A diatomic molecule differs from an atom in having additional degrees of freedom caused by
the ability of the molecule to rotate and vibrate about its center of mass. These motions
can be, in first approximation, considered separately and lead to sets of energy levels,
, but with relatively small values of 
.
Since = h
, the frequencies of
electromagnetic radiation absorbed or emitted by a transition between two of these levels will lie
in the infrared (frequency (nu) > 1013Hz, wavelength (lambda) < 20 
m)
or microwave regions ( < 1011Hz, lambda > 20m) of the
electromagnetic spectrum.
The simplest models for the motion of a diatomic molecule in its electronic ground state are the harmonic oscillator which describes the vibrations of the two atoms along the bond direction and the rigid rotor which approximates the rotation of the molecule in space. Since a diatomic molecule is under investigation, only one bond can vibrate and the rotation of the molecule occurs only about an axis perpendicular to the bond direction and through the center of mass.
Application of Quantum Mechanics to the Simple Harmonic Oscillator (SHO) leads to the energy levels for the vibration of the chemical bond:
where v is a quantum number having only the values v=0,1,2,..., and h is Planck's constant.
is the vibrational frequency (in Hz) given by
where is the reduced mass of the diatomic molecule, equal to
(m1m2)/(m1+m2) (of the order of 10-27kg for HCl),
and k is the force constant of the bond in units of Nm-1 or dynes cm-1.
Note that the heavier the molecule and/or the weaker the bond, the smaller the vibrational frequency.
The vibrational frequency is often expressed in units of cm-1, rather than Hz, through the relation
c =
where c is the velocity of light (3 x 1010 cm s-1). This scaling makes the
vibrational frequency of most molecules a convenient number in units of cm-1, or wavenumbers, i.e.
varies from about 100 to 4,000 cm-1. (The molecule of concern here,
HCl, has ca. 3000 cm-1.) Note from Eq. (1) that the lowest energy level
of a harmonic oscillator has o= h/2 (the zero point
energy) and adjacent levels
are equally spaced with =hc.
Quantum Mechanics also provides the rotational eigen-energies of a rigid linear rotor
where J is a quantum numbercan take only the values 0,1,2,..., and I is the moment of inertia of the dumb-bell.
Here I=m1r12+m2r22 =
R2 where
is again the reduced mass, and R is the interatomic distance (=r1+r2).
If the rotational constant is defined by
B=h/(8
2I)
for the diatomic molecule. The rotational constant can be expressed in wavenumber units
= B/c in cm-1
and varies from about 0.05 to 20 cm-1
among diatomics (for HCl, ca. 11cm-1). From Eq. (3), one sees that the spacing between
adjacent levels for a rigid rotor is
(J)=2J'
where J' designates the upper rotational level. Thus, the rotational levels are not equally spaced,
and they are much closer together than the vibrational levels of the same molecule because of the relatively
small values of 2 relative to that of .
ca. 10 cm-1 in the two states, would be as shown in Fig. 1
(Transitions between levels indicated there will
be referred to later.). Each vibrational state has a set of rotational levels. These rotational levels are not
exactly the same since varies slightly with v, the quantum number
labeling the vibrational state. (This latter effect is called vibration-rotation coupling)
Figure 1: Transitions between the lowest vibrational state, denoted v=0, and the next highrest vibrational state (v=1). Each vibrational state has many rotational levels denoted by a rotational quantum number (J' for the upper vibrational state, and J" for the lower). The spacing between rotational levels of the same vibrational state has been greatly exagerrated for clarity.
Equations (1) and (3) may be combined and if expressed as Term Values of a given state and level of the molecule as labelled by v and J, T(v,J) [in units of cm-1 then
However, these are only a first approximation to the true Term Values since there are small additional terms to be added to account for departures from our simplest models of vibration and rotation:
Here
exe is a small correction for
departure from harmonic
vibrational motion (i.e., a non-parabolic potential energy). This correction is
termed the anharmonicity of the bond. Anharmonicity must occur in all chemical bonds since at a
certain energy all chemical bonds break.
e is due to the interaction between rotation and vibration;
since R changes during the vibrational motion, so does I and therefore .
