J. Phys. Chem. 1996, 100, 16925-16933
16925
Resolution of Stopped-Flow Kinetic Data for Second-Order Reactions with Rate Constants up to 108 M-1 s-1 Involving Large Concentration Gradients. Experimental Comparison Using Three Independent Approaches Brian C. Dunn, Nancy E. Meagher, and David B. Rorabacher* Department of Chemistry, Wayne State UniVersity, Detroit, Michigan 48202 ReceiVed: May 22, 1996; In Final Form: July 26, 1996X
A program has been developed for the implementation of a mathematical treatment which corrects for a concentration gradient within the stopped-flow observation cell for reversible second-order reaction kinetics studied by longitudinal absorbance measurements. This program has been tested using experimental kinetic data for three selected electron-transfer cross reactions with predicted rate constants of 1.1 × 106, 5.8 × 107, and 1.2 × 108 M-1 s-1, respectively. A second gradient-corrected approach has also been applied based on the steady-state absorbance which exists after the flow tube has been filled with the new reaction mixture just prior to the stopping of the flow (a permutation of the continuous-flow method). As a third comparison, the same data were also analyzed using a standard reversible second-order kinetic treatment, without corrections for the concentration gradient, by applying an appropriate time base correction. The experimental kinetic data were obtained using an unmodified commercial stopped-flow instrument with a 2.0 cm observation cell, a measured filling time of 3.8 ms, and a total dead time of 4.6 ms. For reactions with ∆ g 104 M-1 cm-1, all three methods have been shown to be capable of resolving second-order rate constants up to and exceeding 108 M-1 s-1 under conditions where the initial half-life is as small as 600 µs (i.e., about one-eighth the dead time). When the absorbance change becomes extremely small, the steady-state approach appears to generate the most reliable rate constant values. The most surprising observation is that the standard second-order treatmentswhich ignores the existence of a concentration gradientsyields rate constant values which are virtually identical to those obtained when the gradient correction is taken into account. The implications of this discovery are discussed. The demonstrated ability of a standard commercial stopped-flow instrument to yield accurate second-order rate constants up to 108 M-1 s-1 represents at least a 10-fold extension in the previously presumed limits for this method.
Introduction Stopped-flow spectrophotometric measurements represent the most common approach for studying the kinetics of rapid reactions in solution. A number of proposals have been made for developing faster mixing devices which diminish the mixing and filling time, thereby decreasing the time required for initiating reactions.1 For reactions of second order or higher, however, the upper limit on reaction rates which can be studied using any mixing device coupled with monitoring by spectrophotometric absorbance is ultimately limited by the minimum concentration levels which yield reproducibly measurable absorbance changes following the cessation of the flow. This, in turn, is dependent upon (i) the molar absorptivities of the reactants and products, (ii) the length of the observation cell, (iii) the mixing and filling time of the instrument, and (iv) the background noise of the detector. To improve the sensitivity of the monitoring signal, most commercial instruments provide for a long-path configuration in which the solution absorbance is monitored down a length of the flow tube. This approach results in a concentration gradient within the observation cell whenever the half-life of the reaction approaches or is less than the time required to fill the observation cell (Figure 1). Among instruments which are commercially available, typical mixing and filling times for this configuration are on the order of 3-10 ms with observation cells which are 1-2 cm long. It is generally believed that, under such conditions, second-order reactions with less than a 5-10 ms half-life will yield significant errors in the resolved rate X
Abstract published in AdVance ACS Abstracts, September 15, 1996.
S0022-3654(96)01492-X CCC: $12.00
constants since the concentration gradient which develops down the length of the observation cell during the filling process results in a “half-life gradient” as well. This is conceived to place a limit on the fastest second-order reactions which can be determined by the stopped-flow method. As we have noted earlier,2 the most common suggestion for avoiding the problems caused by a concentration gradient is to utilize pseudo-first-order conditionsssince the half-life is then constant during the entire reaction. However, this approach is actually counterproductive since, for a second-order reaction, the implementation of pseudo-first-order conditions requires a large increase in the concentration of one of the reactants which speeds up the kinetics. Monitoring the absorbance through the cell cross section is also useless, since much higher reactant concentrations are required to obtain a viable signal change, further increasing the reaction rate. In commenting on the gradient problem, Chance, one of the pioneers in stopped-flow instrumentation, has previously stated that there is “no solution to this problem.”3 This latter comment clearly refers to the absence of a physical solution, not a mathematical one. If the mixing is efficient and the flow rate within the flow tube is reasonably uniform during the filling interval, it is possible to develop a theoretical mathematical description of the concentration gradient which allows for resolution of second-order kinetic data as long as a measurable absorbance change exists. More than two decades ago, we first proposed a simple mathematical treatment for very fast second-order stopped-flow reactions involving a concentration gradient which consisted of dividing the solution into several discrete segments down the length of the observation cell.2 To simplify the mathematics, © 1996 American Chemical Society
16926 J. Phys. Chem., Vol. 100, No. 42, 1996 the expression developed required the use of stoichiometrically equal initial concentrations for both reactants and the absence of any back-reaction. In recent applications involving redox reactions, however, we have found this approach to be restrictive due to the difficulty of adjusting the two reactant concentrations to the same initial value, since one reactant is commonly subject to air oxidation or autoreduction. We also wished to develop a treatment which has sufficient generality to permit its application to reactions involving a significant amount of backreaction, including the possibility that some product might also be present at the time selected for commencing analysis of the data (the arbitrary “zero” time). On the basis of the foregoing premise, we recently reported the development of a comprehensive mathematical treatment which takes into account the concentration gradient down the length of the stopped-flow observation cell for reversible secondorder reactions involving any concentration ratio in the presence or absence of prior product formation.4 In publishing the mathematical derivation, we expressed the belief that, once a suitable program had been developed, it might prove possible to resolve second-order rate constants as large as 108 M-1 s-1 using a commercially available stopped-flow instrument. We have now developed a program to implement our comprehensive mathematical treatment and have applied it to actual stopped-flow kinetic data. As an alternative approach, we have also developed and applied an iterative steady-state analysisswhich also accounts for the concentration gradientsto the measured absorbance value obtained during the brief interval when the flow tube has been completely filled with the newly mixed reaction solution but just prior to the time when the flow is stopped. Both approaches are shown to yield comparable values of the resolved rate constant for reactions with half-lives much shorter than the instrumental dead time. For comparative purposes, we have also applied a conventional second-order kinetic treatment, which ignores the existence of the concentration gradient entirely, by adjusting the starting concentrations of the reactants and products to agree with the observed absorbance at the selected zero time. The experimental data were obtained with a conventional stopped-flow instrument for which the total dead time is 4.6 ms. The results from all three analytical approaches are shown to be comparable for three different reactions with rate constants ranging from 106 to >108 M-1 s-1 under conditions where the reaction half-life is as short as 600 µs. As the total observed absorbance change diminishes to less than 0.010, the steadystate approach appears to provide the most reliable estimate of the second-order rate constant, under the conditions used in this test. The ability of the standard second-order treatment to resolve extremely rapid kinetic data without correcting for the concentration gradient was totally unexpected. We have now shown by mathematical analysis that the average half-life and the halflife of the average concentration in the cell are similar but not identical. Computer-simulated numerical data reveal that the principal effect of a concentration gradient upon second-order kinetics is to displace the kinetic data on the time axis. The small discrepancies which exist between the gradient-corrected and standard second-order treatments under extreme conditions tend to be smaller than the experimental noise. We conclude that the standard treatment of second-order kinetic data can be applied with reasonable accuracy to systems in which large concentration gradients exist. The experimental data clearly demonstrate that any of the three methods outlined in this work can be used to extend the stopped-flow method to reactions which are up to 10 times faster
Dunn et al.
