THERMAL DISTRIBUTION IN THE HCL SPECTRUM OBJECTIVE

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1 ame: THERMAL DISTRIBUTIO I THE HCL SPECTRUM OBJECTIVE To nvestgate a system s thermal dstrbuton n dscrete states; specfcally, determne HCl gas temperature from the relatve occupatons of ts rotatonal states. PRE-LAB (to be completed before comng to lab) Pror to comng to lab, read through ths wrte-up and perform all the exercses labeled Pre-Lab. Durng the frst 5 mnutes of lab, I wll check that you have done them. Durng the course of the lab, you may fnd that some pre-labs were ncorrect correct them, for I wll grade the fnal versons (provded you d done respectably before lab). OVERVIEW We are gong to look at the nfrared adsorpton spectrum of HCl. The HCl s at some temperature; accordngly, the ndvdual molecules of the sample are statstcally dstrbuted through the dfferent energy states. In Fourer Transform Infrared (FTIR) Spectroscopy, the sample s rradated wth a narrow band of nfrared lght and ndvdual molecules are excted from ther ntal, thermally determned, states to some hgher energy states. Quantum mechancs dctates that these states be dscrete and that the transtons be between only specfc energy states, ths gves an almost one-to-one relatonshp between a transton energy (whch you can measure) and the ntal energy state (whch you want to know). The relatve ntenstes of lght adsorbed n the dfferent transtons reflect the relatve populartes of those transtons and thus the relatve populatons of the ntal states. As you know from Schroeder, ths, n turn, relates to the sample s temperature. Ths lab brngs together a couple of dfferent topcs. I summarze the physcs below. For more detals, you may wsh to revew Schroeder Appendx A, sectons 3 and 4, and Chapter 6, sectons and. HCL Vbratonal and Rotatonal States vbraton rotaton Imagne an HCl molecule as two balls joned by a sprng. Ths system has a number of degrees of freedom (the molecule wholesale translatng n the x, y, or z drecton, the bond bendng back and forth, etc.) We ll be focusng on just the rotaton of the molecule about an axs perpendcular to the sprng and on the vbraton of the sprng, but not ts bendng. Snce we re talkng about ndvdual molecules, quantum mechancs provdes the approprate

2 framework. Leavng ther full justfcaton to your Quantum Mechancs course, here are the pertnent Q.M. facts. The energes assocated wth these two knds of moton are j( j + ) E rot h () I and E vb ( ν + ) hω () where j s an nteger specfc to the molecule s angular momentum, I s HCl s moment of nerta (roughly ev s ), ν s an nteger specfc to the molecule s vbraton, and ω s the fundamental frequency of the molecule s vbraton (roughly rad/s). Transtons Between States Conservaton of angular momentum demands that when a photon nteracts wth a molecule, the molecule s angular momentum change such that ts j number changes by ±. Ths greatly lmts the possble molecular transtons. Schematcally, some of the allowed photon adsorpton transtons are shown below. The llustraton reflects a few expermental aspects: ) t s safe to assume all molecules wll ntally be n a vbratonal ground state (ν 0) snce sgnfcant thermal occupaton of ν corresponds to a temperature of a few thousand Kelvn, and I don t thnk I m gvng anythng away by sayng that your sample wont be that hot. ) The band of IR lght s n the rght range to only excte transtons between rotatonal states n adjacent vbratonal states,.e. ν +. E,4 j 4 Molecular knetc energy E,3 E, E, E,0 E 0,4 j 3 j j j 0 j 4 ν E 0,3 E 0, E 0, E 0,0 j 3 j j j 0 ν 0 Page?j -?j +

3 Transtons and Transmtted Lght Intensty Expermentally, when lght s shown on the HCL gas, some photons wth the energy necessary to nduce transton are adsorbed (and eventually reemtted n random drectons). Lookng at the spectrum of the lght passed through the gas, the ntensty data wll dsplay dps for the photon energes that are adsorbed,.e., for the changes n molecular energes: ε E E E. Relatng ths to j and ν, ph f h E hω + ( j + ) for?j +,?ν + (3) I or h E hω j for?j -,?ν + (4) I ote that each change n energy corresponds to a unque ntal state. The relatve depths of the adsorpton dps should be proportonal to the relatve populatons of those ntal states. One of your tasks wll be to dentfy the peaks wth the correspondng ntal energy states. The other wll be to relate the relatve populatons to the sample temperature. Temperature and State Populaton Ratos The Boltzmann dstrbuton functon relates the relatve (ratos of) populatons for energy states to the sample temperature: n ε ( j ) kt g ( j) e (5) where n s the populaton of state, s the total number of molecules n the sample, k s the Boltzmann constant, T s the temperature n K, g(j) s the degeneracy of the state ( g ( j ) j + for rotatonal states), and E s the energy of the state n queston. As an example, recall the statement that the ntal populaton n vbratonal state ν s neglgble at room temperature. I ll confrm that by calculatng the rato of HCl molecules n ν to those n ν 0. Wth the fundamental frequency of ω rad/s, ν ν 0 ν ν e ( E E ) ν ν 0 / kt 7 0 e hω / kt 0 e evs / s ev / K 300K In ths case, the degeneraces of vbratonal states are the same, g ν g ν0. (Ths wll not be the case when comparng rotatonal states!) So the rato s Or Page 3

4 flppng that, ths means that for every molecule n the ν state there are.0 x 0 6 molecules n the ν 0 state. What about the rotatonal state populatons? Usng the same method, we fnd that there are many dfferent j states populated at room temperature. Or, havng measured the rato of adsorpton ntenstes, thus the rato of populatons n ntal states, you can use ths relaton to solve for the temperature. Experment. Pre-Lab. Usng equatons () and (5), and the expresson for the degeneracy n terms of j, wrte an expresson for the theoretcal populaton of an arbtrary j state n rato to that of j n j 0, both n the vbratonal ground state, n 0:. Re-wrte the expresson so that t solves for T n terms of the rato of populatons, nj/n0. n 0 Page 4

5 T. Pre-Lab. Calculate the allowed adsorpton energy for the j +, ν+ transton out of the j, n 0 state. Snce the data wll actually be plotted as a functon of /λ, not energy, use Emolecule ε photon hc to calculate (/λ) that corresponds to λ ths energy. λ (0, 3) 3. In lab, label ths pont on your plot as ( 0, 3). It should be awfully close to a peak. 4. ow that you have a foot-hold, based on equaton 3), you can see that the next hgher energy (/λ) peak wll be ( 0,3 4 ), the Page 5

6 next lower wll be ( 0, ). Label the remanng peaks approprately. 5. Determne the temperature of your sample usng the relatve populatons (transton ntenstes) of two states of your choce. Repeat wth two other states. Report the average of these two determnatons. otes: ) Most of the ntal states have two allowed transtons, j + and j -; the ntal state populaton s proportonal t the sum of the two transton ntenstes. ) The data s gven as % transmsson, t s the % adsorpton that corresponds to the relatve populaton. Page 6

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