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1288 Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu

Geometry of Benzene from the Infrared Spectrum

Elisabetta Cané, Andrea Miani, and Agostino Trombetti*Dipartimento di Chimica Fisica e Inorganica, Università di Bologna,Viale Risorgimento 4, 40136, Bologna, Italy;*trombet@ms.fci.unibo.it

The rotation–vibration spectra of polyatomic moleculesdo not seem to be a favorite of most laboratory experimenttextbooks in physical chemistry (1–3). The reason is likelyto be the complexity of such spectra and, consequently, thenecessity for sophisticated experimental and theoretical toolsrarely available in student laboratories.

This experiment is designed for an undergraduate physicalchemistry laboratory that permits hands-on student use of aresearch-grade infrared spectrometer to determine the geometryof a polyatomic molecule. The experimental procedure issimple and fast. The one requirement is the availability of aninfrared spectrometer capable of 0.2-cm{1 resolution. Nospecial detectors or cells are needed for the experiment. Themathematical tools for the analysis are very simple, and fairlygood values for the interatomic distances are obtained.

The ν4 fundamental band of a2u symmetry (4) of benzene,a large polyatomic molecule according to the informal classi-fication of molecular spectroscopists, is observed in the IRspectrum at 674 cm{1. The ground-state rotational constantB0 can be easily obtained from the partial rotational analysis ofν4, if a spectrum with 0.2-cm{1 resolution is available. From theB0 rotational constants of C6H6 and another isotopomer, C6D6,the geometry of benzene in the ground state is derived.

Experimental Procedure

For this experiment a 0.18-m cylindrical glass cell, i.d.40 mm, fitted with KBr windows has been used. The cell isfilled with benzene vapor by dropping into it about 5 mg ofliquid C6H6 and C6D6, so the partial pressure of each isotopo-mer is about 7 mbar. C6H6 and C6D6 were purchased fromFisher. CAUTION: the filling must be done with great cautionin fume hoods with adequate ventilation because benzene ishighly flammable, toxic, and carcinogenic, and is a severe eyeirritant. Chemical-resistant gloves and protective glassesshould be worn.

Since the ν4 fundamental vibrations of C6D6 and C6H6are centered at 496 and 674 cm{1, respectively (5), the spectraof the two compounds can be recorded simultaneously. ABomem DA2 FT spectrometer, equipped with a KBr beamsplitter and a DTGS detector, was used to record the spectrumin the region 450–705 cm{1 at 0.2 cm{1 resolution. The timerequired to collect 100 interferograms was about 15 minutes.Pumping down the interferometer is not required because thelines of the ν2 band of CO2 and the rotational lines of H2Opresent in the region of interest do not interfere with theanalysis and can be used to calibrate the spectrum. Thespectrum is shown in Figure 1.

Before proceeding with the analysis of the rovibrationalbands it is necessary to record a second spectrum of thesample cell, after fluxing it with nitrogen for a few seconds

(in a fume hood with adequate ventilation) to remove mostof the benzene. In this spectrum the Q branches of ν4 bandswill be apparent allowing an accurate measurement of theband origin ∼ν0.

Theory

If benzene is assumed to have a regular hexagonal geom-etry, then as far as the rotational behavior is concerned, it isa symmetric rotor (6 ). In this kind of rotor two of the threeprincipal moments of inertia IA, IB, and IC, are equal. Inbenzene the unique moment of inertia is IC, the c-principalaxis coinciding with the C6 rotation. This kind of rotor istermed oblate symmetric.

The rotational energy of an oblate symmetric rotor (6 ),without taking into account the centrifugal distortion effects,is given by the equation:

Fv( J;K) = Bv J ( J + 1) + (Cv – Bv)K 2 (1)

where Bv and Cv are the rotational constants in wavenumber

Figure 1. The fundamental bands of C6H6 and C6D6. The P and Rbranches are clearly seen; the strong Q branches are off scale.

units, and the subscript v means that the value of the constantdepends on the vibrational state. J is the total rotational angularmomentum quantum number and K is the component of Jalong the figure axis, with possible values 0, ±1, ±2, …, ±J.

The rotational constants B and C are related to the prin-cipal moments of inertia :

B = h8π2cIb

; C = h8π2cI c

(2)

where h is Planck’s constant and c is the speed of light in vacuum.According to the selection rules (4 ), the allowed rovibra-

tional transitions in a symmetric rotor are of two types: (i)with ∆J = ±1,0 and ∆K = 0, where the transition moment isparallel to the principal axis of inertia c, and (ii) with ∆J =

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JChemEd.chem.wisc.edu • Vol. 76 No. 9 September 1999 • Journal of Chemical Education 1289

±1,0 and ∆K = ±1, where the transition moment is perpen-dicular to the principal axis c.

In the ν4 fundamental vibration of benzene the vibra-tional transition moment lies along the 6-fold axis and aparallel band is observed for both C6H6 and C6D6. As shownin Figure 2, the six carbon atoms move in phase out of themolecular plane as do the six H (D) atoms, but in the oppositedirection, and during the motion the electric dipole momentchanges along the C6 symmetry axis.

