Determination of rate constants and reaction orders with an open—closed flow-injection configuration

Tuiunra,Vol. 38, No. 2, pp. 125-I32, 1991 Printed in Great Britain

0039-9140/91$3.00+ 0.00 Pergamon Press plc

DETERMINATION OF RATE CONSTANTS AND REACTION ORDERS WITH AN OPEN-CLOSED FLOW-INJECTION CONFIGURATION S. D. KOLEV*,A. RIOS,M. D. LUQUE DE CASTRO and M. VALCAMXL Department of Analytical Chemistry, Faculty of Sciences, University of Cbrdoba, 14004 Cbrdoba, Spain (Received 21 December

1989. Revised 5 July 1990. Accepted 10 July 1990)

Summary-A new, very useful application of open-closed configurations to kinetic studies is reported. The multipeak recordings provided by the manifold used, which features a single conventional photometric detector, were used to calculate the rate constants and reaction orders of a chemical system, namely the l&and displacement reaction between the cobalt(IIkEGTA complex and PAR.

Since its inception, flow-injection analysis (FIA) has gained increasing popularity, both for routine analyses and for research purposes. Among the latter, the determination of different physicochemical constants (e.g., viscosity,’ diffusion coefficients,2 kinetic parameters3) makes the technique a very useful tool for scientific research. Flow-injection analysis combines the advantages of continuous-flow mixing methods and stopped-flow methods4*5 and overcomes their disadvantages, e.g., it allows faster reactions to be investigated and ensures continuous flushing of the measuring cell (it resembles the continuous-flow mixing methods in this respect), and a single experiment provides information on the reaction kinetics through the recorded concentration/time curve. The main disadvantages of the traditional open manifold are the high reagent consumption and the limited amount of information that can be obtained from development of the reaction in a single injection. These drawbacks could be efficiently eliminated by using an open-closed system.6‘8 In this way, multipeak detection could be accomplished without the need to install a series of detectors along the manifold, which could prove rather expensive. An additional advantage of open-closed systems is the possibility of reaching reaction equilibrium, which is not always feasible with open systems. Because of the intensive mixing in open-closed flow systems when *Permanent address: Department of Analytical Chemistry, Faculty of Sciences, University of Sofia, 1126 Sofia, Bulgaria.

the system volume considerably exceeds the injected volume, static reaction conditions (i.e., constant concentration of one of the reactants), which otherwise require sophisticated equipment,4 can be readily accomplished. Open-closed systems have already been applied for determination of partial reaction orders and rate constants of different reactions widely used in chemical analysis.- The results are most likely to be affected by the hydrodynamic regime of the flow system and cannot be considered “pure” chemical kinetic parameters. One approach to the determination of true reaction orders and rate constants irrespective of the flow pattern in the manifold involves finding a relation between the signals obtained under the same hydrodynamic conditions in the absence and presence of the chemical reaction investigated. This paper reports on the theoretical development of such an approach and its application to the study of the ligand-displacement reaction between the cobalt(IIEEGTA complex and PAR. THEORETICAL

FOUNDATION

The dispersion of the analyte in traditional open FIA systems can be described on the basis of an axially-dispersed plug flow model,g which is represented mathematically by the equation: dC/dt = DLd2Cldx2 - udC/dx

(1)

where t is time, x the axial distance, C the concentration, u the mean linear flow-rate and D, the axial dispersion coefficient. This last 125

S. D. KOLEVet al.

126

parameter depends on the extent of convective mixing and on the molecular diffusion, and can be calculated theoretically when this type of mixing is prevalent.‘O When the analyte is involved in a first or pseudo-first order reaction, equation (1) becomes: dC/dt = DLd2C/dx2 - udC/dx - kC

(2)

However, it can be shown that substitution of Co exp( - kt) into equation (2) yields equation (l), with a new dependent variable, Co replacing C. Obviously, Co is the analyte concentration which would be detected in the absence of chemical reaction. On the basis of the reasoning above, if the injected analyte is involved in a first (or pseudo-first) order reaction, the actual rate constant can be determined by processing the data obtained by injecting the analyte into the carrier without [CO(t)] and with reactant [C(t)] according to the equation: C(t) = CO(t)exp( -kt)

(3).

