which will usually
not be an integer.
So what you need to do is you need to take
the nearest integer.
If you get 5.8 from here, then the correct
answer is 6.
If you get 10.2, then the correct answer is
10.
If you get 9.5 then I don't know what the
correct answer is.
Yes Sushnatu?
Actually yes.
So most likely it is 6, well it's somewhere
either 5 or 6.
At least we can zero down upon where the maximum
will come.
But what we've learnt from here is that the
spectrum is going to look something like this.
If it 
follows the population of the levels of origin
of transition, it's going to go up and then
come down.
Of course, that discrete line spectra will
go up and come down.
Okay, so far so good?
Now in the remaining nine minutes that we
have today, let us try and understand a little
more minute detail about the spectra.
To repeat something we have done already,
what is the molecular property about which
we can get information from this spectrum?
Bond strength, bond length, not bond strength.
Bong strength comes from vibrational spectrum,
bond length.
Why, because each of these gaps is 2B and
B is, what is B?
h/8p2ĩr02c.
The only unknown is r0, r0 is basically the
bond length.
So we can actually determine bond length.
Now let us see how this spectrum would be
effected by two different factors.
First of all, continuing with the discussion
that we had a little earlier, suppose I record
a microwave spectrum at room temperature and
then I record microwave spectrum of the same
molecule at, say, 500 degree centigrade.
Will they be same?
Will they be different?
Okay, I am assuming that rigid rotor holds
in bot the cases?
I am assuming bond length does not change.
If the bond length changes as a function of
temperature, then of course, your spacing
will also change.
But if I can assume that bond length does
not change then spacing will remain same.
What will change?
Internal of spectral lines, how will it change?
Yes.
So this maximum, Jmax will move to higher
value.
If I increase the temperature then I can expect
hat Jmax, the J value for which we get the
most intense line will move to higher values.
So I can expect -- what kind of a shift is
it?
So I can expect the entire spectrum to undergo
what is called a blue shift.
Blue means higher energy, red means lower
energy.
So other terms that I used are hypsochromic
and bathochromic shift, but that has been
my Achilles heel.
Ever since my college days I've never been
able to remember which one is bathochromic,
which one is hypsochromic, and why will I
unnecessarily use such big words when I have
simple blue and red at my disposal.
So if you want to use bathochromicand hypsochromic
shift, be my guest; I'll always say blue and
red.
So at higher temperature you expect microwave
spectrum to undergo blue shift; at lower temperature
you expect it to undergo a red shift.
That is point number one.
So we have discussed the effect of temperature.
What do you expect with temperature?
You expect some spectral shift.
Now we'll talk about something else.
If you look at the same microwave spectrum
let us say we are talk about HCl.
We expect a spectrum that looks something
like this.
If the resolution of your spectrometer is
good enough then what you see is each line
looks something like this.
Each line turns out to be a doublet.
I am talking about HCl and I am addressing
a class primarily of chemists.
So can you tell me why?
Exactly.
Chlorine, what is the atomic number of chlorine?
35.5.
Why is it 35.5? because it's a mixture of
two isotopes.
Yes, 37 and 35, no 0.5 there.
So now see what happens.
It is not only r0, you also have ĩ.
So if you have two isotopes present, then
you actually have two kinds of molecules with
two kinds of ĩs.
For that you are going to get different values
of V.
So every line will actually consist of two
lines.
So if you can do a high resolution microwave
spectroscopy then you can get an idea about
isotopic abundance.
Before we discuss non-rigid rotor, that's
where we'll end today's class with, let us
just think for a while with this background,
for what can the spectra be used?
What can be possible applications of microwave
spectroscopy?
One thing I've already said.
You can determine bond length.
What else?
Well, one application is something that is
now absolutely common place.
I am sure everybody has used, if not at home,
then definitely in the hostel.
Everybody has used the microwave oven.
How does a microwave oven work?
No, see, all along so far we have talked about
rotation.
There is no question of vibrating here.
I don't know why but whenever I ask this question
to any audience, including people who have
completed M.Sc. and have come to us to do
Ph.D. everybody talks about vibration, it's
not vibration; it's rotation.
Rotation of water molecules.
So see, the point is this, remember.
One thing that we understood is that the molecule
has to be dipolar and unless you are somebody
who survives on eating steel, all the food
that we eat has 70% water in it as Madhuri
Dixit madam has been telling us for last few
years over TV.
Even we are 70% water.
So what happens when you put the food inside
microwave oven, oven typically has microwave
radiation that is absorbed by water.
So all the water that is there inside the
food, water molecules, they start absorbing
microwave radiation and start rotating, not
vibrating, pleas remember.
But then they cannot rotate forever.
They come to a stop because of dielectric
friction.
Other molecules are there, other water molecules
and non-water molecules around.
They cause these water molecules to come to
a stop, but then all that energy has to be
dissipated.
Where does it go?
All around.
That is why the food gets heated up so quickly
in a microwave oven because now you don't
need a fire from outside.
The fire if you want to call it is within
and as you know, things are more efficient
when the fire is within.
So that's the reason why microwave oven can
provide such efficient heating.
But the point to note is that using a microwave
oven, you can cook fish, but you can never
fry fish.
Why?
Because if you want to fry fish, then you
have to heat a while, you have to go to a
higher temperature.
But if you keep oil even for even 100 years
inside a microwave oven keeping the oven on,
oil will never get heated.
Oil is completely non-polar.
So you cannot fry things.
Now, before we end, hold on.
We are not done yet.
So far we have been pretending that molecules
are rigid rotors.
Such an approximation is always too good to
be true.
We are saying that when molecules rotate we
don't have to worry and all, but actually
we have to worry.
Molecules are usually non-rigid rotors.
Why are they non-rigid rotors, because when
they rotate one thing that can happen is that
due to centrifugal force, the bond can happen
a little bigger, centrifugal distortion.
So what will it effect?
It will affect the average value of r02.
So what happens to the effective value of
B.
If r02 becomes bigger, B becomes a little
smaller, if you think that way.
If you think in terms of perturbation theory,
now we are introducing non-rigid rotor.
This is something that I'll ask you to study
by yourself from Banwell's book.
We'll not do the entire quantum mechanical
treatment.
So you'll only need the results.
That's something you can understand if you
just read Banwell's book, but the point is
?J is now written as BJ(J+1).
Well, this is the expression for the non-rigid
rotor.
Now that is the unperturbed system.
You have studied perturbation theory and you
know that whenever there is a perturbation
-- what is the meaning of perturbation?
A small change.If the change is huge, then
the perturbation theory doesn't work.
If it is small change, then what you can do
is you can start with the original value and
bring in correction terms.
First order correction terms keeping in mind
that the energies have to decrease a little
bit is DJ2(J+1)2.
So you see this correction term becomes more
and more important as you go higher and higher
of the energy ladder which means the molecule
is rotating with higher energy, that is why
centrifugal distortion is more.
So I'll leave it to you to work out the expression
for ?J-->J+1 and I leave it to you to figure
out how will it effect.
So what is this and how does it affect the
microwave spectrum.
What is the spectral signature of non-rigidity?
