Robert 1. Anderson. Ronald s. Nohr, and Larry 0. Spreer
University o f the Pacific Stockton, California 95211
I
I
An Exad Solution to
A Consecutive Reaction Sequence
In general, chemical reactions which occur by mechanisms involving a sequence of mixed first- and second-order reactions are described by nonlinear differential equations for which there are no known exact solutions (1, 2). An important exception to this situation is the irreversible second-order, fint-order reaction sequence
through expressing [B] in terms of [A] plus the stoichiometric excess of B (eqn. ( 6 ) )that the final solutions to the equations have convenient form. Equation (3) can be rewritten as
k
A + B A C
and then rearranged to the following
k
C A D with the rate laws for the consecutive steps Equation (7) can be integrated by the method of partial fractions to yield
The only known solution (3) for systems of this type is of limited usefulness in determining the values of the individual rate constants k~ and kn because it involves an incomplete beta function. This complicated form of the solution was obtained by the method of eliminating time as an independent variable in the rate laws for the consecutive steps (4). We have obtained a novel, more useful form of the exact solution for eqns. (1) and (21, fmm which the rate constants can he obtained in a straightforward fashion.
where 0 is a constant of integration which can be evaluated at t = 0, and is
fl can be called the "stoichiometric ratio" and since B is required to be in excess, e > 0 and fl < 1. The concentration of B as a function of time can he easily evaluated by combining eqns. (8) and ( 6 ) , and rearranging to give
Solution to a System of Differential Equations for a Second-Order, First-Order Reaction Sequence
Consider the stoichiometry for the general overall reaction aA+bB-cC-dD represented by the set of differential equations
Here, the well-known solutions (8) and (9) have been cast in terms of N and p instead of the usual concentrations [Ale and [BJo. Continuing with the second step of the sequence, eqns. (4) and (5) can he combined to give
This is a linear, first-order, inhomogeneous, ordinary differential equation which yields by standard techniques of the operator calculus (5)
The requirement is then made that one of the initial reactants be in excess. Picking B to be in excess, then
or rearranging
Multiplying through by the factor eC"t, and employing the identity dXY/dt = dX1dt.Y + X.dT/dt eqn. (12) is ohtained c2 d cd p* ,[cn = -Ceci,n- dlA1 = --a dt a d t ( e ' Y A ] ) + ;kpkd[A] (E) Integration of eqn. (12) yields
[Bl = $(@% - CAIo + [ A ] ) or finally,
where a = ( a l b ) ([Bl0 - [A],,) and [Blo, [A10 are initial concentrations. The quantity e represents a / b times the concentration of B remaining after completion of the reaction and can be called the "stoichiometric excess." It is
Now multiplying both sides of eqn. (13) by e e k 2 ' pmduces
In eqn. (141, p is a constant of integration which can be evaluated at t = 0, p = [C]o + ( c i a ) (ai3)/(1 - ,5)' (ai3/1 - B = [Ale); Q is an indefinite integral Volume 52, Number 7. July 1975 / 437
first four terms of Q a t t = 30 min were 318, 19.7, 1.43, and 0.117, respectively. By utilizing eqns. (14) and (5). it can he directly verified that the solution for [Dl is given by
where y is a constant of integration, y = kz([D]o + pdlc). Equations (8), (9), (14), and (16) represent an exact solution in closed form for the system of non-linear differential eqns. (3)-(5). Of coune, these solutions can only he useful in obtaining information about the system if the indefinite integral, Q, (eqn. (15)) can be evaluated or effectively approximated. Approximation of Indefinite Integral 0 In this case Q has t h e form 111 - x, x < I where x = @e-bk1=' (@< 1 since B is required to he in excess, and the factor e-b'*m' 5 1 for all values o f t ) . An approximation for 111 - x, x < 1 is given by its well-known Taylor series expansion
+
nbkla for any n = 1,2,. . . , I the solution for Q If ckz then becomes
Determination of k , and k 2 From {Dl and d(Dl/dt It is possible to use our solution to determine the individual rate constants, k l and k ~ even , if only the product, D, can be measured experimentally. All that is needed is the concentration of the ~ r o d u c.t .[Dl. . .. and the rate of its appearance, d[D]/dt, at two times, t l and t2. The basic equation involved in this method is obtained by differentisting eqn. (16) and then substituting (16) into this result. This yields
If we now let (aldap) [- [Dlt,, + [Dlo + pdlc] = SI, and similarly for Sz, then eqn. (18) evaluated a t t l divided by eqn. (18) evaluated at tz gives
All quantities that appear in eqn. (19) are known except k ~ Various . techniques can he employed to facilitate solving this equation. The right-hand side of eqn. (19) can he graphed as a function of kl and the correct value of kl is that which gives the experimental value of R. Another approach might he to select tz = 2tl, and then solve the resulting cubic equation for x R(x - B)(S2(x2-
The approximation for Q has utility because the series is well-behaved and converges after a reasonable number of terms. Since 4 < 1.. .0" will become smaller as n increases, and will'favor rapid convergence. Experimentally, 4 can be made smaller hv increasing the amount of the reactant B which is in excess. The convergence of Q was tested in a particular system in which a = 1, b = 2, c = 1, d = 2, and in which kl and kz were determined indepenM ' dently of the present method to he k l = 2.90 X s-I at 25°C (6); for a particular s - I and kz = 1.17 X experiment in which 0 was equal to 0.12, the values of the 'The solution in the case that ekz = nbkla for same integer n is simply obtained by replacing the term (eICka- """I=" - l)/(ekz - nbkxa) by t ineqn. (17).
438 / Journal of Chemical Education
0) -
1) - ( x 2 -
0) (SI(x - 0 ) - 1) = 0
where x = ebk;e' . Knowledge of the chemically meaningful root, xo, then yields k l = l/batl'"xt. A computer program could also be used to aid in obtaining values of kl from eqn. (19). Once kl is known then eqn. (18) evaluated a t tl yields kz. Acknowledgment Support by grants from The Research Corporation is gratefully acknowledged. LOS Grant C125, RLA Grant C741308. Literature Cited ( l i Benmn. S. W., "The Foundations of Chemical Kinetier," McGraw-Hill, New York.
1960,p.42.
(2) Capollos. C., and Bielrki. 6.. "Kinetic Sy5temr: Mathematical Descripfionn af ChemicalKincticr inSolufion? Wiley-lnt~rscienco.New York. 1912. 131 Kelan. T . A . Phvsik. C h e m NeueFole.. 60. 191 119681. (41 B ~S . w ,~ j cLm. ~ P ~ ~Y Sz. ~. .~1 6 0 5 m . 521. ( 5 ) Noiron. A . Folev. K. W.. and Coral. M.. "Differential Eouations." D. C. Heath and Co.. Boston. 1952. 161 Nohr. R. %.and Spreer. L. O..lnor#. Chem., 13. 123911974,
