This content explains the principles of vibrational spectroscopy, focusing on how diatomic molecules behave as simple harmonic oscillators. It details the quantum mechanical model of vibrational energy levels, the concept of zero-point energy, and the relative populations of these energy states, which are fundamental to understanding molecular structure and dynamics through infrared spectroscopy.
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hello friends today we are going to
learn about
vibrational spectroscopy after studying
this module
you shall be able to conceptualize
vibrational motion
study how a diatomic molecule acts
as a simple harmonic oscillator know the
schrodinger wave equation
for a simple harmonic oscillator learn about
about
the vibrational energy levels in a
diatomic molecule
and know the relative populations in
vibration energy levels of a diatomic [Music]
molecule
let us begin with the introduction we
have already learnt about the rotational
motion of a molecule
considering it as a rigid rotator that is
is
where the bond is rigid and the bond
length is fixed
however this is only an approximation as
there is
always some flexibility or elasticity
associated with the bonds one of the
consequences of such elasticity of
bonds is that atoms belonging to a molecule
molecule
do not remain fixed at their positions instead
instead
vibrate about their mean relative positions
positions
as shown in the video vibrational motion or
or
oscillation consists of to and fro
motion of a particle
along the same path in such a motion
the displacement of the particle from
its mean equilibrium position
varies periodically with time if the
motion has a fixed
time period as well as fixed amplitude
the motion is said to be a simple harmonic
harmonic
motion few examples of simple harmonic motion
motion
in our day to day lives include a simple pendulum
pendulum
showing back and forth motion in small angles
angles
a small mass tied to a weightless spring
that exhibits movements along the same
path etc
figure one depicts these examples since
atoms in a molecules
vibrate about their mean positions along
a given bond
with their bond angles as well as bond lengths
lengths
changing periodically such a motion
can be considered as a simple harmonic motion
motion [Music]
[Music]
let us begin with infrared or
vibrational spectroscopy
we know that electromagnetic spectrum of light
light
consists of different regions with
varying frequencies
and varying energies figure 2 depicts
the electromagnetic spectrum
of light on the extreme right end we
have the radio waves
which have the highest wavelength and
our least in energy
after that we have the infrared region
after that we have uv visible and then
x-rays and then gamma rays in the
increasing order of their frequency
or energy when rays from
infrared region of an electromagnetic spectrum
spectrum
fall on the molecule the vibrational
energy of the molecule
increases this leads to the promotion of
the molecule
from lower vibrational energy levels to higher
higher
vibrational energy levels thus
we can say that a vibrational spectrum
arises by virtue of transitions between
various vibrational energy levels of a molecule
molecule
upon absorption of energy in the infrared
infrared
region the essential criteria for a molecule
molecule
to show such a spectrum is that there
should be occur
a change in the configuration of the molecule
molecule
that is change in the dipole moment of
the molecule
reserves a particular vibrational mode of
of
motion the vibrational spectra of the molecules
molecules
are helpful in obtaining valuable
information about the molecules
in terms of their structure symmetry
interactions priority bond strength
and various other physical parameters
we can also obtain important information
about the elasticity of the bond
undergoing vibrational motion that is
the ease with which we can distort the
bond length
and the bond angle of the molecule from
its mean
equilibrium position when the molecule
absorbs infrared radiation
from the electromagnetic spectrum of light
light
it not only brings about a change in the
vibrational energy
but also in the rotational energy of the molecule
molecule
this is because of the fact that
infrared radiation is of
higher frequency and thus higher energy
than the microwave radiation that brings
about a change
in the rotational energy of the molecule
as a consequence
the rotational spectrum is superimposed
upon the vibration spectrum of the molecule
molecule
making latter complicated to study
one can obtain pure vibrational spectra
by recording the spectra in the liquid
state where
rotation of the molecules is hindered in
the gaseous state
however the molecules undergo free
rotation and vibration
thus giving a joint vibrational rotation spectra
spectra
we shall learn about this in the
subsequent modules [Music]
[Music]
in terms of classical mechanics the
vibrational motion of a molecule
can be represented by taking a model system
system
comprising of two balls of definite masses
masses
m1 and m2 connected to each other
by an elastic spring the two masses represent
represent
two atoms of a molecule and the spring represents
represents
the elastic bond between these two atoms
as discussed above such a system acts
as a simple harmonic oscillator the
situation can be further simplified
by considering a single ball of mass m
attached to a rigid support by an
elastic spring
when the mass is displaced from its
equilibrium position a restoring force
acts in the opposite direction that
tries to bring
the mass back to its equilibrium position
position
if the displacement is small that is the extension
extension
or compression is less than 10 percent
of the bond length of the molecule the molecules
molecules
act as a simple harmonic oscillator and
follow hooke's law that is the restoring force
force
is proportional to the displacement and
it is acting in the opposite direction
as the displacement therefore a negative sign
sign
when we remove the proportionality sign
we get a
proportionality constant that is f is
equals to
minus k x as shown in
equation 1 where f would be my restoring force
force
k will be the spring constant or the
force constant
that is the characteristic of the spring
and measures
