25 Rotational Spectrum
Devendra Mohan
Contents of this Unit
1. Introduction: Rotational Spectrum
2. Molecular Requirement for Rotational Spectra
3. Rotational Spectra of Diatomic Molecules as Rigid Rotor
4. Spectrum of the Rigid Rotator
5. Intensity of Rotational Lines
6. The Effect of Isotopic Substitution
7. Energy Levels and Spectrum for Non-rigid Rotator
8. Techniques and Instrumentation for Microwave Spectroscopy
9. Applications of Rotational Spectra
The students will be able to learn about rotational spectra and its theory.
1. Introduction:Rotational Spectrum Rotational Spectra
Pure rotational molecular spectra arise from transition between rotational energy levels and are observed in far infra red region or microwave region of the electromagnetic spectrum.
Microwave Region –
(3×1010 – 3×1012 Hz) is concerned with the study of rotation of molecules.
Rotation of Molecules
A molecule cannot rotate about some arbitrary axis. However, the principal of conservation of angular momentum dictates that only a few rotations are possible. Generally rotations must be about centre of mass of molecules and axis must allow conservation of angular momentum.
The rotation of a 3D molecule may be quite complex and is conveniently resolved into rotational components about three mutually perpendicular directions through the centre of gravity (of molecule) – the principal axes of rotation.
Thus a body has three principal moments of inertia, one about each axis, usually designated as Ia, Ib and Ic . In general a, b and c are selected in such a way that Ia< Ib < Ic. The 3 principal axes of rotation.:
i. ‘a’- axis:- about the bond axis
ii. ‘b’-axis:- end-over-end rotation in the plane of the paper i.e. the direction in the plane of the paper
passing through the centre of gravity of molecule and perpendicular to a-axis.
iii. ‘c’- axis:- end-over-end rotation at right angles to plane of paper i.e. the axis perpendicular to both a and
b-axis and passing through the centre of gravity of the molecule.
2. Molecular Requirement for Rotational Spectra
The basic requirement for the emission and absorption of radiation by transition between rotational energy states is that the molecule must have a permanent dipole moment.
It is noteworthy that molecular bond lengths remain constant in pure rotation.
The magnitude of electric dipole moment cannot change as electric dipole moment is a vector quantity, therefore rotation can cause a permanent dipole to change direction and hence spectra is observed.
Types of Molecules
The molecules are usually classified into four groups according to the relative values of their principal moment of inertia that is equivalent to classifying them according to their shapes:-
Linear Molecules:-
These are the molecules in which all the atoms are arranged in a straight line. eg. HCl, CO2, HCN, C2H2 etc. In this type of molecules, the moments of inertia along ‘b’ and ‘c’ axes are equal and the moment of inertia along ‘a’-axis is negligible. Thus for a linear molecule Ia=0, Ib = Ic.
Symmetric Tops:-
Here, all the three principal moments of inertia are non-zero but two of them (Ib and Ic) are equal. eg. CH3Cl, BF3 etc. Thus for symmetric tops Ia ≠ 0, Ib = Ic. A molecule of this type spinning about an-axis resembles a spinning top and hence derives its name as a symmetric top.
The molecules of this class are further subdivided as:
i. Prolate symmetric top:- where Ia < (Ib = Ic). eg. CH3Cl, CH3F, NH3 etc.
ii. Oblate symmetric top:- where Ia > (Ib = Ic). eg. BCl3, BF3 etc.
Spherical Tops: –
When a molecule has all the three moments of inertia identical, it is called a spherical top. eg. CH4, SF6, CCl4 etc. i.e. Ia = Ib = Ic. These molecules give no rotational spectrum as they do not have permanent dipole moment due to their symmetry.
Antisymmetric Tops:-
These molecules have all the three moments of inertia different. i.e. Ia ≠ Ib ≠ Ic. eg. H2O, CH3OH, CH2=CH-Cl. The majority of molecules belong to this group.
3. Rotational Spectra of Diatomic Molecules as Rigid Rotor:
In this model, consider the diatomic/linear molecule as a rigid rotator consisting of two atoms taken as point masses m1 and m2 tied at a distance ‘r’ by a weightless rigid rod as shown in the figure. Let the molecule rotates end-over-end (i.e. about an axis perpendicular to its molecular axis) about a point G, the centre of gravity.