Figure 1. Schematic diagram of a stopped-flow tube with longitudinal monitoring of the absorbance. The heavy arrows show the direction of the solution flow prior to the stopping time. (The shading within the tube schematically represents the change in concentration down the length of the observation cell.) For the system utilized in the current study, the specific parameters are as follows: x ) 0.4 cm (the distance from the mixer to the observation cell), y ) 0.3 cm (the diameter of the flow tube), and l ) 2.0 cm (the length of the observation cell). The filling time of tl ) 3.8 ms represents the time required for the newly mixed solution to travel the distance l while the total dead time of 4.6 ms represents the time required for the solution to travel the distance x + l.
than the limits normally assumed to exist. For reactions in which the molar absorptivity differences between reactants and products are on the order of 104 M-1 cm-1, rate constants up to and exceeding 108 M-1 s-1 can, indeed, be resolved. Experimental Section Instrumentation. For the current study, a 25-year-old standard Durrum Model D-110 stopped-flow spectrophotometer was utilized without physical modification. As illustrated in Figure 1, the standard flow tube has an observation path length of l ) 2.0 cm, a diameter of y ) 0.3 cm, and a distance between the mixer and the entry into the observation chamber of x ) 0.4 cm. Standard Hamilton “gas-tight” syringes were utilized, and the driving ram for the mixing system was operated at the recommended pressure of 70 psi (4.8 × 105 Pa). Under these conditions, the time required to fill the 2.0 cm observation cell was experimentally determined to be 3.8 ( 0.4 ms (based on 24 repetitive measurements)5 which represents a flow rate of 5.3 m s-1. At this flow rate, the time required for the solution to flow the 0.4 cm distance from the mixer to the entry point of the observation chamber should be about 0.8 ms, leading to a total dead time of 4.6 ms. The output of the photomultiplier was interfaced to an Eltech Turbo XT computer equipped with a Metrabyte 12-bit A/D board. For the kinetic measurements, the contact on the stopping syringe was set to effect triggering of data accumulation prior to the cessation of flow so that the absorbance readings during the last portion of the flow processsafter the tube had
Kinetic Data for Second-Order Reactions
J. Phys. Chem., Vol. 100, No. 42, 1996 16927
been filled with newly mixed solutionscould be recorded. Prior to use, the voltage settings for the photomultiplier were carefully calibrated so that accurate transmittance measurements could be obtained. The digitized data (in voltage units) were transferred to an Insight 486 PC for conversion to absorbance units followed by final data manipulation and analysis. For all kinetic measurements, the cell block and drive syringes were thermostated at 25.0 ( 0.2 °C using a circulating constant temperature bath. Solutions. Solutions of RuII(NH3)5isn, RuII(NH3)5py, RuII(NH3)2(bpy)2, and NiIII([14]aneN4)(H2O)2 6 were freshly prepared immediately prior to their use in kinetic runs utilizing the procedures previously described.7 All reactions were monitored using the absorbance peak of the relevant Ru(II) reactant in the region of 408-478 nm for which the molar absorptivity is in the range of 104 M-1 cm-1 (Table 1). None of the other reactants or products exhibit significant absorbance in this region. All reactant solutions were standardized by means of absorbance measurements using a Cary Model 17-D dual-beam recording spectrophotometer. All reactions were carried out in 0.1 M HClO4 to control the ionic strength.
