DARLH. MCDANIEL AND CHARLES R. SMOOT
966
Vol. GO
APPROXIMATIONS I N THE KINETICS OF CONSECUTIVE REACTIONS BY DARLH. MCDANIEL AND CHARLES R.SMOOT Contributionf r o m the Departments of Chemistry of the University of Pittsburgh, Pittsburgh, Pa., and Purdue Universitv, Lafayette' Ind. Received Januarg 86, 1.966
A general approximation for the kinetics of the reaction A B C has been developed. The limits of the approximation and the relation between the approximation and the exact solution are established. Comparison of the results of this approximation with other approximations in current use is made. -.f
The increasingly frequent use of steady-state and equilibrium treatment approximations for the kinetic treatment of complex reactions warrants a fresh look a t the assumptions involved. Such a look leads to a more general approximation than has been used previously. Consider the reaction kl ka A Z B + C
dt is negligible in equations 3 and 6. This gives [B] =
~
kz
+
[AI
k1
k3
and The treatment developed here makes use of equations 3, 4 and G and the first approximation that d[B]/dt is negligible in equation 3. As i n the case of the steady-state treatment this assumption and equation 3 yield
icz
Here
k [B] = - _
kz
=
And if we set [AI
kg[B]
+ P I + [Cl
=
(4)
A"
dt
dt
+ d[Cl=
[A]
(10)
k3
A second approximation for d[B]/dt may be obtained by differentiating the above equation with respect to t, i.e.
(5)
Then
d[Al +%I
+
0
If this second approximation for d[B]/dt is used in equation 6, and (10) with (4), then
dt
The exact solution of this set of equations is knoivn112and will be discussed in a later section of which rearranges to the paper. One variation of the equilibrium treatment3 makes use of equations 4 and 6 and the two approxiEquation 12 may be termed the improved steadymations that state approximation. Inspection of equations 7, k (a) [Bl = K,,[AI = [AI 8 and 9 shows that the equilibrium treatment, the kz rigorous equilibrium treatment, and the steady(b) d[Bl -dt is negligible in equation 6 state treatment may all be considered to hold as special cases of (12) where the terms kl and k3, ICs, This gives and ICl, respectively, are negligible terms in the denominator. (7) Limits of the Approximation.-The limits of the improved steady state approximation may be Alternatively, and more rigorously, an expression for d[B]/dt may be obtained from the first ap- roughly set by a third approximation, i.e., (3) and (11) may be equated and d[A]/dt replaced by proximation (a) and used in equation 6, giving kz[B] - k,[A], i.e. (2) giving
+ kz + k.3) [AI + + (kz +
ki(ki
In this paper equation 7 will be termed the equilibrium approximation and equation 8 the rigorous equilibrium approximation. The steady-state treatment3 makes use of equations 3, 4, and 6 and the approximation that d [B]/ (1) T. M. Lowry and W. T. Johns, J . Chem. Soc., 9 1 , 2634 (1910). (2) See also A. A. Frost and R. G. Pearson, "Kinetics and Mechanism," John Wiley and Sons, Inc., New York, N. Y., 1953, pp. 160-104. The treatment given in these references is for a more general case involving a n additional rate constant, kr,a8 a return reaction of C yielding B. The solution for (1) is then a specific case where kd = 0 in the solution given in references 1 and 2. (3) See, for example, ref. 2, pp. 179-183.
IB1 = kz(k1 + kz This reduces to [B] =
+
k3)
~
ks
+ k1
k2
k3
k3)
1-41
when kl >kl (d) ki >k3
(a) kz >>k3 (b) kz >>ki
The Integrated Form of the Approximation.Equation 12 may be integrated to give [A] = Const. exp.
1-
klk3
k~+kz+kr~\
Since the approximation holds only after a transient equilibrium has been established, the boundary condition of [A] = A" a t t = 0 is not useful. Instead an extrapolated boundary a t t = 0 of h)] [A] and [AI [Bl = A" [Bl = [k1/(122 has been used here and gives ,
+
+
and
variation of the concentrations of A, B and C with time calculated from the exact s o l ~ t i o nfor~ the ~~~~ values kl = 2, k2 = 1 and k3 = 1.6 It may be noted that a period of time is required before [B] assumes a conbtant relation with respect to [A]. Until [B] has attained a constant relationship to [A], none of the approximations, (15), (16), (17) and (18) may be expected to hold. This period of time has been termed the induction period. We have taken the' point of inflection in the curve of [B] versus t as a criterion for the length of the induction period, i.e. Figure 2 shows the variation of [C] versus t calculated for kl = 1, k2 = 100, Jc3 = 1 by the approximations and the exact solution. It is expected here that all approximations will be about equally satisfactory since both kl and k3 are small compared to ICp. Figure 3a shows [C] versus t for kl = 1, k2 = 100 and kr = 100. Here it is exDected that the eauilibrium -treatments will be insatisfactory s&ce ka may not be ignored. The steady-state treatment will be about as satisfactory as the improved steady state treatment since kl is small compared t o k2
+
and by difference
k3.