Quantum mechanics also stipulates that only the certain transitions shown
in Fig. 1 are "allowed" via electric dipole radiation. The selection rules for this process
are such that both v and J change by 1 quantum in allowed transitions.
(v=+/-1, J=+/-1) Only molecules with permanent electric dipole moments
may exhibit infrared absorption, v=+1,or emission, v=-1.
J=0 is allowed only for electronic transitions in some diatomic molecules and non-linear
polyatomic molecules. Traditionally, the J transitions are grouped together as branches
(see Fig. 1):
J = + 1, R branch
J = - 1, P branch
J = 0, Q branch (forbidden here)
To a first approximation the length of the arrows in Fig. 1
(proportional to the frequency of the transition)
differ by 2 among the P branch transitions and among the R branch transitions,
but with 4 between the central P(1) and R(0) lines.
If the forbidden Q branch transition v=0, J"=0 to v=1, J'=0 (call it o)
were included in the Figure then all spacings would be 2.
If the very small term involving De is neglected in Eq. (4), the combined rotation-vibration
transitions in Fig. 1 can be written
For least squares fitting the P and R equations can be conveniently combined by letting m=(J"+1) in the R branch and (m = -J") in the P branch, yielding
(Note that there is no value corresponding to o (m=0) for this molecule.)
For example, the transitions in Fig. 1 can be assigned values of m from -5 to +5 (with 0 missing) and a plot
of (m) versus m will yield e
and e. Then by use of Eq. (7) one can also find o.
A typical spectrum is given in Fig. 2.
As illustrated in Fig. 2, all absorption lines are not expected to have the same intensity. The relative intensities are governed largely by the populations of molecules occupying the various rotational levels in the ground v=0 vibrational state (where the rotational levels are designated by J"):
where
"Rot=hceJ"(J"+1).
This is the Boltzmann distribution, where the factor 2J"+1 corresponds to the degeneracy gJ" of a given rotational level. One can see that there would be an exponential decrease in intensity with increasing J" except for the partially compensating factor of gJ" causing a maximum in the relative intensities.
Both the vibrational frequency , (Eq. (2)), and the rotational constant
(through the moment of inertia, I) are affected by a change in the reduced mass
of the molecule. Changing the mass by the introduction of an isotope, such as D for H in HCl, will not change
the force constant k or the equilibrium bond distance Re since the character of the chemical bond
is unaffected. Then from Eq. (2).
Note that Eq. (9) is a first approximation, anharmonicity will change the calculated ratio slightly.
HCl (or DCl) prepared in the laboratory already contains two kinds of isotopomers, H35Cl
and H37Cl, since the natural abundance of isotopes of chlorine is 75.5% 35Cl
and 24.5% 37Cl, but the isotope shift between the chlorine isotopes is much smaller than that
induced by deuteration as can be seen by the much smaller reduced mass ratio of
(H37Cl) / (H35Cl)
relative to (D35Cl) / (H35Cl) (which is
approximately 2).
Since the two dominant parameters, and , in Eq. (4) are different
for the different isotopes,
one can expect a different spectrum for each isotopomer. One can, in first approximation,
calculate the shifts involved from Eqs. (9) and (10). In the case of a mixture of H35Cl
and H37Cl (or D35Cl and D37Cl), the chlorine isotope effect is small
and the spectrometer used in the measurements may have insufficient resolution to detect the
small splittings ( <1cm-1) in the absorption lines in Fig. 2 due to this effect.
Two infrared gas cells are provided with KBr windows, one each for HCl and DCl. Needless to say, the windows are sensitive to moisture so that they should not be touched, and the cells should be kept in a dry environment. These must be filled with each gas to an appropriate pressure to obtain the best infrared spectrum. (Why does the pressure matter?) Rapid scans of the Fourier transform IR spectrometer at various pressures will establish the optimum pressure; then record the spectrum slowly to obtain the best S/N ratio you can in the time permitted.