X0 ) [A]i - [A]0 ) [B]i - [B]0 ) [C]0 - [C]i ) [D]0 - [D]i Similarly, for any subsequent time, t
Xt ) (Abst - Abs0)/l∆
The terms “Abs” and “l” represent the measured absorbance and the length of the observation tube (see Figure 1), respectively, and ∆ represents the difference between the molar absorptivities of the products and reactants, i.e., ∆ ) C + D - A - B. The subscripts “i” and “t” designate the concentrations existing immediately upon mixing of the reactants before any reaction has occurred and at any later time, t, respectively. The value of Absi must be calculated from the theoretical initial reagent solution concentrations immediately following mixing. For the situation in which the zero time is defined as the initial mixing time (X0 ) 0), eq 2 may be modified to either of the following forms:4,10
k12t )
Mathematical Treatment for Reversible Second-Order Kinetic Data Reversible Second-Order Kinetic Behavior. The kinetic expressions which follow are based on a 1:1 reversible reaction of the type k12
A + B y\ zC+D k
where both the forward and reverse reactions are second order, that is, first order with respect to each reactant:
} ]
2aXt + b - r -1 g 2aXt + b + r
q)
)}
2aXt + b - r 2aX0 + b - r [C]e[D]e ln - ln r 2aXt + b + r 2aX0 + b + r
) k12t (2)
In this expression, Xt 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, as represented by X0, and the remaining combined constant terms are defined as follows:4
a ) [C]e[D]e - [A]e[B]e b ) -{([A]0 + [B]0)[C]e[D]e + ([C]0 + [D]0)[A]e[B]e} c ) [A]0[B]0[C]e[D]e - [A]e[B]e[C]0[D]0 r ) xb2 - 4ac The subscripts “e” and “0” refer to the concentrations at equilibrium and at time zero, respectively. If the point in time at which the reactants are mixed is not chosen as the zero time for the data analysis, we can define the extent of reaction which has occurred at the arbitrarily selected zero time as follows:
X0 ) (Abs0 - Absi)/l∆
[{
(3)
(5)
1/q
(6)
The newly introduced parameters in eqs 5 and 6 are defined as follows:
r [C]e[D]e
g)
For such a system, the following integrated expression can be derived:4,8,9
) (
} ]
ek12t )
d[A] ) k12[A][B] - k21[C][D] dt
{(
[{
2aXt + b - r 1 ln - ln g q 2aXt + b + r
or
(1)
21
(4)
b-r b+r
Equations 2-6 are completely general and apply equally well to reactions which go to completion or to those which are reversible, with or without product formation at the zero time. Integral Corrections for the Concentration Gradient. The foregoing expressions assume that, at all points in time, the reactant and product concentrations are uniform throughout the observation cell. As noted in the Introduction, however, whenever the reaction half-life is comparable to or less than the time required to fill the observation tube, a significant concentration gradient develops down the length of the cell during the filling process. An expression can be derived from eq 6 which takes into account the correction for this concentration gradient assuming (i) perfect mixing by the time the solution enters the cell, (ii) a uniform flow rate within the observation path length, and (iii) instantaneous cessation of the flow at the stopping time. The resulting gradient-corrected expression may be written in the form4
X ht )
{
}( ) {
} ( )
r-b 1 geqk12(t+tl) - 1 r-b + - 1 ln qk12t 2aqk12tl g 2a ge -1
(7)
In eq 7, tl represents the time required to fill the observation cell (i.e., the time for the newly mixed solution to flow the length l from one end of the cell to the other) and X h t is the aVerage extent of reaction down the length of the observation cell based on the defined time zero. Thus, X h t is analogous to Xt in eq 4 and may be measured experimentally in the same manner:
16928 J. Phys. Chem., Vol. 100, No. 42, 1996
X h t ) (Abst - Absi)/l∆
Dunn et al.
(4a)
For the treatment of kinetic data, the following parameters must be supplied: (i) all values, (ii) the cell filling time, tl, and (iii) all initial concentration values ([A]i, [B]i, [C]i, [D]i). The absorbance value for each digitized data point is used to calculate an X h t value which is then inserted into eq 7 to permit the evaluation of k12. A nonlinear least-squares subroutine was used to combine the k12 values from all data points to generate a best value. A more complete explanation of the mathematical derivation is provided in our previous paper.4 Analysis of the Steady-State Absorbance. When operating the stopped-flow instrument, it is possible to set the electronic triggering contact so that it activates the initiation of data acquisition prior to the time that the stopping syringe hits the stopping block. (The same feature can be implemented using a recording device equipped with a pretrigger record mode.) This makes it possible to record the absorbance of the reaction mixture between the time when the newly mixed solution has fully displaced the previous solution from the observation cell but while the solution is still in the process of flowing. Even for the fastest reactions, a constant absorbance is observed during this interval which represents a steady-state condition directly analogous to that obtained in continuous flow measurements, except that a concentration gradient exists. Measurements taken during this time interval are independent of any deceleration effect which could affect the subsequent concentration gradient following the cessation of the flow. In analyzing the absorbance value measured during the steady-state condition, we utilized a finite difference approach similar to that utilized in our earlier treatment of the concentration gradient problem.2 The expression generated was then solved by an iterative approach. If concentration measurements are based on the initial mixing time such that X0 ) 0, eq 6 can be rearranged to isolate the value of Xt:
Xt )
r - b + (b - r)eqk12t 2a{1 - geqk12t}
(8)
The absorbance measured during the time of the steady-state flow (Absss) can then be correlated to the sum of Xt values down the length of the observation cell:
Absss - Absi ) ∑Xtn∆λ
(9)
kinetic data, and this value was inserted into eq 8 to calculate the values of Xtn in each segment of the cell during the steadystate period. The value of k12 was then varied systematically until the calculated steady-state absorbance, Absss from eq 9, matched the observed steady-state absorbance value within a specified level of agreement. The value of k12 which satisfied this condition was taken as the best estimate of the rate constant. For each kinetic run, the steady-state approach represents a single-point method and, thus, is not highly sensitive. However, this is compensated by the fact that the steady-state condition can be maintained for a few milliseconds if desired, thereby allowing the absorbance value for a single run to be obtained as a time-averaged signal over a significant duration of time. This tends to average out the effects of the signal noise, an important asset when using reduced concentration levels or when studying reactions which are nearly complete by the time the reaction mixture enters the observation cell so that the signal obtained is small. Moreover, by varying the reactant concentrations on successive runs, the consistency in the rate constant value generated from this approach can be assessed in an adequate manner. Although this approach is