Experimental Verification of a Boltzmann Distribution An Experiment in Spectroscopy and Theoretical Spectrum Simulation Mark Sulkes and Jean E. Osbum Tulane University, New Orleans, LA 701 18 The experiment described provides, as one of its main aims (I),spectroscopic verification of the Boltzmann distrihution. here for rotational deerees of freedom., in easeous HC1 and 12. To appreciate how the spectrum of a diatomic molecule gives information on rotational (and vibrational) temperature, it is necessary first that students understand the spectroscopy of a molecule suchas HCI. This can he done as a separate preliminary experiment or as an adjunct to the one herein described. An equally important goal is (2) to demonstrate that it is in fact possible to simulate spectra theoretically and to compare them with experimental results in order to test the theoretical assumptions; given knowledge of the selection rules for rotational transitions and assumption of the Boltzmann distrihution, i t is rotational temperature that was determined for goal 1without computer simulation. For the HCI spectra, a case simple enough to deterstudents are provided a simulation program that, mine T,,,, throueh "olav". . . ,illustrates the simulation itself and the effect on it of changing temperature. Once these general concepts are illustrated, they are applied to a similar hut more challenging situation. This is provided by the simulation of rotational spectra of ultracold Ip molecules. The general principles of the rotational spectra simulation are similar to the case of HCI, hut now there is the need to account for the broadness of experimentally measured transitions hy means of convolution factors for the simulated transitions. I t is not intended that students fully understand all the computational details of the simulation program provided. The goal is instead to impart an appreciation of the need for applying convolutions-what they are and how they affect simulation
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720
Journal of Chemical Education
results and, further, why the experimental resolution required for the I2 rotational spectra is fairly demanding. These eoals can he attained hv" euided use of a simulation program with appropriate graphics. The implementation of the alaorithms into comouter oroarams for HCI and I? -soec. tra simulation is discussed in ;he appendix.
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Boltzrnann Dlslrlbution According to the classical Boltzmann distribution, given a set of eigenstates of energies Ej and degeneracies gj pertaining to a particular degree of freedom, the ratio of population in the jth level to the O-th level is gj -hE;dkL
-e go
where AEjo = Ej - Eo. In the case of equal degeneracies of the levels, only the exponential factor remains. Derivation of the Boltzmann distrihution is a fundamental result in statistical mechanics and is therefore implicit in many thermodynamic results. However, a derivation is rarely presented in undergraduate physical chemistry courses. The result, though, can easily be experimentally verified in specific instances, such as the rotational temperature of HCI. In this case rotational bands are discretely resolved in IR spectra, and the experimental temperature can he determined using the spectra directly. Alternatively, and more instructive, the experimental spectra can be compared with simulated "stick transition" spectra. "Play" with such a simple computer program, or consideration of a typical rotational spacing relative to kT, quickly demonstrates that HC1
gettinga feel for how the spectrum varies for different values of T a n d B, and why with a B appropriate to HCl(-10 cm-') low temperatures cannot he well determined, for lack of enough "sticks" of sufficient intensity. That is, BIkT >> 1. Actually, separate B+o and B+l figure in the simulation; the foregoing equation is based on the general magnitude of B.
RELATIVE C M ( - 1 ) Comparison of effects of the convolution factor on simulated specua of IS at 5.0 K usin. Gaussian convolution factors n = 0.0 lsticksl. 0.05.0.10.and 0.20 em-' Tne trans8lmn cnosen here was d = 28 uC = 0 and the freqdency mlerval spans a range 01-4 cm-' For s m p clly m e slmu at o m placed sl~cks only accoramg to theor energy separallans relative to one another mls po nt is addressed in lhe discussion of the computer program implementation. rotational snectra cannot .. aive eood - determinations of very low temperitures, well below 100 K. However, n e a r - ~ o l t z mann distributions ior gases at rotational "temperatures" of 10 K or less have commonly heen generated usinp:supersonir rns exoansion twhniaues ( / ) . T h e serond part oithe experiment involves using a computer simulation program to ohtain temperature estimates for very low temperature ( < l o K) rotational I2 spectra available in the literature. The general considerations involved in ohtaining the "stick" transitions are similar to HCI, hut now appreciation of the factor of effective line width, and its convolution in the simulated spectra, is essential. Use of the simulation program greatly elucidates these concepts. Case of HCI Stafford et al. (2) and Shoemaker (3) both discuss the theory of rotational transitions in HCI. (A prior or concurrent experiment involving rotational constant determination in HCI, as outlined in references 2 and 3, is a useful adjunct to this part.) The rotational intensity distrihution for HCl is discussed in great detail and clarity by Herzherg (4). T o make a stick intensity simulation, one simply employs Herzherg's equation 111 (ref 4, p 169) along with the selection rule A J = f 1,with placement of "sticks" along the x axis governed by the frequency of each transition. Some examples of stick simulations are shown in Herzherg's Figure 60. Assuming a Boltzmann distrihution and A J = *l for rotational transitions, i t is simple to show (Herzherg eq 111, ref 4, p 171) that the separation of the R and P branch maxima is AvmmaX = m B T , in em-'
where B FZ B:-,. If one traces ontthe envelooe function of the P and R pe& (in effect converting J into a continuous variable), T can typically he ohtained with a precision of better than 10 K. Although Tcan he ohtained by the foregoine means. i t is nonetheless very instructive to use the stick spectrum simulator. Experimental temperatures can he determined. tcdlously (uncediousls if the experimental spectrum is an alread;digitized computer file) by choosinga T that causes stick intensities to correlate best with the experimental results. The value of the simulation is primarily in
Case of I* B values for 1 2 are roughly 30 times smaller than for HCI. As a result. a rotationallv resolved eas ohase soectrum of 19 should yield good deterGinations ~ f i o ~ t e m p e i a t u rhut e s he relativelv difficult to use for hieher - tem~eratureswhere B l k T
