Jay E. Taylor Kent State University Kent, Ohio 44240
I I
The Stirred Flow Technique Mathematically developed applications for complex kinetic reactions
Limited use of the stirred flow technique for the study of the kinetics of chemical reactions (1-6) has been made. Actually the potential usage is extremely broad, particularly for a wide variety of complex reactions, including enzymatic reactions. The basic theory for the stirred flow technique has been detailed by Denhigh (7-9). An apparatus for precise determination of kinetics of reactions has been developed ( l o ) ,and the several advantages of the technique are enumerated. I n principle, the stirred flow reactor is simply a small reactor which is closed except for the required inlets and outlets. Separate solutions of two or more reactants enter a t a constant rate and are stirred efficiently. A combination of reactants and products flows out a t the same total rate. After attaining the steady state, hoth the concentration and rate of formation of product in the reactor are constant. For a simple bimolecular reaction z/t = k(a - z ) ( b
- x)
(1)
where 2a and 2b represent the original undiluted reactant and x the products formed in the time t. The time is determined from v/u where v is the volume of the reactor and u is the flow rate. A graphical illustration of the significanceof x/t is shown in the figure. Due to the complexity of the apparatus required, there is little advantage in dealing with an uncomplicated first- or second-order reaction. The advantages become apparent only with more complex reactions. Several specific examples follow. Product-Reactant Interaction
K
= Q/(c
+ z - QKa - z - Q)
(5)
whence
It is obvious that an integration of this equation is a formidable task. Under stirred flow conditions
After substitution of eqn. (6) into eqn. (7) there remain two unknowns, 1c and K, since i t is presumed that a, b, e, x, and t are known or are determinable. The values of k and K may he evaluated from two sets of kinetic data, either taken a t different times or at different values of a, b, ore. Equations (6) and (7) may he applied to any reaction meeting the above specifications. If D is [H+]then the situation is ordinarily solved by the use of buffers or the pH stat. Buffers may entail buffer effects and the pH stat is limited only to control of p H ; whereas, the stirred flow reactor controls all aspects of the reaction. For other products, e.g., I- as a product in the iodination of phenols, an excess of reactant may be used; generally, however, this is less than satisfactory. The stirred flow technique does provide a general means of direct measurements of rate data under any conditions. Obviously, i t is adaptable to enzyme studies which may include a combination of effects of the type discussed.
If one of the products undergoes an equilibrium reaction with one of the reactants, an unduly complex kinetic equation is the result. Assume the example
C
+A K
Q (inactive)
(3)
T h u s A = a - x - Q , B = b-x,C = e + x - & , a n d D = x where a, b, and e are the initial concentrations of hoth products and reactants and x is the extent of product formation. D = 0 initially. Thus, in an ordinary reactor
and Supported in part by a grant from the National Institutes of Health.
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Journal o f Chemical Education
Plot of a simple second-order reaction. The curve ir from the integrated kat = x/lo - XI,and the x and t valuer from the stirred flax =/I = kIo - ~ 1 2 . It i, asrvmed that o = 1 and k = 1. The slopes of the dolled lines 5,. S2, SI,S1, give the values of x/t (Lo., the rater) ot the corresponding oinh (XI, t d lxz, t d (xr,Is1 lrr, trl. The theoretical bark for the "re of . xF in the derivations within the text is thus grophicoliy illustrated for the stirred Row technique.
Second-Order Consecutive Reactions (6)
Assuming the example of basic hydrolysis of a diester OHa-2-y
OH-
+ ROIC-COaR b-x
+ ROZC-CO1-
kl
kr
ROnC-COs-
+ Hz0
(8)
Z-Y -0&-COS-
cluded) are similarly used to solve for kz and c in eqn. (17). Substitution in eqn. (16) then gives a value for kl. Obviously a similar procedure may be extended to n equations. Copolymerization in the Stirred Flow Reactor
+ HzO
(9)
a-r-y 2-Y Y and that the reaction is followed by analysis of [OH-] a t specified times t and that a and b are known, then [OH-] = a - x - y (10)
If the above is known to be the correct rate equation, then a minimum of two experimental determinations is needed to evaluate the kinetic constant since there are 3 equations and 4 unknowns. Substitution of the experimental values in eqn. (11) gives k, and x. Then from eqn. (lo), y is obtained and finally /czfromeqni (12). If the above rate equation is not known to be the correct rate expression, a significantly large number of experimental points will be needed to determine if constant values of k, and kz are obtained. Consecutive Reactions of n Steps
Assume a reaction with any given number of steps. It is presumed for the purpose of illustration that each step is second order. This is not, however, a necessary condition.
