In the Laboratory
A Novel Experiment in Chemical Kinetics: The A B → C Reaction System B. R. Ramachandran and Arthur M. Halpern Department of Chemistry, Indiana State University, Terre Haute, IN 47809 A discussion of the kinetics of consecutive reactions belonging to the type A B → C is always included in the undergraduate physical chemistry curriculum; physical chemistry textbooks invariably have a section dealing with such systems. In those discussions, the kinetic treatment is often simplified by invoking the steady-state or preequilibrium approximations. Sometimes, the exact rate equations, which are more complex, are also alluded to. Unfortunately, the associated physical chemistry laboratory course seldom includes experiments illustrating such a reaction mechanism. A number of papers that deal with the kinetics of these processes have appeared in this Journal (1–6), but none of them describe an actual experiment that can be studied in the laboratory. Most of those papers (1–5) only test the validity of the steady-state and preequilibrium approximations in the kinetic analysis. A recent paper (6) nicely demonstrates a method for extracting the rate constants from experimental data obtained for such reaction systems; however, the authors illustrate the method using computer-generated, error-dispersed data, and not data from a “real” experiment. Here, we describe an experiment involving a two-step, reversible, pseudo-first-order reaction that can be conveniently performed in a physical chemistry or biochemistry laboratory course. We will discuss the kinetic analysis of such a general system first, before describing the specific reaction. Kinetics of a Two-Step, First-Order Consecutive Reaction We consider a general reaction scheme in which the reactant R reversibly forms an intermediate I, which in turn is converted to the product P. We assume that each elementary step in the scheme is first order in the corresponding reactant species: R
k1 k{1
2. If (k{1 + k2) >> k 1, then the steady state approximation will apply. Again, the integrated rate equations are single exponential in t (5). 3. If neither of the above conditions is applicable, the exact solutions to the differential equations must be used; they are presented in eqs 4–6.
R t =
[I] t =
d[R] = {k 1[R] + k {1[I] dt
(1)
d[I] = k 1[R] – k {1 + k 2 [I] dt
(2)
d[P] = k 2[I] dt
(3)
We also assume that only species R exists at the beginning of the reaction; that is, [R]t=0 = [R]o, [I]t=0 = 0, and [P]t=0 = 0. Three kinetic scenarios may be considered in such a system, depending on the relative magnitudes of the three rate constants: 1. If (k 1 + k {1) >> k2, then the equilibrium implied in the first step would be established before the second step, and the preequilibrium approximation will apply. The solutions to the differential equations (eqs 1–3) then are single exponential functions in t (5).
R 0k 1 λ2 – λ1
e {λ1t – e {λ2t
(4)
(5) (6)
where
λ1 =
k –λ p–q p +q , λ2 = , and A = 1 1 λ2 – k 1 2 2
(7)
and
p = k 1 + k {1 + k 2 and q =
p 2 – 4k 1k 2
(8)
It should be noted that λ1 is always less than λ 2, since both p and q are positive. From these equations it can be seen that R shows a double exponential decay (with decay constants λ1 and λ 2), and the intermediate I shows an initial buildup followed by a decay. The values of λ1 , λ 2, and A can be estimated from the [R](t) curve using an ‘”exponential stripping” method (6, 7), and then be further refined by a nonlinear regression procedure. k1, k{1, and k2 can be obtained from λ1, λ2, and A using eqs 9:
k1 =
The coupled differential equations for the rate of change in the concentrations of the three species involved are given by
e {λ1t + A e {λ2t
λ2 – λ1
[P](t) = [R]0 – [R](t) – [I](t)
k2
I→P
R 0 λ2 – k 1
Aλ2 + λ1 λλ , k 2 = 1 2 , and k {1 = λ1 + λ2 – k 1 – k 2 (9) A +1 k1
It should be mentioned that a successful determination of the three rate constants depends on the extent of difference between λ1 and λ2, and the magnitude of A. If the values of k1, k-1, and k2 are such that λ1 and λ2 are not too different from each other, and/or if A is very large (or very small) relative to unity, the decay of R approximates to a single exponential function. Thus, both the choice of an appropriate chemical system as well as favorable experimental conditions are very important in being able to perform a complete kinetic analysis in which k1, k{1 , and k2 are obtained. Obtaining the Rate Constants Two methods may be considered for analyzing the observed kinetic data. In principle, all three rate constants can be obtained from an observation of the double exponential decay curve of the reactant, R(t), even if the initial concentration of the reactant is not known precisely. Such a decay curve is characterized by three parameters, λ1, λ2, and A (see eq 4), which may be used to obtain the three rate constants (see eq 9).
