704
WALTERY. WEN
A General Solution for the Sequence of Two Competitive-Consecutive Second-Order Reactions
by Walter Y. Wen Department of Chemistry, University of Oregon, Eugene, Oregon 97403
(Received October 21, 1971)
Publication costa assisted by the National Science Foundation
The kinetic equations for a sequence of two competitive-consecutive second-order reactions have been solved so that both rate constants can be evaluated from runs with any ratio of initial reactant concentrations. The technique has been successfully applied to the data of Ingold for the saponification of dimethyl glutarate by sodium hydroxide. Methods of evaluating the rate constants from kinetic data generated by a series of two competitiveconsecutive second-order reactions have been investigated by Wells,’ recently by Saville12 and reviewed by Frost and Pearsona and Szab6.4 The system is very complex and the kinetic equations cannot be handled by the conventional techniques. Assuming that the initial concentrations of the reactants are stoichiometrically equivalent, Frost and Schwemer6 developed a solution for this system. McMillane showed that the ratio of the rate constants can be determined easily if the simultaneous concentrations of two substances are known. Although Frost and Schwemer’s method6 is computationally convenient, in practice it can be difficult to prepare stoichiometric amounts of reactants. This was found to be particularly true when the reactants are gases and the reactions are rapid.’ To avoid this restriction we have derived a general Frost-Schwemer equation which can be applied to reactant mixtures with any proportions of the two reactants. The chemical system consists of the reactions8
A+B~‘-c+E
(1)
A + C ~ D S E (2) where kl and k2 denote the rate constants of reaction 1 and 2, respectively. Although there are five possible kinetic equations for the system, only two of them are independent. Those for B and D are
dB _ -- -klAB dt
dD = k2AC dt By material balance one obtains
The Journal of Physical Chemistry ,Vole76, No. 6,1972
(3)
(4)
where A , B , C, D , and E are the concentrations of the corresponding substances and the zero subscripts indicate initial concentrations. Since there are only three independent stoichiometric relationships for five variables, two parameters are needed to describe the system completely and one must either measure the concentrations of two species at the same time or the same species at at least two different times (besides t = 0) in order to obtain the two rate constants. The latter technique is deveIoped in the present paper. To simplify the derivations, the following dimensionless variables and parameters are introduced.
A A0
c y -
p = -B
Bo
C Bo
y = -
D
6 = Bo
I n terms of these, C is eliminated from eq 4 using eq 6 and then the ratio of eq 4 to 3 becomes
Equation 8 can be transformed into a homogeneous firstorder differential equation which can be readily solved to giveg (1) P. R.Wells, J . Phys. Chem., 63,1978 (1959). (2) B.Saville, ibid., 75, 2215 (1971). (3) A. A. Frost and R. G. Pearson, “Kinetics and Mechanism,” Wiley, New York, N. Y.,1961,p 178. (4) Z. G.Szab6, “Comprehensive Chemical Kinetics,” C. H. Bamford and C. F. H. Tipper, Ed., Vol. 2, Elsevier, New York, N. Y., 1969,p 61. (5) A. A. Frost and W. C. Schwemer, J . Amer. Chem. SOC.,74, 1286 (1952). (6) W.G.McMillan, ibid., 79,4838 (1957). (7) P. Goldfinger, R. M. Noyes, and W. Y . Wen, ibid., 91, 4003 (1969). (8) For comparison all symbols used in the derivations are adopted from those in ref 3 and 5. (9) W. Y.Wen, Ph.D. thesis, University of Oregon, 1971,p 116.
Two COMPETITIVE-CONSECUTIVE SECOND-ORDER REACTIONS Table I :
01,
a!
y, 6,
= 1.
and E as Functions
- '[z P
(2
p,
K,
705
and p
+ 14 - 0"-1 9
1
(10)
01
=
E
= '[2 P
KP
- 8" (2
+
1
~
E
K - 1
(loa)
-0 In 0
6 = 1 - (1
K - 1
-
- 1-[2 - (2 - lnp)p] P
y =
6-l--
1
= -[2 P
-
- 1np)p
(2
- lnp)P]
(9)
assuming that only A and B are present initially. For a special case of K = 1, direct solution of eq 8 yields 6 = 1 - (1 - In@/?
