12590
J. Phys. Chem. 1994, 98, 12590-12593
Mathematical Treatment for Very Rapid Second-Order Reversible Kinetics As Measured by Stopped-Flow Spectrophotometry with Corrections for the Cell Concentration Gradient Nancy E. Meagher and David B. Rorabacher’ Department of Chemistry, Wayne State University, Detroit, Michigan 48202 Received: March 3, 1994; In Final Form: August 2, I994@
A mathematical approach is described for treating stopped-flow spectrophotometric data for very rapid secondorder reversible reactions. The treatment involves integration of the theoretical concentration gradient down the length of the observation cell, based on an experimentally measured filling time. No restrictions are placed on the starting reagent concentrations, and allowance is made for the presence of finite amounts of product at the selected “zero” time, that is, the treatment allows for reversible reactions with prior product formation. Careful calibration of the stopped-flow instrument is required since absolute, rather than relative, values of the absorbance must be utilized. Using a computer analysis based on the described mathematical treatment, it is anticipated that second-order rate constants as large as lo8 M-’ s-l may be determined with reasonable accuracy for reactions with sizeable absorbance changes (A€ > lo4)using instruments with filling times on the order of 3 ms.
Introduction
Solution from nixe1
General Considerations. For the investigation of rapid reaction kinetics in solution using the stopped-flow technique, the upper rate limit for which successful measurements can be obtained is ultimately dependent upon the ability to observe a detectable signal change following the mixing of the reactants and filling of the observation cell. For second-order reactions, the reaction half-life is dependent upon the initial reactant concentrations and, thus, the lower limit of reagent detectability determines the most rapid reactions which can be studied. Since stopped-flow instruments generally involve spectrophotometric monitoring, the lower limit of detectable reagent concentrations is determined by (i) the difference between the molar absorptivities of the reactants and products (A€) at the selected wavelength and (ii) the length of the observation cell. For most stopped-flow spectrophotometers, the length of the observation cell is increased by bending the flow tube so that the light beam passes down a 1.0-2.0 cm portion of the tube beyond the point at which the reagents are mixed (Figure l ) . l Since a finite time (typically 3- 10 ms) is required for the newly @xed solution to flow down the length of this cell, a significant concentration gradient exists for reactions whose half-lives are short compared to the filling time. This gradient is of no consequence for first-order (or pseudo-first-order) reactions, since the reaction half-life is independent of concentration. However, for second-order reactions, the gradient in “initial” reactant concentrations at the time selected as “time zero” results in large errors in the evaluated rate constant whenever the first half-life approaches or falls below the time required for the newly mixed solution to fill the observation cell. To circumvent the solution inhomogeneity problem when studying very rapid second-order reactions, many manufacturers of stopped-flow instruments suggest that operators either (i) pass the light beam through a cross section of the flow tube so that a homogeneous slice of the solution is observed or (ii) operate under pseudo-first-order conditions. Both approaches are actually counterproductive. The first approach greatly reduces the length of the observation cell and, thereby, requires an increase in reagent concentrations to achieve comparable signal changes. @
Abstract published in Advance ACS Abstracts, October 15, 1994.