The integrated form4 of the equilibrium approximation (7) may be taken as
Figure 3b shows [C] versus t for kl = 100, kL = 100 and k3 = 1. I n this case neither the non-rigorous equilibrium nor the steady-state treatment are expected to be satisfactory since ICl is not small compared to k2 k3. The rigorous equilibrium
and of the rigorous equilibrium treatment (8) as
( 5 ) The values of X may be evaluated readily by means of a table prepared by 8. W. Benson, J . Chem. Phvs., 2 0 , 1605 (1952). He has tabulated ZIand ZP which are the roots of z* - ( 1 K1 Ka)z KlKa where K i ki/ka and Ka = k s / k ~ . From these Xa = ku1 and Xa =
.
~.
.
_
_
(4) It may be noted t h a t [Cl is not particularly affected by the boundary conditions used as long a8 [C] = 0 at t = 0.
+
km.
-
+ +
+
( 6 ) The numerical values of the rate constants in the examples given were arbitrarily selected for ease of calculation and only the relative magnitudes are important.
DARLH. MCDANIEL .4ND CHARLES R.SMOOT
968
1
200 300 400 500 Time, Fig. 2.-Variation of [C] with time for kl = 1, ka = 100, ka = 1 (see eq. 1) comparing the approximations with the exact solution. The solid line denotes the exact solution and (within the limits of the graph) the improved steadystate approximation, eq. 15. The x's denote points calculated by the equilibrium approximation, eq. 16, and the circles denote points calculated by the rigorous equilibrium approximation, eq. 17, and by the steady-state approxim* tion, eq. 18. The induction pericd i e 0.18 time unit. 100
'
Y
4 5 6 7 8 9 Time. Fig. 4.-Variation of [C] with time for kl = 1, kz = 1, k3 = 1 (see eq. 1) comparing the approximations with the exact solution. The solid line denotes the exact solution, the short dashes denote the improved steady-state approximation, the dot-dashed line denotes both the rigorous equilibrium approximation and the steady-state approximation, and the long dashes denote the equilibrium approximation. The induction period is 1.74 time units.
I
1
2
treatment will be about as satisfactory as the improved steady-state treatment since k3 is small compared to k, k2. I n the three examples which were given (Figs. 2 and 3) it was found that the improved steady-state approximation was the only approximation which was satisfactory in all cases. Furthermore, this approximation exceeded the accuracy of the next best approximation although this difference was too small t o represent graphically. This difference is greatly magnified in the cases where the improved steady-state approximation itself begins to deviate noticeably from the exact solution. Figure 4 shows [C] versus t! for kl = 1, kz = 1 and i i s = 1. After the induction period, which covers 40% of
+
3
the reaction, the improved steady-state treatment gives results within 10% of the exact solution whereas the rigorous equilibrium treatment and the steady-state treatment have a maximum error of 40%. For example, a t t = 4 time units the value of [C] in terms of A O is 0.747 by the exact solution, 0.735 by the improved steady-state approximation, 0.87 by the rigorous equilibrium and steady-stateapproximations, and 0.98 by the equilibrium approximation. Finally, the case where kz = 0 and ICl and k3 become equal may be considered. This is the case where the improved steady-state treatment is expected to behave most poorly. Figure 5 illustrates
I
3 4 5 Time. Fig. 3a.-Variation of [C] with time for kl = 1, kz = 100, k 3 = 100 (see eq. 1) comparing the approximations with the exact solution. The solid line denotes (within the limits of the graph) the exact solution, the improved steady-state ap roximation, and the steady-state approximation. The &roken line denotes the equilibrium approximations, eq. 16 and 17. The induction period is 0.06 time unit. 3b. Variation of [C] with time for kl = 100, k2 = 100, ks = 1 (see eq. 1) compar'ing the approximations with the exact solution. Within the limits of the graph the solid line denotes the exact solution, the improved steadystate approximation, and the rigorous equilibrium approximation. The broken line denotes the equilibrium approximation and the steady-state approximation. The induction period is 0.06 time unit. 1
2
Vol. 60
-
1 I/'
Y
o.2 4'
1
2
3 4 5 6 Time. Fig. 5.-Variation of [C] with time for kl = 1, k2 = 0, ka = 1 (see eq. 1) comparing the approximations with the exact solution. The solid line denotes the exact solution, the short dashes denote the improved steady-state approximation, the dot-dashed line denotes both the rigorous equilibrium approximation and the steady-state approximation, and the long dashes denote the equilibrium approximation. The induction period is 2.00 time units.
