Infrared spectroscopy of acetylene

Transcription

1 Infrared spectroscopy of acetylene by Dr. G. Bradley Armen Department of Physcs and Astronomy 401 Nelsen Physcs Buldng The Unversty of Tennessee Knoxvlle, Tennessee Copyrght Aprl 007 by George Bradley Armen* *All rghts are reserved. No part of ths publcaton may be reproduced or transmtted n any form or by any means, electronc or mechancal, ncludng photocopy, recordng, or any nformaton storage or retreval system, wthout permsson n wrtng from the author. A: INTRODUCTION Molecules form an nterestng brdge between larger, classcal objects and smaller systems such as atoms. Because of ther sze and structure, we can observe motons we have a classcally ntutve noton of n the quantum regme. In partcular we ll get some nsght nto rotatonal and vbratonal motons at the quantum level. In ths laboratory we wll study the nfrared (IR) absorpton of acetylene molecules, relatng the observed spectrum to the molecule s structure. Furthermore, we ll see very drectly the consequences of Boltzmann s thermal dstrbuton, whch s fundamental to many areas of physcs. To explore all ths, we ll make use of an nterestng and mportant laboratory tool: the Fourer transform nfrared (FTIR) spectrometer. As physcsts, the spectrometer s of nterest to us smply because of ts nature: It s a drect applcaton of the famous Mchelson nterferometer (of ntroductory physcs courses) to very practcal problems. Whle the FTIR s a standard laboratory tem, one should apprecate all of the very dffcult techncal problems that needed soluton before t could become an off-the-shelf pece of equpment. In ths laboratory you should come away wth 1.) Some of the basc deas of molecular spectroscopy n the IR regme.

2 .) Experence n usng, and an understandng of, the FTIR spectrometer. 3.) Some detaled understandng of H C moton at the quantum level, and therefore 4.) an ncreased knowledge of molecular physcs, and maybe some ncreased nsght nto quantum mechancs n general. B: THEORY Ths secton outlnes some of the bascs of molecular physcs. Molecular structure and dynamcs s an mportant branch of physcs, although n recent years the subject has been preempted by the physcal chemsts. It s mportant to us because t hghlghts some very basc and fundamental quantum mechancs. In partcular, descrbng vbratonal moton provdes a good ntroducton to the basc concepts behnd feld theory. Addtonally, rotatonal spectra demonstrate n a very drect way the statstcal nature of quantum systems n equlbrum at room temperature. The followng dscusson s ntended to gve us a basc apprecaton of what we re seeng n (and practcal help n analyzng) the acetylene FTIR spectra. 1. Overvew The theory of molecular structure begns by a separaton of motons occurrng on very dfferent tme scales. To solve for the quantum structure of a molecule, one begns by specfyng the coordnates 1 for each nucleus ( R =1,.. N ) and each electron ( r =1,.. Ne ). The Hamltonan can be wrtten as n H = K + V V. nuc nuc nuc + Kel + Vnuc el + el el The nuclear Hamltonan K + contans the knetc and nuclear-nuclear Coulomb nuc V nuc nuc repulson energes. The remanng terms descrbe electronc moton: Knetc ( K ), el 1 We should also nclude spn coordnates to be complete. We ll do so later when t s needed.

3 electron-nucle Coulomb attracton ( Vnuc el ), and electron-electron Coulomb repulson ( Vel el ). It quckly becomes obvous that, snce the nucle are so much more massve than the electrons, a reasonable approxmaton (Born-Oppenhemer) s to frst consder the nucle fxed n some confguraton R. We can then solve for the moton of the speedy electrons. The resultng electronc wave functon ψ el depends on the nuclear coordnates parametrcally. The total wave functon s thus approxmated by the separaton ψ = ψ R, t) ψ ( r, t : R ), T nuc ( el j where ψ nuc s the wave functon descrbng the actual nuclear moton. Wth ths separaton we can fnd statonary electronc egenstates whose energes also depend parametrcally on the R : H ψ el el, n H el ( rj : R ) = En( R ) ψ el, K el + V el el + V nuc el n ( r ( R ) j : R ). Now, usng the Born-Oppenhemer scheme we assume that as the nucle move,.e. R R + δr, the electrons n ther egenstate ψ ( r : R ) nstantly accommodate to el, n j the new nuclear confguraton, remanng n the same egenstate level n: ψ r : R ) ψ ( r : R + δr ). el (, n j el, n j Ths s the adabatc approxmaton, referrng to the evoluton of a wave functon undergong slow changes n ts potental. The electronc energy En( R ) appears as an added potental n the nuclear equaton of moton ( Knuc + Vnuc nuc + En ) ψ nuc = ψ nuc. t Now we can solve for the dynamcs of the nuclear moton gven that the electronc moton s constraned to a certan statonary state n. The electronc state of the molecule profoundly affects the nuclear moton. Fgure 1 llustrates a typcal example:

4 Fg 1. Effectve nuclear potentals for two cases of electronc confguraton. The fgure sketches a plot of the nuclear potental energy for a datomc molecule as a functon of the nuclear separaton R. Wth the electrons n a bondng state,.e. localzed between the two nucle, they can screen the nuclear-nuclear repulson, lowerng the energy. However, at smaller separatons the repulson wns out. The potental thus has a mnmum at a separaton R mn. If the nuclear moton sn t too energetc, t can exst n a stable confguraton around R mn. On the other hand, the electrons can form an antbondng state (perhaps localzed away from the nter-nuclear regon), the screenng fals and there s no mnmum. To fnd statonary states of the molecule we can now fnd egenfunctons of the nuclear wave functon. Nuclear moton can (deally) be separated nto two types: rotatonal n whch the molecule rotates rgdly about ts center of mass, and vbratonal n whch the relatve nuclear confguraton oscllates. For molecules these two types of moton also occur on very dfferent tme scales and we can assume another Born- Oppenhemer-lke separaton ψ = ψ ψ. nuc vb rot In the next two subsectons we ll explore each of these separated motons. However, t s always mportant to keep n mnd these are approxmate wave functons. At the fundamental level all motons are correlated to some degree: the electronc confguraton certanly dctates nuclear moton, but the reverse s also true. Lkewse, the state of vbraton of the nucle affects the molecular moment of nerta, and so the rotatonal

5 moton. None-the-less, f we confne ourselves to low-lyng states the separaton scheme works qute well.. Vbratonal moton The molecular theory of vbratonal moton s the quantum extenson of our famlar classcal dynamcs of small oscllatons. Whle the soluton for datomc molecules s straghtforward, for more complcated, polyatomc molecules such as acetylene the problem s best attacked usng the method of normal modes. Consder a molecule consstng of N atoms. Suppose t s n some specfc electronc bondng state that supports a stable confguraton of the nucle that s there s a potental mnmum when the nuclear confguraton s at R, mn, = 1,, N. Let s frst try to fnd a more sutable coordnate system for the problem. How many coordnates do we need? We wsh to nvestgate moton nvolvng only small dsplacements from equlbrum, excludng any collectve translatonal or rotatonal moton. There are ntally 3N degrees of freedom. To exclude translatonal moton we can gnore 3 of these (.e. the locaton of the center-of-mass). To exclude collectve rotatons we can exclude three more, say the three angles defnng a unque rotaton about the CM (except for lnear molecules such as acetylene, whch need only two angles). Thus for an N-atomc molecule we are free to choose a set of M = 3N 6 (or M = 3N 5 for lnear molecules) generalzed coordnates q ( = 1,..., M ) to descrbe vbratonal moton. For ths set of coordnates (some of whch may be angles) there s some equlbrum pont q, eq correspondng to the confguraton R = R, mn. The next step s to ntroduce coordnates that reflect moton about ths equlbrum pont, the most natural beng ζ = q q, eq. Now, for small enough dsplacements about ζ = 0, we can expand the potental n a Taylor seres, keepng only the lowest (quadratc) terms: M 1 V ( ζ ) V + ζ ζ, mn V, j, j j where The lnear terms are zero of course, snce the potental s a mnmum.

6 V, j V. ζ ζ j 0 The approxmaton s llustrated n Fg.. We must bear n mnd throughout what follows that the quadratc (harmonc) approxmaton wll only be vald (f at all) for the lowest lyng states of moton. Classcally, t s always possble to have an energy close enough to equlbrum for the approxmaton to be vald. Ths may not be the case quantum mechancally. Fg. Approxmatng the potental near equlbrum. Wth the harmonc approxmaton establshed, the classcal soluton proceeds lke clock work: Lagrange s equatons of moton result n a set of M coupled dfferental equatons for the tme dependence of the ζ. Dependng on the energy and ntal condtons, the functons ζ (t) need not be perodc. The dea of normal coordnates arses when we look for a new set of coordnates η whch are. To do so we look for collectve motons whch have a perodc tme dependence. t Wthout gong nto detals, the procedure nvolves substtutng e ω nto the equatons of moton and lookng for solutons. Ths leads to an egenvalue problem wth the result that there are M frequences ω (some possbly degenerate) assocated wth the collectve (normal) coordnates η = β k k a ( k ) ζ,

7 (k ) where the terms a are the egenvector assocated wth the egenvalue ω k, and β k s a scale parameter for the k th mode. In ths coordnate system the equatons of moton are uncoupled and the soluton s smply η ) ωkt k ( t) = ηk (0 e. Any moton the molecule can go through can be represented by a superposton of normal modes. Should the boundary condtons be such that only a sngle normal mode s present, the condton wll persst forever. To make all ths more concrete, let s examne the normal modes of vbraton for acetylene. Acetylene (C H ) has a lnear (classcal) equlbrum structure: Fg. 3 Equlbrum structure of acetylene. The energy stored n the trple bonds are what supply welders wth such a hot flame when acetylene burns wth oxygen. Snce there are N = 4 nucle, and the structure s lnear, there are M = = 7 normal modes of oscllaton. Two of these modes are doubly degenerate, gvng fve dstnct frequences. Table 1 outlnes the acetylene vbratonal modes. Note that the frequences of oscllaton are gven n cm -1. Ths s confusng but very common n spectroscopy, and now s as good a tme as any to sort t out. The wavenumber σ = 1/ λ s a tradtonal pseudo-unt of ether energy or frequency. The followng denttes help wth convertng wavenumbers nto the mpled unts. ω πν, ν = c / λ = cσ, E = ω = hν = hcσ. Here are some conversons between the dfferent quanttes: 1cm 1 9,979 Mhz ev.