similar to the premise utilized in generating eq 7, the steady-state condition vastly simplifies the iteration process for resolving the k12 value. Results For the purpose of providing an adequate test for the newly developed computer program based on eq 7 as well as the steady-state method represented by eqs 8 and 9, we proposed to study the kinetics of a series of reactions which were independent of pH (to eliminate at least one possible source of experimental error) and which also involved reagents with reasonable absorbance parameters. A brief survey of the literature indicated that almost no reliable kinetic data are available for reactions with rate constants above 107 M-1 s-1 which meet these criteria. Therefore, we selected a series of outer-sphere electron-transfer reactions, involving well-characterized metal complex reagents, for which approximate rate constant values could be predicted from theory. For the selected reactions, the predicted second-order rate constants span 2 orders of magnitude (106-108 M-1 s-1) with ∆ values of about 104 M-1 cm-1. The specific reactions selected for study were three closely related reactions for which the reactant properties are well characterized, all reactions being of the general type
n
where Xtn represents the extent of reaction in segment “n” and λ represents the length of the individual segments (in centimeters). For example, if the observation cell for our instrument is divided into 20 segments, each a millimeter in length, and if we account for the solution contained in the 4 mm (x) between the mixer and the entry of the reaction mixture into the observation tube, eq 9 can be treated in the specific form 24
Absss - Absi ) ∑Xtn∆λ
(9a)
n)5
The time assigned for the extent of reaction in each individual segment, Xtn, is equal to
tn ) nλ/V
(10)
where V is the flow rate of the solution in the flow tube (in cm s-1) during the filling process as determined from the measured filling time.5 In practice, an initial estimate of k12 was obtained from a normal second-order treatment of the stopped-flow
k12
RuII(NH3)xLy + NiIII([14]aneN4) y\ z RuIII(NH3)xLy + k 21
NiII([14]aneN4) (11) where RuIII/II(NH3)xLy represents RuIII/II(NH3)2(bpy)2, RuIII/II(NH3)5isn, or RuIII/II(NH3)5py.6 The potentials, self-exchange rate constants, and effective radii for all four reagents are listed in Table 1 along with the calculated equilibrium constants and ∆ values associated with each reaction. In each case, the only absorbing species at the selected wavelength is the Ru(II) complex for which the molar absorptivity value is in the range of (0.78-1.2) × 104 M-1 s-1 (Table 1). Based on the Marcus cross relationship11 (including corrections for the nonlinear and work terms), the predicted values of k12 for the three Ru(II) reagents reacting with NiIII([14]aneN4) are 1.1 × 106, 5.8 × 107, and 1.2 × 108 M-1 s-1, respectively (see Table 2).12 This was judged to provide a sufficient range of rate constants to evaluate the newly developed approaches. Reaction Involving RuIII/I(NH3)2(bpy)2. The oxidation of RuII(NH3)2(bpy)2 by NiIII([14]aneN4) was chosen for the initial
Kinetic Data for Second-Order Reactions
J. Phys. Chem., Vol. 100, No. 42, 1996 16929
TABLE 1: Physical Properties of Reagents Used in This Work As Reported in Aqueous Solution at 25 °C, µ ) 0.10 M reagent oxidant NiIII([14]aneN4) reductants RuII(NH3)2(bpy)2 RuII(NH3)5isn RuII(NH3)5py
Ef,a V
k11,b M-1 s-1
r × 108,c cm
0.95f
1.0 × 103f
3.6f
0.89g 0.387i 0.32k
8.4 × 107h 1.1 × 105j 1.1 × 105h
5.6h 3.8j 3.8h
λMax,d nm
∆ × 104,d M-1 cm-1
log K12e
488 478 408
0.934 1.19 0.778
1.01 9.52 10.7
a All potentials are referenced to NHE. b Electron self-exchange rate constant. c Effective radius in forming the outer-sphere complex preceding electron transfer. d The wavelength maxima and the molar absorptivity values are for the Ru(II) complexes; all other species do not absorb appreciably at these wavelengths so that the values listed here are equal to -∆. e Calculated equilibrium constant for reaction with NiIII([14]aneN4). f Haines, R. I.; McAuley, A. Coord. Chem. ReV. 1981, 39, 77-119. Cf.: (i) McAuley, A.; Macartney, D. H.; Oswald, T. J. Chem. Soc., Chem. Commun. 1982, 274-275. (ii) Fairbank, M. G.; Norman, P. R.; McAuley, A. Inorg. Chem. 1985, 24, 2639-2644. g Seddon, E. A.; Seddon, K. R. The Chemistry of Ruthenium; Elsevier: New York, 1984; p 444. h Brown, G. M.; Sutin, N. J. Am. Chem. Soc. 1979, 101, 883-892. i Stanbury, D. M.; Haas, O.; Taube, H. Inorg. Chem. 1980, 19, 518-524. j The value of k11 is assumed to be identical to that for the corresponding pyridyl complex, RuIII/II(NH3)5py. k Yee, E. L.; Weaver, M. J. Inorg. Chem. 1980, 19, 1077-1079.
TABLE 2: Resolution of Experimental Second-Order Stopped-Flow Kinetic Data for Selected Test Reactions As Treated by Three Independent Methods (All Data at 25 °C, µ ) 0.10 M HClO4) reaction conditions
[NiIII],a µM
[RuII],b µM
Absic (calcd)
Absssd (obsd)
theor k12,e M-1 s-1
A-1 A-2 A-3 B-1 B-2 B-3 B-4 C-1
9.32 90.0 229 10.1 10.5 22.6 44.0 6.27
3.06 28.1 73.4 6.39 6.97 6.97 6.97 5.55
0.057 0.52 1.37 0.152 0.159 0.159 0.159 0.086
0.057 0.26 0.38 0.020 0.019 0.007 0.008 0.006
1.1 × 106 1.1 × 106 1.1 × 106 5.8 × 107 5.8 × 107 5.8 × 107 5.8 × 107 1.2 × 108
resolved k12,f M-1 s-1 gradt
corrh
2.3(1) × 106 3.00(7) × 106 4.8(4) × 106 4.5 × 107 4.3 × 107 1.1 × 107 1.2 × 107 2.5(1) × 108
steady statei
uncorrj
calcd initial t1/2,g ms
(not applic) 2.2(1) × 106 2.0(1) × 106 6.0 × 107 5.0 × 107 4.7 × 107 2.5 × 107 2.09(7) × 108
2.2(1) × 106 3.1(1) × 106 5.2(2) × 106 4.1(6) × 107 4.8(5) × 107 1.1(2) × 107 0.99(4) × 107 2.6(1) × 108
37 3.8 1.5 1.9 1.8 0.75 0.37 0.60
a The nickel oxidant for all reactions is NiIII([14]aneN4) (i.e., NiIII(cyclam)). b The ruthenium reagents used are as follows: reaction A, RuII(NH3)2(bpy)2; reaction B, RuII(NH3)5isn; reaction C, RuII(NH3)5py. c Absi represents the calculated initial absorbance based on the initial concentrations immediately after mixing. d Absss represents the observed absorbance during the steady-state condition just prior to stopping the flow (see text). e The theoretical k12 value represents the second-order rate constant as calculated using the Marcus cross relation, including nonlinear and work term corrections, based on the parameters listed in Table 1; such values could be in error by up to a factor of 3 or so. f Except when italicized, the values in parentheses represent the standard deviation for the last digit shown based on repetitive kinetic trials: e.g., 2.3 (1) × 106 represents a standard deviation of 0.1 × 106; only a single kinetic trial, representing the average of 6-8 kinetic runs was conducted for reaction conditions B-1 through B-4 so that the standard deviations shown for the gradient uncorrected data treatment (in italics) merely represent the level of agreement of the individual data points to the overall kinetic fit. g The calculated initial half-life values shown here are based on the experimentally determined rate constants of 2.2 × 106, 4.5 × 107, and 2.5 × 108 M-1 s-1 for reactions A, B, and C, respectively. h The gradient-corrected calculations are based on eq 7. i The steady-state calculations are based on eqs 8 and 9. j The uncorrected calculations are based on eq 2 and ignore the existence of a concentration gradient.
