Hello everyone welcome to this lecture in
the last lecture we have discussed how we
can obtain an absorption spectrum.
We had mentioned that if you plot absorbance
obtained from a spectroscopy experiment against
wavelength lambda we can obtain an absorption
spectrum. Although conventionally absorbance
is plotted against wavelength we can also
plot absorbance against frequency as these
are interchangeable. So, we can write h new
equals hc by lambda. Moreover we can also
plot absorbance against wave number.
Wave number is denoted by new bar a new bar
is related to frequency because h new equals
hc new bar or we can write new equals c new
bar. So, the next question is what is the
unit of new bar because we can write new equals
c new bar or rearranging this equation we
can write new bar equals new by c. So, we
know the unit of the left hand side is equal
to the unit of the right hand side. So, in
order to find the unit of new bar we have
to find the unit of the right hand side and
the unit of the right hand side is unit of
new by unit of c that is 1 by second by centimeter
per second this gives 1 by centimeter or centimeter
inverse.
So, centimeter inverse is the unit of new
bar and also lambda is normally represented
in the unit of nanometers. So, let’s look
into the relation between nanometer and centimeter
inverse. So, let’s say the wavelength lambda
equals x nanometer, so in centimeter inverse
the wave number will be 1 by lambda that is
1 by x nanometer that is 1 by x times 10 to
the power - 7 centimeter equals 10 to the
power 7 by x centimeter inverse.
Later we will discuss about different forms
of spectroscopy like UV visible spectroscopy,
infrared spectroscopy, microwave spectroscopy
and we will see that we get different spectroscopy
depending on the wavelength of the light or
the frequency of the light interacting with
the matter. Conventionally when we plot a
UV visible spectrum we use wavelength in the
x axis however when we plot an infrared spectrum
conventionally we plot wave numbers on the
x axis this is just a convention used for
convenience such that the values plotted on
the x axis are not very large numbers.
So, let’s see what do I mean by that for
example 1000 wave numbers is a typical IR
frequency so if we convert this number into
nanometer it will be 10 to the power 7 by
1000 equals 10 to the power 4 nanometer, so
10 to the power 4 is a larger number than
1000. Similarly 10 wave numbers is a typical
energy gap involved in a microwave transition,
so this amounts to 10 to the power 7 divided
by 10 that is 10 to the power 6 nanometer
which is a very, very large number.
So, far we have explained the absorption process
by saying we have a molecule in a stationary
state of energy E1 and when light of energy
equals h new 12 interacts with the molecule
then the molecule goes from the stationary
state with energy E1 to another stationary
state of energy E2 such that Delta e or E2
- E1 equals h new 12. apparently only one
frequency is involved that is new 12 in the
transition process or in other words in terms
of experiments if we use a polychromatic light
source that is light having many, many frequencies
we can expect that only one frequency that
is new 12 should be absorbed by the matter.
So, in all other frequencies where light is
not absorbed absorbance is given by A equals
log I 0 by I T and because in the other frequencies
light is not absorbed by the matter so I T
for those frequencies will be equal to I 0,
so absorbance we can write log I 0 divided
by I 0 equals log 1 that is 0. The absorbance
should only be non0 at the frequency new 12
so this can be represented in a spectrum so
again we are plotting absorbance versus frequency
and let’s say this is my frequency new 12.
So for all other frequencies the absorbance
will be 0 and only at new 12 the absorbance
will be non0. However it has been seen from
experiments that the spectroscopic lines are
not as infinitely sharp as shown in the plot.
But they are more or less broad additionally
we should realize that as the light sources
polychromatic that is the light consists of
many frequencies thus multiple transitions
can be obtained from the same sample.
So, this is a spectrum where the lines are
associated with the anti symmetric stretching
mode of carbon dioxide. At high resolution
the spectrum seems to consist of multiple
lines from multiple transitions so if we expand
the scale or if we zoom into one particular
line the line with apparent narrow feature
is observed to have a definite width and a
characteristic shape. The shape of the line
spectrum is known as line shape.
As this shape is a function of frequency the
line shape can be represented using a line
shape function which is given by f new - new
0 where new 0 is the frequency corresponding
to the maximum absorbance. So, new 0 is also
known as the peak position for a single peak
corresponding to a transition the shape of
this spectrum is symmetric with respect to
new 0. So, the observation of the width and
line shape from the carbon-dioxide spectrum
immediately raises two questions.