8 mode comments frequency (cm -1 ) ω 1 Symmetrc CH stretchng ω Symmetrc CC stretchng ω 3 Ant-symmetrc CH stretchng ω 4 Ant-symmetrc bendng (crcles and crosses represent out-of and nto the paper respectvely) 61.9 ω 5 Symmetrc bendng Tab. 1 Normal modes of acetylene. Now, to go to the quantum mechancal descrpton of vbraton we need one more fact from classcal theory. Ths s that, by choosng normal coordnates for the problem, the vbratonal Hamltonan s transformed to the smple form M H vb = p = 1 + ω η, where p s the momentum conjugate to the coordnate η. To fnd the wave functon we replace the ( p, η ) wth approprate operators and solve Schrodnger s equaton. Snce H vb s just a sum of M ndependent harmonc oscllators, the vbratonal egenfunctons separate as ψ η ) = ψ ( η ) ψ ( η )... ψ ( ). ( η vb n1 1 n nm M

9 Here, each ψ ( η ) s the usual harmonc oscllator egenfuncton (of angular frequency n 1 ω ) wth quantum number n = 0,1,,. A vbratonal egenstate s specfed by the set { n } of M quantum numbers assocated wth each normal mode. The energy of such a state s M E{ n } = ω ( n + 1/ ) = 1. For a gven state, we lke to thnk of each mode as contanng n partcles, each of 3 energy ω. Ths s the orgn of our deas of phonons n solds (let M 10 ) or of photons n space (the normal modes are the Fourer components of the free classcal feld). The lowest vbratonal level s the ground state { 0,0,0,...0} wth a zero-pont vbratonal energy of E M 1 { 0} = ω = 1. Recall that the vbratonal energes depend on the ω whch are n turn derved from the potentals V, j. These of course depend on the electronc confguraton, and so the vbratonal spectra for dfferent electronc states of a molecule can be dfferent. (In ths lab we wll always be n the electronc ground state.) Recall also that we had some reservatons about the harmonc approxmaton. The egenstates ψ vb{ n } are approxmatons to the true state of affars. If we wsh to be more precse we can use these states as a bass for further calculatons, ncludng the hgher-order (anharmonc) terms as a perturbaton. We ntend to look at the absorpton spectrum of acetylene. Usng IR radaton we are assured that the molecules wll always be n the ground electronc ground. However, what knds of vbratonal ntal states wll there be? Boltzmann statstcs tells us that f the molecules are n equlbrum wth a heat reservor of temperature T, the probablty of fndng the molecule n a state of energy E, relatve to that of a state E s 3 P ( E) g( E) ( E E )/ kt = e P( E ) g( E ), 3 k here s of course Boltzmann s constant, not to be confused wth ts earler use as an ndex or wth ts tradtonal use as a wavevector n Appendx B.

10 where g(e) s the degeneracy of the state E. The probablty of a partcular vbratonal state, relatve to the ground state, s then P rel ( E { n }) = g( E{ n e }) nω / kt. -1 At room temperature (300 K) we have kt = ev 08.5 cm. Table shows the results for some of the lower energy vbratonal states of acetylene. We see that, apart from the ground state, only the lowest bendng modes have any apprecable populaton. The hgher-energy stretchng modes are essentally unpopulated. Vbratonal state n 1 n n 3 n 4 n 5 g energy (cm -1 ) P rel Tab. Relatve populatons of the lowest-excted vbratonal states (300 K). Wth all ths sad about molecular vbraton, we need only to understand the one vbratonal transton we wll be studyng. Ths s the transton from ground { 00000} to the state { 10100} n whch both the CH stretchng modes are excted by one quantum smultaneously. In the jargon of the feld ths s referred to as the ν 1 + ν 3 combnaton -1 lne. We choose ths transton to study snce ts energy s near = 6686 cm, about at the peak of our detector s response. The actual transton energy s, as you wll see, somewhat lower than that expected from our theory. Ths s due to the falure of our harmonc approxmaton, as mentoned prevously. We are also avodng the subject of vbratonal selecton rules, whch also break down for non-deal cases.

11 3. Rotaton moton Fnally, we nvestgate the nature of the rotatonal part of the wave functon ψ rot. In keepng wth our separaton-of-varables scheme, we assume the molecule to be n some specfc electronc and vbratonal state. Because rotaton s so much slower than vbraton, we approxmate the molecule to be n some averaged, fxed confguraton R (dependent on the partcular vbratonal state the molecule s n). Snce we re dealng wth lnear acetylene, excted only n ts stretchng modes, the problem of rotatonal moton s equvalent to that of a datomc molecule: We need only consder rotaton about the center of mass perpendcular to the molecular axs. The total 1 energy s all knetc ( K rot = Iω ) 4. Snce the molecular angular momentum s just L = Iω the rotatonal Hamltonan s 1 L I H rot =. We mmedately know the egenstates: they are the sphercal harmoncs Each rotatonal egenstate (quantum number J) thus has energy Y,. J M E( J ) = J ( J + 1), J = 0,1,, I and s ( J + 1) -fold degenerate due to the possble projectons L z = M wth ( M = J,, + J ). Furthermore, each egenstate has a specfc symmetry wth respect to a coordnate reflecton J Y Ω) = ( 1) Y ( ), J, M ( J, M Ω so that states of even J have even party, and of odd J have odd party. Ths symmetry wll turn out to mportant: an nverson Ω Ω n the rotatonal coordnates s equvalent to exchangng one carbon nucle wth the other, and smlarly for the hydrogen nucle. 4 Here, I s the moment of nerta about the CM and now ω denotes angular velocty!