If the reaction product rather than the analyte itself is monitored, then we may write: dC,/dt = DLpd2Cp/dx2 - udCJdx where C, is the concentration

+ kC

(4)

of the product and

DLP its axial dispersion coefficient.

R,+R,&P

If, in addition, we assume the axial dispersion coefficients of the analyte (DL) and the product (DLP) to be similar, then the following relationship will hold: c,+c=c;=co

(5)

where CE is the concentration of product that would arise if all the analyte reacted immediately after injection. By using the equations c =(C”,-

C,) = CO,exp(-kt)

(6)

and C, = Cz [l - exp( - kt)]

cesses in the flow system are governed by convective mass-transfer, since any difference between the diffusion constants of the analyte and the product would result in no substantial differences between the corresponding axial dispersion coefficients. In fact, the flow pattern of an open-closed flow system is convectively controlled because of the intensive mixing resulting from the use of the peristaltic pump. According to these considerations, the concentration of the monitored chemical species (i.e., the analyte or the product) at the detection point in the absence of a chemical reaction can be taken as the initial or final (steady-state) value under batch conditions, while the concentrations monitored in the event of a chemical reaction can be regarded as the transient concentrations corresponding to these initial or final concentrations. It may be assumed that, if the condition involving the equality of the axial dispersion coefficients of all the chemical species taking part in the reaction is valid, this reasoning could also be applied to reactions other than those of first or pseudo-fist order. In this respect, the following reaction stoichiometry and kinetic equation are a representative example:

dC,/dt = kc, C,

(8) (9)

where C, and C, are the concentrations of reactants R, and R2, respectively. If the concentration/time curves for at least two of the chemical species involved in the reaction in the absence of chemical interaction (e.g., Cy and Ci) are known, then, taking into account that C, = Cy - C,, and C, = Ci - Cr,, the corresponding solution of equation (9) allows the rate constant to be calculated. Assuming the product is the chemical species monitored:

(7)

equation (4) can be transformed into a form of equation (1) in which the variable Ci replaces C. Further, if the product is monitored, the rate constant can be calculated from equation (6) provided the shape of the concentration/time curve obtained by injection of the product or the analyte in the absence of chemical reaction is known in advance. The assumption that the axial dispersion coefficients of the analyte and the product are very similar imposes no substantial limitations as long as the dispersion pro-

If only the stoichiometry is known and both the rate constant and the reaction orders are to be determined, then the following approach could be applied: (i) the time-dependence of Cy, Ci and Cp should be obtained; (ii) among the kinetic equations with solutions similar to equation (lo), i.e., f (Cy , C;, C,) = kt, that yielding the same k value for all time points should be selected as the most likely kinetic equation; (iii) such an equation should be

Determination

of rate constants and reaction orders

0

CARRIPR

w Y

R2

75cm

Vl

-

,..,!C,,.,

I

I

I

360 cm Fig. 1. Scheme of the open-closed FIA manifold.R, , R,, reagents; P, peristaltic pump; V, , selecting valves for channels (1) and (2); V,, injection valve.

checked at different initial concentrations Cy and Ci should be varied).