studies since its predicted rate constant of k12 ) 1.1 × 106 M-1 s-1 is well within the normal stopped-flow range. Moreover, the predicted equilibrium constant for this reaction is only 10 (Table 1) so that the back-reaction becomes important in the later reaction stages (the reaction proceeding to only 75% completion under equimolar conditions). Initial measurements were carried out using reactant concentrations averaging 6 µM (reaction A-1 in Table 2) for which the predicted initial halflife of approximately 70 ms indicates that the concentration gradient is insignificant. The steady-state method is not applicable for these conditions since Absss ) Absi. The standard second-order treatment (eq 2) yielded k12 ) 2.2 × 106 M-1 s-1, in acceptable agreement with the predicted value in view of the uncertainties in the various parameters utilized in the Marcus relationship. The concentration-gradient-corrected method yielded a similar value of k12 ) 2.3 × 106 M-1 s-1, confirming the viability of eq 7 for the limiting condition where no concentration gradient exists. Two higher reagent concentration conditions were also utilized (reactions A-2 and A-3) to generate predicted initial half-lives of 3.8 and 1.5 ms based on the experimentally derived rate constant value obtained at the lowest concentration level (A-1). In both cases, the experimental kinetic data were treated using all three programs based on eqs 7, 8-9, and 2, respectively, to determine the trends in the resolved rate constant as the reaction rate increased. For condition A-2, only eqs 7-9 were expected to treat the data adequately; however, all three
approaches generated k12 values which were within acceptable agreement with the value obtained at low concentrations (A1)salthough only the steady-state approach was in complete agreement. For condition A-3, the gradient-corrected and -uncorrected treatments are obviously in significant error, being 2.1-2.4 times larger than the value obtained for (A-1). Examination of the experimental data shows that, following the cessation of flow, the absorbance is decreasing much more rapidly than the computer simulation would predict for a rate constant of 2.2 × 106 M-1 s-1. As noted above, this specific reaction does not proceed to completion, and the experimental kinetic data obtained under reaction condition A-3 represent only the last portion of the reaction where the back-reaction should be important. However, the back-reaction does not appear to account for the observed error. The error also cannot be attributed to incomplete mixing since the steady-state approach should be similarly affected, and this latter method is seen to generate an accurate k12 value for both conditions A-2 and A-3. In fact, the accuracy of the k12 value generated by the steadystate approach is particularly intriguing in view of the difficulties observed in the kinetic data following the cessation of flow. Reactions Involving RuIII/I(NH3)5isn and RuIII/I(NH3)5py. Although the kinetic measurements made on the RuII(NH3)2(bpy)2 oxidation reaction with the Ni(III) reagent, as described above, provide strong evidence of the ability of all three
16930 J. Phys. Chem., Vol. 100, No. 42, 1996
Figure 2. Curve I represents a typical averaged signal for eight successive stopped-flow runs, after conversion to absorbance, for the reaction of NiIII([14]aneN4) with RuII(NH3)5isn in aqueous solution at 25 °C under reaction conditions B-1 (see Table 2). The initial constant absorbance value represents the steady-state absorbance, Absss, during the last part of the flow. Zero on the time axis represents the point at which the flow was stopped. The solid circles (curve III) superimposed on the experimental signal represent computer-simulated absorbance data, based on k12 ) 4.5 × 107 M-1 s-1 for these same reaction conditions taking into account the concentration gradient down the length of the observation cell (with corrections for the concentration gradient prior to the entry of the reaction mixture into the observation cell). The superimposed solid curve (curve II) represents a similar computer simulation ignoring the concentration gradient. The time scale for the latter curve has been shifted 2.5 ms to the left to permit the exact overlay of the latter “theoretical” curve to the experimental data.
approaches to generate reasonable rate constant values under conditions involving half-lives as short as 3.8 ms, particularly when using the steady-state approach, the high concentration levels required to shorten the half-life of this reaction down to the millisecond level necessarily resulted in signal changes which were exceptionally large. Thus, it seemed wise to attempt similar measurements on reactions with much larger rate constants which would require smaller concentrations and, therefore, yield much smaller signals. For this purpose we studied the corresponding oxidation reactions for RuII(NH3)5isn and RuII(NH3)5py for which, as noted above, the predicted rate constants were 5.8 × 107 and 1.2 × 108 M-1 s-1, respectively. Unfortunately, these rate constants cannot be independently measured under conditions where there is no concentration gradient within the observation cell. Thus, in judging the accuracy of the resolved rate constants, we are forced to rely more heavily upon the veracity of the theoretically calculated k12 values as well as the consistency of the values obtained by applying the three independent methods. As a result of the very small absorbance changes observed for reactions B and C, the signal-to-noise ratios were very poor. To diminish the noise levels, the signals from 6-8 repetitive runs were computer-averaged before data analysis was attempted. A noise with a frequency of about 800 Hz persisted even after signal averaging (Figures 2 and 3). This noise is attributed to mechanical Vibrations arising from the activation of the drive ram since the periodicity appeared to be relatively uniform from one run to the next. For the oxidation of RuII(NH3)5isn, conditions B-1 and B-2 represent two independent measurements made on different days
Dunn et al.
