Catalysis of Organic..

224 Cavitating Ultrasound Hydrogenationtemperatures, we can rigorously apply transition-state theory to describe the u 3 C3(-H) and C3(-D) elimination processes. In this scenario, the ratio of rate coefficientsfor C-D and C-H elimination from the surface intermediate is given by [16],k Dk H= (Q t,D /Q t,H ) (Q † D / Q† H ) (1)where Q t,D and Q t,H are tunneling terms and Q † D and Q † H are the partition functionsof the transition state. For a sufficiently large barrier to reaction, which isreasonable for our system, the tunneling correction is close to unity [16-18]. Herewe approximate the Q † D and Q † H terms of equation (1) by the two bending modes ofboth C-C-H and C-C-D, respectively. In particular, we estimate these terms usingdata for ethane and deutero-ethane [19] where: w 1 (D)=503, w 2 (D)=661, w 1 (H)=671,and w 2 (H)=849 cm -1 , where w 1 is the out-of-plane CH bond bending mode. Usingexpressions for the vibrational partition function results in k D /k H =0.39, or atransition-state theory predicted deuterium number (via k D /(k D +k H )) of 0.28 at 298K.These predictions can now be compared to experiment. Examining our datasetof Figure 2 leads to extrapolated c→0 trans-olefin deuterium numbers of 0.20 (MS)and 0.46 (US). The former value is close to that predicted by transition state theoryjust described. We propose, therefore, that the less energetic conventional (thermal)processing approach yields trans-olefin primarily through the C3-H/D eliminationgiven by transition-state theory employing the reaction temperature of 298 K.Conversely, the results for the more energetic cavitating ultrasound system is alsoexplainable using transition-state theory except that a much higher, non-equilibrium(compared to the thermal bath), vibrational temperature is required to model thedeuterium number of 0.5. This is because transition-state theory predicts that thedeuterium number asymptotically approaches 0.5 as the temperature approachesinfinity. A temperature of 800 K results in a computed deuterium number of 0.40,thus this is likely the minimum vibrational temperature that describes the cavitatingultrasound system.This assignment is supported, indirectly, by the measured activities of theseexperiments. For example, the activities were measured to be (in M/g-catalyst hour):0.98 (MS-H 2 ); 0.61 (MS-D 2 ); 180 (US-H 2 ); and 190 (US-D 2 ). Hence, the ~250-foldgreater activities of the ultrasound systems is consistent with the expected, morerapid, statistical C-H/D dissociation process as compared to the conventional (e.g.,stirred/silent) mediated systems. Additional support for this model arises from astudy of gas phase cis-2-butene isomerization to trans-2-butene [15] at 291 K. Herethe c→0 extrapolated trans deuterium number of ~0.27 is supportive of C3-H/Delimination predicted by transition-state theory in this system at thermal equilibrium(e.g., vibrational temperature equal to translational temperature).In the context of the accepted olefin isomerization mechanism, our resultsillustrate that transition-state theory can accurately model the competition betweenC-H and C-D activation for olefin exchange (isomerization) for the case of