(94
which also can be obtained by taking the limit of eq 9 as K + 1. It can be shown, using eq 5 , 6, 7, and 9 or 9a, that CY, y, and B can be expressed as functions of p , K, and p. Results of these derivations are summarized in Table I. It should be noted that if p = 2, eq 10 reduces to the Frost-Schwemer equation,6 that eq 11 is exactly that obtained by Mcil/lillan,6 and that both y and 6 are independent of p . I n terms of the dimensionless variables and parameters, the integrated form of eq 3 becomes .I
.2
.3
.4
.5
.6
.7
.8 .9
ID
B
By substituting eq 10 into eq 13, r can then be evaluated if p , K, and the limit of P are known. For p = 1.5 Figure 1 shows r and CY plotted against P for various K . r was calculated from eq 13 by means of the trapezoidal formula using a step size for /3 of 0.01. Since for a particular P and K, r has only one value, plots similar to Figure 1 for y, 6, and E can be constructed using eq 10, 11,9, and 12. Because T is defined as k&t, a plot of /3 vs. log t for a single run with p = 1.5 should yield a curve parallel to one of the family of ,8 vs. log r curves. But K can be more accurately determined by the time ratio ( t ratio) method.6 It should be clear that the ratio of time for 50% of B reacted to time of 10% reacted, t 6 O / t 1 0 , equals r50/r10 which is a function of K. Several results of K as functions of t ratio, with p = 1.5, for various ratios of B reacted are shown graphically in Figure 2. Similar t-ratio plots based on A, C, D, or E also can be obtained from the related equations listed in Table I. Using plots analogous to Figures 1 and 2, or the corresponding tables if desired, kl and k2 can be determined from the results of B single run following any of
Figure 1. Plots of r and K, with p = 1.5.
01
us. p for various values of
I-ratio b a s e d on
%I)
reacted
Figure 2. Plots of K us. t ratio for 60% and lo%, 60% and 20%, and 50% and 10% of B reacted, with p = 1.5.
the five species. The calculational procedure is described by Frost and Schwemer.6 However, because the relations between @ and CY, y, 6, or E are quite The Journal of Physical Chemistry, Vol. 76, No. 6,1078
WALTERY, WEN
706 Table I1 : Calculations of the Rate Constants for the Saponification of Dimethyl Glutarate by Sodium Hydroxide a t 20.3' Using the Data of Ingoldlo ( p = 1.0)
t , min
A , 10-8 mol/l.
a
7
ki, I./(mol min)
0.0 2.5 4.2 6.0 10.0 12.0 14.1 16.0 18.0 20.0 23.0 26.0 29.0 32.0 35.0
1.999 1.929 1.883 1.839 1.743 1.699 1.655 1.619 1.577 1.539 1.487 1.435 1.387 1.345 1.307
1.000 0.965 0,942 0,902 0.872 0.850 0.828 0.810 0.789 0,770 0.744 0.718 0.694 0.673 0.654
0.0366 0.0615 0.0877 0.1465 0 1760 0.2065 0.2340 0.2635 0.2930 0.3368 0.3810 0.4255 0.4680 0.5210
7.324 7.324 7.320 7.329 7.337 7.326 7.316 7.323 7.329 7.319 7.331 7.340 7.316 7.446
I
I I
a
0.95 0.90 0.85 0.80 0.75 0.70 0.65
3.6 7.6 12.0 16.9 22.2 28.2 35.7
K
35/05 35/10 35/15 35/20 30105 30110 30/15 30120
a
9.917 4.697 2.975 2.112 7 * 833 3.710 2.331 1.669 Average =
0.152 0.142 0.152 0.145 0.154 0.156 0.156 0.152 0.151
~
Table I11 : Comparison of the Rate Constants Obtained from the Data of Ingoldlo by Different Methods of Calculation Method
Ingold9 Wideqvist 10 This work
hi, I./(mol min)
k t , l./(mol min)
7.24 7.277 7.334
1.12 1.12 1.107
The Journal of Physical Chemistry, VoL 76, No. 6,1978
4
5
6
7
8
Figure 3. Plots of t ratio us. p for various values of the ratio of 50% to 10% of B reacted.
__
t ratio
3
9
IO
P
Average = 7.334 kz = 0.151 X 7.334 = 1.107 -Calculation of comparison of % a 1, min reacted
2
K,
with
complicated, data analysis becomes very tedious if the reaction is monitored by following the concentration of a species other than B. Our calculations of a and r us. /3 covered the ranges of 0.2-20 for p and of 0.02-50 for K . The cases of K = 0 and K = m were also evaluated. Figure 3 shows plots of the ratio of the time required for 50% reaction of B to that required for 10% reaction us. p for various values of K , which indicate that the t-ratio technique is quite sensitive for values of p between 1 and 4. If p is very small, then B is in large excess and one has to sacrifice experimental precision of determining B. When p is in the range of 1-2, the t-ratio technique can be most conveniently applied. The above method has been employed to interpret the data of Ingoldlo for the saponification of dimethyl glutarate by sodium hydroxide at 20.3'. The results of the calculations are shown in Table 11. In Table 111, the calculated values of the rate constants are compared with those obtained by two other approximation techniquesl0l1l in which kz had to be determined by independent kinetic measurements. Acknowledgment. The author wishes to thank Professor R. M. Noyes and Dr. R. J. Field for enlightening discussions. This research has been supported by the National Science Foundation. (10) C. K. Ingold, J. Chem. SOC.,2170 (1931). (11) S. Wideqvist, Acta Chem. Scand., 4,1216(1950).