Figure 3. Experimental absorbance curve (curve I) for the reaction of NiIII([14]aneN4) with RuII(NH3)5py in aqueous solution at 25 °C (C-1 in Table 2) based on the average of six repetitive stopped-flow runs. All symbols and curve designations are the same as in Figure 2. Note that the steady-state absorbance, Absss, is only 0.006 and decays to half that value within 2 ms after the flow is stopped. The time scale for the computer-simulated curve (curve II) which ignores the concentration gradient has been shifted 1.3 ms to the left to permit the exact overlay of the latter “theoretical” curve to the experimental data.
with different stock solutions to determine our ability to generate consistent data. For these reaction conditions, the anticipated initial half-life (based on the theoretical second-order rate constant of 5.8 × 107 M-1 s-1) was 1.4 ms so that a very large concentration gradient should exist. The rate constant values obtained for both the (B-1) and (B-2) data using all three methods are within reasonable agreement as shown in Table 2. When the reactant concentration levels were increased to the point where the reaction half-life was considerably less than 1 ms (B-3 and B-4), the calculations based on the time-dependent absorbance data, with or without corrections for the concentration gradient, yielded k12 values which were in error by a factor of 4 or more. Examination of the data suggests that this is due to the fact that the total observable absorbance change is only 0.007 (Table 2), and the absorbance falls to zero very quickly after the flow is stopped. Under these circumstances, it is somewhat surprising that the steady-state approach yields an excellent k12 value for (B-3). For the (B-4) reaction conditions, where the initial half-life should be on the order of 370 µs based on a rate constant of 4.5 × 107 M-1 s-1 (or 8% of the instrumental dead time), essentially the same Absss value was observed so that the steady-state k12 value is then in error by a factor of 2. We suggest that, at this point, we have exceeded the ability of even the steady-state approach to yield a reasonable rate constant value. The oxidation of RuII(NH3)5py was utilized as the ultimate test of our kinetic approach.12 Data were obtained only for reactant concentrations of 6 µM, for which the calculated initial half-life is about 600 µs and the steady-state absorbance was only 0.006. Further decreases in the reactant concentrations resulted in a signal which was too small to measure accurately while larger concentrations resulted in reaction times which were too fast to monitor. Six individual runs were averaged before treating the data by each of the three methods. With such a tiny signal change, the observation that all three approaches yielded the same average rate constant value within experimental
Kinetic Data for Second-Order Reactions
J. Phys. Chem., Vol. 100, No. 42, 1996 16931
error is remarkable. The calculated rate constant, based on three such averaged kinetic runs, was (2.1-2.6) × 108 M-1 s-1, depending upon the method employed. These values compare favorably with the predicted k12 of 1.2 × 108 M-1 s-1, the difference between the predicted and experimental rate constant being nearly identical to that observed for reaction A. Discussion Viability of Resolved Rate Constants. The studies on the RuII(NH3)2(bpy)2 oxidation reaction permit an evaluation of the second-order rate constant under classical nongradient conditions (A-1) so that a reference value is accurately established. When the reaction rate is accelerated to the point where the initial half-life is less than the instrumental dead time (A-2), an acceptably accurate k12 value is still obtained by all three methods. Even under conditions where the half-life is reduced to less than one-half the dead time (A-3), the steady-state approach continues to provide an accurate measurement of k12. For the two fastest reactions, no independent check can readily be made on the accuracy of the resolved rate constants since the reactions cannot be monitored under conditions where the concentration gradient within the cell is absent. For reaction conditions B-1 and B-2, the rate constant values obtained by all three methods are very close to the theoretically predicted rate constant, suggesting that there is validity to these measurements. The consistency in the k12 values obtained by the steadystate method, even when the calculated half-life is on the order of 750 µs (B-3), suggests that this latter approach continues to be applicable even for extremely fast reactions. The errors obtained for the other two methods at the higher concentration levels is at least partly attributable to the fact that the total observable absorbance change is only about 0.007 (or about 1.5% transmittance change), and most of that change occurs within the first 2 ms after the cessation of the flow. Thus, there is almost no signal change to monitor. The absorbance measurements obtained for reaction B-1 are illustrated in Figure 2 where curve I represents the averaged signal from eight successive runs for which the calculated initial half-life is 1.9 ms or about 40% of the time required for the solution to travel from the mixer to the far end of the observation cell. The superimposed solid curve (III) represents the computersimulated absorbance values for k12 ) 4.5 × 107 M-1 s-1 with corrections for the concentration gradientsassuming a flow rate of 5.3 m s-1 and accounting for the 4 mm distance from the mixer to the entry point of the observation cell. The solid circles (II) represent similar computer-simulated data ignoring the concentration gradient. In order to achieve this latter overlay, however, the time scale for this curve has been shifted by 2.5 ms to the left. For the fastest reaction studied (C-1), the remarkable agreement among all three methods must be considered to be somewhat fortuitous given the fact that the initial half-life is only 13% of the dead time, and the total observable absorbance change is only 0.006. The experimental absorbance curve (I) for one data set, representing the computer-averaged signal from six successive runs, is illustrated in Figure 3. The computersimulated absorbance curves are analogous to those in Figure 2. In the case of the solid circles (II), representing the simulation in the absence of a concentration gradient, however, a shift in the time axis of 1.3 ms to the left was required to achieve a perfect overlay. Effect of the Concentration Gradient upon the Experimental Data. The difference in the k12 values obtained by application of eqs 2 and 7, as listed in Table 2, is largely attributed to the fact that, in using eq 2, the data were truncated
Figure 4. Solid curves shown here represent simulated kinetic data, as generated from eq 7, with various concentration gradient corrections representing filling times from tl ) 0 to 32 ms as indicated. All data are calculated for a hypothetical second-order reaction for which k12 ) 5.00 × 107 M-1 s-1; Keq ) 1000 with starting reactant concentrations of [A]i ) 10.0 µM and [B]i ) 10.1 µM (yielding an initial half-life of 1.97 ms). Absorbance values are based on the following parameters: ∆ ) 1.00 × 104 M-1 s-1, l ) 2.00 cm, and x ) 0.40 cm. The solid circles represent simulated kinetic data generated from eq 2 (i.e., ignoring the concentration gradient) for the same hypothetical reaction and are seen to superimpose exactly on the corresponding solid curve for tl ) 0 as anticipated. The solid squares represent the same hypothetical data generated from eq 2 except that the time base has been shifted 1.8 ms to the left in an attempt to overlay these data on the solid curve for tl ) 4 ms (representing conditions similar to those of our instrument). The solid triangles represent a 5.7 ms shift in an effort to overlay these same data on the solid curve for tl ) 16 ms; significant deviations are observed in this last case, illustrating the nonequivalence of the half-life of the aVerage concentration and the aVerage half-life of all concentrations (see text).
