Anal. Chem. 1987, 59, 283-291 (11) Scypinski, S.; Cline Love, L. J. Anal. Chem. 1884, 56, 322. (12) Turro, N. J.; Cox, G. S.; Li, X. Photochem. Photobiol. 1883, 37, 149. (13) Turro, N. J.; Okubo, T.; Chung, C. J. J . Am. Chem. SOC. 1882, 104,
1709.
(14) Bolt, J. D.; Turro, N. J. Photochem. fhotobiol. 1982, 35, 305. (15) Kano, K.; Takenoshita, I.; Ogawa, T. J."Chem. SOC.Jpn. 1882, 321. (16) Kobayashi, N.; Saito, R.; Hino, H.; Hino, Y.; Ueno, A,; Osa, T. J . Chem. SOC.,ferkin Trans. 2 1983, 1031.
283
(17) Yorozu, T.; Hoshino, M.; Imamura, M. J . Phys. Chem. 1882, 86,
4426.
RECEIVED for review July 10, 1986. Accepted September 16, 1986. Both T.C*W*and W.R.S. acknowledge financial support from Instrumentation Laboratory while on sabbatical.
Rapid Rate Measurements by the Pulsed-Accelerated-Flow Method Mark T. Nemeth, Kimber D. Fogelman, Thomas Y. Ridley, and Dale W. Marperurn* Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
First-order rate constants as large as 124 000 8-' ( t 1,2 = 5.6 ps) and secondorder rate constants as large as 2 X 10' M-l 8-l are measured with a pulsed-accelerated-flow (PAF) spectrometer. The method uses a variation of flow velocity during data collection to resolve reaction rate constants from mixing rate constants. The valldlty of the method is demonstrated by calibration reactlons under pseudo-first-order and second-order (equal and unequal concentration) conditions. The performance and llmltatlons of a twln-path mixing/observation cell and the PAF method are reported.
Many irreversible reactions are too fast to study by stopped-flow methods (1-3) and cannot be studied by relaxation methods, which require reversibility. A recent paper (4)described a pulsed-accelerated-flow (PAF) spectrophotometer that measured pseudo-first-order rate constants in the range of 300-12000 s-l and consumed only 6 mL of each reagent per push. In the present work we extend the range of pseudofirst-order rate constants that can be measured by the PAF method to include 200-124000 s-l. We also show that second-order rate constants as large as 2 X lo9 M-' s-l (values that approach the diffusion controlled limit) can be determined. The PAF method is based on the continuous flow method with integrating observation (CFMIO), in which the reaction is observed along the direction of flow from the point of mixing to the exit of the solution from the observation tube (2,3,5). The observed absorbance is a function of the rate of mixing, the rate of chemical reaction, the length of the observation tube, and the flow velocity, as well as the molar absorptivities and concentrations of the reactants and products. Gerischer and Heim ( 5 , 6 ) derived equations that describe the absorbance as a function of velocity for a CFMIO experiment when mixing is ideal. These equations are given in Table I for first-order and second-order (equal and unequal concentration) reactions. In the PAF method solutions experience a constant acceleration during each push and, therefore, have a range of flow velocities. Figure 1 illustrates the linear velocity ramp that permits 250 instantaneous velocities between 3 and 11 m s-l to be sampled for solutions in the observation tube. All velocities in the range are large enough to give turbulent flow in the observation tube ( 2 , 3 ) . We have shown for first-order conditions that the absorbance measured in a constant velocity
experiment is equal to the absorbance at the same instantaneous velocity in a PAF experiment (4). For the time interval of each measurement, the solution in the observation tube experiences a very small range of velocities, and therefore the reaction profile is similar to that developed during a constant velocity experiment. The equations in Table I apply to systems where the mixing process can be neglected. This requires complete mixing in a segment of the observation tube that is negligible compared to the length of the tube required for the reaction. Faster chemical reactions are complete within shorter segments of the observation tube so that this assumption is not valid. Hence, for fast reactions the mixing process results in a bias in the measured absorbance values. An apparent rate constant (kapp),which is less than the actual rate constant, can be calculated by using the ideal equation and the measured absorbance values. We have demonstrated (4) that both a mixing rate constant (k-, s-l) and a first-order reaction rate constant (kr, s-l) can be resolved from kappbased on successive firstorder processes (eq 1). The first step is the physical mixing A
+B
L x
(A*B),ix
k, +
P
(1)
of the reactants A and B. The mixing rate constant depends on the flow velocity, u (m SI), in the mixing/observation tube where k , is a proportionality constant (m-l) (eq 2). The
kmix = k,v
(2)
