8 The Vibrating Diatomic Molecule -I
Contents:
1. Simple Harmonic Oscillator
2. Energy Levels
3. Spectrum
4. Population of Energy Levels
Summary
The students will be able to learn about Vibrating molecule as Simple Harmonic Oscillator, Its Energy Levels and Spectrum.
1. Simple Harmonic Oscillator (to calculate the Frequency of vibration):
It is assumed that the atoms of diatomic molecule are vibrating along the direction of the bond due to which there is a periodic lengthening and shortening of the bond length.
Thus, a vibrating diatomic molecule can be approximated to a linear harmonic oscillator, whose frequency of vibration can be calculated using Newton’s equation of motion. Let the two atoms of the molecule with masses m1 and m2 are joined by a string having spring constant k. As there is no external force, there is no effect on the centre of mass due to oscillations o the atom. The two atoms vibrate back and forth with respect to centre of mass.
According to Hook,s law, force exerted by the two atoms of a molecule on each other (when these are displayed from equilibrium position) is proportional to the change in the inter nuclear distance. Now, suppose the bond distorted from its equilibrium length re to a new length r, then restoring force on each atom of diatomic molecule are
Where k is known as force constant and is measure of the stiffness of the bond, r1 and r2 are the positions of atom 1 and 2 relative to the centre of mass of molecule. We know that
Putting the value of r1 in first equation of motion, one gets
where x represent displacement of the bond length from its equilibrium position.Therefore equation (2) gives
2. Energy Levels
Vibrational energies, like other molecular energies are quantised and the permitted vibrational energies for any particular system can be calculated from Schrodinger equation. The Eigen values for the energy of a linear harmonic oscillator are of the type
Where v is the vibrational quantum number, equal to zero or an integer, and ω is the vibrational frequency of the oscillator expressed in wave numbers. We shall now derive it using Schrodinger wave equation.
A vibrating diatomic molecule is approximated as a harmonic oscillator.
The potential energy function under the influence of which nuclei vibrate is then parabolic and is of the form given by
Where x is the displacement from the mean position or equilibrium position. Then Schrodinger wave equation can be written as
and is valid only for v=0,1,2,………the restriction on v also restricts energy values E.
finally, writing
Where ω is the vibrational frequency of the vibrating diatomic molecule expressed in wavenumber. The above equation gives the allowed energies for the harmonic oscillator. Significance of above equation lies in predicting the existence of zero point energy, equal to 1/2 (hcω(v=0)).
If we transform energy value to term value (on dividing by hc), we obtain for vibrational terms
Thus we have a series of equispaced discrete vibrational levels(figure),the common separation being ωcm-1.the spacing between vibrational levels is considerably larger than the spacing between rotational levels of a molecule.
3. Spectrum:
Suppose a transition occurs from an upper vibrational level, in which the quantum number is v′ to a lower state with quantum number v′′. The change in vibrational energy will be
Thus vibrational spectrum is expected to consist of a single band at ω cm-1. Thus an intense band in infrared spectrum is to be concluded as vibrational spectrum, owing its origin to harmonic vibrations of the nuclei along internuclear axis. However, infra-red spectrum also consists some weak bands (called overtones) at frequencies slightly lesser than 2ω,3ω,…..etc. Their appearance suggests that vibrations deviate from being harmonic and analysis should be made by treating the vibrating diatomic molecule as an anharmonic oscillator.
The vibrational spectra are known only in absorption. Electromagnetic radiations can induce transitions among the vibrating molecule, an electrical coupling must be present. If the vibrating molecule produces an oscillating dipole moment, then the desired coupling results due to the interaction of this dipole moment with electric field of radiation. Consequently, homonuclear diatomic molecules like H2, N2 and O2 that possess a zero dipole moment for any band length will not interact with the radiation. On the other hand, molecules like HF,HCL,HBr,HCN have a dipole moment, which is some function of internuclear distance,(and consequently gives rise to an oscillating dipole moment )will exhibit vibrational spectra.
4. Population of Energy Levels
Considering the case of HCl molecule. where, the frequency of spectral line arising due to transition between = 0 = 1 states is
( v)1,0=2,890 cm-1
So that (∆ E )1,0=hcν10
=6.62 X 10-27 X 3 X 1010 X 2890
=5.75 X 10-13 erg.
Representing the energy of a molecule in = 0 state, the lowest state, is much greater than the population N1 in = 1 state, or in words, only a large small fraction of the molecules populate the vibrational levels at ordinary temperature. This means that most of the molecules are in the lowest allowed vibrational state. In a spectroscopic study, therefore, one investigates the absorption of radiation by these r=0 states molecules.
Thus main vibrational transition in absorption is v = 1← = 0.
Transition rule
For the probability of any given transition, it is essential to assume that the diatomic molecule has a permanent dipole moment. For a linear harmonic oscillator the eigen functions are of foam
We find that the result differs from zero only if the change in the vibrational quantum number in the two states, between which transition is to occur, is equal to unity. Therefore for a harmonic oscillator, selection rule is
Δv=±1
Putting this condition in equation (3)
Vv=w
Predicting that for a harmonic oscillator the frequency of the radiation emitted or absorbed should be equal to the mechanical frequency, ω, of vibration of the system. Thus we find that, like classical theory, quantum mechanically the frequency of radiated light is equal to the frequency Vwc= (w )of the oscillator, no matter what the value of the initial state is. In fig below the allowed transitions are indicated by vertical lines. It is obvious from the figure that they all give rise to the same frequency.
Assignments:
1. HCI has a single intense band at 2882.9 cm-1 in the near infrared spectrum. If this is a vibration spectrum, find out the vibrational frequency.
2. Find the force constant for the H-CI bond if the vibrational frequency of H1CI35 is 8.9 x 1013 Hz. Also calculate the reduced mass of the molecule The reduced mass of H1CI35 is
m1 is mass of Hydrogen and m2 that of Cloride
Now
3. The fundamental vibrational frequency of HCI is given to be 2990 cm-1 Calculate the fundamental frequency of DCI assuming same force constant.
4. Atomic weight of each atom of the CI2 molecule is 35. The fundamental vibrational band of is at 2940.8cm-1. Find out the corresponding fundamental vibration band CI2 molecule in which one atom has atomic weight 35 and the other37. Also find the separation of spectral lines?
Therefore,the separation of the spectral lines is =40cm-1.
5. Calculate the ratio of the number of molecules in v=1 to v=0 vibrational states at 298K if the spacing between levels is 2×10-13 erg/mole.
The number of molecules Nυ in the υ vibrational state w.r.t. υ=o is
which is less than 1% of the molecules are in the v=1 state.
6. For HCI molecule, the separation between adjacent vibrational states is 2885.9cm-1. Calculate the ratio of the number of molecules in v=1 to v=0 vibrational states at 1000 K.
7. What will be the ratio of HCI molecules in the first excited rotational state to those in the first excited vibrational state at 1000K. The rotational constant is 1.32×10-3eV while the separation between adjacent vibrational levels is 2990cm-1.
Summary
- The atoms of the diatomic molecule have been considered to be vibrating along the direction of the bond, due to which there is lengthening and shortening of bond length.
- The vibrational energy levels and allowed transitions between the,m for a diatoimic molecule
- Potential energy function and vibrational energy levels for a diatomic molecule considering the anharmonic oscillator
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