The reaction used is
D2SO4(l) + 2KCl(s) -> 2DCl(g) + K2SO4(s)
and the apparatus for the preparation of DCl and for filling the gas cell is shown in Fig. 3.
The procedure is as follows:
About 6 g of solid KCl is placed in the 125 ml flask and 3 ml of D2SO4 (from a small ampoule) placed in the dropping funnel. Stopcocks 1 and 2 remain closed, but 3, 4, 5 and 6 are opened, and the system pumped out through the stopcock 6. Check for leaks. If the system is tight, close stopcock 6 and add the D2SO4 dropwise, with constant stirring, until the pressure gauge reads the desired pressure. Record the pressure and close stopcocks 4 and 5 to the gas cell. [Note: do not add all of the liquid D2SO4 since the air could enter the system through stopcock 2]. Remove the gas cell and run the infrared spectrum to see whether the DCl pressure is satisfactory to obtain an optimum spectrum.
Two gas cells, 10 cm long with 32 x 3 mm KBr windows. Nicolet 5 PC Fourier transform infrared spectrometer, 2 cm-1 resolution.
The advantages of FTIR spectroscopy are (1) large energy throughput relative to the small slits and small apertures of older conventional monochromators. (2) obtaining data at all frequencies simultaneously (multiplex advantage) instead of scanning the frequency range. These advantages result in the ability to obtain an infrared spectrum in the range of 500 to 4,000 cm-1 with a resolution of 6 cm-1 in less than 1 second of instrument time, where a comparable spectrum by older scanning methods would require perhaps 10 minutes. In FTIR an interferogram is obtained which is then converted by a Fourier transform to intensity versus frequency, i.e., to an infrared spectrum as we usually view it. (It is understood that the interferogram contains all the information contained in the spectrum but in a different form.) The conversion of the interferogram via the Fourier transform is rapidly done by computer. The integral to be evaluated is
where S() is the spectral function and I(x) the interferogram function.
The interferogram may be obtained by the Michelson interferometer depicted in Fig. (4).
Radiation from a hot source (globar) falls on a beam splitter which allows half of the energy to pass to the moveable mirror and half to the fixed mirror. If no sample is present, the two reflected beams then recombine and interference of the two waves leads to the interferogram measured by the detector. The moveable mirror, moving constantly back and forth through a displacement x, leads to constructive and destructive interference of the two waves. A continuous broad-band source, containing many closely-spaced frequencies, will yield a strong pattern at x=0 where all waves are in phase, but the signals will decay rapidly as the mirror displacement changes and the various waves destructively interfer. A sample placed between the interferometer and the detector absorbs some of the frequencies present, thereby modifying the interferogram. We note here also that the larger the displacement that the mirror can achieve, the higher the resolution, i.e., the better the ability of the instrument to discriminate between closely lying absorptions. However, there are practical limits to the extensive motion of the mirror.
This has, of course, been only a brief introduction to the field of Fourier transform spectroscopy which has produced dramatic improvements in infrared, Raman, and particularly nuclear magnetic resonance (NMR) spectrometry. A few references for additional reading are listed below.
(cm-1) as accurately as possible,
tabulate with the corresponding m values, and calculate (m).
e-3e) and (2e),
and then e and e.
o from Eq. (7).
o(DCl)/o(HCl)
and e (DCl)/ e(HCl) and
compare them with the "harmonic" values given by Eqs. (9) and (10).
1. P.W. Atkins, Physical Chemistry, (5th edition), Freeman, NY, 1994, pages 574-6.
2. J.D. Graybeal, Molecular Spectroscopy, McGraw-Hill, NY, 1988, Chapter 12, page 332ff.
3. The authoritative sources for the spectroscopy of all diatomic molecules (up to about 1979) are the two books: G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, (2nd edition), Van Nostrand, New York, 1950 and K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979.
4. P.R. Griffiths, Fourier Transform, Infrared Spectrometry, Wiley, NY, 1986.
5. L. Glasser, J. Chem. Educ., 64, A228, 260 (1987).
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© 1996 // Innovative Teaching Labs // 18.9.1996