Vol. 74 No. 8 August 1997 • Journal of Chemical Education
975
In the Laboratory In the first method, which has been referred to as “exponential stripping” (6, 7), the double exponential decay curve is broken down into two time regimes: long and short. Since in eq 4, λ 2 > λ1 , it follows that the contribution from the faster component [i.e., the exp({λ 2t) term in [R](t)] becomes less significant at long times. Thus, in this time regime [R](t) approaches single exponential decay. The decay constant, λ1 , can be obtained from the negative slope of the ln [R] vs. t plot. An extrapolation is then made of this straight line to earlier times, and the difference between the observed values of [R](t) and the extrapolated ones is determined for several time points. These differences, ∆[R], are then cast into a semilogarithmic plot, that is, ln ∆[R] vs. t, from which slope λ2 is obtained. The amplitude ratio A is then found from the t = 0 intercepts of the long-time regime and the subtracted semilogarithmic plots (i.e., along and ashort, respectively) using the relationship A = exp(ashort – along). Note that knowledge of [R]0 is not required in this analysis. This is particularly significant because, since the reaction described here is followed spectrometrically, one does not have to carry out a separate determination of the molar absorptivity of the reactant. In the second approach, optimized values of the parameters λ1 , λ2, and A can be obtained globally from a four-parameter (λ 1, λ 2, A, and exp along) nonlinear regression analysis of the double exponential [R] vs. t curve using eq 4.1 However, such a procedure requires initial “seed” values of the four parameters to be optimized. These initial guesses are important because one desires the regression algorithm to locate the absolute minimum in a highly complex error surface that is a function of the parameters λ 1, λ2 , A, and exp along. This result ensures that the minimization algorithm returns the “true”, optimized values of the parameters. In practice, however, the regression procedure may become trapped in a local minimum in the error surface, which may be far removed from the global minimum, and consequently return spurious values of the parameters. In such a situation, the parameters returned will depend on the values chosen for the initial estimates. This difficulty can be circumvented if the initial seed values are reasonably close to their true values. The values obtained from a preliminary, exponential stripping method can be used as excellent seed values for the nonlinear regression. Creating the semilogarithmic plots, and performing linear regression on them, as described in the exponential stripping method, can be carried out easily using common software utilities such as EXCEL, LOTUS, QUATTRO PRO, etc. However, more advanced, yet accessible, software products such as RS/1, Sigma Plot, or Mathcad are required for nonlinear regression analysis. Some of these utilities will also give the standard deviations of the optimized parameters. In this work, we used RS/1, a scientific and statistical spreadsheet (BBN Domain Corporation, Cambridge, MA 02140). Application to a Specific Reaction In illustrating the principles outlined above, it is important to identify a chemical reaction that cleanly exhibits reversible, consecutive kinetics. We describe one such reaction here that satisfies several pedagogical purposes. It is the reduction of the chromate ion by glutathione at nearneutral pH’s. In this redox reaction, the tripeptide glutathione, GSH, (γ-L-glutamyl-L-cysteinylglycine), is oxidized to glutathionyl disulfide, GSSG, and Cr(VI) is reduced to Cr(III), as shown in eq 10. 2 CrO42{ + 6 GSH + 10 H+ → 2 Cr 3+ + 3 GSSG + 8 H2O (10)
976
Since this reaction is believed to be involved in the toxicity and carcinogenicity of chromium(VI), its kinetics and mechanism have been the subject of numerous research investigations (8–10). The reaction involves the reversible formation of a chromium(VI) thioester intermediate from chromium(VI) and GSH, and a subsequent redox step between this intermediate and a second molecule of GSH that results in the ultimate products, Cr(III) and GSSG (8). Under excess GSH, all three kinetically important steps (the forward and reverse reactions in the first step, and the reaction in the second step) are pseudo-first-order in nature, and the reaction system is described by the general scheme A B → C. Thus in the previous discussion, R, I, and P refer to Cr(VI), the Cr(VI)-thioester intermediate, and (probably) Cr(III), respectively. A