second step is the chemical reaction rate process of the mixed reactants, (AsB),~,, to give products (P). Previously, pseudo-first-order conditions were used with one of the reactants in large excess. Equation 3 gives the double-reciprocal de-
-1-
1 1 - -+ -1
kapp
kmu
kr
(3)
pendence that is found experimentally. Plots of l / k a p pvs. l / v are linear with the intercepts equal to l / k , and the slopes equal to l / k m . In the present work we demonstrate that eq 3 also can be applied to reactions studied under second-order equal and unequal concentration conditions. In this work a new twin-path mixer with 10 radial input jets is developed. The point of mixing is at the center of the cell, and the reactants flow in opposite directions away from this point. Consequently, light passes through twin segments of identical mixing and reaction profiles. This mixing cell is used to study reactions under pseudo-first-order conditions
0003-2700/87/0359-0283$01.50/00 1987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
284
Table I. Equations for the Calculation of Rate Constants Using the Continuous Flow Method with Integrating Observation" type of reaction
Mcalcd
Mexptlb
first-order R i products irreversible second-order equal A + B products
(A, - A J / ( A o - A,)
-
(A,
CO = CA = CB
-
Am)/(Ao- A,)
variableC
(1 - e-Y)/y
Y = kb/u
In (X + 1 ) / X
X = k,aCob/v
irreversible second-order uneaual A + B h?,products
* CB
CA
'References 5 and 6. b M = reaction progress variable; A = absorbance. ' k = first-order rate constants s-l; k,, = second-order rate constant, M-'sd; b = path length, m; v = solution flow velocity, m s-l; C, C,, and Cb = initial concentrations of reactants, M. 13
I
I
I
lo)
11
E
9 -
x
7 -
-
c, 4
5 -
U
0
3 -
-1
'
0
i
'L-1 1
0.1
0.2
0.3
0.4
0.5
0.6
time, s Flgure 1. Flow velocity in the observation cell during an accelerated
push with the PAF spectrophotometer.
with half-lives as short as 5.6 ps and under second-order conditions with initial half-lives of only 5.2 p3. The twin-path cell also is used to study reactions with second-order rate constants as large as 1 X l@M-l s-l under pseudo-fiist-order conditions and to investigate the mixing process through the study of very fast (12, > lo6 s-l) neutralization reactions. Two additional cells similar to previous designs have been tested.
EXPERIMENTAL SECTION The electronics, motor, syringe ram, and data acquisition system of the PAF instrument have been described previously (4). The spectrophotometer portion of the instrument, separated on a vibration isolated granite table from the syringe ram, consists of a monochromatic light source, transfer, focusing and beam splitting optics, the mixing/observation cell, and signal and reference photovoltaic detectors (EG&G Model HUV-4000B). A monochromator illuminator housing (Oriel Model 7292) is mounted on a 3/s in. aluminum plate with an f / 3 monochromator at the source focal length. High-intensity visible light is provided by a 100-W tungsten halogen lamp powered by a remote programmable dc power supply (HP 6267A) in the constant current configuration. The light is passed through a single grating monochromator (Oriel Model 7240) with a ruled grating (Oriel 7270) and variable entry and exit slits (Oriel Model 7250) and then through a narrow band-pass filter (10 nm bandwidth at 0.5 peak) centered at the desired wavelength. This final filter is necessary to limit stray light to a greater extent than the monochromator can provide. The monochromatic light is collimated through an f / 3 lens and focused through an f / 5 lens to the center of the observation cell. Immediately prior to the observation cell, a 45' quartz beam splitter directs a portion of the light to a reference detector. A feedback circuit samples the reference detector signal and adjusts the power supply current to maintain a highly stable monochromatic light source (drift < 0.0004 A/h). This optical configuration provides a usable wavelength range of 350-700 nm with sufficient sensitivity to observe absorbance changes as small as 0.002 over the velocity range of an accelerated push. Unlike most UV-vis spectrophotometers, which are shot noise limited, the PAF spectrophotometer has been limited by a constant level of noise from dark current at the detector. This noise