this case. Here the induction period takes up 61% of the reaction. Yet after the induction period is over the improved steady-state treatment gives values for [C] within 5% of the values calculated by an exact treatment. The rigorous equilibrium treatment and the steady-state treatment again have maximum errors of about 40'%. For example at t = 3 time units the value of [C] in terms of A" is 0.80 by the exact solution, 0.78 by the improved steady-state approximation, 0.95 by the rigorous
REACTION BETWEEN A Q U O FERROUS IONAND CUMENEHYDROPEROXIDE
July, 1956
equilibrium and steady-state approximations, and 1.OO by the equilibrium approximation. Approximations for the Kinetics of Higher Order Reactions.-The degree of success of equation 15 in approximating the exact solution prompts us to look a t higher order reactions where an exact solution has not yet been found and one must resort to an approximation. For the general case nA
zB + kl
k3
C
kz
both the non-rigorous equilibrium and steady-state treatments yield the same form of the rate con-
960
stant as in the previous case whereas our approximation yields’ nkika [AI“ _ -- n2ki [A]”+ kz + ks
dA dt
1
We are currently investigating these higher order reactions as well as studying the implications of these results on the calculation of equilibrium constants from the forward and reverse rate constants of complex reactions. NOTE ADDED riv PRooF.-Equation 15 may be derived in a somewhat different manner than has been used here; see XI. J. Laidler, Can. J . Chem., 33, 1614 (1955). (7) The rate constants refer to production or disappearance of B
THE REACTION BETWEEN AQUO FERROUS IRON AND CUMENE HYDROPEROXIDE’ BY W. L. REYNOLDS~ AND I. M. KOLTHOFF Contribution from the School of Chemistry, University of Minnesota, Minneapolis, Minnesota Received February I , 1068
The rate constants of the reaction between ferrous iron and cumene hydroperoxide have been found to be 3.53 X lo8 exp (-9970/RT) and 9.25 X 108 exp (-10840/RT) liter mole-’ sec-1 in HzO and DzO media, respectively. The main products of the reaction are ferric iron, acetophenone and ethane. A reaction mechanism is postulated and discussed. The possible nature of the rate-determining reaction step is also briefly discussed.
The ability of hydroperoxides to furnish free oxides. Wise and Twigga have investigated the radicals upon reduction with a suitable reducing stoichiometry of the reaction between aquo feragent has led t o their widespread use as initiating rous iron and cumene hydroperoxide (CHP) and agents for p~lymerization.~It has been observed determined some of the products formed. I n this that different hydroperoxides yield widely different report the results of a more thorough study of the rates of polymerization in emulsion polymerization kinetics and mechanism of the latter system me systems. Differences in the rates of polymerization presented. The reaction was also studied in D10 with different hydroperoxides may result from dif- since it was hoped that the results might provide ferences in the rates of reaction of the hydroperox- information about the mechanism of the rate-deterides with a given reductant, from differences in the mining step of the reaction. reactivity of the free radicals produced in the reExperimental duction of the hydroperoxides, and from differences Determination of Rate Constants.-The method by which in the distribution of hydroperoxide between the the reaction between aquo ferrous iron and CHP waB aqueous and organic layers. Since an iron(I1) followed and by which the rate constant was determined species frequently is used as the reducing agent for already has been described.’ When the rate of reaction the hydroperoxide a study of the kinetics of the re- in Dz0 was determined, rate measurements were made in DzO and HzO media to eliminate the possiactions between various hydroperoxides and various alt,ernatively bility that the DqO results were affected by incorrect tunciron(I1) species will provide information whether tioning of the rotated platinum wire electrode. The electhis factor is responsible for the different behavior trolysis cell employed for.DzO media had a capacity of 10 of the hydroperoxides as initiating agents. Kolthoff ml. but otherwise was similar in construction t o the cell used HzO media. The Ih.0 was obtained from Norsk Hydroand Meddial4 Fordham and william^,^ Orr and for electrisk Kvaelstofaktieselskab and was used both without Williamse and Kolthoff and Reynolds’ have re- further treatment and after distillation from alkaline perported results of kinetic studies on rates of reaction manganate and dilute sulfuric acid, similar results being between aquo ferrous iron and various hydroper- obtained with both samples. (1) This work was carried out under the sponsorship of the Federal Facilities Corporation, Office of Synthetic Rubber, in connection with the Synthetic Rubber Program of the United States Government. (2) Abstracted from the theais of W. L. Reynolds presented to the Graduate School of the University of Minnesota in partial fulfillment of the requirernents for the Ph.D. degree, 1955. (3) I. M. I