its stiffness x is the displacement from the
the
equilibrium position the negative sign
shows that the restoring force
f acts in the direction opposite to the displacement
displacement
in order to restore it to its original
equilibrium position when the mass is displaced
displaced
from its equilibrium position some work
is done
this work gets stored in the body in the
form of potential
energy v that is we can write
f is equals to minus dv by dx
that is our equation 2 substituting
equation 1
in equation 2 we get dv
is minus of minus kx dx
or we can write dv is equals to
kx dx when we integrate this equation
we get v equals to half kx
square this is our equation three if we
plot this potential energy
we against displacement x from the
equilibrium position
we get a parabolic curve as shown in figure
figure
3. figure 3 shows the potential energy
of a simple harmonic oscillator
it can be seen that the variation of
potential energy with displacement
is a continuous function this means
that as per the principle of classical mechanics
mechanics
the body can vibrate with any amount of
potential energy
the above behavior can also be
correlated with bond formation
in a molecule when two atoms approach
each other during bond formation their occurs
occurs
repulsion between the two positively charged
charged
nuclei and also between the negatively
charged electron
clouds of the two atoms in addition
there occurs attraction between the
nucleus of
one atom and the electrons of the other
and vice versa at an internuclear distance
distance
where these attractive forces balance
out the repulsive forces
the energy of the molecule will be minimum
minimum
and the said internuclear separation
will be termed
as equilibrium bond length or simply
bond length if we try to compress the bond
bond
and bring the atoms close to each other
that is decrease the bond length the
repulsive forces will increase
rapidly similarly pulling the atoms apart
apart
in order to increase the bond length is
also not
favored due to loss in the attractive
forces any change in the bond length
therefore requires an input of energy
thus extension or compression of the bond
bond
can thus be easily compared to the
behavior of a spring
that acts as a simple harmonic
oscillator and
obeys hooke's law figure 4 shows the
distortion in a diatomic molecule
in terms of extension and compression of
a spring
from its equilibrium position we already know
know
that the force required to displace the atoms
atoms
from their equilibrium position is given by
by
newton's second law as given by equation
4 as f is equals to m
a which can be written as f is equals to
m a that can be written as m
d square x upon dt square
where m is the mass of the body a
is its acceleration and x is the
displacement from the equilibrium position
position
from equation 1 we already have f is
equals to minus
k times x combining the hooke's law that
is equation 1
and newton's equation of motion that is
equation 4
we get m d square x upon dt square
is equals to minus kx or
d square x upon dt square
plus k by m times x is equals to 0.
let this be our equation number five the
general solution for
above equation five is given by
x is equal to a sine omega t
where x is the displacement from the
equilibrium position
a is the maximum displacement of the
vibrating particle
from its mean position or the amplitude of
of
vibration and omega is the angular frequency
frequency
of the vibrating particle this can be
further written as
x is equals to a sine 2 pi
new classical times t where
new classical is the classical frequency
of the vibrating particle
and t is the time period of vibration
this can be further written as x is
equals to
a sine under root k by m times
t where m is the mass of the body
let this equation be our equation number six
six
from the above equation six we see that
vibrational motion of a simple diatomic molecule
molecule
executing a simple harmonic motion varies
varies
sinusoidally with time the time period
taken for one oscillation is given by t
equals to 2 pi under root m by k
as shown in equation number seven the
frequency of vibrational motion
is defined as the number of oscillations
taking place
per unit time the classical expression
for the vibrational frequency
is given by new classical is equals to
1 by 2 pi under root k by m
as shown in equation 8. it was observed
that the laws of classical mechanics
could not explain the behavior of
microscopic particles
therefore the correct treatment and
correct expression
for vibrational energy of a diatomic molecule
molecule
had to be derived using quantum mechanics
mechanics
on solving the appropriate schrodinger
wave equation
we get the vibrational energy of a
diatomic molecule
executing simple harmonic motion as
e v is equals to v plus half
h nu classical as shown in equation
9 where ev is the vibration
energy v is the vibrational quantum number
number
which can have values from 0 1
2 3 4 and so on
h is the planck's constant equals to 6.626
6.626
into 10 to the power minus 34
joule second and new classical as
earlier told
is the classical frequency of the
vibrating particle
substituting the value of classical
vibrational frequency
from equation 8 in equation 9
we get ev becomes equal to
v plus half h by 2 pi under root
k by m let this be our equation number
10. since energy is given by the falling expression
expression
also ev is equals to hc
nu bar as shown in equation 11
where h is the planck's constant equals to
to
6.626 into 10 to the power
minus 34 joules second c
is the speed of light equals to 2.999
into 10 to the power 8 meters per second
and nu bar is the wave number usually
expressed in centimeter inverse units
the expression 11 can be substituted in
equation 10
to get vibration energy in wave number units
units
as e bar v in centimeter inverse
that is the vibrational energy in wave
number units
that is i centimeter inverse is equals to
to
v plus half h by 2 pi into
1 by hc into under root k by
m or this can be written as
e bar v in centimeter inverse is equals to
to
v plus half into 1 by
2 pi c under root k by m
as shown in equation 12.
or e bar v in centimeter inverse is
equals to
v plus half nu c bar as
shown in equation 13 where my nu c bar
is equals to 1 by 2 pi c
under root k by m as shown in equation 14.