level exceeds that of the source due to a combination of the requirement of strongly attenuating band-pass filters and lower sensitivity of the solid-state photodiode detector compared to a conventional photomultiplier. The choice of such photodiode detectors is favored based on their excellent stability and ease of vibration-free mounting. Reduced intensity of light to the signal detector, due to the smaller diameter of the observation tube in the twin-path cell, required some later modifications to the spectrophotometer. Initially, fixed diameter diaphragms were used to block a portion of the light to the feedback detector in order to balance the light leveh at the two detedon. The majority of data has been collected with the instrument in this confiiation. Once the performance of the twin-path cell was well established, however, several changes in the electronics and optics were made to optimize signal throughput. In particular, the focusing lens of the optical transfer box was changed from f / 5 to f / 7 and the preamplifier gain stages to the ADC and feedback electronics were reduced to allow a greater signal intensity compared to the detector dark current noise level. In this configuration, data were collected for the most rapid reactions of W(CN)84- with IrClB2-. Fabrication of New Cells D and E and Corrected Description of Previous Cella A, B, and C. Cells A, B, and C, which were described previously (4),all had slits in the entrance cell face in order to provide multijet mixing. The dimensions that were given for these jets were based on a cell window that fit tightly against the entrance cell face. However, a design error allowed the windows to be held 0.23 mm from the cell face. This created a thin layer between the cell face and the window through which reagent solutions could flow. As a result, a cylindrical gap existed a t the entrance to the observation cell (2 mm diameter by 0.23 mm height). This permitted reactants to enter the observation tube from the area of the cylindrical gap as well as from the slits. Laminar flow in the thin space between the window and the cell face appears to have created pie-shaped wedges of reactants that flowed to the observation tube with minimal premixing. The cylindrical gap led to significantly lower injection velocities than those calculated from the slit dimensions. Mixing and the subsequent conversion from laminar to turbulent flow occurred primarily in the observation tube. The wall area of the cylindrical gap was approximately three times the cross-sectional area of the slits at the entry to the observation tube, so that 75% of the reactants entered the observation tube from the pie-shaped wedges. The realization that this design error had occurred led to the development of two additional cells, a no-slit cell (cell D) and a 26-jet cell (cell E), which contained no gap between the window and slits. Continued testing of these cells, as well as cells A, B, and C, led to problems of solution lag as the flow was accelerated. Also the cells became increasingly difficult to rinse between runs. The delivery lag was traced to compression of thin layers of air trapped in the delivery manifold and between the manifold and O-rings. Although this problem could be minimized by very tight fitting end caps, this approach in turn made it difficult to disassemble the cell for cleaning. The tolerance of these caps degrades with continued cleaning, which limits their useful lifetimes. The rinsing problem became more serious with continued use until inordinately large volumes of rinse solutions were required to remove all reactants from the cell. This problem also is caused by the internal delivery manifold, where the reactant solutions under pressure are forced between the parts of the cell.
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
285
E, and F are available as supplementary material. See paragraph at end of paper regarding ordering information. Pressure Transducer. Pressure measurements were made with a National Semiconductor LX1830GBN 0-300 psig backward gauge pressure transducer. The gauge was mounted in a PVC "T" adapter and installed in the flow system at the points indicated by the numbers 1-4 in Figure 3. The transducer output voltage was recorded with a Tektronix 7313 oscilloscope triggered at the start of a push with the trigger pulse available from the PAF master controller (4). Reagents. The preparation and standardization of Na21rC16.6H20, (NH,),Ce(NO&, and K4Fe(CN)6solid reagents and t solutions of Fe(C104),and Ru(bpy)gS+have been described previously (4). Solid K,W(CN), was prepared by the method of Figure 2. Top view and center cross-section view of the mixing reglon Leipoldt (7), and its purity was checked by elemental analysis. of the 10-jet, twin-path (cell F) mixer: W, windows; BR, brass alignment Solid Cu(dmp),.CF3S02was prepared by the method of McMillin ring; A and 8 , reactants; 18, O-ring grooves. (8) from CuC03 (Baker), CF3S03H(Aldrich), 2,9-dimethyl-1,10phenanthroline (dmp) (Sigma),and ascorbic acid (Sigma). MES LP KHP (Fisher), NaN03, NaH2P0,.H,0, and K2S04 ,(Aldrich), (Millinckrodt) and the thymol blue indicator (Kodak) were all used as received. Solutions of W(CN)t- were made from the potassium salt just prior to use in order to