12 That s all we need to know, however t s worth pontng out that I wll actually ncrease wth J (a non-rgd rotor) whch produces a small addtonal energy proportonal to [ J ( J + 1)]. We ll gnore ths, and any other hgher-order effects. Let s estmate the energy coeffcent of the rotatonal states of acetylene. To do so we frst need to calculate the moment of nerta C C H H I = ( M r + M r ). The rad of the carbon and hydrogen nucle (relatve to the CM) are easly establshed from the bond lengths of Fg. 3: masses, we know that r m and r m. For the C M =1.0 u and M = 1.0 u, where the atomc mass unt s C 7 1u = kg. As one mght expect, the result s pretty small H H 46 I kg m. Wth ths result, the rotatonal energy constant s then I J = ev 1.19 cm -1. We see that, unlke the vbratonal energes, the rotatonal energes are much less than kt at room temperature. The probablty of encounterng an acetylene molecule n an excted rotatonal state s therefore large, and our experment wll gve a drect vew of how these probabltes are dstrbuted. 5. Ro-vbratonal spectra. Consder a transton n whch a molecule n a lower ro-vbratonal state absorbs a photon and s excted to a hgher state. The total wave functon undergoes a change characterzed by the change n quantum numbers { n } J { n} J. The transton energy s just the dfference n energes of the two states:

13 hν = E,{ },{ } ( vb n E + J J + 1) J ( J + 1) vb n. I I Recall that for our ν 1 + ν 3 combnaton lne the acetylene molecule remans lnear. Thus, any dpole moment must be parallel to the molecular axs. In ths case the dpole selecton rules for IR absorpton are ΔJ = ± 1. Thus the ro-vbratonal spectra dvde naturally nto two branches (.e. sets of lnes) wth wavenumbers: R branch: J = J + 1 wth J 0. σ ( J ) = σ + B ( J + 1)( J + ) BJ ( J + 1). R vb P branch: J = J 1 wth J 1. σ ( J ) = σ + B ( J 1) J BJ ( J + 1). P vb We ve used the obvous shorthand σ = Δ hc and B = h / 8π ci (and smlarly for B ). Wth the defntons vb E vb / B B + B and δ B B, a lttle algebra allows us to rewrte the transtons n the form σ ( J ) = σ R σ ( J ) = σ P vb vb + B( J + 1) + δ ( J + 1) BJ + δj for for J 0. J 1 Snce the upper (prmed) state s (by defnton) n a hgher vbratonal mode, we expect that I I and so δ s negatve and probably small. We see that the R-branch forms a seres of lnes (ascendng wth J) at frequences hgher than σ vb, the P-branch at lower frequences (descendng wth J), and there s no transton at σ vb. Furthermore, the spacng of adjacent lnes are

14 σ ( J + 1) σ ( J ) = B + (J + 3) δ R R for J 0 σ ( J ) σ ( J + 1) = B (J + 1) δ P P for J 1 σ (0) σ (1) = 4B. R P Snce δ s small, lnes for low J wll all have an approxmate spacng of B, wth a gap of twce ths between the hghest P-branch and the lowest R-branch lnes. Snce we expect δ 0, the spacng between adjacent P-branch lnes ncreases wth ncreasng J (decreasng frequency). Alternately, the R-branch lne-spacng decreases as J ncreases (frequency ncreasng). If we contnue to ncrease J to large values, the R-branch spacng would go to zero somewhere near Jmax B / δ, thereafter the seres actually turns around and can mx wth the P-branch. In our case δ s suffcently small (along wth our observable values of J) so that ths won t be a worry. Fgure 3 schematcally llustrates the basc lne confguraton. Fg.3 Ro-vbratonal lnes. There s a clever trck, called the method of combnaton dfferences, whch wll allow us to drectly extract the ground-state rotatonal constant B. One fnds (try t) σ ( J ) σ ( J + ) = (4J + 6 B. R P ) Once we ve assgned our J values to the two branches, we can take the approprate dfferences between a number lne pars to get a good measure of B. Two further, more obvous, relatons we wll use are σ ( J ) σ ( J + 1) = 4( J + 1 B and R P )

15 ( J + J + = + J + σ R ) σ P ( 1) σ vb δ ( 1). Next, consder the ntensty of the absorpton lnes. For the most part, these smply reflect the populatons of the ntal states so we can use Boltzmann statstcs to predct lne ntensty. However, there s one more complcaton to consder frst. Ths complcaton nvolves the degeneracy g(e) of our energy states and yet another degree of freedom we need to consder nuclear spns. Acetylene s made of carbon and hydrogen. Naturally occurrng carbon s almost all (98.9%) of the sotope 1 C whch has nuclear spn s = 0. Hydrogen 1 1 H on the other hand has a spn s = (Deuterum H s very rare n nature). The hydrogen spns on ether sde of the molecule have a slght nteracton we can gnore, except for the fact that any nteracton at all makes the total (coupled S = s 1 + s ) spn states the approprate nuclear-spn states to consder. As we all know, two spn ½ states couple to gve ether a snglet ( S = 0 ) or a trplet ( S = 1) total angular momentum state wth g ( S) = S + 1 degenerate egenstates of S z. As wth the rotatonal wave functons, these states of total nuclear spn (ns) also have a defnte party under exchange of spn coordnates: S + 1 ψ ns s, s ) = ( 1) ψ ( s, ), ; S ( 1 ns; S 1 s.e. the trplet s symmetrc under coordnate nterchange, and the snglet s antsymmetrc. Now we must apply quantum mechancs at ts most fundamental level. Snce the electrons and hydrogen nucle are all fermons (ntrnsc spn ½) the total molecular wave functon must be ant-symmetrc under nterchange of any par of ther coordnates. The ground electronc and vbratonal states are symmetrc functons, so the symmetry of the state s determned by the rotatonal and nuclear-spn degrees of freedom. Under nterchange of the hydrogen coordnates the total wave functon undergoes ψ ( 1) S + 1 ( J 1) ψ whch must be ant-symmetrc. Thus, ground molecular states of even J must have S = 0, and odd J states requre S = 1. Because of the extra nuclear-spn degeneracy g (S) odd-j lnes wll be about three tmes as ntense as adjacent even-j lnes. More precsely we can wrte the lne ntenstes as

