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welcome back to the lectures on molecular
spectroscopy and introductory chemistry in
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this short lecture let us look at the rotational
and vibrational line intensities from an elementary
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point of view using the maxwell boltzmann
statistics it was mentioned to you some lectures
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ago that there are three parameters which
are of importance in any ah study using spectroscopy
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namely the line positions which are dealt
with by quant mechanics the line intensities
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which require both quant mechanics and also
statistical mechanics since we deal with populations
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of molecular system in different energy levels
and the third was the line ah width we shall
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not consider the line width in this course
but let us look at the line intensities here
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little more closely and particularly for the
rotational microwave and infrared infrared
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ah frequency is an transitions ok now the
ah basic ah formula or the basic prescription
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that you should remember is that molecules
the number of molecules in any energy state
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means assume that we have n naught n one n
two etcetera molecules in different energy
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levels e zero e one e two etcetera there is
distribution in thermal equilibrium distribution
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of molecules in different energy levels
due to thermal equilibrium
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what we need to know is the boltzmann formula
which was also mentioned to you earlier in
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the course the boltzmanns man formula tells
you that the number of molecules n at any
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given energy level e is approximately proportional
to the degeneracy of that energy ah often
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it written using omega of e and to the boltzmann
factor which tells you ah the waiting for
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that number for that given energy e by k b
t in the case of rotational spectroscopy the
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molecular energy levels for the ah simple
system of a diatomic they are proportional
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to ah the j in to they they are basically
j into j plus one you recall the energy level
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formula for rotation for a given quantum number
j it is b j in to j plus one times h c the
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quantum number j corresponds to two j plus
one states which are given by different values
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of of the projection of j on the molecular
axis and these are called the k quantum numbers
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and they go from minus j to plus j in integer
steps
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in the case of diatomic molecule we have the
symmetry axis which is the axis of the molecule
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about which there is no movement inertia therefore
there is no rotational degree of freedom ah
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which would show up in that for that motion
the other two axis which are perpendicular
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to the ah molecular axis the moment of inertia
above both of those axis are identical therefore
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there is one moment of inertia and the moment
of inertia is built in this constant b as
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h by eight pi square i c you have already
solved such problems therefore what you have
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here is are there is only one energy level
in the sense one energy value for a given
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j so if you write e j its b j in to j plus
one h c but there are two j plus one states
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which are d generate because they all have
the same value and these are indicated by
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the wave function the two quantum numbers
j and k and the wave functions are usually
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written as using the dirac notation
ket notation is called the bracket notation
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ah what do you see is that the states are
represented as the states j k and for molecular
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systems where the rotational quantum numbers
are integers these j k states can be identified
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with specific representation through spherical
harmonics
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y j k of theta phi which you had come across
in the case of hydrogen atom as contributing
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to the angular distribution of the wave function
therefore the molecular wave functions for
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the rotational states or two j plus one whole
the degenerate because all the sates starting
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from k is equal to j to j j minus one j j
minus two and all the wave down to j minus
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j all have the same energy in the absence
of any external field or in the absence any
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other consideration like movements of inertia
being different and so on for a diatomic molecule
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the true movement of inertia are equal the
third movement of inertia is zero therefore
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we have all the energy levels for a given
j b ah degenerate therefore what does this
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tell you this tells you that if you are calculating
the molecules the number of molecules in a
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given energy state corresponding to the quantum
number j energy value corresponding to the
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quantum number j and the number of molecules
in another energy corresponding to a different
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value j prime this tells you that is ratio
is two j plus one by two j prime plus one
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and then you have e to the exponential to
the minus the energy corresponding to the
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quantum number j and the energy corresponding
to the quantum number j prime the difference
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between the two divided by the boltzmanns
constant k b t so this is the formula this
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is what is called the fundamental formula
for the calculation of elementary form of
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micro wave intensity without other consideration
like ah ah reactions and ah i mean the interaction
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between rotation and vibrations and all those
things not being considered pure micro wave
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transmission of rigid by atomic molecule that
number of molecules in any given energy state
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j to the number of molecules in another energy
state j prime is given by the ratio of the
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degeneracy's and the exponential or the boltzmanns
man factor given by the energy difference
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so what is this if you write that you recall
that e j is b j in to j plus one h c and e
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j prime s b j prime into j prime plus one
h c and if you calculate for example n one
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by n zero n one meaning e one with j is equal
to verses e with j is equal to zero e zero
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the grounds rotational state if you do that
then this is three the two j plus one here
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is corresponding to the value j is equal to
one and this is j is equal to zero therefore
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the degeneracy is here is one three by one
and what do you have is the energy between
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e e one with the difference minus e one minus
e not by k b t and that you know is two b
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h c therefore what you have is three exponential
minus two b h c by k b t ok
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now this energy two b h c is very small compared
to the factor k b t the thermal energy for
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any temperatures like greater than ah say