nice feature of this reaction is that Cr(VI) (the reactant, R) and the thioester intermediate (the I) have reasonably different absorption spectra, rendering the spectrometric study of the reaction very easy and convenient. The decay of Cr(VI) can be followed spectrometrically by monitoring its absorbance in the near UV at 370 nm. The evolution and fall of the intermediate can be followed at 430 nm; however, since Cr(VI) has a small, but finite, absorption at that wavelength, the A 430 nm vs. t curve shows a nonzero t = 0 intercept (contrary to the zero intercept expected from eq 5). Consequently, a quantitative analysis of that curve requires a correction, which will not be discussed in this paper. The three pseudo-first-order rate constants (k1, k{1, and k2 , above), besides being dependent on the concentration of GSH and reaction temperature, are also highly sensitive to pH, nature of buffer, and buffer concentration (10). Thus, the reaction conditions must be chosen judiciously in order to observe a well-resolvable double exponential decay (see eq 4) of Cr(VI). These experimental details are presented below. Experimental Details Glutathione (Aldrich), K2Cr 2O 7 (J. T. Baker, “Baker Analyzed”), and K2 HPO4 (Merck, Reagent Grade) were used without further purification. Spectrometric data were obtained using Spectroscope, a miniaturized, modular, fiber optic, diode array spectrometer (Ocean Optics, Dunedin, FL). This apparatus permitted us to monitor the reaction at 370 and 430 nm simultaneously. The dip probe, which is connected to the rest of the Spectroscope through a fiber optic cable, is immersed in the analyte solution taken in a reaction vessel. A standard, temperature-controlled, 1-cm cell may be used in a conventional spectrophotometer. Since aqueous solutions of glutathione are susceptible to slow oxidative degradation, they must be prepared just prior to use. The following reaction conditions may be used to obtain satisfactory kinetic data for the reactant and the intermediate: [Cr(VI)]0 = 1.0 × 10{4 M; [GSH]0 = 5.0 × 10{3 M; [K2HPO4 ] = 5.0 × 10{2 M; pH = 6.0; temperature = ambient. In our kinetics runs, we used a 4-dram vial with a onehole screw cap as the reaction vessel. Five milliliters of 8.0 × 10{3 M GSH, 1.0 mL of 0.4 M K2HPO4 , and 1.5 mL of 5.0 × 10{3 M HCl were placed in the vial and the pH of the contents was adjusted to 6.0 by adding minute amounts of NaOH or HCl. The vial was clamped in a constant-temperature bath (25.0 ± 0.1 °C in our experiments), and the dip probe of the spectrometer was introduced into the solution. After allowing about 10 min for the attainment of thermal equilibrium, 500 µL of a thermally equilibrated 1.6 × 10{3 M K2 Cr2O7 stock solution was quickly injected into the vial, and data acquisition was simultaneously started. The contents were kept stirred with a tiny magnet before and during the reaction. We collected the absorbance data for about
Journal of Chemical Education • Vol. 74 No. 8 August 1997
In the Laboratory 40 min; about 1100 data points were acquired during that period. We recommend that the data should be collected at least for about 2 half-lives of the reactant (as monitored by the absorbance at 370 nm). Results and Discussion The time dependence of the Cr(VI) absorbance, monitored at 370 nm (A370) is shown in Figure 1. The same curve is presented in the semilogarithmic form, ln A370 vs. t, in Figure 2, to show more clearly the double exponential nature of the decay. Figures 1 and 2 also portray the formation and decay of the thioester intermediate, monitored at 430 nm. As described earlier, the exponential stripping method was used to determine the initial estimates of the parameters λ 1, λ2, and A. A straight-line fit on the long-time, linear portion (t > 1000 s) of the ln A370 vs. t plot returned values of along = {1.458 (σ = 1.9 × 10{3 ) and λ 1 = 4.35 × 10{4 (σ = 1.0 × 10{6 ) s{1, respectively; along is the t = 0 intercept of this line. The standard deviation and the correlation coefficient of regression were 9.7 × 10{3 and .996, respectively. This fit is also shown in Figure 2. The decay due to the fast component alone was then calculated by subtracting exp ({1.458 – 4.35 × 10{4 t) from A370 in the short time regime (t < 350 s) of the decay. A straight-line fit of the logarithm of this fast decay component gave the values ashort = {1.127 (σ = 3.1 × 10{3) and λ 2 = 6.45 × 10{3 (σ = 1.7 × 10{5) s{1, respectively; ashort is the t = 0 intercept of the fast decay component. The standard deviation and the correlation coefficient of regression