16 P( J ) E( J )/ kt hcbj ( J + 1)/ kt = g( J ) e = (S + 1)(J + 1) e, wth the connecton between J and S mplct. Because both σ (J R ) and σ (J P ) orgnate from the ntal state J they should have about 5 the same ntensty. Lower-J states have a lower energy so the exponental term favors them, whle the degeneracy factor ncreases wth J. The trade off s a maxmum lne ntensty for J max, somewhere around kt J max hb. -1 For T = 300 K and B =1cm we have J max 08.5 / 10. Fgure 4 dsplays a synthetc spectrum wth energes and ntenstes calculated -1 from our dscusson. The parameters used n the calculaton are B =1.15 cm, -1 δ = 0.0 cm, ν vb = 0, and T = 300 K. One see s that both the P and R branches have a maxmum at about J = 10. Because δ 0, the R-branch s compressed at larger J whle the P-branch spreads out at lower frequences. Ths choce of parameters s not too far off from what we wll observe n the lab. Fgure 5 dsplays a close up of the spectrum around the center frequency. Ths wll provde a gude when assgnng J values to our actual spectrum. 5 The absorpton probabltes to the R and P branches are actually dfferent. Hence, the R and P branch ntenstes are dfferent, but only for low values of J. Ths s dscussed n appendx A.

17 σ (cm -1 ) Fg. 4 Calculated ro-vbratonal spectrum. C: PROCEEDURE Fg. 5 Detal of the spectrum near the center frequency. You should (of course) have studed ths wrte-up before comng to lab!

18 1. Acqurng data To start, spend some tme famlarzng yourself wth our FTIR spectrometer: Your lab nstructor wll gve you the basc tour and ntroducton to the machne. Addtonally, appendx C s ncluded as a quck reference to the MIR 8000 software. The FTIR needs to run for about an hour to stablze, so ths tme can be used n practce takng low-resoluton data and manpulatng t wth the spectral vewer. Once you re famlar wth the FTIR operaton, take a good spectrum of acetylene. A resoluton of 1 cm -1 s adequate for our purposes, averagng a total of 1000 to 1500 scans. Remove the acetylene cell and take a blank spectrum wth the same resoluton and total number of scans. It s a good dea to save data n several fles rather than a sngle (see appendx C). A total scan of 1500 passes wll take about 45 mnutes wth ths resoluton. Instead of surfng the web whle ths s gong on, you can begn the analyss part of the lab by preparng some of the Excel datasheets such as exercse III. Once both scans are completed, use the spectral vewer software to add up total acetylene and blank spectra. Dvde these spectra to get an acetylene transmsson spectrum and export the area of nterest 6 as a.txt fle. Copy ths to a floppy dsk and transfer to your own computer.. Analyss Import your text fle nto an Excel spreadsheet and plot the acetylene transmsson data n the range around our ν 1 + ν 3 combnaton lne. Adjust the scale of the graph so you can see the frst 0 or so lnes of both rotatonal branches. At ths pont you need to assgn J values to the lnes. Ths s not straghtforward, the oscllatng background can be confused wth the weak central lnes. The best way s to par up the most ntense lnes of the R and P branch (these wll be odd J) and work nwards, usng Fg. A1 as a gude. Locate the frst 0 peak postons of both the P and R branches. For our purposes, t s suffcent to smply locate the data pont of a peak wth maxmum ntensty. You can do ths easly by clckng on the graph and movng the ponter from data pont to data pont: a box wll appear wth each pont s x and y values. In your excel fle make 6.e. what s dsplayed currently n the spectral vewer. If you choose too wde a range the data fle wll be unmanageable n Excel.

19 columns of J, σ (J P ), and σ (J R ) values. We re now ready to extract the ro-vbratonal parameters by calculatng approprate combnaton dfferences. Begn by calculatng σ R ( J ) σ P ( J + )]/(4J + 6). If you ve assgned the J values correctly these quanttes should all be smlar and equal to the ground-state parameter B. Compute the average and standard devaton of these values ths s our expermental value and error for B. Repeat the procedure, ths tme calculatng [ σ R ( J ) σ P ( J + 1)]/ 4( J + 1). Ths gves us an expermental value and error for B = ( B + B ) /. Wth the values of B and B, fnd the values and errors 7 of B and δ. Notce that ths sn t a very accurate method for determnng δ. To overcome ths lmtaton, we use our fnal dfference relaton: Compute columns for ( J +1) and [ σ R( J ) + σ P( J + 1)]/. A plot of these quanttes should gve a reasonably straght lne. Fttng a straght lne to ths data gves us σ vb (the ntercept) and δ (the slope). You can do ths by addng a trend lne to the graph, however ths gves us no dea of the error. To get the standard error n the ft parameters, use the regresson functon under the excel tools/data analyss menu. Usng ths value and error of δ, go back and determne B agan. For your fnal results, choose the best values of B, B, and σ vb. 3. Exercses I. From your values of B and B determne the moments of nerta for the ground and excted states. Snce the ν 1 + ν 3 combnaton lne only exctes symmetrc and antsymmetrc CH stretchng modes, assume the CC separaton remans unchanged. Usng the ground-state confguraton of Fg. 3, determne the CH separaton n the excted vbratonal state. Recall that these separatons are the postons of the nucle averaged over the vbratonal moton. Now, n a harmonc potental, the average poston s the same for all states. Convnce yourself that the actual potental must be somethng along the lnes of the potental curve of Fg.,.e. openng out to larger separatons. (Sketch n excted wave functons). 7 If you don t know how to propagate errors, t s tme you should. Consult the UT Prmer on the subject located on the class web ste.