(i.e.,

EXPERIMENTAL

Reagents

Aqueous solutions of ethyleneglycol bis-(/?aminoethyl ether)-N,N,N’,N’-tetra-acetic acid 4-(2-pyridylazo)resorcinol mono(EGTA), sodium salt monohydrate (PAR), cobalt(H) nitrate (guaranteed reagent grade) and boric acid/sodium hydroxide buffer” (pH = 8.8) were used. Apparatus

A Hewlett-Packard 8452A diode-array spectrophotometer furnished with a Hellma 178.12QS flow-cell (18 ,ul) and interfaced to a Hewlett-Packard Vectra ES/12 personal computer was used. A Gilson Minipuls-2 peristaltic pump, Tecator L 100-l and Rheodyne 5041 injection valves, and a Tecator TM-III chemifold were also used. Manifold

The manifold used is depicted in Fig, 1. Valve V, is an ordinary valve that can be switched between channels 1 and 2, while the other channels lead to waste. The sample containing both reagents, mixed at a confluence point upstream of the injection valve V,, is introduced into the flow system previously closed by valve V, . The reagent concentration in the carrier stream, and the pH, are the same as in the injected sample. Pump P ensures the flow circulation in the closed system. The lengths of the two reactors are 75 and 360 cm, respectively. Teflon tubes of 0.5 mm bore are used. The flow-rate in the closed system is measured from its total volume (2.08 ml) and

127

the mean residence time is taken as the time difference between two adjacent peaks. The volume of sample injected is 77 ~1. All volumes are determined from the weight of doubly distilled water required to occupy them. The volume of the sample loop is also checked on the basis of the injected analyte concentration, the volume of the closed system and the equilibrium analyte concentration in the absence of chemical reaction. Chemical reactions and spectral data

Under the experimental conditions, viz. pH = 8.8, the predominant species of PAR is the singly charged species, HR-,12 and the complex of cobalt(H) with EGTA occurs in the forms CoY2-, CoHY-, and CoH2Y, the stability constants of which are 1012.3,1W9 and lo’.‘, respectively.i3 Cobalt(I1) forms a red complex with PAR with a metal-ligand ratio of 1: 2 in slightly alkaline media.i4 The stability constants of the complexes CoHR+ and Co(HR),+ are 10’“.oand 107~‘,L5 respectively. It should be noted that magnetic susceptibility measurements have revealed that cobalt occurs as cobalt(II1) in its complexes with PAR and PAN.16 According to Funahashi and Tanaka,i3 the oxidation of cobalt(H) occurs immediately after the incorporation of the second PAR molecule. The ligand-substitution reaction can be represented schematically in two stages: CoY’+HRLCoHR+Y CoHR + HR 2

COG

(11) (12)

where Y’ denotes all forms of EGTA, irrespective of their extent of protonation. The overall reaction can be obtained by adding equations (11) and (12): Coy’ + 2HR -%

Co(HR), + y

(13)

The electrical charges of all chemical species in equations (11)-(13) have been omitted for clarity and simplicity. The proton mass-balance has not been taken into account, because all the chemical reactions take place in a buffered medium. The kinetics of reaction (11) on the one hand, and of reactions (12) and (13) on the other, can be elucidated if COY’ is in excess in the first case, and PAR in the other two. Calculations based on the stability constants of CoHR and

S. D. KOLEW ef

128

200

300

LOO

500

al.

600 WAVELENGTH

700

600

(nm)

Fig. 2. Spectra of 2.50 x 10m5Msolutions OE (1) PAR, (2) COY’; (3) CoHR, (4) EGTA (coinciding with the spectrum of COY’); (5) Co(HR),. The spectrum of Co(HR), was recorded with excess of PAR present.

Co(HR), showed that if equal volumes of equimolar solutions of cobalt(I1) and PAR are mixed, the 1: 1 complex obtained is predominant ( > 99.9%), whereas if the concentration of PAR is twice that of cobalt(I1) the Co(PAR), complex should predominate. On the basis of this, solutions of CoHR and Co(HR), were prepared. By mixing equal volumes of equimolar solutions of cobalt(I1) and EGTA, the corresponding complex (COY’) was obtained. The spectra of COY’, PAR, CoHR, Co(HR), , and EGTA are shown in Fig. 2. All solutions were prepared in boric acid/sodium hydroxide buffer (pH 8.8). The ligand-displacement reaction was investigated by monitoring the absorbance of the flowing solution at 410 nm (A,, for PAR), 448 nm (isosbestic point for PAR and CoHR), 505 nm (maximum for CoHR) and 515 nm, where the absorbance was due almost exclusively to CoHR. Concentration DS.absorbance plots obtained at these wavelengths over the concentration range O-5.00 x 10eSM for PAR, CoHR, and Co(HR), were all linear. Procedure

