268
Anal. Chem. 1984, 56,268-270
(21) Weetall, Howard H. "Methods in Enzymology"; Mosbach, K., Ed.; Academic Press: New York, 1976; Voi. 44, Chapter 10, pp. 134-148. (22) Kipiniak, Walerian J . Chromafogr. Sci. i981, 19, 332-337. (23) Patka, Johann, personal communication Biochemie GmbH Kundl, Austria, Jan 25, 1983. (24) White, Roderick E.; Carroll, Margaret A.; Zarembo, John E; Bender, Douglas A. J . Anfibiof. 1975, 28, 205-214. (25) Kaiser, R.; Gottschalk, G. "Elementare Tests zur Beurtellung von Messdaten"; Bibliographisches Institut A G Mannhelm, 1972; Chapter 9.
(26) "Wissenschaftliche Tabellen", 7th ed.; Ciba-Geigy Ltd.: Basel, 1971; pp 32-35.
(27) Knecht, J.; Stork, G. Z . Anal. Chem. 1974, 270, 97-99. (28) Abraham, E. P.; Waley, S. G. "Beta-Lactamases"; Hamilton-Miller, J. M. T., Smith, J. T., Ed.; Academlc Press: London, 1979; Chapter 11, p 334. (29) Abraham, E. P.; Waley, S.G. "Beta-Lactamases"; Hamilton-Miller, J. M. T., Smith, J. T., Ed.; Academic Press: London, 1979; Chapter 11, p 324. (30) Whlte, Roderlck E.; Zarembo, John E. J . Anfibiot. 1981, 3 4 , 836-844.
RECEIVED for review August 2, 1983. Accepted October 6, 1983.
Determination of First-Order Reaction Rate Constants by Flow Injection Analysis Joseph T. Vanderslice,* Gary R. Beecher, and A. Gregory Rosenfeld' Nutrient Composition Laboratory, Beltsville H u m a n Nutrition Research Center, United States Department of Agriculture, Beltsville, Maryland 20705
It Is shown that flrst-order rate constants can be easily determlned by flow Injection analysis by conslderlng the additlonal term In the laminar flow dlffuslon-convectlon equation. The derlvatlon of the necessary equation Is staightforward and the resultlng expression is used to determine the pseudoflrst-order reaction rate for the permanganate oxldatlon of benzaldehyde. With KMnO, as the rate-llmltlng reagent and all other reactants In excess, the rate constant was determlned to be (1.05 f 0.02) X 10-2s-', In agreement wlth the rate determlned by a manual method.
The technique of flow injection analysis, as originally developed (1-4)) was designed for multiple, rapid analysis of unknown samples. However, with sufficient attention to experimental details, it is possible to determine molecular parameters, such as diffusion coefficients, from such measurements (5). It is the purpose of this paper to show that unimolecular rate constants can also be obtained with this method, provided that the experimental design is made to conform to the restraints imposed by laminar flow theory. In an attempt to establish mathematically sound guidelines for the design of flow injection systems, the laminar flow diffusion-convection equation
where D is the molecular diffusion coefficient, C is the concentration at any point in the tube a t a given time, x is the distance along the tube, r is the radial distance from the tube axis, a is the radius, t is the time, and uo is the maximum velocity at the center of the tube, was numerically integrated in regions applicable to most flow injection analysis work. It was found that the numerical solutions for the initial appearance time of a peak and the base line to base line times could be represented in the regions of interest by relatively simple analytical expressions. In addition to tubing radius and length as well as flow rate, the expressions contained the Present address: University of Maryland Medical School, Baltimore, MD 21201.
molecular diffusion coefficient and it was shown how the data could be treated to obtain this molecular parameter. These derived expressions have since been used by Gerhardt and Adams (6) to obtain accurate values of the diffusion coefficients of biogenic amine neurotransmitter-related compounds. The derived expressions were also found to give remarkably good agreement with experiment and the numerically calculated curves agreed well both in times and shapes with those actually observed (5). This indicates to us that eq 1gives an accurate description of behavior in flow injection systems when no tube-coiling effects are present. A further look at the convection-diffusion equation reveals that flow injection systems can be used to determine first-order rate constants very simply. If it is assumed that the concentration of the rate-limiting solute is being monitored and that this solute is undergoing a first-order reaction in the carrier stream, then the equation becomes
D (a2c' - + - + -a2c' ax2
ar2
1 ac!) = r dr
where C' is the concentration of solute undergoing reaction. With a simple transformation, C' = Ce-k*,eq 2 reduces to the identical form of eq 1. The solutions to this are then identical except that the concentration profile, C'(x,r,t), with chemical reaction is related to that without chemical reaction, C(x,r,t), by the experimental expression (3) It should then be possible to compare curves with and without chemical reaction at various flow rates (reaction times) and determine a first-order rate constant. For eq 3 to hold, it is essential that the diffusion coefficient of the substance being monitored is independent of the different carrier streams (Le., the ones with and without reactant). The reaction rate has to be first order or pseudo first order and, most importantly, the injected reactant must be premixed to an identical environment that it will "see" in the carrier stream. To check the above results, the reaction chosen was the permanganate oxidation of benzaldehyde (PhCHO). The stoichiometry of the reaction is apparently (7)
0003-2700/84/0356-0268$01.50/00 1984 American Chemical Society
12H+
ANALYTICAL CHEMISTRY, VOL. 56, NO. 2, FEBRUARY 1984
-
+ 5PhCHO + 4MnO46 H z 0 + 4Mn2+ + 5PhC(=O)OOH
(4)
-
which is fast compared to the following two steps:
+ PhCHO 2PhCOOH (5) + Mn2+ + HzO M,Oz + PhCOOH + 2H+ (6)
PhC(=O)OOH PhC(=O)OOH
Reaction 4 appears to be approximately first order under the chosen experimental conditions. Here, the concentration of KMn04 was chosen to give the necessary sensitivity for detection with the colorimeter and the H+and PhCHO concentrations were adjusted so as to be 10 times the KMn0, concentration.