minimize acid decomposition of the tungsten complex. For the reaction of Cu(dmp),+ with IrCls2-, the Cu(dmp),+ solutions were prepared in 0.010 M MES at pH 6. The IrC162solutions for this reaction were kept slightly acidic, to prevent decomposition of the iridium complex. The ionic strength of both Flgure 3. Block diagram of solution handling system for twin-path solutions was adjusted to 0.1 M with K2S04. The concentration mixer: DS, drhre syringes; TV, three-way slider valves; DB, distributor of Cu(dmp),+ was determined from the absorbance at 454 nm (t blocks; MC, mixing cell; LP light path; D, detector; J, joiner; RS, re7950 M-' cm-') (9). The IrCb2- concentration was determined ceiving syrlnge. Ntlmbers refer to the points at which pressure was measured. from the absorbance at 487 nm (e 4075 M-I cm-') (IO). The of electron-transfer reactions were monitored at or near the A, the following species: Cu(dmp),+; IrCls2-;and Ru(bpy):+ (A, It is a severe design requirement to have a flat window (tol453 nm, e 13800 M-' cm-I). The reactions with Ce'" were monerance of *0.013 mm) held against a flat cell surface (tolerance itored away from ita A, (315 nm), at 350 nm ( e -3600 M-' cm-') of *0.013 mm), which in turn contains many entrance ports with and 395 nm (t -1000 M-' cm-'). The indicator solutions for slits leading to an observation tube as in cell E. The window must acid-base reactions were prepared just before use and adjusted not lift off the surface or flex under high pressure, yet it cannot to pH -10.5. All solutions were passed through cellulose acetate be pressed too tightly against the surface without danger of filters (0.45 pm, Millipore) and degassed before use. breaking the window or distorting the slit dimensions. The twin-path cell is designed to overcome these difficulties. RESULTS AND DISCUSSION Twin-Path Cell F and the Solution Handling System. A In the PAF method the performance of the mixing cell is totally new design for the mixing/observation cell has been fabricated and tested on the PAF instrument. The side view and of critical importance to achieve successful rate measurements. the center cross-section view of the mixing region of the twin-path The first objective is to obtain a high degree of mixing in as cell F are shown in Figure 2. The cell is made from PVC rod small a volume and time as possible. PAF measurements (Harvel) 34.9 mm 0.d. and 39.6 mm long. The diameter of the require that the instantaneous flow of each reactant solution observation tube is 1.4 mm, while in cells A to E the diameter is the same (Le., neither reactant flow rate can lag the other was 2.0 mm. Since there is bidirectional flow in the twin-path as the push velocity increases). It also is desirable to minimize cell, the diameter of the observation tube was reduced to maintain reagent consumption by use of small volumes to rinse reagents the same total cross-sectional area that was used previously. from the mixing cell. The twin-path cell F is a new design Therefore, the same relationship exists between the linear disthat gives superior performance compared to the other cells. placement transducer measurement and the velocity in the obAdvantages of the Twin-Path Cell. In the twin-path cell, servation tube. Ten input jets are arranged radially around the observation tube (Figure 2), 36O apart. Each inlet port consists the internal manifold is replaced with external distributor of three stages: '/, in.-28 flat bottomed to a depth of 6.35 mm; blocks (DB of Figure 31, which lead to external entrance ports 0.8 mm diameter to a depth of 14.3 mm; and a 0.33 mm diameter around the center of the cell. This external manifold design hole, 2.5 mm long, that enters the observation tube. The cell is eliminates the problems of volume delivery lag and rinsing. aligned, and the windows (19.1 mm diameter X 6.35 mm thick, The inlet jets are drilled holes rather than slits, and the mixing T-19 Suprasil 1,Amerisil, Inc.) are held in place (Figures 2 and region is in the center of the cell (Figure 2), rather than at 3) by PVC tipped brass rings (BR in Figure 2). the front window. Slight displacements of the window from A block diagram of the solution handling system for the the cell face or small irregularities in its surface have negligible twin-path cell is shown in Figure 3. The drive syringes consist effects on the cell performance because the reagents contact of brass rods with Teflon plungers seated within a dual cylinder Kel-F syringe block (14.4 mm i.d.) with 2-013 Viton O-rings. Each the windows only after the observed reaction. The k , values reagent is pushed from a drive syringe (DS)through a three-way range from 900 to 2300 m-', which gives 2 to 5 times better slider valve (TV, Altex 201-54) into a distributor block (DB) with mixing than was achieved with cell A. Two to three pushes one entrance port and five exit ports. The exit ports of the blocks from the drive syringes (6 mL/push) are sufficient to fully are connected to alternate input jets of the mixing cell. All rinse the system. The cell itself requires much less rinsing connections are made with flanged 1.6 mm i.d. Teflon tubing with volume than the distributor blocks and tubing. Repetitive in.