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Unformatted text preview: one chooses a ‘‘reference reaction rate’’ designated
by rI, rII,…,rK. These reference reaction rates are used to
express the elementary rate of production of species A as
AIChE Journal (43) in which MAk is a component of the mechanistic matrix.8
This representation of the elementary rate of production can be
used with Eq. 42 to express the net rate of production of
species A as (40) This result is essential for the analysis of macroscopic systems because it speciﬁes the global net rates of production
that must be inferred from measurements of the composition,
selectivity, conversion, and yield. In Appendix B, we show
that there is an equivalent pivot theorem for atomic species
balances as opposed to the molecular species balances associated with Eq. 40. For atomic species, this same pivot theorem appears in the transient analysis of batch reactors and
the steadystate analysis of continuous stirred tank reactors
(CSTR). Elementary Stoichiometry: R k ¼ MA k rk ;
A Published on behalf of the AIChE (47)
DOI 10.1002/aic 543 This partition leads to the following two matrix equations
2 32
3
R1
M1 I M1 II M1 III : M1 K
6 R2 7 6 M2 I M2 II M2 III : M2 K 7
676
7
6 R3 7 6 M3 I M3 II
:
:
:7
6 7¼6
7
6:7 6 :
:
:
:
:7
676
7
4:5 4 :
:
:
:
:5
RN
MN I MN II
MN K
ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} 2 3
rI
6 rII 7
67
6 rIII 7 (48a)
67
4:5
rK stoichiometric matrix 2 3
rI
MNþ1 I MNþ1 II : : MNþ1 K 6 rII 7
RNþ1
67
4 : 5¼4 :
:
::
: 5 6 rIII 7
67
4:5
:
: : MM K
RM
MM I
ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
rK
Bodenstein matrix
2 3 Elementary kinetic schema IV: 2 3 kIV H þ HBr À H2 þ Br
!
(52d) Elementary kinetic schema V: kV 2Br À Br2
! (52e) It is important to note that the schemata listed in Eqs. 52
are not linearly independent. In fact, there is no physical
requirement that this be the case, thus, the stoichiometric matrix S that appears in Eq. 49 may have a rank less than N.
This list of schemata represents a set of pictures and what
we need for analysis are equations. We indicate how to
develop these equations in the following paragraphs.
When the rules of mass action kinetics11 are applied to
the picture given by Eq. 52a, we obtain the following equation (48b)
In terms of compact notation, we express the ﬁrst of these
two results as
R ¼ Sr (49) in which R is the same column matrix of net rates of
production that appears in Eq. 13. We refer to S as the
stoichiometric matrix9; however, we must be careful to note
that the pivot matrix appearing in Eq. 23 is also composed
of stoichiometric coefficients. Because of this, there is a
possibility of confusion when referring to ‘‘stoichiometric
coefficients.’’ The second of Eqs. 48 can be expressed as
RB ¼ Br I
Elementary chemical reaction rate equation I: RBr2 ¼ ÀkI cBr2 ; (53)
and on the basis of this expression, we choose the ‘‘reference
reaction rate’’ to be
rI kI cBr2 Elementary reference reaction rate I: (54) Next we need to make use of elementary stoichiometry,
thus, we recall Eq. 41 with k ¼ I to obtain
A¼M
X
A¼1 (50) NJA RI ¼ 0;
A J ¼ Br (55) in which RB is the column matrix of net rates of production
for the Bodenstein products, and B is the Bodenstein matrix.
The classic resolution of this set of equations is to impose the
condition of local reaction equilibrium expressed as As this process involves only Br2 and Br, we see that Eq.
55 provides Local reaction equilibrium : and this immediately leads to RB ¼ 0 (51) and use the result from Eq. 50 to eliminate the concentration of
the Bodenstein products from the column matrix of reference
reaction rates in Eq. 49. In this manner, Eq. 49 becomes a key
element in the design of a chemical reactor.
At this point, we are ready to return to Eq. 43 and indicate
how one applies elementary stoichiometry in the process of
determining the elements of the mechanistic matrix. To
accomplish this, we consider the classic example10 of the
reaction of hydrogen with bromine to form hydrogen bromide. The elementary chemical kinetic steps associated with
the reaction of H2 with Br2 to form HBr are illustrated by:
kI !
Elementary chemical kinetic schema I : Br2 À 2Br (52a) 2 RI 2 þ RI ¼ 0
Br
Br Elementary stoichiometry I: Br þ H2 À HBr þ H
! H þ Br2 À HBr þ Br
!
544 DOI 10.1002/aic RI
Br
2 (57) SCHEMA I
ð52bÞ Elementary chemical kinetic schema III:
kIII I
RBr2 ¼ À This represents a mathematical statement that the atoms
of bromine are conserved during the process illustrated by
Eq. 52a; however, this result is usually considered to be
intuitively obvious and thus is not identiﬁed as elementary
stoichiometry. One can think of Eq. 57 as an example of
‘‘counting atoms’’ in the same manner that led from the picture given by Eq. 24 to the equation given by Eq. 25. Howeve...
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