Vibrational Spectroscopy

Introduction to Vibrational  Spectroscopy

• The rotational spectra of a diatomic molecule were considered with rigid bond, but the Co – valent bond is not perfectly rigid, it is somewhat flexible this flexibility gives rise to one more type of motion to the molecule which is vibrational motion.
• For vibrational motion absorption takes place in the IR region. After absorbing radiation in the IR region molecule vibrates at many rates of vibrations giving rise to closely packed absorption bands called “ IR- absorption spectra”.
• Various bands correspond to characteristic functional groups and bands present in a molecule. This IR spectroscopy is the most powerful technique to identify the chemical compounds OR IR spectra provide information about the nature of the bond and the structure of the molecule.
• The diatomic molecular system is stable and the atoms are separated by a distance known as bond length or internuclear distance.
• The stability of the molecule is due to two opposite forces,

1. Force of repulsion between electrons of both the atom and nucleus of both the atoms.

2. Force of attraction between the electron of one atom and the nucleus of another atom and vise-versa.

• When these two forces are counterbalanced at a certain distance it gives the stable molecular system the change in energy of the atoms in a molecule is created by the absorption of IR radiations.

The equilibrium distance is disturbed and the atoms will experience vibrations. These energy changes are observed in the form of vibrational spectra.

Modes of vibrations

Various types of vibrations in IR spectroscopy

There are two types of fundamental vibrations for a molecule.

1. Stretching vibration
2. Bending vibration

Stretching vibration requires more energy than bending vibration

1)  Stretching vibration: –

      In these vibrations, the distance between the two atoms increases or decreases, but the atoms remain in the same bond axis. It is of two types

a. Symmetrical Scratching

In this stretching, the movement of atoms with respect to the central atom in a molecule is in the same direction.

b.   Unsymmetrical Stretching: –

In this stretching Vibration, one atom approaches the central atom while the other departs from it.

2)  Bending Vibration: –

      Bending Vibrations are of four types.

a)   Scissoring: –

      In the Scissoring type two atoms move in opposite directions.

b)   Rocking: –

     In the Rocking type, the movement of atoms takes place in the same direction.

c)   Wagging: –

      In the Wagging type two atoms move either up or down the plane with respect atom.

d) Twisting: –

In this type, one of the atoms moves up the plane while the other moves down the plane with respect to the central atom.

Examples

Introduction to infrared spectra of simple molecules:-

1. CO₂ :-  It is a linear triatomic molecule. Therefore, modes of vibration are

3n – 5, n = 3,

3n – 5 = 4. Two modes of vibrations are scratching and two are bending (asymmetric scratching).

In the case of  CO₂, the symmetrical stretching   O =C= O does not involve any change in dipole moment and I.R. is Inactive.

In asymmetric stretching, one bond is stretched and the other is compressed.

              O =C= O this produces the dipole moment of the molecule with the asymmetric stretching is I.R. Active. Two bending are identical and differ only in one direction.  

 a)            O =C= O                             b) O =C= O  

2. H₂O: -It is a triatomic non-linear molecule. The modes of vibration are 3n – 6 = 3, n=3.

   Two stretching and one bending mode of vibration. All are IR active.

Zero–Point Energy

Zero – Point Energy represents the lowest vibrational energy levels in a molecule. The Zero – Point Energy ‘ Eo’ of the molecule can be obtained from the equation,

Eo = (0 + ½)hc  = ½ hc

It implies that the molecule is always vibrating and is never at rest even at absolute zero temperature when transitional and rotational motions are absent.

Thus, from the above equation, it is seen that this zero point energy depends on the vibrational frequency (ŵ) of the molecule and hence on the strength of the chemical and the atomic masses.

Force Constant.

The Force Constant (k) of a bond is a measure of the rigidity or stiffness of a bond i.e., the force required to stretch or compress a bond

The greater the value of the force constant strong is the bond and the higher the stability of the bond.

Vibrational or IR spectroscopy offers a convenient and reliable tool to determine the force constants of different types of bonds in the molecule.

Bond energy is the energy required to break a bond. Hence, the force constant can be directly related to the bond energy of a bond.