0022-3654/94/2098- 12590$04.50/0
I
Figure 1. Schematic of a stopped-flow cell defining the parameters x
and 1. For example, if the observation tube diameter is 0.3 mm and the length is 2.0 cm, each reactant concentration would need to be increased by approximately 6.7-fold to achieve an equally sensitive signal using the smaller path length. This would result in a 44-fold increase in the initial reaction rate. As a result, the half-life is further decreased and no advantage is gained. The second suggestion is based on the fact (as noted above) that the half-life of first-order reactions is independent of the initial concentration and, thus, will be uniform throughout the observation cell, thereby avoiding any problems associated with the concentration gradient. However, this latter approach requires using a large excess of one of the reagents which then results in a corresponding decrease in the reaction half-life. In contrast to these two suggested “solutions” to the inhomogeneity problem, it has been our experience that, for very rapid second-order reactions, “better” results can be obtained by using the longer observation cell and operating under conditions where the two reagents are approximately equal in concentration. The initial data, obtained during the time increment when the inhomogeneity of the solution within the observation cell is significant, can be discarded, and the remaining data can be utilized to calculate the characteristic second-order rate constant. This latter approach presumes that a measurable signal can still be obtained beyond the point where the concentration gradient is significant. Theoretically, an improved treatment of the solution inhomogeneity problem could be achieved if the concentration gradient down the length of the observation cell were modeled mathematically, taking into account the time necessary for the newly mixed solution to flow from one end of the cell to the other.2 Such an approach has been previously ~onsidered,~ and
0 1994 American Chemical Society
Very Rapid Second-Order Reversible Kinetics several obstacles have been noted. In general, (i) mixing is not complete at the time the solution enters the observation cell, (ii) problems associated with laminar flow may result in a lack of uniformity in the flow rate across the diameter of the flow tube, and (iii) the newly mixed solution actually enters and exits the observation region at right angles (Figure 1). Thus, a smooth integration down the entire length of the cell may not be representative of the complete concentration gradient. Nonetheless, the use of an experimentally-determined filling time should permit the development of an approximate characterization of the concentration gradient within the observation region, which is vastly superior to the alternate assumption that no such gradient exists. Based on the foregoing considerations, we previously developed a simplified mathematical model for treating second-order data in which the concentration gradient within the observation cell was treated as a series of homogeneous segments (as contrasted to a continuous integrati~n).~In developing that model, we imposed the restriction that the initial concentrations of the two reagents be identical (in the stoichiometric sense) and assumed that the reaction proceeded to completion with no contribution from the back reaction. Using the resultant mathematical expression which we developed, we were able to increase the upper limit for resolvable second-order rate constants to about lo7 M-' s-l, even with reactant concentrations as high as 2 x M where the calculated half-life was 700 ,us. Recently, we have been studying very rapid second-order electron-transfer reactions in which the reduced reagent is often subject to some air oxidation prior to reaction initiation. As a result, it has been difficult to maintain exactly equal concentrations of the reagents at the time of reaction initiation. Moreover, some of these reactions do not proceed to 100% completion and the contributions of the back reaction have been complicated further by the presence of a finite amount of product at the selected zero time. To address these complexities, we have now developed a completely general mathematical treatment for second-order reactions in which the contribution of the back reaction is taken into account, including an allowance for the presence of some product(s) at the selected zero time. We have also removed the restriction for equal initial concentrations of the reagents. In the current treatment, the cell gradient is treated by integrating down the length of the observation cell. Although the final mathematical expression is exceedingly complicated, it can presumably be solved by computer methods to generate reasonable rate constants for second-order reactions having halflives significantly shorter than the filling time. In using an instrument with a 2.0 cm observation cell and a filling time of 3.5 ms, it may be possible to resolve second-order rate constants as large as lo8 M-' s-l with reasonable a c ~ u r a c y . ~
Generalized Treatment for Reversible Second-Order Kinetic Data In generating an expression to correct for cell inhomogeneities in stopped-flow kinetic measurements on second-order reactions, we have attempted to make the treatment as general as possible. Thus, we have chosen to incorporate into our treatment the possibility that the reaction under consideration may be reversible (i.e., may not proceed to virtual completion) and may also involve the presence of a finite amount of product at the time of reaction initiation. Since textbooks on chemical kinetics do not generally cover the case for reversible second-order kinetic processes of this type, a brief derivation of the applicable expression for such a reversible second-order process in the
