INDEPENDENCE OF CHEMICAL REACTIONS R U T H E R F O R D A R l S A N D
R. H . S. M A H
Department of Chemical Engineering. C'niversity of Minnesota. Minneapolis, M i n n .
A systematic treatment i s given for evaluating the stoichiometry of a chemical reaction system on the basis of limited experimental observations. It i s shown that the number of independent reactions may be determined b y a procedure which does not demand any knowledge of the composition of the reacting species and that Gibbs's rule of stoichiometry may be used as an independent check. The matrix of stoichiometric coefficients may then b e computed to within a nonsingular linear transformation. The advantages of such an approach are clearly demonstrated in the analysis of a stirred tank reactor in which the governing equations are reduced to a minimum. The analysis also results in the segregation of reaction and transport effects and simplifies the studies of stability.
THE
RELATION of stoichiometry to chemical kinetics, both pure and applied, is somewhat the same as that of kinematics to fluid mechanics. I n the latter discipline, it is essential to start with a thorough understanding of the nature of continuum motion per se before the notions of dynamics are introduced. Still later are the dynamical equations solved under restraints representing particular physical conditions. By analogy, bve take stoichiometry to be the study of possible concentration changes or the kinematics of the change of chemical species. Pure reaction kinetics is concerned with the dynamics of these changes, with the mechanisms and rates of reaction, while applied reaction kinetics considers what happens Lvhen the resulting equations are solved within the physical conditions of a particular type of reactor. Like all analogies this one cannot be pressed too far, but it does give a certain insight into the structure of the subject and shows the fundamental importance of stoichiometry. Just as tensor analysis is the proper language of kinematics, so linear algebra is the proper language of stoichiometry. Its notions of independence and rank express precisely the nature of the independence of chemical reactions and lead to the minimum number of necessary equations in any given situation. We wish to comment on some of these points here, showing how the independence of reactions may be established and how this approach illuminates the physical picture. The chemical engineering literature does not contain much on these lines. Brotz ( 6 ) . in his treatment of stoichiometry, uses an algebraic, but not specifically matrix, notation. Grutter and Messikommer, in their discussion of isothermal reactors. have used a stoichiometric matrix ( 7 0 ) . Perhaps the fullest treatment is in de Groot's text on thermodynamics (.9), but here the context and bearing are somewhat distant from chemical engineering applications. A notable exception to the general neglect is Beek's review of the design of packed catalytic reactors ( 3 ) .where he clearly shows the importance of reducing the equations to an independent set. By using a matrix identity, he shows that all formulas contain reference only to the independent subset of the possible reactions. \\le shall prefer to work from the beginning with a linearly independent set of reactions and start by giving a test for this independence. A recent article making use of the idea of rank and the canonical reduction of a matrix is by Ames (2). In this, a minimal set of kinetic differential equations was developed by matrix elimination methods. In another recent publication:
90
l&EC FUNDAMENTALS
Wei and Prater 1 7 1 ) have made a full and penetrating use of the concepts and methods of matrix analyses. Independence of Simultaneous Reactions
Consider a system in which there are S distinguishable chemical species, which partake in R chemical reactions. If these species are denoted by B I B Y . the rth reaction may be ivritten in the form:
where pra is the stoichiometric coefficient of B , in the rth reaction. It is convenient to give positive values to the stoichiometric coefficients of species that are to be regarded as the products of a reaction. The stoichiometric coefficients are defined u p to a constant factor, but there is no accepted convention as to how they should be normalized. The simplest convention is to take p,]. . . p7,* to be integers with no common factor other than 1.
Pings (73) has recently suggested
' 1 p,..i
s=l I
= 1, and de Groot has stoichiometric coefficients @,,Ma ( M , being the molecular weight of B,) and takes Zp,, M , = 1,
for the sum of the positive coefficients. The mathematician lvould probably be inclined to orthonormalize the rows of B . If the species (B1. . . Bs) are written as a row vector b ', then Equation 1 may be written in matrix notation as:
Bb
=
0
(2)
Here, B is the matrix { p , $ ) and b is not a numerical vector but the symbolic one of the species. (A prime denotes the transpose of a vector or matrix.) Linear independence of reactions means that there is no way of expressing any of the reactions as a linear combination of the others, and hence there is no nontrivial vector y such that: ~
B'r
=
0
(3)
The term row-rank is used for the greatest number of linearly independent rows of a matrix. It is a theorem of linear algebra (7,8, 77) that the row-rank and column-rank, which is similarly defined in terms of columns, of a matrix are equal, and their common value is called the rank. Thus, the condition for the independence of the R reactions is that the rank of the matrix B should be R. Each of the chemical species B , is composed of elements. If the set of elements .4n,n = 1 . ..V,includes all the atomic
species in the compositions of B I . . . BS and aIhis the number of atoms of A,, in species 19,. then: b
14)
Aa
=
\\.here A is the matrix of atomic coefficients c y S , , and a is the symbolic vector of atomic species A , . Substituting for b in the equations for the reacrions of Equation 2 qives: 15)
BAa = 0
T h e atomic species must be conserved, so that every coefficient of a in these equations is zero : i6)
BA = 0
IVritten in full? this is:
include such inerts among the B's and give them stoichiometric coefficients of zero. Ifj,(t) is the amount of B , present in the system as determined a t time t, the differences A y s ( t ) = y 8 ( t ) - ~ ~ ( are 0 )not all zero. IVith R independent reactions, the! are determined by the R extents of reaction e , ( t ) :
or Ay(t) = B'e(t)
+
Let y be measured a t S 1 instants t o = 0 < f l < . . . < t s . An S X S matrix can be formed ~ v i t hcolumns A Y ( ~ , , ~ )I72, = 1. . .S; denote this by: AY
In general, S may be greater than. equal to: or less than .\.>but R is not greater than (2; - 1) since there is a t least one relation
/3,,~M,= 0. Holvever,
between the columns of B. namely
the matrix A may not be of the highest rank possible. For example. xvith -4, = K. .-I2 = C1.