were 1.8 × 10{2 and .999, respectively. Figure 2 shows this fit also. From the values of ashort and along, the value of A, which is given by the relation exp (ashort – along ), was calculated to be 1.39 (σ = 5.0 × 10{3 ). These estimates (λ1 , λ2 , A, and exp along ) were then used as the seed values in a nonlinear regression analysis of the overall decay, A370 vs. t, using eq 4. The following globally optimized values of the three parameters of interest were obtained: λ 1 = 4.38 × 10{ 4 (σ = 1.0 × 10 { 6) s {1 ; λ 2 = 6.51 × 10 {3 (σ = 1.9 × 10{5) s{1; A = 1.40 (σ = 3.1 × 10{3). The standard deviation of regression of this fit is 1.6 × 10{3. This optimized fit is also included in Figures 1 and 2. Equation 9 was then used to calculate the three pseudo-first-order rate constants of interest in this reaction, k1, k{1, and k2 . We obtained k1 = 3.98 × 10 {3 (σ = 1.2 × 10{5) s{1; k{1 = 2.25 × 10{3 (σ = 2.2 × 10{5) s{1; and k2 = 7.17 × 10{4 (σ = 3.0 × 10{6 ) s{1. These results are summarized in Table 1. The table also includes the values of the three rate constants calculated from the estimates of λ1 , λ2 , and A from the exponential stripping method. It can be seen that the values agree very well with those obtained from the global nonlinear regression analysis. It was mentioned above that the Cr(VI)-thioester intermediate (I) shows an initial evolution followed by a decay, as described by eq 5. To evaluate this complex decay curve quantitatively, we used the values of λ1 and λ 2 obtained above and carried out a fit (after incorporating the correction for the absorption of Cr(VI) at 430 nm). This result is superimposed on the experimental evolution–decay curves shown in Figures 1 and 2. Conclusion The reduction of Cr(VI) by the tripeptide glutathione follows a two-step, consecutive reaction mechanism and exhibits a fascinating kinetic behavior. Despite its complex kinetics it is highly amenable to analysis, and is thus suitable as an undergraduate physical chemistry or biochemistry experiment. It offers the student an opportunity to study in the laboratory an interesting and important biochemical
Figure 1. Absorbance vs. time plots in the reduction of Cr(VI) with GSH at 25.0 °C in K 2 HPO4–HCl buffer at pH 6.0. [Cr(VI)]0 = 1.0 × 10{4 M; [GSH]0 = 5.0 × 10{3 M; [K 2HPO4] = 5.0 × 10 {2 M. A: Decay of Cr(VI) monitored at 370 nm. — Experimental data; ... best fit obtained using eq 4 and the global nonlinear regression analysis. B: Initial buildup followed by decay of Cr(VI)-GSH thioester intermediate monitored at 430 nm. — Experimental data; ... fit using eq 5.
Figure 2. Ln (absorbance) vs. time plots illustrating application of exponential stripping method to analyze kinetic data shown in Fig 1. A: Decay of Cr(VI). — Experimental data; ... best fit obtained using global nonlinear regression analysis. B: — Initial buildup followed by decay of thioester intermediate; ... fit using eq 5. L: ... Best straight line for long-lived component of the decay of Cr(VI). S: Short-lived component. — Subtracted decay, ∆R vs. t ; ... straight line fit.
Table 1. Results for Decay Parameters and Pseudo-FirstOrder Rate Constantsa Method Parameter
Exponential Stripping (SD)
Global Nonlinear Regression (SD)
λ1/s{1
4.35 × 10{4 (1.0 × 10{6)
4.38 × 10{4 (1.0 × 10{6)
λ2/s{1
6.45 ×
6.51 × 10{3 (1.9 × 10{5)
A k 1/s{1 k {1/s{1 k 2/s{1
1.39 (5.0 ×
10{3)
3.93 ×
10{3
2.24 ×
10{3
7.13 ×
10{4
10{3
(1.7 ×
10{5)
1.40 (3.1 × 10{3)
(1.1 ×
10{5)
3.98 × 10{3 (1.2 × 10{5)
(2.1 ×
10{5)
2.25 × 10{3 (2.2 × 10{5)
(3.0 ×
10{6)
7.17 × 10{4 (3.0 × 10{6)
a Values
for k1 , k{1, and k 2 were calculated from values for decay parameters using eq 9.
Vol. 74 No. 8 August 1997 • Journal of Chemical Education
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In the Laboratory reaction that clearly illustrates the principles presented in lecture and in the text book. Note 1. It can be seen that eq 4 can also be written as [R](t ) = {[R]0/( A + 1)} {exp({λ1t ) + Aexp({λ2t )}. Thus, if [R] 0 (obtained from the reactant absorbance at t = 0) were known precisely, a threeparameter fit (λ1,λ2, and A) is sufficient in the data analysis. Usually, however, there is a time lag between sample mixing and data acquisition that prevents the true absorbance at t = 0 from being measured directly. For example, the first data point in the experimental run illustrated in this paper corresponds to t = 12.1 s. Thus the coefficient in eq 4 is treated as an additional parameter.
978
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Journal of Chemical Education • Vol. 74 No. 8 August 1997