reagent concentration in the sample loop of valve V2 (Fig. 1) equal to that in the carrier. The experiments were repeated for three different carrier flow-rates (1.7, 2.5 and 3.1 ml/min). RESULTS AND DISCUSSION

In the first series of experiments the liganddisplacement reaction was investigated under pseudo-fist-order kinetic conditions. For that purpose, solutions of CoHR (Fig. 3B) and COY (Fig. 3C) were injected into carrier solutions in which PAR was in large excess. The axial dispersion coefficients of all reagents and products of the reactions studied were assumed to be equal, and the absorbances for individual species were assumed to be additive. The dispersion of CoHR was calculated from the recordings shown in Fig. 3A. The recordings in Fig. 3B allowed calculation of the extent of the reaction CoHR + HR e Co(HR)*, by solution of the simultaneous equations for the total absorbances at 510 and 550 nm, and the corresponding molar absorptivities of CoHR and Co(HR),. The extent of the displacement reaction was similarly calculated from Fig. 3C. Under the experimental conditions above, only reaction (12), the interaction of CoHR with excess of PAR, and reaction (13), the overall ligand-displacement reaction, again in excess of PAR, were investigated. In both cases, the rate law could be expressed by means of first-order kinetic equations.

During the experiments, the following injections were made: (i) 5.00 x 10e4M solutions of PAR and CoHR into a carrier containing only buffer. (ii) 5.00 x 10W4M solution of CoHR into a carrier containing 2.50 x 10m4MPAR or 1.25 x 10W5,2.50 x 10e5, 5.00 x 10m5, 1.25 x 10e4 or 2.50 x 10-4M Coy’. (iii) 5.00 x 10v4M solution of COY’ into a (14) dCc~,,ldt = - kz CcoHR carrier containing 1.25 x low4 or 2.50 x 10m4M dC,,. /dt = - k,,zCcou, (15) PAR. This was supported by the good linearity of In all cases, the corresponding confluent solutions of COY’ and PAR were twice as the corresponding first-order plots (Fig. 4). The concentrated as the carrier. This resulted in a k2 and klez values, determined by the least-

129

Determination of rate constants and reaction orders

260 TIME

400 [SEC]

0.8000 5 f m g 0.4oOc : 4 300

800 TlME

900

1500

[SEC]

0.8000 ti f a 8 0.4000 2 u o.oOC0 0

300

600 TiME

900

1200

1500

[SEC]

Fig. 3. Recordings obtained upon injection of CoHR into a carrier containing buffer only (A) or 2.50 x lo-‘M PAR (B), and of CoY’ into a carrier containing 2.50 x 10e4M PAR (C). The flow-rate was 1.7 mi/min in all cases.

squares method, were 8.85 x 10m3 and 2.41 x 10m3set-‘, respectively. A PAR solution (5.00 x 10b4M) was injected into a carrier containing only buffer or COY’at various con~ntratio~s (1.25 x 10-5-2.50 x 10-4M) for investigation of reaction (11). The concentration/time curve obtained by injecting PAR into a carrier containing buffer only was alrnost identical to that obtained on injection of CoHR into the same carrier. This shows that the assumption involving the equality of the axial dispersion coefficients of the reagents, on which the proposed method relies, is valid for the CoHR/PAR/CoY’/EGTA system. Con-

centrations of COY’ in the carrier above 2.50 x 10V5Mensured completion of the liganddisplacement reaction in a few seconds, and the recordings showed only the dispersion of the already obtained CoHR complex (Fig. 3A). The absorbance/time recordings obtained for 2.50 x lo-’ and 1.25 x 10-SM COY’ in the carrier solution are shown in Fig. SA and B, respectively. In both cases, the concentrations of CoHR, COY, and PAR were of the same order of magnitude, so the pseudo-tit order simplification may not be applied. The concentrations of PAR and CoHR at each peak maximum were determined from the molar absorptivities of

