HigginsWhitaker_AICHEJ_2012

For single independent reactions this analysis can be

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Unformatted text preview: V (74) Next we direct our attention to Eq. 71 and use that result with Eq. 74 to extract the following representation for the net rate of production of hydrogen bromide: RHBr ¼ kII cBr cH2 þ kIII cH cBr2 À kIV cH cHBr 546 DOI 10.1002/aic RH \\ 2 kI cBr2 =kV ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RH \ kII cH2 2 kI cBr2 =kV \ ð77Þ Experimental verification of these restrictions remains as a challenge. Conclusions (72a) kII cBr cH2 À kIII cH cBr2 À kIV cH cHBr ¼ 0 (76) which is exactly the same form as the reaction rate expression determined experimentally by Bodenstein and Lind.12 This suggests that the series of kinetic steps illustrated by Eqs. 58 through 62 are consistent with the reaction kinetics; however, one must always remember that the route to Eq. 76 is not necessarily unique. Thus, there may be other schemata that lead to essentially the same result given by Eq. 76. A key assumption associated with Eq. 76 is that of local reaction equilibrium illustrated in a general sense by Eq. 51 and for this particular case by Eqs. 68. In reality, the net rates of production of the Bodenstein products will never be zero; however, they may be small enough so that Eq. 51 represents an acceptable approximation. For the special case of the hydrogen bromide reaction one can show (See Problem 9-12 of Ref. 3) that small enough means that the following restrictions are satisfied: RBr \ 2 kI cBr2 =kV ; \ 3 kI cBr2 rI 6 rII 7 6 kII cBr cH2 7 6 76 7 6 rIII 7  6 kIII cH cBr2 7 6 76 7 4 rIV 5 4 kIV cH cHBr 5 kV c 2 rV Br kI cBr2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁ pffiffiffiffiffiffiffiffi 2kII 2kI =kV cH2 cBr2 ¼ 1 þ ðkIV =kIII Þ ðcHBr =cBr2 Þ À (68) (75) In this article, we have examined two applications of stoichiometry. The first of these is the classic situation in which the conservation of T atomic species is used to constrain the net rates of production of N molecular species. For single independent reactions, this analysis can be carried out by counting atoms and balancing chemical equations; however, real-world applications rarely involve single independent reactions and matrix methods are required to obtain reliable results. For either local or global stoichiometry, the single application is represented by the pivot matrix that maps the net rates of production of the pivot species onto the net rates of production of nonpivot species. This mapping does not allow one to determine the absolute values of the net rates of production. However, it is essential for the analysis of chemical reactors to determine the experimental values of the net rates of production of the pivot species, RTþ1, RTþ2,…, RN. Two key matrices appeared in this analysis. The first of these is the atomic matrix that provides for a precise statement of Axiom II, whereas the second is the pivot matrix that maps the net rates of production of the pivot species onto the net rates of production of the nonpivot species. In Appendix B, we show that this same pivot matrix appears in the solution of the atomic species balances for batch reactors and flow-through reactors operating at steady state. For batch reactors, the pivot theorem provides constraints on how the moles of atomic species change during the reaction. For flow-through Published on behalf of the AIChE February 2012 Vol. 58, No. 2 AIChE Journal reactors operating at steady state, the pivot theorem gives the constraints on the net atomic species molar flow rates that are possible. When charged species (ions) need to be considered, the principle of conservation of charge is easily imposed on the analysis as indicated in Appendix C. In the study of reaction kinetics, elementary stoichiometry and mass action kinetics are used to develop expressions for all the net rates of production. In this case, the conservation of atomic species is imposed on each elementary step of a set of chemical kinetic steps. These elementary steps are so simple that conservation of atomic species is generally accomplished by ‘‘counting atoms’’; however, the application of a more rigorous method is described in this article. In this analysis, the mechanistic matrix provides a mapping of reference reaction rates onto all the net rates of production; thus, absolute values of RA, RB,…,RN are produced that allow for the design of chemical reactors. When charged species (ions) need to be considered, the principle of conservation of charge is easily imposed on each individual kinetic step as suggested in Appendix C. Both the pivot matrix and the mechanistic matrix are composed of what are commonly known as stoichiometric coefficients; however, the two sets of stoichiometric coefficients are different, and they serve different functions. Because of this, we need to be precise when we speak about stoichiometry and stoichiometric coefficients. Acknowledgments This work began many years ago when the authors were challenged by Ruben Carbonell to develop a clear and rational treatment of stoichiometry. Ramon Cerro helped the authors to meet that challenge, and the authors are grateful for the contributions of these two colle...
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