14.
here e bar v is the vibrational energy
in wave number units
v is the vibration quantum number which
can have values from
0 1 2 3 4 and so on
c is the speed of light which is equals to
to
2.999 into 10 to the power
8 meters per second k is the force constant
constant
and m is the mass of the body
for a system with two masses m1
and m2 as in the case of a diatomic molecule
molecule
the mass in the above equation 14 is replaced
replaced
by the reduced mass of the system
that is we get nu c bar is equals to 1
by 2 pi c
under root k by mu in equation 15
where mu is the reduced mass of the
system given by
m1 into m2 divided by m1 plus
m2 as shown in equation 16.
equation 13 gives the expression for
vibrational energy
of a diatomic molecule acting as a
simple harmonic oscillator
and highlights one very important result
that vibrational energy of a harmonic oscillator
oscillator
is always quantized this was contrary to
the results obtained
from classical mechanics where the
vibrational energy of harmonic oscillator
oscillator
can vary continuously the quantization emerges
emerges
from the solution of schrodinger wave equation
equation
due to the presence of vibrational
quantum number
v which can take up only discrete values as
as
0 1 2 3 and so on
and not any random values
let us now look at the energy level
pattern for a diatomic molecule
the various vibrational energy levels
for a diatomic molecule
acting as a simple harmonic oscillator
can be obtained
by substituting the values of
vibrational quantum number
v equal to 0 1 2
3 and so on in the equation 13.
substituting the value of v equals to 0
in equation 13 will give us the lowest
vibrational energy
as e bar v is equals to
0 plus half nu c bar
or e bar v is equals to
half nu c bar as shown in equation 17
thus we observe that the lowest
vibrational energy
is half nu c bar and not zero
as observed for a vibrating body using
classical mechanics this value of lowest energy
energy
is termed as zero point energy for a
simple harmonic
oscillator the fact that zero point energy
energy
is non-zero means that atoms
can never be at rest relative to each other
other
in fact at absolute zero when
all the translational and rotational
motion of the molecule
such an observation is in agreement with
the heisenberg
uncertainty principle which states
that it is impossible to determine
the precise values of position and momentum
momentum
of a moving subatomic particle
simultaneously that is delta x
into delta p x is greater than
equals to h by 4 pi as depicted in equation
equation
18. if the atoms are not
moving at all and are at standstill
the uncertainty in position will be zero
this will give uncertainty in momentum as
as
infinity as per the equation 18.
in order to avoid such a situation the atoms
atoms
must have some uncertainty in position
which means that the atoms cannot be at
standstill they ought to be vibrating
at their mean positions this behavior is
contrary to that obtained from the principles
principles
of classical mechanics where the atoms
of a diatomic molecule
can be at standstill and can possess
zero vibrational energy substituting the
other values of vibrational quantum
number we as 1
2 3 and so on in equation 13
we get the energy of higher vibrational
energy levels
as shown for v equals to 1
e bar v comes out to be 3 by 2
nu c bar for v equals to 2
e bar v is equals to 5 by 2
nu c bar for v equals to 3
e bar v is equals to 7 by 2
nu c bar for v equals to 4
e bar v is equals to 9 by 2
new c bar and so on as shown in equations
equations
19 the above results show that the
energy values of a diatomic molecule
acting as a simple harmonic oscillator
are quantized
and they cannot vary abruptly in fact
it is the occurrence of these quantized
vibrational energy levels
that gives rise to vibrational spectra
of the molecules
which again could not be explained by
the principles of classical
mechanics on plotting these energy values
values
on a parabolic curve we get the various
vibration energy levels
as shown in figure 5 below figure 5
shows the vibrational energy levels
in a simple harmonic oscillator we can conclude
conclude
that a diatomic harmonic oscillator
consists of equally spaced vibration
energy levels
the distance between any two consecutive
vibration energy levels can be calculated
calculated
by the difference in their energy as
delta e bar from 0 to 1 is
equals to e bar 1 minus
e bar zero that is the vibrational energy
energy
for v equals to one level
minus vibration energy for v equals to
zero level which on substituting their
values would be 3 by 2 nu c bar
minus half nu c bar and delta e bar from
0 to 1
comes out to be nu c bar
similarly if we calculate delta e bar from
from
1 to 2 vibrational energy level
that would be equal to the difference of
e bar
for v equals to 2 minus e bar for v
equals to 1 and on substituting their
respective values as 5 by 2 nu c bar
minus 3 by 2 nu c bar delta e bar
for 1 to 2 comes out to be again
nu c bar similarly delta e bar
for 2 to 3 would again come out to be nu c
c
bar and so on it can be seen
that the distance between any two consecutive
consecutive
vibration energy levels remains the same throughout