20 II. The frst correcton to the harmonc approxmaton s to nclude an extra term to the energy expresson for a vbratonal mode: E = ω ( n + 1 / ) + ωχ( n + 1/ ), where the parameter χ s called the anharmoncty constant of the th mode. For the Morse potental these are exact egenvalues. Usng your measured value of σ vb and the values n Table 1, what s the average χ = ( χ 1 + χ3) /? III. Make an excel spreadsheet that plots a synthetc ro-vbratonal spectrum as n Fg. 4. As parameters, use your measured values of B and δ. Make the temperature T an adjustable parameter, and calculate lnes up to at least J = 60. Don t worry about the relatve transton ampltudes dscussed n appendx A. Explore how the spectrum changes wth temperature. Prnt out spectra for T = 100, 300, and 900 K. REFERENCES FTIR: Fundamentals of Fourer Transform Infrared Spectroscopy, by B.C. Smth (CRC Press, 1996). Modern Fourer Transform Infrared Spectroscopy, by A.A. Chrsty, Y. Ozak, and V.G. Gregorou (Elsever, 001) [Comprehensve Analytcal Chemstry vol. 35] Some ntroductory physcs of molecules: Atoms and Molecules, by M. Wessbluth (Academc Press, 1978) Molecules and Radaton: An Introducton to Molecular Spectroscopy, by J.I. Stenfeld (The MIT Press, 1981) Advanced, ncludng detals of acetylene s normal modes: Molecular Spectra and Molecular Structure II. Infrared and Raman spectra of Polyatomc Molecules, by G. Herzberg (Van Nostrand Co., 1945)

21 APPENDIX A: Relatve absorpton strengths. Whle the ro-vbratonal lnes σ (J R ) and σ (J P ) both have ntenstes proportonal to the populaton of ther common ntal state J, to be more precse we should also consder ther absorpton probabltes. These depend on whether J J + 1 or J J 1. The treatment of ths problem rests on the algebra of angular momentum. Snce ths s a lttle more advanced and the physcs non-ntutve, we ve relegated the subject to ths bref appendx where the deas are smply outlned and results presented. For absorpton from an ntal state J to another state J, the transton probablty s proportonal to 1 W ( J J ) = JM R J M. g( J ) M, M Here, we calculate the dpole probabltes of each transton JM J M, summng over all fnal substates M and averagng over all ntal M substates. Ths double sum s called the lne strength of the transton. Usng some technques standard n atomc physcs, the summaton can be resolved to an M- and M -ndependent quantty: the reduce matrx element of the renormalzed sphercal harmonc of rank 1. In symbols 1. (J + 1) (1) W ( J J ) = J C J Whle these functons have an analytc form, all we care about s ther numercal values. In partcular, we want to determne the rato of ntenstes of the R and P branch lnes orgnatng from the J th ground-state rotatonal level. Snce the lne ntenstes are I R ( J ) = P( J ) W ( J J + 1) and I P ( J ) = P( J ) W ( J J 1), we see that the ntensty rato of the R-branch lne to the P-branch lne s smply

22 I I P ( J ) W ( J J + 1) = ( J ) W ( J J 1) R. Wth no further ado, we smply present the results: J I R ( J ) / I P ( J ) One sees that the effect des off rapdly and so, except for the lowest values of J, t s permssble to neglect the absorpton lne strengths as we have n Fg. 5. However, t s the low-j regon we use to dentfy the lnes. For ths purpose, we revse Fg. 5: Fg. A1 Low-J synthetc spectra, corrected for transton strengths.

23 APPENDIX B: Fourer transform spectrometers. A. Overvew Wth the advent of several technologes (partcularly computers) Fourertransform spectroscopy has become a practcal tool for studyng the transmsson propertes of substances n the IR regme. Fourer-transform nfrared (FTIR) spectrometers are standard tems n the laboratory, used extensvely for chemcal analyss. In general FTIR spectroscopy s much more effcent than more tradtonal spectrometers based on dspersve elements such as prsms or gratngs. Instead of dspersve optcs, FTIR s based on the use of an nterferometer. Fgure B1 below shows a block dagram outlnng the spectrometer s operaton. Before lookng at detals, let s consder the overall scheme of the spectrometer: Fg. B1. FTIR operaton. A Broadband IR source, often a tungsten-halde lamp, s shone through the sample under study. The source has a dstrbuton of ntensty over frequency I ( ), and after traversng the sample assumes a new dstrbuton I s ( ν ) = Ts ( ν ) I0( ν ). The sample transmsson T (ν ) s our quantty of nterest, related to the absorpton ( A =1 T ) and so s ultmately to the sample s photo-absorpton cross secton va Beer s law. The sample ntensty s drected nto the heart of the spectrometer, the nterferometer. Here, as we shall see, the entre spectral range of the sample ntensty s employed n the measurement. It s ths use of all frequency components smultaneously whch makes FTIR spectroscopy so effcent: In dspersve methods we sngle out a sngle frequency at a tme, throwng away all others for that measurement. 0 ν