toward the end of the reaction whereas eq 7 (which was solved iteratively) was applied to the data up to the point where equilibrium was attained. As shown by the computer-simulated curves in Figures 2 and 3, the time dependence of the absorbance data appears to be virtually identical, with and without corrections for the concentration gradient, except for a displacement of the time axis. This observation would appear to suggest that, for second-order reactions, the aVerage half-life down the length of the concentration gradient is virtually identical to the halflife of the aVerage concentration within the obserVation cell. To test the actual impact of the concentration gradient upon second-order kinetic behavior in the absence of experimental variables, computer-simulated data were generated using eq 2 (standard second-order treatment) and eq 7 (gradient-corrected treatment) based on a hypothetical reaction in which k12 ) 5.00 × 107 M-1 s-1 and Keq ) 1000 with initial concentrations (upon mixing) of [A]i ) 10.0 µM and [B]i ) 10.1 µM (∆ ) 1.0 × 104 M-1 cm-1). The solid curves shown in Figure 4 represent the theoretical trend in absorbance for this reaction as predicted by the gradient-corrected expression (eq 7) based on several different cell filling times ranging from 0 to 32 ms. (Increasing the filling time corresponds to the development of a larger concentration gradient.) The solid circles in Figure 4, representing the absorbance behavior as predicted by the standard secondorder treatment (eq 2), are seen to superimpose exactly on the solid curve for the case involving zero filling time (top curve)
16932 J. Phys. Chem., Vol. 100, No. 42, 1996 as would be anticipated if eqs 2 and 7 are both valid. The curve represented by solid squares in Figure 4 is identical to the solid circle curve except that the time base has been shifted to the left by 2.5 ms so that it overlays the gradient-corrected curve for a 4 ms filling time (which essentially represents the experimental condition for our instrument). The fact that the latter two curves appear to superimpose exactly indicates that the standard second-order treatment (eq 2) can be applied to experimental data under these conditions provided that an adjustment is made in the initial starting concentrations of the reactants and products in accordance with the absorbance value which is observed at the time when the flow is stopped. Further displacement of the standard second-order curve (by 5.7 ms to the left) has been made in the case of the solid triangle curve in Figure 4 in an attempt to overlay this curve on the gradient-corrected curve corresponding to a 16 ms filling time. Careful examination reveals the fact that these curves do not superimpose exactly. Thus, as the concentration gradient continues to increase, the standard second-order treatment eventually deviates perceptibly from the gradient-corrected curve. Mathematical analysis of the half-life behavior predicted for simplified second-order reactions, in which the initial concentrations of the two reactants are stoichiometrically equal and for which there is no back-reaction, reveals that the average halflife within the concentration gradient is not identical to the halflife of the average concentrationsas would be required if data generated by eq 2 could simply be displaced in time to generate eq 7.13 Nonetheless, the two half-life behaviors are similar as suggested by Figure 4. In fact, it appears likely that, under most experimental conditions, the second-order rate constant obtained by application of eq 2 (in which the concentration error is ignored) will agreeswithin the limits of experimental errorswith the value obtained using the concentration-gradientcorrected approach (eq 7). Sources of Error. For reaction conditions A-2 and A-3 (Table 1), the rate constant values obtained with and without gradient correction (eqs 7 and 2, respectively) tend to deviate together as the half-life decreases while the value obtained by the steady-state approach (eqs 8 and 9) remains relatively constant. This suggests that the first two methods are subject to a common error which does not affect the steady-state approach. It is possible that this error may result from the deceleration which occurs as the flow is stopped, since this process cannot be carried out instantaneously. Such a deceleration would increase the concentration gradient down the length of the observation cell upon stopping the flow, relative to the gradient which exists during the period while the flow is continuing. Although a similar trend was not observed for system B, no definitive conclusions can be formulated relative to this latter system due to the extremely small absorbance change observed for conditions B-3 and B-4. The foregoing discussion has also assumed that the reactants are perfectly mixed at the time they emerge from the mixer. Dickson and Margerum14 have carefully examined the problems of incomplete mixing and have formulated an expression to make suitable corrections for mixing as applied to first-order (or pseudo-first-order) reactions. These authors have, in fact, reported that small mixing corrections can become necessary for reactions with half-lives below 7 ms in the case of some instruments. Presumably, a similar correction could be made in the case of second-order reactions. In the studies described in the current work, however, the consistency of the rate constants generated by the steady-state approach suggests that mixing errors are not significant with our instrument for reactions with initial half-lives longer than about 1 ms.