J. Phys. Chem., Vol. 98, No. 48, 1994 12591 absence of inhomogeneity corrections is provided below. For simplicity of presentation, this derivation assumes both one-toone stoichiometry and a reaction order which parallels the stoichiometry. For second-order reactions in which this is not the case, suitable modifications can easily be made. In deriving a complete kinetic expression, we consider a general reversible reaction of the type k12
A+B;;;;- C + D
(1)
for which the applicable rate expression is second order in both directions: W d t = k12[AI [Bl - k21 [Cl [Dl
(2)
In eq 2, the quantity X represents the extent of reaction at any time, t,
x = [AI,
- [AI = [BI, - [BI = [CI - [CI, = [Dl - [Dl0 (3)
where the bracketed terms without subscripts represent molar concentrations at any time, t, and the quantities with zero subscripts represent the corresponding molar concentrations at the time defined as t = 0. (Note: If no reaction has occurred prior to t = 0, then [C]O= [D]o = 0 and eq 3 is still valid.) Using the subscript e to represent the concentration values existing at thermodynamic equilibrium, it is apparent that dXJ dt = 0 and, from eq 2, kl2[Ale[Ble = k21[Cle[Dle
(4)
or
Substitution of eq 5 into eq 2 yields, upon rearrangement
The relationships in eq 3 can then be substituted into eq 6 to generate an expression in the single variable X which can then be integrated in the form
Integration then yields the following expression:6
+ b - l/b2 - 4uc 2uXr + b + l/bz-4ac 2uX0 + b - .Jbz-4ac 2uX0 + b + 2/b2-4ac 2uXr
l-
= k12t (8)
where X , represents the change in molar Concentration of each of the reactants and products at any time, t, relative to their values at the selected zero time and the combined constant terms in eq 8 are defined as
a = [ClePle - [Ale[Ble
Equation 8 can be simplified by recognizing that XO = 0 by definition; and subsequent concentration gradient corrections are facilitated by further combining some of the constant terms
Meagher and Rorabacher
12592 J. Phys. Chem., Vol. 98, No. 48, I994
also be a function of the average concentrations of each species at the time defined as t = 0:
as follows:
r = J b 2 - 4uc
Abso = l(rA[A]O
+ EB[B]O + Ec[C]O + €@]o)
Therefore, %,, the average extent of reaction at any time, t, relative to the selected zero time is related to the difference in absorbance:
b-r ‘ = K r
%, =
Abs, - Abs,
ZAE
to yield the following modified form:
+b - r P t = In{ 2uX, + b + .) - In
where
2uX,
(9)
or 2uX,
(12)
+b +r
It should be emphasized that eqs 9 and 10 are completely general and are applicable to reactions which proceed to completion (i.e,, where [Ale or [B], 0) or to reversible reactions in which no product is formed at the defined zero time (Le., [Clo = [Dlo = 0) as well as to reversible reactions with prior product formation.
A€ = EA
+
where the E values represent the molar absorptivities of the corresponding subscripted species at the selected wavelength. The observed absorbance at the selected zero time, Abso, will
- EC
- ED
The values of [A],, [B]o, etc., in eq 12 are not independently known but can be calculated relative to the theoretical concentrations of each species immediately following mixing by taking the difference between Abso and the theoretical initial absorbance, Absi, to provide a “delta” correction, XA,for the extent of reaction prior to the point of the selected zero time:
-
XA =
Absi -Abs,
~AE
= [AIi - [A], = [BIi - [B], =
Cell Concentration Gradient Corrections As noted in the Introduction, for reactions in which the initial half-life approaches, or is shorter than, the filling time of the observation cell, a concentration gradient exists at the time the flow is stopped and data measurement is initiated. There are two basic approaches for treating the concentration gradient down the length of the observation cell. The first approach involves a numerical solution in which the cell is divided into n “homogeneous” segments. This approach was used in our previous paper which treated only the specific second-order case in which [A], = [B]o with no kinetic contribution from the back r e a ~ t i o n .For ~ those limiting conditions, a reasonably straightforward solution for correcting for the concentration gradient was derived. However, for the more general case being considered here, the second-order reversible rate equation is sufficiently complex to cause much greater difficulty in applying this “simplified” approach. A second approach is to view the concentration of the reaction variable, X , as a three-dimensional surface involving X , t, and d, where d represents the distance along the length of the observation cell (0 5 d I1, where I represents the total length of the cell). If we consider X to be a smooth function of both t and d, an exact solution can be obtained. This approach can be simplified by recognizing that the concentration in terms of d can also be related to the relative time required for the mixed solution to reach any point in the observation cell during the filling process. Thus, integrating over the path length at any given instant in time is equivalent to averaging the value of X over a time interval equivalent to the filling time of the cell. For the case in which the reaction is monitored using absorbance measurements, the absolute absorbance value (Abs) at any time, t, is a measure of the average co_ncentrationsof all species in the cell designated as [AI, [B], [Cl, and [Dl:
€*
Absi - Abs, [AI, = [Ali -
IAE
, etc.