Parallel transitions of symmetric rotors may appear similarto the rovibrational bands of diatomic molecules with simpleR and P branches, with the addition of a strong central Qbranch (4 ). For the ∆J = ±1 selection rule (R and P branches,respectively) the wavenumbers of the transitions are given bythe equation

∼ν = ∼ν0 + (B1 + B0)m + (B1 – B0)m2 + ∆(C – B)K 2 (3)

where B1 = Bv4 = 1; B0 is the ground state rotational con-

stant; m = {J for the P branch (∆J = {1) and m = J + 1 for the Rbranch (∆J = +1); K ≤ |m|; ∆(C – B) = (C1 – B1) – (C0 – B0);and ∼ν0 is the band origin wavenumber.

The Q-branch transitions (∆J = 0; ∆K = 0) are given by∼ν = ∼ν0 + (B1 – B0)J ( J + 1) + ∆(C – B)K 2 (4)

For a fundamental vibration i the change of the rotationalconstants in passing from the ground to the vi = 1 state isvery small. Therefore transitions with the same J but differentK fall nearly at the same wavenumber in the P and R branches;and in the Q branch all transitions have approximately thesame wavenumber, since (B1 – B0) is usually very small.

For the ν4 band of C6H6, ∆(C – B) ≅ {1.31 × 10{4 cm{1

and (B1 – B0) ≅ {1.35 × 10{4 cm{1 (7 ). Therefore the ν4 has asimple and symmetrical shape about the band center P and R

structure, with a very strong Q branch for both isotopomers,as is shown in Figure 1. In the region around the Q branchare observed several weaker Q branches due to the hot-bands,i.e. transitions of type (νi + ν4 ) ← νi, where νi is a low-fre-quency vibration of benzene. These transitions are slightlydisplaced with respect to ν4 because of the anharmonicity.

The analysis of the rotational structure can be done bythe combination difference technique (8) or, as reported inref 9 for the analysis of the CO rotation–vibration spectrum,by a polynomial fitting to the measured R and P transitions.The J- numbering of the P and R transitions (shown in Fig.3) is obtained from a good estimation of ∼ν0 and ascertainingthat the approximately equal spacing between adjacent R-branch or P-branch lines is equal to 2B, where B = B0 = B1.

As shown above, the wavenumber of the line-like Qbranch is a good approximation to ∼ν0. In the spectrum ofFigure 1 the Q branches are too strong to allow the measureof the band origin. Greatly reducing the amount of benzeneby flushing the cell with nitrogen for a few seconds leads tovery narrow Q branches.

The data reported in Table 1 were obtained with thepeak-finder provided in the spectrometer software. Table 2lists the rotational constants B0 and Bv4

= 1, and ∼ν0 values forC6H6 and C6D6 obtained by a 2nd-degree polynomial fit-ting of data of Table 1. The values of B0 and Bv4

= 1 cal-culated with the combination difference technique (8) arethe same as those listed in Table 2, within the standard de-viation of the constants. The unknown interatomic distancesof benzene are easily derived from the principal moment ofinertia in the plane of the molecule:

IB = 3mC rC–C2 + 3mH(rC–C + rC–H)2 (5)

for C6H6, with a similar equation for C6D6. IB of C6H6 andC6D6 are obtained from the experimentally determined B0values. The difference of about 3 × 10{4 cm{1 between B valuesobtained in this analysis and those in the literature is due tothe neglect of the centrifugal distortion effect in the rota-tional energy expression. In fact, with the inclusion of theDJ centrifugal distortion constant (4 ) in the analysis resultsconsistent with the literature data are obtained. The rC–C andrC–H derived in this work and those taken from literature arereported in Table 2.

Further Considerations

These measurements can be analyzed further by studentswho have a specific interest in molecular spectroscopy, to ex-tend the the experiment by:

1. including the centrifugal distortion in the expressionof the rotational energy and determining DJ;

2. verifying the Redlich–Teller product rule, an isotopeeffect on the vibrational frequencies (4 );

3. measuring the intensity of ν4 and calculating the valueof the transition moment (10);

4. measuring the wavenumbers of the satellite Q branches(see Fig. 1), due to the hot bands, and determiningsome anharmonic constants (11).

Literature Cited

1. Shoemaker, D.P.; Garland, C. W.; Steinfeld, J. I.; Nibler, J. W.Experiments in Physical Chemistry; 4th ed.; McGraw-Hill: NewYork, 1981.

Figure 2. Normal mode ν4: black arrows indicate the displacementof the carbon atoms and dotted ones indicate the displacement ofthe hydrogen atoms out of the molecular plane.

Figure 3. J numbering of P and R branches of the ν4 parallel bandof C6H6 .