=
3, P
=
++ 2C02 + 4C02 +
= 0 2H?O = 0 4H20 = 0
0. and a t most tivo independent reactions
are possible. A n Experimental Approach
Gibbs's rule only gives a n upper bound to the number of independent reactions. To discover hou many are actually taking place in any given system. its composition changes must be studied. I t will be supposed that any chemical species for which the total amount present does not bar) is not reckoned among the ,E?&. In other contexts it is convenient to
I\.e therefore have the S X S observational matrix AY expressed as the product of a n S X R matrix B' and a n R X S matrix E. By a theorem of linear algebra (7. 8. 7 1 ) . this is a standard representation of an S X S matrix \\.hose rank is R. I t follo\vs that if the rank of I Y is determined, this will sho\v hoiv many independent reactions are accounting for the observed composition changes. Sotice that from this approach no upe has been made of the information inherent in the atomic matrix. Gibbs's rule. therefore. gi\,es an independent check o n R. Determining the Rank of a Matrix with Errors
'The rank of 1Y may be determined by the procedure described beloiv ivhich. in essence? makes use of (elementary) ro\v operations \virh some elementary precautions against the gro\vth of errors intrinsic in AY, IVe recall that the rank of a matrix is not changed by the interchanges of it5 rolvs and columns. Such operations may, therefore, be employed to bring the numerically largest element to the top left hand corner of the matrix. This procedure is usually knolvn as the selection of a '.pivot" by row and column interchanges. T h e top row is then divided through by this pivotal element yielding a leading element of unity. hText.this new top row is multiplied by the leading element of the second rorv and this is subtracted from the second ro\v: giving a ne\v second rolv Ivith the leading element of zero. This elimination is continued to the last row, so that the first column of the modified matrix VOL. 2
NO. 2
M A Y 1963
91
is (1 ,0:0 ,. . O ) '. Now the pivotal selection and elimination are performed with the (S - 1) X (S - 1) submatrix which does not contain the first row and the first column. As this process is repeated with the successive submatrices. elements of unity are developed down the diagonal with zeros beneath them. If the rank of the matrix, R: is less than the original number of roivs: S,and if errors are absent, this procedure will yield a new matrix with the last (S - R ) rows identically zero. T h e procedure just outlined is indeed the same as the triangularization entailed in matrix inversion, and the selection of pivots minimizes the growth of errors during the elimination (4,5). T h e important difference is that unlike the matrix inversion here, AY is not expected to be a full rank matrix. I t is: therefore. not only desirable but also essential that the element Lvith the largest numerical value should be used as the pivot a t each stage of elimination. This point should become evident on Lvorking through the numerical example. \Vith the corruption introduced by experimental error, the determination can no longer be expected to be so decisive. Instead of vanishing identically, the most we can hope for, in general, is that the last (S - R ) rows will be manifestly small. -4full statistical analysis would require a derivation of the distribution function for the error of each element a t each stage of the process in terms of the distribution of errors in the original observations. I n the case of a 2 X 2 determinant, even if the elements a r e random variables normally distributed about their means ivith the same variance? the full distribution is a very formidable affair [see (72), for example]. Instead of attempting a full treatment: the upper bounds of the errors will be examined here and used to keep some track of the latter. Suppose. then. each term r n i j of a matrix M can have an E, absolute error of a t most E - i.e., f i t j - E < m,, where mi, is the true value. If E is small enough for its square to be neglected and certainly smaller than lA,,I the absolute error in the reciprocal of mij is e / f i i j 2 . Consider then the four typical terms involved in the process of the rank analysis:
+