-28 connectors. The solutions mix at the center of the cell runs of the same reactants do not require any rinses. and travel in opposite directions away from the center of mixing Apparent first-order rate constants are calculated from eq toward the windows. A space of 0.254 mm between each window 4 (4-6), where Melpdrepresents the degree of reaction in the and the cell interior allows the solution to flow to the exit porta. The products flow into a PVC joiner block (J)with two entrance porta and one exit port. The output of the joiner flows into the receiving syringe (RS).The physical characteristics of cells D, Products
1
8
,Products
1
286
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987 300
Table 11. Pressure Measurements for Twin-Path Mixing Cell position= 1c 2c 3c 4c
1D 3D 4D
1RS
slope,bpsig
s2 m-*
3.88 (f0.09) 4.0 (fO.l) 1.88 (f0.03) 1.52 (f0.04) 3.28 (fO.09) 0.95 (f 0.02) 0.63 (f0.02) 1.80 (k0.07)
I
,
I
250
intercept: psig 29 (f4) 26 (f6) 29 (f2) 28 (f2) 4 (f3) 10 (A21 7 (fl) 30 (f4)
w'
5
150
v) v)
W
100
C, receiving syringe connected; D, receiving syringe disconnected; RS, cell bypassed, direct connection to receiving syringe; see Figure 4 for position information. bSlope and intercept of Dressure vs. the sauare of the velocity.
50
0
,
0
observation path. The observed absorbance, A,, at a given velocity in the observation tube is the integrated value expressed by eq 5.
25
50
75
100
125
v', m?s2
Flgure 4. Pressure dependence on velocity squared for the twin-path and 4C (0). See Figure 3 and Table 11 cell at points 2C (0),3C (A), for positionsand conditions. The values of v refer to the flow velocities in the observation tube rather than the velocities at points at which pressure was measured.
where n is the number of identical flow paths, fi is the molar absorptivity of the ith species, M-l cm-', b is the reaction path length, m, and C, is the distance-dependent concentration of the ith absorbing species, M. A. and A , refer to the initial and final absorbances of the reaction. For mixing cells A to E, n is unity and b is 0.0200m, whereas, for the twin-path cell, n is 2 and b is 0.0100 m. The latter cell has one half the observation tube cross sectional area to allow similar velocity profiles to those of earlier designs without the need of increased reagent volumes. Under these conditions no changes are required in the velocity analysis programming. The fact that the solution flows in opposite directions has no effect on the data treatment. However, it provides two identical paths to observe reaction and, thus, doubles the observed signal change for fast reactions (i.e., reactions that are essentially complete in 0.01 m). Pressure Measurements. Substantially higher pressures in the flow system are needed to achieve the desired velocities in the PAF method compared to a typical applied gas pressure of 60 psig (pounds per square inch, gauge) that is used to initiate the push in stopped-flow mixing. Our measurements for the PAF apparatus show that the pressure is proportional to the square of flow velocity plus the back pressure, d (eq 6), which agrees with observations for flow through a pipe (11).
P = cu2 + d
(6)
We measured the pressure at various locations in the flow system (numbered points in Figure 3) in order to locate the main pressure drops. Three plots of eq 6 are shown in Figure 4. Table I1 gives data at various points for twin-path cell F. For ease of comparison, the velocity in eq 6 refers to the velocity in the observation tube rather than at the point measured. With the twin-path cell, the values of c and d at point 1C (near the drive syringes) are 3.88 f 0.09 psig s2 m-2 and 29 i 4 psig, respectively. Hence at lC, a flow velocity of 11 m s-l produces a pressure of 498 psig, which is close to the limiting pressure of the instrument. The limit of 500 psig is determined by three factors: (1) The present drive-syringe O-rings do not seal against the Kel-F syringe block above this pressure. (2)This is the rated pressure limit of the slider valves. (3) It is difficult for the motor to provide sufficient torque at high revolutions per minute to achieve linear velocity profiles above this pressure. The intercept, d, in eq 6,is due primarily to the receiving syringe attached to the velocity transducer, which has an internal tension spring that provides back pressure.
