Indian Journal of ChemistryVol. ~2A. May 199~. pp. ~H 1-~H2

Exact solution of rate equations for consecutive first andsecond order reactions

R D Murphy

Department of Physics, University of Missouri.ik.ansas City, Missouri 041 110, USA

Received 20 July 1992; revised and accepted II January 1993

The differential equations describing a first order reaction followed consecutively by a second orderreaction are analyzed. The solutions involve modified Bessel functions. Numberical results and a meansof obtaining the rate constants from the concentration-time data are presented.

In a recent paper Casado et al:' developed aprocedure for obtaining rate constants for a kinet-ic system of first order reactions. In this paper wediscuss a first and second order system which car.be solved exactly. Although the existence of theanalytic solution was pointed out some time ago ',it is little known or used: only a few textbooks [e.g.refs (3) and (4) 1 note it in passing and it is missingfrem major compendia of chemical kinetics re-sults. Our purposes here are to present the exactsolution in a more convenient form, to draw atten-tion to useful computational algorithms, and topresent a method of extracting rate constants forsuch a system from the concentration versus timedata.

FormulationWe consider an irreversible first order reaction

followed consecutively by a second order reactionwith rate constants k, and k2' respectively. Thereaction is:

A~B (1)

B+BtC (2)The concentrations [A], [BJ and [C] obey the equ-ations:

d[A]/ dt= - k, [A]

d[B]/ dt= k, [A] - k2[Bj2

d[C]/ dt= (k/2)[BF

... (3)

... (4)

... (5)

The solution of Eq. (3) is elementary:

[Al=~ exp( - kit) ... (6)

where AI) is the initial value of [A]. Substitution ofEq. (6) into Eq. (4) and introduction of appropri-

ate dimensionless parameters leads to the follow-ing equations:

dy/ dx = aexp( - [3T)- y2

dz/ di = il2 = (I /2)[aexp( - [3T)- dy/ dT.]

... (7)

... (H)

where T=kzt, a=A.lk/k2, [3=k/k2, and y(T) andz( T) are the values of lB] and [Cj as functions ofthe reduced time parameter T. The nonlinearity ofEqs (3 )-(5) prevents complete scaling to dimen-sionless variables; the effect is that [A], [B], [C] andthe rate constants are in molar units.

Solution and numerical resultsThe solution of Eqs (7) and (H) will be ex-

pressed here in terms of the modified Bessel func-tions In and Kn, which are the regular and irregu-lar, respectively, solutions of the modified Besselequation. The notation and analytic properties ofthese functions are those used in ref. (5); while itis true that In and K, can be expressed as ordinary(cylindrical) Bessel functions of imaginary argu-ment, this fact is not helpful for practical comput-ations. We must define two parameters, namely~(J =[2 J(a)j![3 and1;=~oexp( - [3T/2). Then thesolution of Eq. (7) is:

The quantity c is the constant of integration oneexpects because Eq. (7) is a first order differentialequation; the relation between c and y", the initialvalue of y, is:

C;;; - [Yolo(;o) + (B~ol2)II(;O)J/lYIIKII(~II)

- (f3~II/2)KI\;II)l· ... (10)

INDIAN J CHEM. SEe. A. MAY 1993

1 Ol,-~~~~~~~~~~~~~~~~~

----- ----- -----.

4

Fig. I-Curves of Eq. (9). yrt ) versus ,=k,tfor various valuesof the ratio of rate constants ~ = k.tk-: ~~_ ~ = 1.0; - --

~ = 2.0; and -'-'- ~ = 0.5.

If Yo = 0, as would usually occur, then c = Kt(1;o)1I,(~o)·

Figure 1 contains curves of y(T) for three differ-ent values of the ratio of the rate constants, B = k/k'., under the assumptions that Yo = 0 and A) = 1;as noted above, this represents a choice of molarunits. The plots were made using the algorithms inref. [0 I. As one would expect, the solutions all riseto a maximum and fall asymptotically to zero.

Equation (8) can be readily integrated to yield:

z(T) = z., + (}\/2)[1 - exp( - BT)]- (1I2)y(T) ... (11)

Curves of z(T) (for the same values of B used inFig. 1) under the assumptions that A) = 1 . andZu = 0 are displayed in Fig. 2. All three curves ap-proach an asymptotic value of A/2.

Determination of rate constantsThe rate constants k j and k2 as well as the

quantity A) can be determined from concentra-tion-time data by the following simple procedure.First, A) and k, can be determined by a linearleast squares analysis of a plot of [A] against thetime t (not the dimensionless time parameter Tused in this paper). Secondly, k2 can be obtainedfrom the long time behaviour of [B] as a functionof time, using the following argument. The asymp-totic behaviour of [B] in time, i.e, y(T) for large va-lues of T, corresponds to small values of ~. Usingthe analytical forms of the limiting behaviour ofthe modified Bessel functions In and K, found inref. (5), it is easy to show that as t --> 00, y --> 1/(const, + k2 t). Thus, a plot of 1I [B] will be linear

0.5i,-~~~~~~~~--~~-----------;

0.4 ----- --»: --

0.3 /'/' ...--- .

./

N

./ »>/ /

/ -:0.2 //

-:/ /

O. j/ //

/ // /

0.0 /..~

5 0 j 2 3 4 5

Fig. 2-Curves of Eq. (11), zf t.) versus, = k,t for various va-lues of the ratio of rate constants ~ = k11k2• ~ = 1.0;

--- ~=2.(r and -'-'- B=O.5.

for large enough time and the slope of the line isk2• Then a check on the consistency of the resultscan be obtained from the maximum in the plot of[B] versus t, where from Eq. (4) it can be seen that[B]=[kjAuexp( - kjt)/k2r:!. Theoretical calculationsof [A], [B] and [C] versus time can then be per-formed on a microcomputer using Eqs (6), (7) and(8) and the algorithms of ref. (6) and comparedwith experimental data. If necessary, the parame-ters kj, ~ and Al can be refined using a non-line-ar least squares algorithm such as is contained inref. (()).

AcknowledgementIt is a pleasure to acknowledge with thanks sup-

port for this work from the United States ArmyResearch Office through Contract No. DAAG-29-85-K0064. The assistance of Mr. M.D. Murphy inpreparing the graphs is greatly appreciated.

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Hill, New York) (1955) P 54.5 Handbook of mathematical function, edited Iby M Abram-

owitz & I Stegun. NBS Applied Mathematics Series No. 55,(U.S. Government Printing Office, Washington, D.C.)(1972).

6 Press W H, Flannery B P, Teukolsky S A & Vetterling WT, Numerical recipes, (Cambridge University Press, Cam-bridge) (1986).