ten kelvin and suddenly t three hundred kelvin
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this energy b h c is much smaller than k b
t therefore you see that the number of molecules
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in the higher energy state to the number of
molecules in the energy state in this particular
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case is actually that ratio is greater than
one or n e one divided by n e not this factor
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is almost equal to one or very close to one
therefore its is greater than one so what
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you see in the micro wave transfusion is since
the degeneracy increases as j increases there
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is this accommodation of the higher energy
state of the system in to many more levels
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all of which are degenerate verses the accommodation
of the rotational states into that lower j
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where the degeneracy's are slightly lower
therefore what you see is that the number
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of molecules in e one is usually greater than
the number of molecules in the energy state
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e zero and the number of molecules in the
energy state e two is again greater than the
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number of molecules in the energy state e
one and so on so you can say n e j is greater
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than n e j minus one so on until greater than
n e zero this is go on like thats forever
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no it doesnt because you see the rotational
energy levels also increase very very fast
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because you see the lowest energy level is
zero the next one is two b the next one is
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actually six b four b six b and the third
is twelve b and this one is twenty b and so
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on therefore the energy gap between successive
rotational quantum numbers the states of ah
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states with successive rotational quantum
numbers the gap increases therefore at some
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point of time the numbers start decreasing
therefore if you plot the micro wave rotational
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intensity as a function of j you start seeing
for j is equal to say zero one two three etcetera
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you start seeing that well lets keep the zero
out of picture start with one with respect
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to zero this is some number two increases
three the j three increases and so on but
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after some j when it reaches some maximum
it comes down to it starts becoming less and
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less and so on and what you see is that the
micro wave intensities are like an envelope
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bit increasing and there is a maximum for
a some value of j
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and that can be easily calculated because
if you write the ah the number density or
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the number for that particular j b in proportional
to two j plus one times exponential minus
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h c b j into j plus one by k b t because this
is ah energy and the the correct unit is the
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h c times the b which is only a wave number
unit
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so if you have this you want to find out what
is the value of j for which the number is
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maximum you can do a simple calculus by taking
the derivative of the number ah numbers with
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respect to j and then set that equal to zero
you know its an proximate calculation j is
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not continuous therefore you know to take
the derivative doesnt make much sense but
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to get a feel for this the energy levels ah
gaps of skill very small compared to thermal
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energies therefore if you take a derivative
like this you can see that immediately this
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gives you the following that the number density
gives you the derivatives you take the derivatives
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of this it is two times e to the minus h c
b j into j plus one by k b t thats the first
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term of the derivative the second is the derivative
of this term which if you take it is minus
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two j plus one and this will give you the
ah expression e to the minus h c b j in to
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j plus one by k b t times this derivative
which is one by k b t and the minus sign is
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already here it will give you h c b in to
two j plus one this is what you will get and
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if you set that equal to zero its easy to
show that the j max is approximately square
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root of k b t by two h c b minus one half
we call that this is a if you set this equal
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to zero the exponential factor goes away so
its ah you can see immediately that it is
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two minus two j plus one whole square in to
h c b by k b t thats equal to zero and that
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gives you the solution ok thats the j max
the corresponding value of j is the j max
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and you can see that this is the ratio
therefore microwave intensity is actually
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peak for some middle value of ray verses ah
the starting from j is equal to zero to some
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other value j what about the infrared intensity
for a diatomic molecule there is no problem
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because all vibrational states are non degenerate
single vibrational states ah the degeneracy
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is one therefore in the case of vibrational
motion if you write just n v ah for n v prime
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where v and v prime are the vibrational quantum
numbers for two different states for states
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with ah two different states and then you
can see that this is nothing other than simply
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exponential minus e v minus e v prime by k
b t therefore there is no factor in front
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of the exponential which counters the decrease
of the exponential you can see that vibrational
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intensities usually or ah maximum for the
v is equal to zero to one transition or the
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v is equal to zero state itself and then its
slightly lower and so on and lower and so
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on secondly with respect to microwaves and
rotation spectroscopy the energy gap is very
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small compared to the thermal energy whereas
in the case of vibrational states the energy
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gap is comparable or even more than thermal
energy therefore the number of molecules even
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in the first excited vibrational state is
growing to be a lots smaller fraction of the
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number of molecules in the ground state this
is not so in the case of rotations in the
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case of rotation a large number of j values
are populated fairly well whereas in the case
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of vibration unless the vibrational energies
are very close to each other most molecules
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will be at any given temperature in the ground
state and few were in the first exited state
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and even few were in the second exited state
therefore you can see the drop in the intensity
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very very directly so these are things that
you have to keep in mind in observing the
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intensity and also some of the spectra that
i will show in the in one of these lectures
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or through the lecture notes we will continue
this with the spectroscopy of the micro wave
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spectroscopy of poly atomic molecules in the
next lecture until then
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thank you very much