Matrix Formulation of Chemical Reaction Rates A Mathematical Chemical Exercise Lionello poglianil and Mario Terenzi Dipartimento di Chimica, Universita della Calabria, 87030 Rende (CS), Italy Some years ago, L. Papula2presented a method to obtain the kinetic equations of a consecutive reaction scheme involving only irreversible fwst-order steps. The method was based on matrix algebra. In fact, matrix algebra offers an elegant way of solving the differential equations for simple kinetic systems. The use of matrix algebra has an advantage in that it allows the simultaneous solution for the rates of the reaction for the different chemical species. Accordingly, we present here an application of matrix algebra to first-order rate equations. One of the cases treated (side opposing reactions) is also treated in a different way by N a q ~ i who ,~ presents a more general treatment on the similarity between chemical kineties and vibration modes. First-Order Reactions Consider the reaction
yields the eigenvalues, w{s wl = 0 and wz = --k
To the eigenvalue wl = 0, belongs the following system of homogeneous linear equations
solutions of which are Al = 0 and Az that can be regarded as a free parameter and be chosen equal to A, withA some nonzero constant. Eigenvector C1, which belongs to eigenvalue w,, then has the following form
.. To the eigenvalue wz = -k, belongs the following system. X-+Y
Let the molecular concentrations of X and Y at time t be Cx(t)= Cx and Cdt) = Cy
whose solution is A,=-Az=B
respectively The rate of formation of X and Y are thus and C;
where B is a nonzero constant. Thus, eigenvedor Cz,which belongs to eigenvalue w2,is
where
Then this reaction obeys the following system of homogeneuos linear differential equations of the first order with constant coefficient.
k)= e qq
(I)
This first-order rate equation can be rewritten in the following compact matrix form. C'=AC
(2)
The general solution of the fxst-order kinetic problem is then the followine linear combination of the two indeoendent C1 and C2 eigenvedors.
-
With the aid of the initial conditions, Cx(0)=a and Cy(0) = b, we obtain B = a and A = a + b The final chemical kinetics equations will be
which has a general solution
with C' = wC. By the aid ofthis solution, eq 2 leadus to the following matrix eigenvalue problem. wC=AC (4) Since wC = wIC, where I is the unit matrix (A-wI)C=O
Opposing First-Order Reactions Consider the reaction
(5)
The matrix form of the differential rate equations has the following form.
For nontrivial solutions to this equation to exist, the determinant I A - wI I must be zero. Solution of
(13)
'Author to whom correspondence should be addressed. 'Papula, L. Mathernatik fur Chemiker; Ferdinand Enke Verlag: Stuttgattl975. 3Naqvi,K. R. J. Chem. Ed. 1989, 66(9),703.
Here the coefficient square matrix is the result of the sum of two coefficient square matrices of the fust-order reaction type. Note that the two diagonals have opposite signs. This equation lead us t o the same kind of matrix
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eigenvalue problem of the first-order reaction, with the same kind of nontrivial solutions. The wi eigenvalues are found by solving the following determinantal equation. (14)
Solving, we obtain
Diverging Side Reactions
Consider the reaction
e x + z
Y
The coefficient square matrix that describes this system has the following form
w1=0
and w2 = i k , + k,)
k, 0 0
The C1 and C z eigenvetors calculated from
(25)
with k=k,+kz
With the aid of the same kind of mathematical approach and with the initial conditions
have the following form
Cx(0)= a
CdO) = b
CZ(0)= e
we obtain the following rate equations Cx = a exp-kt)
.
.
where
where A and B are some nonzero constant and where k K = A and h = k f + k , k,
and
For the general solution to this problem c = R ] = (KA A +-BBexp(-fit) exp-ht)
CZ= e +K" ( 1- exp-kt))
where (18)
with the aid of the initial conditions
Converging Side Reactions
Cx(0)= a and CdO) = b
Consider the reaction
and with
Y(akf- bk,) m= h
X
e-z
The coefficient square matrix that describes this system has the following form.
we get the final chemical rates (29)
The final rate equations are Cy=-K(a b, - m exp-ht) +
(l+K)
where
Consecutive First-Order Reactions
Consider a simple, consecutive reaction involving only irreversible fust-order steps
and Cy = b exp(-k',t)
X + Y + Z
Papula's original treatment starts with the following matrix.
where Cx(0)= a
(21)
(31)
CdO) = b
Cz(0)= c
Opposing Side Reactions
Consider the following hypothetical reaction.
Then, with The resulting coefficient square matrix is the sum of the two previous square matrices.
it yields the following final kinetic equations
c, = a kl exp(-kit) k
+ (bk - akl)k exp-k2t)
(23)
-k
k',
k,
0 -k;
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(33)
279
For kl=k,=kd
(d stands far divergent)
and k', = k', = k c
(C
A rapid comparison between Cx and Naqvi's3 Bz shows the similarity of the two different approaches.
stands for convergent)
with CAO) = a
Cy(0)= b
Conclusion
CZ(0)= c
and with
the fmal rate equations will be (34)
Ks
b-c
+ 2 -P(-@) cv = ZK+~
280
a-b-c - X2(2K+ 1)
Journal of Chemical Education
We notice that the number of rows in our matrices eauals the number of chemical species in the chemical ecpacons. Also. the coefficientmatrix is a sauare matrix that can be used to fully characterize the chekcal reaction. While the simple formalism presented applies fairly well for chemical equations with first-order steps only, a more elaborated matrix method was developped very recently for kinetic systems composed by steps of any order? Berberan-Santos. M. N.; Maltinho, J. M. G. J. Chem. E h c .
-(Fht'
(35)
1990, 67(5),375.
