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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
912 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, realworld 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 ﬂowthrough 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 ﬂowthrough 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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