One what are the possible functional forms
of these shapes and number 2 what physical
processes are responsible for these shapes.
Line shape functions fall into 1 of the 2
general categories one is homogeneous and
the second is inhomogeneous. A homogeneous
line shape occurs when all the molecules in
the system have identical line shape functions.
Here we should understand that we have an
ensemble of molecules in the sample solution
on which we shine light during the spectroscopy
experiment. So, let’s say we have made a
sample solution of concentration equal to
0.1 molar and let’s say the volume of the
container which is placed in the direction
of light so that the light is passed to the
container and goes through the detector.
So, the volume of this container is 1 milliliter
so we know that for 1000 milliliter of sample
with concentration equal to 1 molar we have
a Avogadro number of molecules which is approximately
equal to 10 to the power 23 molecules. But
now in our solution we have a volume of 1
milliliter and a concentration of 0.1 molar
so the number of molecules we have during
the experiment is 10 to the power 23 divided
by 10 times 1000 equals 10 to the power 19
molecules.
Thus if all the molecules are affected the
same way all are homogeneously affected during
the light matter interaction they will have
the same line shapes and we will have homogeneous
broadening or homogeneous line shape function.
For example if the absorbing species in the
gas phase is subjected to high pressure then
all the molecules are found to have an identical
pressure broadened line shape for a particular
transition. Pressure broadening of a transition
is thus said to be a homogeneous broadening.
In contrast if the sample is dissolved in
a liquid solvent then this order inherent
in the structure of the liquid provides different
solvent environments for the solute. As each
solute experiences a slightly different environment
they have a slightly different spectrum. The
observed absorption spectrum from the experiment
is made up of all the different spectra for
the different molecular environments and it
is said to be inhomogeneously broadened.
In general the homogeneously broadened spectrum
can be represented by a Lorentzian. So, the
Lorentzian line shape function represents
homogeneously broadened spectrum. The Lorentzian
line shape function is given by f new - new
0 equals 1 by pi gamma by 2 new - new 0 squared
+ gamma squared by 4 so here gamma is the
parameter specifying the width of the spectrum.
The inhomogeneously broaden spectrum is given
by a Gaussian line shape function.
And the Gaussian line shape function can be
written as f new - new 0 equals 1 by Sigma
root over 2 pi E to the power - new - new
0 squared by 2 Sigma squared, here Sigma is
related to the width of the spectrum. So,
we can quantify width of the observed spectrum
in terms of full width at half maximum. This
is also written as FWHM from the initial alphabets
of full width at half maximum. So, let’s
try to understand how to obtain full width
half maximum or FWHM from a spectrum.
So let’s say we have an absorption spectrum
so we are plotting absorbance versus frequency.
So, first thing we have to find out what is
the absorbance at new equals new 0 because
at new 0 the absorbance is maximum. Once we
find the maximum value of absorbance we have
to find what is half of that maximum value
and once we find that half of the maximum
value of absorbance we have to find the values
of the frequency that corresponds to half
of the values of the absorbance.
So because we are considering a spectrum which
is symmetric about new 0, so we will have
two such frequencies one on the left and one
on the right of your new 0. So, now the frequency
difference between these two points is the
full width half maximum. Now instead of taking
this frequency difference if we take the frequency
difference between new 0 and one of these
frequencies which has half the maximum absorbance
then that is given by half width at half maximum
and because the spectrum is symmetric half
width at half maximum equals full width at
half maximum by 2.
So, if the functional form of the line shape
is known we can find the functional form of
the full width at half maximum. So, let’s
look into the Gaussian functional form. So,
we have to evaluate this function at new equals
new 0 so at new equals new 0 f new - new 0
becomes 1 by Sigma root over 2 pi E to the
power 0 because E to the power 0 is 1 so this
is 1 by Sigma root over 2 pi. So, now that
means the maximum absorbance is 1 by Sigma
root over 2 pi.