The pressure drop across the system is due to friction, constriction, and kinetic energy changes. The pressure drop across the Teflon tubing due to friction is proportional to length and can be predicted from eq 7 (12),where Ap is the
(7) pressure drop, psi, f is the friction factor, L is the length of tubing, m, d is the diameter, m, u is the density, kg m-3, g is the acceleration due to gravity, 9.8m 8 ,u is the velocity in the tube, m s-l, and C is the proportionality constant, 0.00142 m2 kg-' psi. Friction factors of 0.032 and 0.028 have been determined for 1.6 mm i.d. and 0.8 mm i.d. tubing, respectively, from pressure measurements of the tubing alone. These factors cause changes which are small compared to the pressure drop across the cell. The main pressure drop, 2.20 psig s2 m-2 (P2, - Pk) is across the mixing/observation cell. In order to determine the primary point of this pressure drop within the cell, the windows were removed, so that the solution sprayed directly out of the observation tube, and the pressure was measured at point 1. The slope of this pressure vs. velocity squared plot is 2.26 f 0.06 psig s2 m-2, which, is, within experimental error, equal to the pressure drop across the assembled cell. Therefore, no back pressure is associated with the passage of the solution through the exit porta of the mixing cell. Hence, the pressure drop of the cell is due to constriction and friction across the input jets and the right angle redirection as the solution enters the observation tube from the jets. The magnitude of this pressure change is not surprising because the velocity in each input jet reaches 40 m s-l. Consequently, the opposing jets have a net change in velocity of 80 m s-l (or 180 mph) at an observation tube velocity of 11 m s-l. The pressure limitation determines the maximum flow velocity and therefore it affects both the magntiude of the observed signal change and the maximum rate constant resolvable by the PAF instrument. Design changes in the three factors that limit the pressure might double the pressure limit of the system (to the 1000 psig limit of Teflon tubing). However, due to the velocity squared dependence,the limiting velocity would increase to no more than 15 m s-l. Hence, a practical velocity limit for the twin-path cell is 11-15 m s-l. Second-Order Reactions. Previous CFMIO studies (3, 13) with second-order reaction conditions have been somewhat restrictive in range and application due to the need for calibration curves to account for mixing effects. These curves
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
287
Table 111. Summary of Reactions Studied under Second-Order Equal and Unequal Conditions rate constant 10-8k12,M-'
reaction
104 x initial
io4 x initial
(medium, hobs&
t l l z (mad, s
t l p (mid, s
measd
lit.
W(CN)84-+ IrClz(0.5 M H2S04,497 nm) W(CN):- + CerV (0.5 M H2S04,497 nm) Cu(dmp),++ IrCh2(0.033 M K2S04, 0.005 M MES, pH -6,450 nm)
5.34
0.423
1.02 f 0.07
0.924
0.329
1.9 f 0.3
(1.02 f 0.05)O 0.8gb 2.3b
2.17
0.220
This work.
18 f 2
14c
Reference 16. Reference 15.
Table IV. Summary of Reactions Studied under Pseudo-First-Order Conditions k12, M-'
reaction (medium)
+ Ru(bpy):+ (1 M HCI04)
Fe,?
Fe,2+ + CerV (0.5 M H2S04) FeSm2* + IrCla2c6.5 M HCiO,) W(CN):IrCb2(0.5 M H2S04)
+
1 0 - 3 k ~ ( ~ ~s-1) ;
m-l
iO+k,,
measured
predicted
1.1 f 0.02
(8.6 f 0.1) X lo5
14.0
1.3 f 0.1
(1.31 f 0.01) X
lo6
39.8
1.7 f 0.2
(3.73 f 0.05)
X
lo6
(3.5 f 0.2) x 106b
2.0
(1.02 f 0.05)
X
lo8
8.9 x 107~
8.82
124
f
0.2
(8.7 f 0.7) X 1.3 X lo6'
Okr(-) is the largest rate constant measured for system. bReference4. CDulz,G.; Sutin, N. Znorg. Chem. 1963,2, 917-921. dReference 16. eReference 13.