The wave number of the radiation absorbed for vibration is given by,                           

 v =1 / 4 π² c² . (k/π)

 Therefore K = 4π² c² π v²      

            Where π = reduced mass

                        v= wave number

The unit of force constant ‘k’ is Newton’s per meter (N.M^-1)

Group Frequencies

The most practical use of IR spectra is in the field of organic chemistry. In a molecule, the bonds of the groups have definite vibrational frequencies and they are not affected by the rest of the molecule. Such frequencies are known as Group Frequencies.

Group Frequencies lie above and below the fingerprint region.

Groups with light atoms like ^–CH₃, ^–OH,^–CN, ^–C=O, etc. absorb above 1400cm^-1

whereas the groups containing heavy atoms absorb below 700 cm^-1

Vibration – Rotation Spectra

Vibrating Rotor:-

As changes in both rotational and vibrational energy take place, the expression for the energy of a vibrating rotor should include both expressions for the vibrational as well as rotational energy.

Selection Rule:-

1. When radiation is absorbed or radiated by a molecule, it is found theoretically as well as experimentally transition occurs between certain energy levels as a result of what is known as the selection rule.
2. For harmonic oscillator –rigid rotor, these require that vibrational quantum number ‘v’ and rotational quantum number J.
3. Thus, under condition, if the spectral lines obtained are clumped together in what is known as R-branch lines and when transitions form another clump in the spectrum known as P-branch lines.

• Vibrational Energy level:-

The vibrational energy is quantized. The expression for the vibrational energy of a molecule is,

Where  ⱳ is the frequency of oscillation,

            v is the vibrational quantum number ( V = 0, 1,2,3……)       

By assigning different values to the vibrational quantum number different energy values of the vibrational energy can be obtained.

The Vibrational energy level diagram of the molecule is shown above.

P and R branches with reference to IR spectra

When a molecule behaves as a harmonic or inharmonic oscillator there is a simultaneous change in rotational energies. This combined spectrum is called a vibrational, rotational spectrum. Vibrational energy changes are 100 times greater or more than rotational energy changes. They are considered independent & there is no interaction between them. It is given by,

Where B is the rotational constant.

For harmonic oscillations  (v – v’) = 1

That is no Peak.

This frequency is known as the bond center which gives a flat curve.

The spectral line corresponding to this is called P-branch and R-branch.

This vibrational–rotational spectrum is called P – R spectra which is equally spaced with a flat curve or a gap in the center. Eg HBr

Frequencies of the fundamental first and second overtone bands are approximately in the ratio 1:2:3.

(a) Molecules whose vibration satisfies the selection rule ∆v =  are called Harmonic. In this case, the stretching and bending of the bond is small and they absorb little energy for vibration.

(b) The molecules which can absorb large amounts of energy and show distortions are called Anharmonic oscillators. (c) The vibrational energy anharmonic molecule is given by the equation,

Where, v = vibrational quantum number having values 0,1,2……

h = Planck‘s constant.

= oscillatory frequency in wavenumber

But, A = (v + ½)²hcѿx

Where x = Anharmonicity constant.

Therefore, on substituting valve of A in eq (i) becomes

(d) Let the vibrational quantum number change from v = 0 to v=1. The energy change corresponding to this transition can be given as,

This absorption band is known as the fundamental band or first harmonic band.

(e) When the transition occurs from V = 0to V = 2_, the frequency of spectral lines is given by,

This is known as the first overtone or second harmonic band

(f) Similarly, The frequency of spectral lines of transition from V₀ to V₃ is given by,

This is known as the second overtone or third harmonic band.

(g) Therefore It is observed, the frequencies of eq (v), (vi) & (vii) i.e., fundamental, first overtone, and second overtone are in proportion,

ѿ (1-2x) : 2 ѿ(1-3x) : 3 ѿ(1-4x)

Here, x is small which can be neglected.

ѿ : 2 ѿ : 3 ѿ i.e. 1:2:3.

The frequencies of the fundamental first and second overtone bands are approximately in the ratio 1:2:3 And in terms of wavelength they are 1 :1/2: 1/3

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