So, the absorbance at half the maximum absorbance
value is 1 by 2 Sigma root over 2 pi. So,
now we have to find the frequency at which
the absorbance is 1 by 2 Sigma root over 2
pi, so we can write 1 by 2 Sigma root over
2 pi equals 1 by Sigma root over 2 pi E to
the power - new - new 0 squared by 2 Sigma
squared. So, we will cancel out this so we
have or we can write ln half equals - new
- new 0 squared by 2 Sigma squared.
So, we can also write from this equation ln
2 equals new - new 0 square by 2 Sigma squared
or new – new 0 squared equals 2 Sigma squared
ln 2 or new - new 0 equals + - Sigma root
over 2 ln 2. So, now we needed to find new,
so new equals new 0 + - Sigma root over 2
ln 2. So what does it mean so I have new 0
so this frequency is new 0 + Sigma 2 ln 2
and this is new 0 - Sigma 2 ln 2 so the full
width half maximum equals two Sigma root over
2 ln 2.
So, now let’s look into the physical processes
that lead to broadening of the spectrum one
of them is due to quantum mechanical effects
this arises from the Heisenberg's uncertainty
principle between energy and time which is
given by Delta E delta t is of the order of
h cross, so Delta E is the uncertainty in
energy and delta t is the uncertainty in time.
The expression tells us that if the system
survives in a quantum state for a finite time
the energy of the state in principle cannot
be known with accuracy better than Delta E
which is given by h cross by delta t or h
by 2 pi delta t.
Now let’s think about spontaneous emission
when we excite a molecule to a higher energy
state Einstein hypothesized that the molecule
will spontaneously decay from the higher energy
state. If the molecule on average say decays
after time tau so tau is the time the molecule
survives in the excited state then tau is
known as a lifetime and we can write delta
E equals h by 2 pi tau and for Delta E we
can write h Delta new equals h by 2 pi tau
can cancel out h so Delta new equals 1 by
2 pi tau.
So here Delta new is the uncertainty in frequency
which gives rise to a broadening of the spectrum
along the frequency axis. Thus the spectral
broadening or delta new is inversely proportional
to the lifetime if the life time is short
the spectrum will be broad and vice versa
we have discussed that A is Einsteins coefficient
for spontaneous emission. The unit of A is
second inverse that is inverse of the unit
of time.
It can be shown that A is equal to 1 by lifetime
or 1 by tau. Moreover lifetime can be affected
by interactions between the quantum state
and the surrounding particles that is collisions.
In the condensed phase the occurrence of collisions
are greater than that in the gas phase. As
these collisions are inelastic in nature they
reduce the lifetime of the excited state.
A similar case happens when pressure is increased,
increase in pressure increases the number
of collisions and thus reduces the lifetime.
In all these cases the spectral line shape
is Lorentian and thus these factors lead to
homogeneous broadening. One of the most common
examples of gas phase inhomogeneous broadening
occurs due to Maxwell Boltzmann distribution
of molecular velocities and is called Doppler
broadening. When viewed from the frame of
the atom the Doppler effect results in light
energy shift or shift in the frequency of
light when the source is moving either toward
or away from the observer.
When a source emitting radiation with frequency
new 0 moves with a velocity V and the observer
is placed in the Z direction that is the detector
in is in the Z direction then the observer
detects the radiation with a frequency not
equal to new 0 but new which is equal to new
0 1 + - Vz divided by c but this plus and
minus sign is related to an approaching or
receding source. It can be shown that the
full width half maximum Delta new equals 2
new 0 by c root over 2 ln 2 k T where T is
the temperature divided by M.
So we see that Delta new or full width half
maximum of the Doppler broadening has an expression
similar to what we had obtained for a Gaussian
line shape. Thus Doppler broadening is inhomogeneous
broadening and gives rise to a Gaussian line
shape. We can see that the Doppler width is
proportional to the square root of temperature
and inversely proportional to the square root
of mass. For example the Doppler width is
1 gigahertz for a UV visible transition at
room temperature.
However for hydrogen the mass is small and
the Doppler width is approximately much bigger
that is approximately 30 gigahertz. So, finally
I would like to end by mentioning that all
these processes the Doppler effect, the collisions,
the quantum mechanical effect that is lifetime
happens simultaneously and thus the spectral
line shape is never perfectly Gaussian or
Lorentzian but is a convolution of Gaussian
and Lorentzian line shape function and is
given by the voigt function or the profile
of the spectrum is known as voigt profile.