24 The output of the nterferometer I s ( Δx) turns out to be the Fourer transform of I s (ν ). A computer s than used to nvert the transform. Ths s accomplshed usng the hghly effcent fast Fourer transform (FFT) algorthm wthout the FFT routne FTIR would be mpractcal. The sample transmsson s a combnaton of the transmsson of the sample cell and that of the substance we are actually nterested n, say T X, so that T s = TX Tcell. To get T X one takes two measurements, one of the sample and the other of a blank cell. If you ensure that these two measurements collect the same total rradance (e.g. same number of scans) then the substance s transmsson can be obtaned by numercally dvdng the two spectra T X ( ν ) = I s ( ν ) / Iblank ( ν ). Often ths procedure s useful smply to remove the frequency varaton of the source (and detector). B. How t works Let s now turn to the detals of the nterferometer. FTIRs employ some verson of the basc Mchelson-Morley nterferometer shown n Fg. B. Fg. B. Mchelson-Morley nterferometer. Consder a plane wave of ampltude 0 E and frequency ν enterng the nterferometer from the left. The wave propagates along a lnear path as

25 E( x, t) = Es cos( kx πνt), where x s the dstance traveled and the wavevector k = π / λ = πν / c 8. The average ntensty of ths wave s I s = E s. When the wave reaches the beam spltter t s separated nto two waves each wth ½ the orgnal ampltude; one s transmtted (blue) and the other (red) s reflected 9. The reflected wave travels upward untl t s reflected by the fxed mrror, fnally travelng down through the beam spltter and nto the detector. The transmtted wave travels on to the rght where t s reflected back by the movable mrror, then back off the beam spltter where t fnally recombnes wth the frst wave at the detector. The poston Δx of the movable mrror s measured wth respect to the pont x 0, the dstance at whch both mrrors are equdstant from the beam spltter. Consder the mrror set at some poston Δ x. The ampltude of the recombned wave at the detector (D) s Es E( D, t) = [ cos( kd πνt) + cos( kd + kδx πνt) ], the second wave havng traveled an extra dstance Δx. Ths dstance s called the optcal path dfference (OPD) whch we denote as, and can be ether postve or negatve. The sum of the two oscllatory terms must tself oscllate at the same frequency, so we can wrte n general that cos( α + ωt) + cos( β + ωt) = Acos( δ + ωt). Snce we are nterested only n the average ntensty of the recombned wave, whch s what the detector responds to, all we need fnd s A. Ths s easly accomplshed by representng the cosnes as phasors and then employng trgonometry to solve the vector sum. The result s A = + cos( β α). 8 Note that (the magntude of) the wavevector = π wavenumber, lke ω = π ν. Also, c and λ are really not the vacuum values, but those wthn the meda (.e. ar) fllng the spectrometer. 9 For smplcty, we gnore any phase changes assocated wth reflectance etc. real FTIRs nclude extra optcal elements to compensate for these effects.

26 The average ntensty at the detector s therefore I I s = [ 1 cos( k ) ]. D + Now, consder a broadband spectrum of waves enterng the nterferometer. The average ntensty of each such wave s I s (k). We use an ncoherent source (.e. an ncandescent lamp), whch means there s no steady phase relatonshp between waves of dfferent frequency. If ths s so, then the detector responds to the (ncoherent) sum of ntenstes of the ndvdual waves, so that for a gven mrror dsplacement the total detected ntensty s I D 1 1 ( ) = IT + I s ( k)cos( k ) dk, 0 where the total ntensty s I = I ( k) dk = I (0). T 0 s D Hence, f we scan the mrror back and forth around the zero poston we are essentally recordng the Fourer (cosne) transform of the ncdent ntensty dstrbuton. The wavenuumber k and the OPD naturally arse as a complmentary par of varables for the transformaton. FTIR spectra are usually output as a functon of k n cm -1. A small HeNe laser s used to accurately measure : The HeNe beam s smultaneously passed through the nterferometer and montored wth ts own detector. Countng nterference frnges as the mrror moves locates ts poston to wthn a laser wavelength ( 633 nm 0.6 μm ). Note that when = 0 waves of all frequences add constructvely, but for larger the ntegrand oscllates wth k and so the ntegral becomes small. The transformed data I ( ) D s therefore strongly peaked about = 0, and ths peak s often referred to as a center burst. However, the small-ampltude data away from the center burst provdes most of the nformaton needed to reconstruct the spectrum. Fgure B3 shows a typcal center-burst plot. The natve data plotted s somethng proportonal to f ) = I D ( ) I (0). ( D

27 To recover the frequency spectrum n terms of k, the nverse transform s computed usng the FFT algorthm. 0 k) = f ( )cos( k ) 0 I ( d mrror poston (counts) Fg. B3. Typcal natve data. Whle we have gnored most of the techncal detals, there are a few ssues basc to ths type of spectroscopy worth consderng. Frst, because and k are related as a Fourer transform par, there s an uncertanty relaton between them. If we want a hgh-resoluton spectra (to dstngush spectral features close n k), we need to be able to dstngush broad features n. Thus, the resoluton of an FTIR spectrometer s dependant on how large an OPD s achevable. -1 A spectrometer wth a maxmum OPD of cm can attan a resoluton of Δk = 0.5 cm. Secondly, our ablty to measure short wavelengths (or hgh energes) s lmted to how small a Δ can be acheved. The Nyqust samplng theorem states that to reconstruct a sampled wave, we must sample t at a frequency at least twce that of the wave. If the mrror poston can be located to half of the 0.6 μm HeNe wavelength, then the shortest wavelength we can hope acheve s 0.6 μm. Ths s why FT spectroscopy s generally lmted to the IR regon.