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1290 Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu

2. Matthews, G. P. Experimental Physical Chemistry; Clarendon:Oxford, 1985.

3. Sime, R. J. Physical Chemistry—Methods, Techniques, and Experi-ments, 1st ed.; Saunders: Philadelphia, 1990.

4. Herzberg, G. Infrared and Raman Spectra of Polyatomic Molecules;Van Nostrand: Princeton, NJ, 1945.

5. Cabana, A.; Bachand, J.; Giguere, J. Can. J. Phys. 1974, 52,1949–1955.

6. Levine, I. N. Physical Chemistry; 4th ed.; McGraw-Hill: Singapore,1995.

7. Hollenstein, H.; Piccirillo, S.; Quack, M.; Snels, M. Mol. Phys.1990, 71, 756–768.

8. Stafford, F. E.; Holt, C. W.; Paulson, G. L. J. Chem. Educ. 1963,40, 245–251.

9. Mina-Camilde, N.; Manzanares, C. I.; Caballero, J. F. J. Chem.Educ. 1996, 73, 804–807.

10. Bertie, J. E.; Keefe, C. D. J. Chem. Phys. 1994, 101, 4610–4616.11. Cané, E.; Miani, A.; Trombetti, A. J. Mol. Spectrosc. 1997, 183,

204–206.

oitatoR–noitarbiVlatnemirepxE.1elbaT n fosnoitisnarT ννννν4CfosdnaBlatnemadnuF 6H6 Cdna 6D6

C6H6 C6D6

J P ( J mc/) {1 J R ( J mc/) {1 J P ( J mc/) {1 J R ( J mc/) {1

21 93.966 6 26.676 32 69.884 31 95.00531 89.866 7 10.776 42 86.884 41 78.00591 17.666 8 83.776 52 23.884 51 91.10502 33.666 9 47.776 62 99.784 61 05.10512 59.566 01 31.876 72 66.784 71 58.10522 65.566 11 15.876 82 23.784 91 54.20532 71.566 21 88.876 92 40.784 02 77.20542 08.466 31 02.976 03 17.684 12 50.30552 24.466 41 36.976 13 24.684 22 24.30562 40.466 51 10.086 23 70.684 32 17.30572 56.366 61 83.086 33 57.584 42 99.30582 52.366 71 57.086 43 54.584 52 13.40592 88.266 81 21.186 53 21.584 62 26.40503 84.266 91 15.186 63 87.484 72 39.40513 11.266 02 68.186 73 64.484 82 22.50523 37.166 12 92.286 83 51.484 92 45.50533 33.166 22 36.286 93 58.384 03 58.50543 49.066 32 20.386 04 25.384 13 41.60553 55.066 42 83.386 14 81.384 23 24.60563 91.066 52 97.386 24 88.284 33 28.60573 87.956 62 21.486 34 55.284 43 70.70583 24.956 72 15.486 44 42.284 53 14.70593 10.956 82 78.486 54 19.184 63 07.70504 26.856 92 52.586 64 95.184 73 10.80514 32.856 03 85.586 74 72.184 83 13.80524 78.756 13 89.586 84 69.084 93 16.80534 54.756 23 53.686 94 06.084 04 29.80544 11.756 33 27.686 05 23.084 14 32.90554 76.656 43 90.786 15 79.974 24 25.90564 92.656 53 64.786 25 26.974 34 48.90574 09.556 63 28.786 35 13.974 44 41.01584 05.556 73 22.886 45 99.874 54 34.01505 37.456 83 95.886 55 66.874 64 47.01515 23.456 93 39.886 65 73.874 74 30.11525 79.356 04 13.986 75 40.874 84 63.11535 84.356 14 76.986 85 67.774 94 36.11545 71.356 24 40.096 95 04.774 05 89.11555 67.256 34 53.096 06 70.774 15 22.21565 63.256 44 87.096 16 57.674 25 45.21575 19.156 54 61.196 26 24.674 35 78.21585 06.156 64 94.196 36 21.674 45 61.31595 02.156 74 19.196 46 08.574 55 64.31506 18.056 84 42.296 56 34.574 65 77.31516 93.056 94 16.296 66 61.574 75 60.41526 30.056 05 69.296 76 58.474 85 63.41536 36.946 15 53.396 86 84.474 95 07.41546 32.946 25 66.396 96 31.474 26 65.51556 78.846 35 50.496 07 18.374 36 09.51566 25.846 45 24.496 17 15.374 46 61.61576 60.846 55 87.496 56 25.61586 36.746 65 61.596 66 97.61507 68.646 75 15.59617 54.646 85 19.59627 70.646 95 42.696

06 46.69616 49.69626 23.79636 76.79646 20.89656 83.89666 57.89676 51.99686 74.996

NOTE: The missing experimental transition lines are overlaid with CO2or H2O lines present in the region.

~ν0 /cm{1

dna,snigirOdnaB,stnatsnoClanoitatoR.2elbaTenezneBfosecnatsiDcimotaretnI

B0 mc/ {1 r C–C mp/ r H–C mp/

C6H6

)2(34981.0 )3(03981.0 )4(679.376 )1(1.931 )6(4.801

)2(5377981.0 a )2(4836981.0 a )3(56479.376 a )2(46.931 b )1(3.801 b

C6D6

)3(97651.0 )3(07651.0 )4(132.694 — —

)2(50751.0 b )2(69651.0 b )1(412.694 b — —

NOTE: Numbers in parentheses are standard deviations in units ofthe last digit.

aFrom ref 7. bFrom ref 5.

B v4 = 1/cm{1