EXPERIMENTAL SECTION The flow injection system used is shown in Figure 1. Two ”reactant” streams, one containing H+ and KMn04 and one containing either H+ in HzO (no reaction) or H+ and PhCHO in HzO are brought together in a mixing “T” before entering the inject loop. These streams are continuously flowing and the transit time from the mixing “T” through the inject loop is approximately 1-2 s. In the no reaction case, the sample is injected into a carrier stream of H+ in H20. In the reaction case, the sample is injected into a carrier stream of H+ and PhCHO in HzO. The H+ concentration in the two “reactant” streams and the carrier stream was kept at 37.5 mM (HCl). The PhCHO concentration, when in the carrier stream, was 16.6 mM, while in the reaction stream it was held at 33.2 mM (1:l dilution occurs on mixing). The KMn0, concentration in the reactant stream was kept at 1.33 mM. The “reagent” streams were pumped through 2 mm. i.d. Tygon tubing with a polystatic pump (Buchler Model 2-6100, Fort Lee, NJ) into the mixing “T”(0.40 mm i.d. (8)).The inject loop (100 pL) and the carrier stream tubing ( L = 700 cm, uncoiled) were constructed of 26 gauge (AWG) Teflon tubing. The flow cell volume was 10 pL. The carrier stream pump was a depulsed Milton Roy mini pump (Riviera Beach, FL). The detector consisted of a photomultiplier photometer (American Instrument Co., Silver Spring, MD) that was modified (details of modification available upon request) to use a 10-mm path flow cell (Kratos Analytical Instruments, Inc., Westwood, NJ), a one-half inch round interference filter (550 nm peak transmission), and a small endlens tungsten lamp (9). The output from the colorimeter was processed through a linear logarithmic converter and displayed on a strip chart recorder. The reaction was also studied manually by rapidly mixing the reactants in a cuvette and monitoring the absorbance of KMnO, as a function of time at 550 nm with a spectrophotometer (Beckman Model 25, Brea, CA). The initial H+ and PhCHO concentrations were the same as in the flow injection carrier stream and the initial KMn04concentrations were varied from 1.33 mM to 1/40thof this value.
RESULTS AND DISCUSSION The observed peaks are shown in Figure 2 for both the no reaction and the reaction cases. If eq 3 is valid, then the natural logarithms of the ratio of absorbances at peak height, for example, should be equal to -ktpk, where tpk is the time a t peak height. In Table I, the values of k obtained a t the different flow rates are shown. The average value of k of 1.05 X was then used to calculate, again using the e-kt factor, random points on the “reaction” peaks from the observed absorbances of the nonreaction peaks. These calculated points are shown as circles in Figure 2. The times, t , at each point were determined from the distance from inject and the known chart speed (10 cm/min). The only discrepancy occurs for q = 1.06 mL/min because the flow rates differed between the two runs at q = 1.06 mL/min by 3%. If this difference is corrected for, the dashed curve becomes the “observed” peak and again the agreement is very good. It is also necessary to mention that the base line to base line times are the same for
269
Table I. Rate Constants Determined from Ratio of Absorbances at Peak Heights
min
ratio of absorbance
b h
1.06 2.26 3.5 4.7
0.363 0.599 0.700 0.758
102 49.5 33 25.5
q , mL/
k , s-’ 0.99 x 1.04 X 1.08 x 1.09 X (1.05 k
10-2 10.’ 10.’ 0.05) x
lo-’
Table 11. Rate Constants Determined by Manual Methods Ci( KMnO,),
mM
k , SS’
c,
0.88 1.47 1.81 1.97
c,i4 CJ20 C,i40
X X
X X
10.’
lo-’ 10.’