have been established for known systems from plots of Melpd vs. the ideal half lives at fixed velocities. Use of these Calibration curves has assumed that the effect of the mixing process is independent of the reaction system; however, this is not entirely true. It is desirable to eliminate the need for calibration working curves and to resolve the mixing rate and reaction rate contributions from data for each reaction, as we have done for pseudo-first-order reactions (4). Although eq 3 was derived for first-order conditions, Toor has suggested the same type of treatment may be applied to integrated models of higher order (14). We have extended the treatment to include second-order reactions under both equal and unequal concentration conditions. Values of klzapp, the apparent second-order rate constant, are calculated iteratively from the absorbance data and the equations for ideal mixing given in Table I for irreversible second-order reactions. The double-reciprocal relationship in eq 8, yields a straight line
00 000
010
0 20
0 30
l/v,s/rn
+
with the intercept equal to 1/kl2, where k12is the second-order rate constant after correction for mixing. Three sets of electron-transfer reactions were used to evaluate the PAF performance under second-orderconditions. There is good agreement between the average values and the predicted values (Table 111). Tables of individual rate constants are available aa supplementary material. The original rate constant for the oxidation of Cu(dmp),+ by IrClsZ-was determined under second-order equal conditions in this laboratory (15) by pulsed-flow CFMIO with a calibration curve (3). The reactions of Cu(dmp),+ and of W(CN)*" with IrCle2were studied under both second-order equal and second-order unequal concentration conditions. The ratio of reagent concentrations varied from 1:l to 5:l. The second-order rate constants for both these reactions are independent of initial half-life and the initial ratio of concentrations. The plots for reactions of Cu(dmp),+ with IrCle2-under two different second-order-equal concentration conditions are shown in Figure
Flgure 5. Double-reciprocal plots for the reaction of Cu(dmp),+ IrClt- under second-order, equal concentration conditions: A, initial t,,, = 28 ps; B, initial t,,2 = 108 ps. The plots are least-squares lines, weighted in proportion to velocity. The reciprocal of the intercept gives the second-order rate constant, (1.8 f 0.2) X lo9 M-' s-'.
5 and demonstrate that the double-reciprocal plot can be used to correct for the mixing process for second-order reactions. This model predicts a constant intercept, and experimental error accounts for the difference seen in Figure 5. Secondorder rate constants calculated from the intercepts (1.75 and 2.06 X lo9 M-l s-l) agree with the average value of (1.8 f 0.2) X lo9 M-'s-l for eight sets of conditions. The double-reciprocal plot can therefore be used to correct for the mixing process for second-order equal and unequal reactions even with initial half-lives as short as 5.2 ps. Pseudo-First-Order Reactions. Table IV summarizes the performance of the twin-path mixing cell with four sets of electron-transfer reactions. Most of the data were analyzed by the double-reciprocal relationship (eq 3). The reactions of excess with R ~ ( b p y ) , ~in+ 1.0 M HC104 gave k, values in the range of 1890 s-l to 8820 s-l (Table V) that were in excellent agreement with the predicted values
288
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
Table V. Kinetic Data for Reactions Studied under Pseudo-First-OrderConditions, Twin-Path Cell, T = 25 "C
1 0 3 x [W(CN)l'],M 40
20
+ R~(bpy),~+, 1 M HC104, X 450 nm
Fe,? 124,000 s-'
103[Fe,,2+], M
40
106[Ru(bpy)33+], M
2.06 4.13 6.18 8.24 10.3
10+k,, 5-1
0.86 0.89 1.03 1.31 1.4
1.89 3.58 5.21 7.42 8.82
4.2 4.2 4.2
4.2 4.2
W(CN):-
+ 1rCb2-,0.5 M H2S04,X 497 nm
30
10
00 00
20
lo3 x
40
60
SO
100
00 120
Concentration of Excess Reagent, M
Figure 6. Observed firstorder constants vs. concentration of excess reagent for four electron transfer reactions: (O),Few2+i- Ru(bpy):+; i- Ce(1V); (A),FeaqZ+ IrClt-; (0 and M), W(CN),& iIrClt-.