Now, as in Classical Mechanics, the energy of rotation is given as:
The rotational energy levels allowed to this rigid diatomic molecule can be obtained by solving the Schrodinger Wave Equation. Its solution gives the energy eigen values and the eigen wavefunction as:
The allowed rotational energy levels for diatomic molecules (rigid rotator) are shown in figure below.
Comparing equation (10) with classical result of equation (5), one finds
Thus only certain discrete values of angular momentum are allowed for rigid rotator. Then it follows from equation (4) those only certain rotational frequencies, υrot. are possible.
4. Spectrum of the Rigid Rotator
According to Classical Electrodynamics, an oscillating dipole moment is associated with the emission of an electro- magnetic radiation of frequency equal to the frequency of oscillation. For the diatomic molecule consisting of unlike atoms, during the course of rotation, the component of a dipole in a fix direction changes periodically with a frequency equal to the rotational frequency i.e. classically, an e.m. radiation of frequency υrot. is to be emitted and for molecules consisting of two like atoms, no dipole moment arises and therefore no light is emitted. Further, as υrot. can have any value, so classically the absorbed or emitted spectrum should be continuous. However, according to Quantum Mechanics the rotator emits radiation during the course of transitions from a higher to a lower rotational energy level and the frequency of this radiation corresponds to the energy difference between these levels. The rotational spectra are generally considered in terms of wave number. The wave number of the emitted quantum photon due to transition from higher level (E’) to lower level (E”) is:
In order to calculate the frequency/ wave number of the emitted or absorbed lines, it is necessary to drive the selection rules for such transition and application of Quantum Mechanics shows that Selection Rules for Rotational Spectrum are:
The spectrum of rigid rotator thus consists of equidistant lines, the first line being at 2B (cm-1) which corresponds to J” = 0,
2nd line will be at 4B (cm-1) for J” = 1 and so on. Thus a series of spectral lines with wave number values 2B, 4B, 6B,….. (cm-1) is expected for the rotational spectrum of a rigid diatomic molecule. The separation between any two adjacent lines is 2B. The energy levels and the allowed transitions for emission spectrum are shown schematically.
Important points –
- The energy of ground state (J=0) is zero because the molecule is not rotating.
- As J increases the molecules rotate more and more quickly as a result the energy levels are more widely spaced apart
- The energy level separation are compatible with the microwave region of the electro magnetic spectrum
Energy Levels and allowed transitions
5. Intensity of Rotational Lines
The pure rotational spectrum of a diatomic molecule consists of a series of equally spaced lines corresponding to transitions J = 0 1, J = 1 2, J = 2 3, and so on. Although all such transitions (with ΔJ = ±1) are equally likely to occur, this doesn’t however, mean that all spectral lines will be equally intense. In fact, the line intensities will be directly proportional to the initial number of molecules present in each level (i.e. population of each level). The population of these levels is governed by Boltzmann Distribution Law, according to which if N0 is the number of molecules in J = 0 state then the number of molecules in the Jth level will be
Also each energy level is (2J +1) – fold degenerate. Thus taking degeneracy into account, the total (relative) population at the energy level EJ will be:
For maximum intensity, NJ should be maximum & to find the value of J for which the population is maximum, we put dNJ / dJ = 0. This leads to
That is, the level nearest to J value will have the maximum population. Hence, transition originating from the level having integral ‘J’ value which is nearest to ‘J’ value given by equation (23) will have the maximum intensity.
6. The Effect of Isotopic Substitution
When a particular atom in a molecule is replaced by its isotope, the resulting substance / molecule is identical chemically with the original molecule. In particular there is no appreciable change in inter-nuclear distance on isotopic substitution. There is, however, a change in total mass and hence in the moment of inertia (because I = mr2) and B value (because B = h / 8 π2 I c) for the molecule.