S. D. KOLES et al.

130

2

A 1

[SEC]

TIME

Fig. 4. First-order plots corresponding to equation (14) (0) and equation (15) (a), where A =ln(C’&,,,/C,,,) and E = In (C&,/C,,.).

PAR and CoHR (Table 1) at the wavelengths monitored and the absorbance values at the same wavelengths. Only the first five peaks were processed. As only the data for 2 wavelengths were required, 410 and 448 nm were chosen. Good agreement with the data found at 510 and 550 nm was observed. The data obtained were processed by using the kinetic equations listed in Table 2. First, an initial COY’concentration of 1.25 x 10e5M in the carrier was used, as it allowed the first four peaks to be processed. The

reaction was completed before the appearance of the fifth peak. At an initial COY’ concentration of 2.50 x lo-“M, the reaction was completed by the time the third peak was recorded, so only the first two peaks could be used. For all calculations, the initial concentration of PAR at each peak maxims was determined by injection into a carrier containing buffer only, while the initial concentration of COY was assumed to be that in the carrier prior to intr~u~tion of the PAR sample. The results obtained are plotted in Fig. 6. Parameter R is the term on the left-hand side in the solution of a given kinetic equation (Table 2), divided by its maximum value (i.e., that co~esponding to the last peak used in the calculations). Parameter T is the time, divided by that of the appearance of the last peak maximum used. According to this normalization, the co-ordinates of the last peak used in the calculations will be T = 1; R = 1, irrespective of the kinetic equation used. The following kinetic equation provided the best fit to the experimental data at a COY’concentration in the carrier of 1.25 x lo-‘M: dCJdt = k m

(16)

o.aooo-

E f 5: 0.1000m *

g

5tOnm 550 nm nnnnn _..

xm

1.

0

100

200

TIME

300

1

400

[SEC]

8

e

2 L 0

0.8000

0.4000 510 nm 550nm 448nm 4lOnm

0x)000 0

200

TlME

300

400

[SEC]

Fig. 5. Recordings obtained for injection of 5.00 x 10-‘M PAR into a carrier containing COY’ at a concentration of 2.50 x lo-‘M (A) and 1.25 x 10-sM (B). The flow-rate was 3.1 ml/min.

Determination

of rate constants and reaction orders

131

Table I. Molar absorptivities (l. mole -I. cm -I) of PAR CoHR and Co(PIR),

PAR CoHR Co(HR),*

410

448

510

550

3.23 x 10’ 6.31 x IO”

1.54 x 10’ 1.59 x 10’

1.26 x lo) 2.84 x 10’ 5.57 x 10’

650 1.67 x lo’ 2.36 x 10’

*The spectral features of Co(HR), were investigated with excess of PAR present. Under such conditions, the molar absotptivities of Co(HR), could not be determined at 410 and 448 nm because of the very high absorbance of the PAR. Table 2. Kinetic equations used for the calculations No.