where
For normal stopped-flow instruments in which equal volumes of the two reactant solutions are mixed, the theoretical initial concentrations of each species, [Ali, [Bli, etc., can be assumed to be one-half the values of the original concentrations in the drive syringes (assuming equal diameter syringes), thereby allowing Absi to be calculated.’ If we first rearrange eq 10 to isolate the term X,,
x, = ( r - b)(eP‘ - 1) 2u(geP‘ - 1)
the average extent of reaction within the observation cell, %,, can be calculated by integrating eq 10a over the time interval t t tl (where tl represents the filling time, Le., the time required for the solution to flow the length of the observation cell, I):
-
+
The term on the right may be factored out and integrated in parts to yield:
Equation 16 can be solved by writing a suitable computer program to provide the best fit to the (nonlinear) experimental
Very Rapid Second-Order Reversible Kinetics data. The experimental rate constant, k12, is evaluated from the parameters p and r as previously defined:
As we have recently shown8 in treating electron-transfer kinetic data with eq 8 (i.e,, without correcting for the concentration gradient), the values of [Alo, [Blo, [Clo, and [Dlo may require small adjustments to account for slight air oxidation or other degradation prior to initiating the stopped-flow measurement. Trial values can be based on the observed “Abs” value at the time selected as t = 0, and iterative corrections can be made to improve the kinetic fit. Similar approaches can be applied when utilizing eq 16 as well.
Conclusions The mathematical approach described above assumes that mixing is complete at the time the solution enters the observation cell. That such is generally not the case has been admirably demonstrated by Dickson and M a r g e r ~ m .These ~ investigators have shown that the treatment of mixing as an additive firstorder term is sufficient to correct for mixing problems in the case of rapid reactions run under first-order conditions. A review of their work is recommended. For second-order reactions, however, the necessary correction for incomplete mixing within the reaction cell is undoubtedly more complex and, in combination with the concentration inhomogeneity problem, would be extremely difficult to characterize adequately. To the extent that incomplete mixing in the reaction cell is a problem, the derivation given above is incomplete. However, it is our contention that, for well-designed instruments, the concentration inhomogeneity problem predominates in the case of rapid second-order reactions. Ultimately, problems of mixing
J. Phys. Chem., Vol. 98, No. 48, 1994 12593 and laminar flow will place an upper limit on the reaction lifetimes for which eq 16 is applicable. The veracity of eq 16 can be tested for any specific instrument by utilizing a reaction with a known second-order rate constant (e.g., lo6 M-’ s-l) and thereafter increasing the initial reactant concentrations to reduce the half-life to values of 1 ms or less. For some instruments, it may be necessary to make an arbitrary adjustment in the value of tl based on these measurements.
Acknowledgment. This work was supported in part by the National Science Foundation under Grant CHE-92 18391. References and Notes (1) Obviously, lengthening the cell permits a lowering of the reagent concentrations and a consequent decrease of the reaction half-life. (2) The cell filling time can be accurately established by filling one of the reagent syringes with water and the other with a dye solution; by first flushing the observation cell with water and then activating the mixing device, the time required to fill the observation cell can be observed by recording the ramp signal for the absorbance increase. (3) Roughton, F. J. W.; Chance, B. In Technique ofOrganic Chemistv; Vol. VIII, Part II, 2nd ed.; Friess, S. L., Lewis, E. S . , Weissberger, A., Eds.; Interscience: New York, 1963; pp 703-792. (4) Lin, C.-T.; Rorabacher, D. B. J . Phys. Chem. 1974, 78, 305-308. (5) The limits given are based on typical systems. The actual rate constant limits which can be achieved are dependent on a number of factors including the background noise level and the mixing efficiency of the instrument utilized and the absorbance difference between reactants and products. (6) Smith, J. M. Chemical Engineering Kinetics, 2nd ed.; McGrawHill: New York, 1970; pp 60-65. (7) An experimental check on the actual concentrations of each reagent in the drive syringes can be obtained by filling the observation cell with the solution from each drive syringe in turn. (8) Meagher, N. E.; Juntunen, K. L.; Salhi, C. A.; Ochrymowycz, L. A,; Rorabacher, D. B. J. Am. Chem. SOC.1992, 114, 10411-10420. (9) Dickson, P. N.; Margerum, D. W. Anal. Chem. 1986, 58, 31533158.