This technique provides a means for direct determination of the several rate constants. The following is a mechanism for a typical copolymer reaction; for example, ethylene with propylene. The formation of polymer begins with the reaction of the initiator I with monomer MI or Mz forming a reactive end group MI' or Mzl. This is turn reacts with more MI or whereby the 14,' or M2' is converted to the corresponding polymeric unit which becomes the next outermost molecule in the polymer chain. The process continues with growth of the polymer chain and PI and P2 represent the total polymer formed. New chains are formed with the introduction of new molecules of I. I, is the concentration of initiator which is continuously introduced into the stirred flow reactor. The monomers and initiator flow into the reactor a t a constant rate and the equal outflow contains polymer, monomers, initiator, and intermediary species. Inside the reactor I, MI, Mz, PlP2, and A h ' and 14%'remain constant with time due to the steady state conditions. The lo becomes I MI' Mzl upon its entrance into the reactor; lo is deemed to be very small with respect to MI MI. The basic equations are given below.
+
+
+I M1 + Z M,
I, = I
+
k1
kl
MI'
(assume excess M I )
M1'
(assume excess M I )
(19) (20)
+ MI' + M,'
MI'
+ M , - M,' + PI
MI'
kr + M, M,' + P*
(21)
kll
Assume that ao, bo, &. . ,yoare the init,ially known concentrations of the reactants, that e, e. . .x are the undetermined total concentrations of the intermediates, and that z is the final product which is analytically determined. The values of z are dependent only upon the initial reactant concentrations and the rate constants. Then,@-c, bo-c, e-e, &-e. . .x-z, y,-~, and z represent the actual concentrations of A, B, C, D, X, Y, and Z which exist at any time t. Under the steady state conditions of the stirred flow reactor c / t = kl(aa
- c)(bo - c) - e)
ell = kdc - e)(d,
(25)
MI' and MZ'may each be assigned an average reactivity regardless of the length of chain. The following rate equations may be written
(161 (17)
If i t is attempted to eliminate without approximation MI' and Mz' from among the four equations, a very complex algebraic expression is the result. However, if it is assumed that kl and kz are large There are then n equations and 2n - l unknowns. A minimum of n - 1 kinetic points are needed to evaluate the unknowns, kl. . .k, and c, e. . .x. For example, if it is assumed that n = 3 and x = e, two sets of experimental data with duplicate values of z, yo, and tare suhstituted in eqn. (18) to give values for x (or e) and k, (or ks). With this value of e two more sets of experimental data (one corresponding to the above may be iu-
M,'/t
=
k,(Io - MI' - M I ' )
Ml'/t
=
kz(Io - MI' - Ms')
and
+
+
If (kl kJt >> I, then IO = MI' M*'. The following equations are derived by substituting 14,' = I, - MI' in Volume 46, Number 1 1, November 1969
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eqn. (26), solving for MI' and substituting for MI' in eqn. (28). Following a similar procedure MI1 = lo MZ1is substituted in eqn. (27), etc.
Upon combining eqn. (32) and eqn. (33)
I n eqn. (34) let ([PI]/ [PJ) [l&lz = a; ([PI]/ [Pel) [MI][M2] = b; [MIl2= c; and [ M I ][M,] = d. Upon using two sets of experimental data, a], a,eto., i t is seen that
stirred flow reactor if the initial monomer concentrations are known. These examnles illustrate the versatilitv and wide applicability of the stirred flowreactor. Itis seen that a reaction for which a differential rate equation can be written for the mechanism, either proposed or known, is amenable to the stirred flow technique. Certain other advantages of the stirred flow technique in addition to those cited above should be mentioned. Since all reactants and products remain constant during the course of the reaction the solution is meohanically buffered against changes in pH, salt concentration, product-reactant interaction, etc. Reactant or product decomposition is held constant. Short-lived intermediates mav be studied at length - through constant regeneration by inflow of the reactants. With a, small reactor relatively small volumes of reactants are needed.
Literature Cited
where u is indicated from eqn. (35). Correspondingly
Then, upon substituting klz = ekll into eqn. (32), the rate constants kll and kal may be specifically evaluated by substituting two additional experimental points. Finally kzzand klz are then obtained from eqn. (36) and eqn. (38). Therefore, an analysis of only PI and Pzor M, Mz is needed in the solution flowing out of the
+
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(1) Buss, W. C., AND TAYWR,J. E., J. Am. Chem. Sac., 82, 5991 (1960). (2) YOUNG, H. H., JR., AND HAMMETT, L. P., J . Am. Chem. Soe., 72,280 (1950). (3) SALDICK, J., AND HAMMETT, L. P., J . Am. Chem. Sac., 72, 283 (1950). (4) RAND,M. J., AND HAMMETT,L. P., J. Am. Chem. Sac., 72, 287 (1950). (5) HUMPHREYS, H. M., A N D HAMMETT, L. P., J. Am. Chem. Soc., 78,521 (1956). 161 BURNEW.R. L., A N D HAMMETT, L. P., J . Am. Chem. Soc., 80, 241'5 (195i). DENBIGH, K. G., Trans. Famday Sac., 40,352 (1944). DENBIGH, K. G., HICKS,M., AND PAGE, F. M., Trans. Faraday Sac., 44,479 (1946). DENBIGH, K. G., A N D PAGE,F. M., Di~cussionsFaraday Sac., 17, 145 (1954). TAYLOR,J. E., AND ARORA,S. L., Chem. Instr., 1 , 353