28 APPENDIX C: MIRMAT 8000 FTIR Here s a pctoral outlne of usng the MIRMAT software. We wll use the program for two separate tasks: A) acqurng and storng an FTIR spectrum and B) manpulatng these spectra. The program s a lttle puzzlng wth some (what I thnk are) bugs. A) Acqurng data Make sure the FTIR power s on. On the computer desktop, clck on the MIRMAT 8000 con to launch the program. The openng screen wll appear: Openng screen Double clck on the MIRMAT start opton. A box wll appear askng f you want to ntalze the nstrument. Clck yes. The setup screen then appears. There are only three settngs we need monkey wth: Detector: Choose InGaAsP. Resoluton: 64 cm -1 s fast for practce, and 1 cm -1 s enough for resolvng the H C lnes. Savng Method: Ths determnes how the data s stored. If you are just foolng around, leave the settng at Prompt Me. Otherwse choose Loop Mode. A box wll appear tellng you t has reset the trgger mode.

29 Setup screen Once, you ve made your choces, clck on accept whch opens the data screen: Data screen If you are foolng around (prompt mode) just leave all the settngs and clck fresh start. If you are acqurng data for real (loop mode) you need to gve some nformaton: In ths mode the spectrometer adds each scan to the last up to the number specfed by Coadd# of scans (n ths example 51 scans). Ths result s then saved to fle n a bnary format (.mat). Then the process s repeated. The Number of Loops can be adjusted to control ths (here 3 loops). Each of these fles wll be placed n the drectory you choose (here n C:\BRAD). The drectory must already exst. (t s easest to make

30 your own n the root drectory). Each loop of data wll be stored wth the Fle Name you specfy, appended by the loop number,.e. here CH_1.mat, CH_.mat, and CH_3.mat. If you want, you can also fll n the comment lnes, however ths won t be of much use to us. When you are ready to scan, clck on Fresh Start. The program wll begn acqurng data: Fast acquston screen In the loop mode, the actual nterferometer data s dsplayed n real tme, but the Fourer transformed spectrum s not. In ths example we ve already taken 14 scans of the frst (0 th ) loop. If at any tme you want to look at the spectrum acqured, you must clck the nterrupt button whch halts the acquston. If you then clck auto scale Y you should see somethng lke Halted acquston

31 To contnue takng data clck contnue. If you want to explore the spectrum n more detal, once you have nterrupted the acquston, clck on the lower SPV (spectral vewer) button: Spectral vewer Wthn the spectral vewer you can look at varous portons of the spectrum, change unts and manpulate data. (Next secton). To contnue takng data, clck close and then contnue on the acquston screen. When the fnal loop fnshes, the program dles and you can ext. Notce the absorpton band around 7000 cm-1 n the above example. Ths s due to water vapor the spectrum here s a blank. B) Manpulatng spectra. Once you ve collected data fles you wll need to manpulate them n several ways. Ths s accomplshed usng the spectral vewer. The spectral vewer allows you to read n.mat format spectrum fles, store them n varous temporary regsters, perform mathematcal functons on ndvdual spectra (such as takng the logrthm) and on pars of spectra (such as addng and dvdng). To get good data, we wll want to add up separate data fles, and to get rd of the source/detector varatons we wll want to dvde the total acetylene spectrum by the blank spectrum. From the MIRMAT openng screen, clck on SPVewer. The openng screen wll appear showng a generc center burst. To load a prevously-saved.mat fle, clck on the Load button. A box wll appear, and you can choose the fle you want to load (the browse button s easest). A prmtve.mat fle contans a lot of nformaton. We want the plot of the resultng spectrum verses wavenumber.

32 Openng screen Loadng a fle to vew To get ths data, we need to tell SPV what to plot. To do so, clck on wav (for wavenumber) n the fle contents box, then on the arrow button drectng the data nto the x axs. Then smlarly drect the spec data nto the y axs. If you are dealng wth.mat fles that have been manpulated and resaved, the choces of data to plot wll be x and y nstead of wav and spec. Once you are ready clck OK. You wll then see the data dsplayed as:

33 Pushng to stack A There are 6 memory stacks or regsters n whch spectra can be stored. The above fgure shows the data beng pushed nto stack A; smply pull down the Mem Stack selecton arrow and clck on the stack you want. Now you are free to load another fle wthout loosng the current data. The screen below shows the stuaton n whch there are two stacks loaded, A and B: You can tell whch stacks are loaded from the buttons n the lower left corner stacks that have somethng n them are n bold face type. Pushng on any of these buttons dsplays the contents. Summng two spectra Now, to operate on the stored data clck on the Math button. The left hand wndow wll open. Important to us s the Functon(A?B) button. push t and the lower wndow opens. Ths wndow allows us to perform operatons on two spectra, savng the

34 result n another stack. In ths example we re addng spectrum A to spectrum B and savng the result to stack C. Note here the notch near 6500 cm -1, whch s due to acetylene absorpton. Now, when you have somethng you want to work wth you need to export t as a text fle. We need to make ths fle as small as possble, so use the x-axs boxes to reduce ncrease the scale around your area of nterest. In the example below, we re lookng at the reduced range of 6000 to 8000 cm-1. (You can now see the R and P branches of the acetylene transtons.) For your transmsson data, only export data regon that ncludes the R and P branches there are a lot of data ponts here, especally usng hgh resoluton. To export what s dsplayed as a text fle, clck Export. A typcal fle-handlng wndow wll open and you can enter a fle name. The only trck s that you are not offered a.txt choce of fle type. However, f you smply nclude the extenson wth the fle name the program knows what to do. For more nformaton (but not much) you can consult the MIR 8000 manual.