the respective “reaction” and “no reaction” peaks indicating that the diffusion coefficient of KMn0, was not changed by the addition of the benzaldehyde reagent. In contrast to the above results, Painton and Mottola (IO) recently found that base line to base line times decreased when a reaction occurred in their flow-injection system. However, one reactant was injected into another without premixing and there is a resulting difference in solvents within and without the bolus resulting in unusual boundary layer effects. The application of eq 1 is not clear-cut when it is applied to one of the reactants. In their case the reaction (and solvent diffusion) is occurring primarily at both ends of the bolus and is not uniform throughout the entire bolus. The rate constants obtained for this reaction from the manual method are given in Table 11. For each initial concentration of KMnO,, the absorbance decreased exponentially implying first order with respect to this reagent. However, as the initial concentration of KMn04 was varied, the experimentally determined rate constant varied according to the expression
k = (0.88
(
+ (1.12 X lo-’) co;o“)
X
(7)
where Co is 1.33 mM and Ci is the initial concentration at various dilutions. This indicates, of course, that the reaction is not truly first order in KMn04. If, however, one takes into account the average dilutions that are occurring in the flow injection procedure for the different flow rates, the average value of k obtained from eq 7 is 1.05 X IO-’ s-l in perfect agreement with the flow injection results. As a final note, we remark that the rate constant can also be determined by a ratio of the two areas under corresponding peaks. Because of the exponential relationship
A, _ -A,,
kAt/2
kat 2
sinh -
Where t, is the midpoint between the two base line times and At is the difference between the two base line times. Since lim
Ax-0
sinh x X2 1 + - + ... x 3!
this term does not cause a problem in iteration procedures.
270
ANALYTICAL CHEMISTRY, VOL. 56, NO. 2, FEBRUARY 1984
Table 111. Determination of Rate Constant from Ratio of Areas flow rate,
mL/min 1.06 2.26 3.50 4.70
t,, s 106.8 52.9 35.4 27.6
At, S 79.2 28.0 22.2 18.6
[sinh (kAt/2)]/ ( k At/ 2) 1.03 1.01 1.01 1.00
A,
All,
240
705 308 196 148
181
137 110
k , s-!
1.04 X lo'* 1.02 x 1.04 X 1.08 X l o - * av (1.05 i 0.03) X
lo-*
Recorder
Carrier q = 1 to 5 mLlmin Stream
Sampie Inject Valve
q =2.5mLlmin.
Mixing " T '
q = 2.5 mLlmin. Colorimeter Peristaltic Pumo Reservoir (a): H +
+
KMnO, In H 2 0
Reservoir (b): H + in H20 or H
+
+
6 CHO in H20
Figure 1. Diagram of flow injection systems for the measurement of rate constants. See text for details.
be exact and fit the experimental data perfectly. It is evident, from our experience, that accurate measurements of inject times are necessary and that the flow rates must be carefully controlled. The procedure, however, as well as the required apparatus, appears to be simpler than such other techniques as stop-flow analysis, although the latter can measure the rates of faster reactions (11). Registry No. KMn04, 7722-64-7; PhCHO, 100-52-7.
LITERATURE CITED
9 lmllrn\nl
Figure 2. Absorption peaks (550 nm) obtained with and without chemical reaction. The dashed curve is an "experimental" curve generated from the observed solid curve because of a 3% dlfference in flow rate.
Shown in Table I11 are the values obtained for the rate constant from the ratio of the different corresponding areas. The results agree with the value obtained in the earlier procedure. In summary, then, rate constants for first-order reactions can be easily obtained from a properly designed flow injection system. The appropriate theoretical expressions appear to
(1) Stewart, K. K.; Beecher, G. R.; Hare, P. E. Fed. h o c . Fed. Am. SOC. Exp. Elol. 1974,33, 1439. U.S. Patent 4013413. (2) Beecher, 0. R.; Stewart, K. K.; Hare, P. E. I n "Proteln Nutritional Quality of Foods and Feeds, I"; Friedman, M., Ed.; Marcel Dekker: New York, 1975;pp 411-421. (3) Ruzicka, J.; Hansen, E. H. Anal. Chlm. Acta 1975, 78, 145-157. Danish Patent Appllcatlon No. 4846/84,Sept 1974; U.S. Patent 4 022 575. (4) Stewart, K. K.; Beecher, 0. R.; Hare, P. E. Anel. Blochem. 1978,70, 167-173. 15) ~, Vandersllce. J. T.: Stewart.. K. K.:. Rosenfeld. A. G.: Hloos. D. J. Talanta 1981,28, 11118. (6) Gerhardt, 0.;Adams, R. N. Anal. Chem. 1982,54,2618-2620 errata Ibid. 1983,55,816. (7) Brewster, R. Q. "Organlc Chemlstry"; Prentlce-Hall: New York, 1953. (8) Hooley, D. J.; Dessy, R. E. Anal. Chem. 1983,65,313-320. (9) Beecher, G. R. Adv. Exp. Med. B o / . 1978, 105,827-840. (10) Palnton, C. C.; Mottoia, H. A. Anal. Chem. 1981, 53, 1715-1717. (11) Krottlnger, D. L.; McCracken, M. S.; Malmstadt, H. V. Am. Lab. (Fairfleid, Conn .) 1977,9 (3),51-59.
--
RECEIVED for review August 9, 1983. Accepted October 21, 1983.