Dunn et al. Conclusions and Recommendations As Dickson and Margerum have noted, some stopped-flow instruments are much worse than others in terms of the mixing error. Therefore, when studying very fast second-order reactions via stopped flow, the characteristics of the instrument should be checked carefully using selected test reactions. As outlined above, initial data analysis can be carried out using the standard second-order treatment afforded by eq 2, assuming that the mixing and flow characteristics of the instrument can be ignored. The consistency of the data obtained with any given stoppedflow instrument should then be checked by utilizing a test system with a second-order rate constant on the order of (1-4) × 106 M-1 s-1. The reactant concentrations should be incrementally increased until the initial half-life is diminished to the level which may be encountered with other reactions for which the same instrument is to be utilized. The comparable results obtained in this study with all three methods show that any one of the three approaches described herein may be used to resolve stopped-flow kinetic data for second-order reactions with half-lives that are less than onehalf the instrumental dead time or smaller. In view of these results, there appears to be no advantage in utilizing eq 7 (despite the thoroughness of its approach) for resolving kinetic data involving concentration gradients in the stopped-flow cell, except as a check on the veracity of rate constant values obtained by the other two approaches. As noted above, the computersimulated data in Figures 2-4 demonstrate that the concentration gradient can virtually be compensated by shifting the time axis for the standard second-order treatment so that the initial reactant concentrations are adjusted to agree with the absorbance value observed at the point where the flow is stopped. Any suitable time after the cessation of flow may be selected as the zero time for commencing data analysis. As the initial half-life approaches or becomes less than the instrumental dead time, it is recommended that the steady-state approach also be applied to the data as a check on the veracity of the rate constant obtained using eq 2. This requires a determination of the flow rate during the filling process; the distance from the mixer to the entry to the observation cell must also be known. When the rate constants generated by these two methods begin to diverge, the steady-state method may be regarded as the more reliable approach. However, to preserve confidence in the results obtained with the steady-state method, a variety of initial reactant concentrations should be tested insofar as the signal-to-noise limitations may permit. The failure of the steady-state method may imply the onset of mixing problems. As in the case of reaction B-4 in the current studies, the observable absorbance change may also become so small for reactions with half-lives below about 1 ms that the error in the rate constant resolved by the steady-state approach may simply be attributed to the obvious signal-to-noise limitations. We have demonstrated that the foregoing approach may allow for the evaluation of second-order rate constants up to and exceeding 108 M-1 s-1 with surprising accuracy. This raises the limit for stopped-flow measurements of second-order rate constants by at least an order of magnitude over the limits which have heretofore been assumed to apply. As a result, a vast array of second-order reactions, which were previously thought to be beyond the normal stopped-flow range, should now be accessible for kinetic investigations. Acknowledgment. Financial support of this work by the National Science Foundation (Grant CHE-9218391) is gratefully acknowledged. The authors express their appreciation to Professor Paul T. Young of the City College of Charleston, SC,
Kinetic Data for Second-Order Reactions for his informative discussions regarding the general integration properties of linear and nonlinear continuous functions. The authors also acknowledge the contributions of Qiuyue Yu for assistance in synthesizing the ruthenium reagents used in this work. References and Notes (1) (a) Several novel mixing approaches are discussed in: Rapid Mixing and Sampling Techniques in Biochemistry; Chance, B., Eisenhardt, R. H., Gibson, Q. H., Lonberg-Holm, K. K., Eds.; Academic Press: New York, 1964. (b) Jacobs, S. A.; Nemeth, M. T.; Kramer, G. W.; Ridley, T. Y.; Margerum, D. W. Anal. Chem. 1984, 56, 1058-1065. (c) Nemeth, M. T.; Fogelman, K. D.; Ridley, T. Y.; Margerum, D. W. Anal. Chem. 1987, 59, 283-291. (d) Regenfuss, P.; Clegg, R. M.; Fulwyler, M. J.; Barrantes, F. J.; Jovin, T. M. ReV. Sci. Instrum. 1985, 56, 283-290. (e) Demyanovich, R. J.; Bourne, J. R. Ind. Eng. Chem. Res. 1989, 28, 830-839. (f) Engh, S. A.; Holler, F. J. Anal. Chim. Acta 1989, 224, 211-224. (2) Lin, C.-T.; Rorabacher, D. B. J. Phys. Chem. 1974, 78, 305-308. (3) Chance, B. In Techniques of Chemistry. Volume VI: InVestigation of Rates and Mechanisms of Reactions. Part II: InVestigation of Elementary Reaction Steps in Solution and Very Fast Reactions, 3rd ed.; Hammes, G. G., Ed.; Wiley-Interscience: New York, 1974; p 14. (4) Meagher, N. E.; Rorabacher, D. B. J. Phys. Chem. 1994, 98, 12590-12593. (5) The cell filling time was determined by filling one drive syringe with 0.01 M HCl plus phenolphthalein indicator while the other syringe was filled with 0.05 M NaOH. After flushing the cell with either solution, the drive cylinder was activated, and the time required for the transmittance to decrease from 100% to its final constant value was monitored. For recording such measurements the photomultiplier was connected directly to a recording oscilloscope to permit pretrigger record so that the entire filling sequence could be observed. (6) Ligand abbreviations are as follows: isn ) isonicotinamide, py ) pyridine, bpy ) 2,2′-bipyridine, [14]aneN4 ) 1,4,8,11-tetraazacyclotetradecane (cyclam). (7) Meagher, N. E.; Juntunen, K. L.; Salhi, C. A.; Ochrymowycz, L. A.; Rorabacher, D. B. J. Am. Chem. Soc. 1992, 114, 10411-10420. (8) Smith, J. M. Chemical Engineering Kinetics; 3rd ed.; McGrawHill: New York, 1981; pp 64-67. Cf.: (a) Pladziewicz, J. R.; Lesniak, J.
J. Phys. Chem., Vol. 100, No. 42, 1996 16933 S.; Abrahamson, A. J. J. Chem. Educ. 1986, 63, 850-851. (b) King, E. L Int. J. Chem. Kinet. 1982, 14, 1285-1286. (9) If the reaction goes to completion, an even simpler second-order expression may be utilized as presented in any standard kinetics textbook:
([A]0 - [B]0)-1 {ln([B]0[A]t) - ln([A]0[B]t)} ) k12t (see, e.g.: Steinfeld, J. I.; Francisco, J. S.; Hase, W. L. Chemical Kinetics and Dynamics; Prentice-Hall: Englewood Cliffs, NJ, 1989; p 9). (10) If the reverse reaction is not significant, ln g ) 0 and g ) 1 in eqs 5 and 6 (see ref 9). (11) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265322. (12) A fourth reaction involving the oxidation of CoII(bpy)3 with RuIII(NH3)2(bpy)2, for which the calculated rate constant was 2.5 × 108 M-1 s-1, was also investigated. For this latter reaction, we were unable to obtain a measurable absorbance signal even during the steady-state region. This suggests that the rate constant for this latter reaction is actually g5 × 108 M-1 s-1, which exceeds the limits of detection for the current instrumentation. (13) The aVerage half-life down the length of the concentration gradient can be expressed as
ht 1/2 )
k A0 (a - b) 2
2
)]
[(
(A0kb + 1)(A0ka + 1)
ln
A0ka + 1 A0kb + 1
whereas the half-life of the aVerage concentration within the obserVation cell is equal to
t1/2(A h t) ) ln
b-a [A]0kb + 1
(
)
[A]0ka + 1
where a and b represent the times required for the solution to flow from the mixer to the near and far ends of the reaction cell, respectively. The two expressions are seen to contain similar terms but cannot be made identical. (14) Dickson, P. N.; Margerum, D. W. Anal. Chem. 1986, 58, 3153-3158.
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