Solution

Equation

(1) dC,/dt = kc, C, (2) dC,/dr = /CC,C; forA >O:

(3) dC,/dt = kC,C;‘2 A=@-C:

x=cy-c, (4) dC,/dt = kC;‘2C2 A=C:-C; x=c;-c,

for A < 0:

&[arctanE-arctiE]=

-kt

the same as the solution for equation (3) but Cq replaced everywhere by Cy

(m-fi)(fi+&)

(5) dC,/dt = kC;‘2C;

(FX

=_kt

+ fi)(fi

- fi)

A=Cp-C;

x=cq-c, Subscripts 1 and 2 refer to PAR and CoEGTA, respectively.

where C, and C, are the concentrations and COY’ respectively. However, the equation dC,Jdt = k &C;

of PAR

(17)

cannot be completely discarded as a possible approximation to the true kinetic equation for the ligand-displacement reaction studied. The values of both rate constants obtained by a least-squares method were 6.53 lo,‘. male-0.5. set-’ for equation (16) and 1.15 x 10” Fs. male-1,5. set-’ for equation (17). Processing the results obtained with a COY’ concentration of 2.50 x 10-SM in the carrier solution yielded the following values for the two rate constants above: 5.11 1°,5.mole-0.s . set * for equation (16) and 2.55 x lo5 F5. mole-‘.5. set-’ for equation (17). These results allow us to conclude that the most likely kinetic equation for reaction (11) is equation (16). Accordingly, the reaction must

be much faster in the presence of excess of COY than excess of PAR. This was experimentally confirmed, which is an additional proof for the validity of equation (16). 7

1.0

R 0.5

I 0

0.5

1.0

T

Fig. 6. plots of R DS.T for equauons(1) 0; (2) .; (3) 0; (4) 0; and (5) 0.

S. D. Ko~nt et al.

132 CONCLUSiONS

The proposed method for the determination of kinetic constants and reaction orders of chemical reactions can be successfully applied to kinetic studies, and can be implemented most efficiently in open-closed flow-injection systems where several peak maxima can he obtained from a single injection, and where the course of the chemical reaction from the beginning to eq~~b~urn can be monitored with a single detector. These two major advantages make open-closed systems superior to traditional open flow-injection systems for this purpose. Acknowledgement-The authors are grateful to Pergamon Press (7?&rntu) for financial support accorded to Dr. Spas D. Kolev by the Ronald Belcher Memorial Award. REFERENCES 1. D. Betteridge, W. C. Cheng, E. L. Dagless, P. David, T. 8. Goad, D. R. Deans, D. A. Newton and T. B. Pierce, Analyst, 1983, IOS, 1.

2. D. Betteridge, E. L. Dagless, B. Fields, P. Sweet and D. R. Deans, Anal. Proc., 1981, 18, 26. 3. J. M. Hungerford, G. D. Christian, J. Rti6ka and J. C. Giddings, Anal. Chem., 1985, 57, 1794. 4. H. A. Mottola, Kinetic Aspects of Analytical Chemistry, Wiley, New York, 1988. 5. D. Perez-Bend&o and M. Silva, Kinetic Methot& in Analytical Chemistry, Horwood, Chichester, 1988. 6. A. Rios, M. D. Luque de Castro and M. ValcPrcel, Anal. Chem., 1985, 57, 1803. 7. I&m, Anal. Chim. Acta, 1986, 179, 463. 8. J. M. Fern&de&Romero, M. D. Luque de Castro and M. Val&cel, ibid,, 1989, 219, 191. 9. S. D. Kolev and E. Pungor, Anal. Gem., 1988,68,1700. 10. G. Taylor, Proc. Roy. Sot. A, 1953, 219, 186. 11. D. D. Perrin and B. Dempsey, Buffers for pH and Metal Zon Control, Chapman & Hall, London, 1974. 12. W. J. Geary, G. Nickless and F. H. PolIard, Anal. Chim. Acta, 1962, 26, 575. 13, S. Funahashi and M. Tanaka, B&l. Chem. Sot. Japart, 1970,43, 763. 14. D. Nonova and B. Evtimova, Anal. Chim. Acta, 1972, 62, 456. 15. W. J. Geary, G. Nickless and F. H. Pollard, ibid., 1962, 27, 71. 16. T. Iwamoto and M. Fujimoto, ibid., 1963, 29, 282.