Consider the example of carbon monoxide (C12O16 ≡ CO). If 12C is replaced by 13C isotope, there is an increase in mass and hence a decrease in B value. If ‘B’ value for C13O16 molecule is designated by B´, then B´ < B (where ‘B’ is for C12O16 molecule). This change will be reflected in rotational energy levels of the molecule. As shown in the figure, the spectrum of the heavier species will show a smaller separation between the lines (2B’) than that of the lighter one (2B). The wave number υ for spectral line in normal molecule is υ = 2B (J+1) and the wave number for isotopically substituted molecule will be υ’ = 2B’ (J+1)
Hence the shift in the wave number due to isotopic substitution is:
7. Energy Levels and Spectrum for Non-rigid Rotator
In the spectrum of a rigid diatomic molecule, the spectral lines are equidistant with a separation of 2B between each two lines. But experimentally, it has been found that the separation between successive lines decreases steadily with increasing J values. The reason for this decrease is obviously due to the decrease in the B value. All the bonds are elastic to a certain extent and thus our assumption of a rigid bond is only an approximation. Plainly, the bond length increases with ‘J’. Thus, for a correct treatment, we need to consider the diatomic molecule as a non-rigid rotator. The Schrodinger wave equation may be set up for a non rigid rotator and the rotational energy levels are found to be:
From equation (29) it is clear that centrifugal distortion effects are greatest for molecules with small amounts of inertia (or μ) and small force constants. In a given molecule, the effect of the centrifugal distortion constant is to decrease the rotational energy which increases rapidly for higher rotational levels and thus has been shown in the figure:
The Selection rule for transitions for the case of non – rigid rotator is again ΔJ = ±1. The wave number of the transition J → J+1 is:
where υJ represents equally the upward transition from J to (J+1), or the downward transition from (J+1) to J. In equation (31), the first term is same as that of the rigid rotator and additional term gives the shift of lines from that of rigid molecule which increases with J as (J+1)3.
8. Techniques and Instrumentation for Microwave Spectroscopy
(Source:-https://www.slideshare.net/khemendrabhardwaz/rotational-spectra-microwave-spectroscopy)
The basic requirements for observing pure rotational spectra in absorption are a source of continuous radiation in the proper infrared region, a dispersive device and a detector. Radiation from the source is taken, which passes through the HCl vapour. The transmitting beam falls on a condensing mirror. The collimated beam passes through a rock salt prism and is brought to a focus at the thermal detector by means of a focusing mirror.
9. Applications of Rotational Spectra:
1 To determine the molecular structure in gas phase molecules.
2 When fine and hyperfine structure can be observed, the technique also provides information on electronic
structure of molecules
3 In connection with radio astronomy, the technique has a key role in exploration of chemical composition
of interstellar medium. Useful for studying how stars and planetary system born.
4 Rotational spectra of a molecule can be examined accurately at room temp.
Summary:
In general, the rotational properties of any molecule are expressed in terms of the moments of inertia about three perpendicular axes set in the molecule Ia,Ib,Ic (convention Ic≥Ib ≥Ia )
It is assumed that the molecules are rigid rotors ( that is ,no distortion under stress of rotation).
Spherical rotors: 3 equal moments of inertia (CH4, ???4, ??? ??6).
j2=square of the magnitude of the angular momentum, and we established that:
J2 = (J+ 1)ℏ2
J=0,1,2,………………….
Rotational energy levels of a spherical rotor are confined to:
Separation of adjacent levels is:Ej-Ej-1 =2BJ
Because B decrease as I increases, large molecules have closely spaced rotational energy levels.
Symmetric rotors: 2 equal moments of inertia( (??3, ??3??, ??? ??3?l ).
Linear rotors:1 moment of inertia (molecular axis) equal to zero ( ??2, ???, ???, ?? = ?H ).
nuclei treated as mass points
→rotation occurs only about an axis ⊥ to the line of atoms →zero angular momentum about the line
→similar to the case of a symmetrical rotor but with K=0.
Rotational energy levels of a linear rotor are confined to:
For a linear rotor: K=0 , but the angular momentum may still have 2J+1 components on the laboratory axis ,
so its degeneracy is 2J+1.
Asymmetric rotors:3 different moments of inertia (?2?, ?2??, ??3?).
Centrifugal distortion
Atom of rotating molecules are subject to centrifugal forces
- → distortion of the molecular geometry
- → stretching of the bond
- → moments of inertia increase
- → rotational constant decreases
- → energy levels are slightly closer than rigid predictions Treated empirically by subtracting a term from the energy :
EJ=BJ(J+1)-DJJ2(J+1)2
DJ=Centrifugal distortion constant =a measure of a bond stiffness . Large when the bond is easily stretched.
For a diatomic molecule:
Hence the observation of the convergence of the rotational levels as J increases can be interpreted in terms of the rigidity of the bond