(O), Fe,;+
10-,km,
+
from the second-order rate constant, klz = 8.7 X lo5 M-I s-l ( 4 ) . The mixing proportionality constant, k,, averaged 1100 m-l. Reactions of excess Fea:+ with Ce(1V) in 0.5 M HzS04 yielded a maximum value of k, of 14 000 s-*. The measured and predicted k12 values, (1.31 f 0.01) X lo6 M-' s-l and 1.3 X lo6 M-ls-l, were in excellent agreement. In this study, a first-order rate constant as small as 200 s-l was measured (Table V). The reactions of excess Fe,q2+ with IrCb2- in 0.5 M HC104 were also tested extensively with the twin-path cell (Table V). Values of k, ranged from 3840 s-l to 39 800 s-l. The measured k12 value was (3.7 f 0.2) X lo6 M-l s-l, which was in excellent agreement with the previously measured klz value of (3.5 f 0.2) X lo6 M - ' d (4). The reactions of excess W(CN):- with IrCls2- in 0.5 M H2S04were tested and gave k, values that ranged from 4590 s-l to 124000 s-l (Table V). Figure 6 shows the excellent linearity of plots of the pseudo-first-order rate constants vs. excess reagent for this reaction and for the other three redox reactions that were tested in the twin-path cell. Sutin measured a value of 6.1 X lo' M-' s-l (16) for the reaction of W(CN)s4- with IrCls2- under second-order conditions with a stopped-flow instrument. Holzwarth later measured a value of 8.9 X lo7M-' s-l under second-order equal conditions with a CFMIO instrument (13). It has been demonstrated ( 2 , 3 )that, with stopped-flow measurements, the mixing process will result in low measured values for fiit-order rate constants larger than 300 s-l. Similar effects have been noted for very fast reactions studied under second-order conditions by stopped-flow methods. Mixing time errors and concentration inhomogeneities in the monitoring light path occur because the time required to fill the observation cell cannot be neglected (17, 18). Hence, Sutin's rate constant appears to be slightly low. Our study of the oxidation of W(CN)a- by IrC12- was carried out under pseudo-first-order conditions with initial concentration ratios of 1 0 1 or 5:l. The measured second-order rate constant, (1.02 0.05) x lo* M-' s-l, is in reasonable agreement with the value obtained by Holzwarth. The ability to study reactions under pseudofirst-order conditions, with rates that are beyond the capabilities of the stopped-flow instrument even under secondorder-equal conditions, demonstrates the potential of the PAF method with the new twin-path cell. The rate constant of
*
0.408 0.886 1.26 1.92 2.18 2.70 3.10 3.16" 4.53" 5.03" 6.OZa 7.55" 9.04" 10.7" 12.9"
0.52 0.94 1.38 2.20
4.59 9.93 12.9 21.2
1.41
1.71 1.81
1.79
2.28
22.7
2.21 2.12
2.95 3.35 6.01 9.01 10.5 12.0 15.0
28.3 29.7 26.7 39.8 43.8 51.4 66.7 76.9 106 124
2.03 1.97 1.86 2.08 1.97 2.10 2.06 2.33 2.30
18.1
22.4 27.6
Feaq2++ Ce'", 0.5 M H2S04,X 350 nm 1O3[Fe,?1, M
105[Ce'"], M
0.136 1.27 5.32 8.01 10.7
1.5 7.5 7.5 7.5 7.5
Fe,? 103[Fea?], M 1.00 1.32 2.03 2.50 2.55 2.55 2.57 2.58 2.58 2.58d 3.06 4.04 5.06 5.07 6.63 7.58 8.24 9.18 10.2 10.2
10-3km,
io-%,,
rn-l
S-l
1.54 1.35 1.28 1.19
0.20 1.75 7:19 10.4 14.0
+ IrC12-, 0.5 M HCI04, X 497 nm
105[IrC1,2-], 10T3km,W3k,,
M
rn-l
s-l
AAtotalb
AAobadC
3.0 3.0 3.0 7.7 3.0 3.0 1.5 0.75 0.51 0.3 3.0 7.7 9.1
1.46 1.49 1.55 1.83 1.56 1.24 1.19 1.15 1.07 1.13 1.62 1.86
3.84 5.89 7.80 9.40 9.77 10.7 10.8
0.0414 0.0270 0.0204 0.0444 0.0164 0.0153 0.0076 0.0033 0.0026
7.7
1.93 1.97 1.71 1.97 1.89 1.65 1.88
0.2092 0.2089 0.2089 0.5497 0.2104 0.2158 0.1077 0.0521 0.0361 0.0197 0.2092 0.5488 0.6679 0.5496 0.5503 1.0981 0.5509 0.6707 1.0981 0.6711
7.1
15.0 7.7 9.1 15.0 9.1
1.88
12.1
10.4 16.6 11.9 15.0 18.6 19.1 24.5
26.5 31.1 35.2 35.1 39.8
0.0009
0.0139 0.0278 0.0273 0.0219 0.0171 0.0315 0.0135 0.0145 0.0238 0.0128
a Reaction run under 5:l conditions. AAtatal = total absorbance change of reaction. AAobsd= calculated absorbance change across velocity range of analysis (2.9-10.5 ms-I). dSignal change too small for accurate measurement of k..
124000 s-l is a factor of 400 greater than the limit of stopped-flow methods. This performance represents a significant advance in the ability to study rapid reactions in solution. Resolution of k,Values. The double-reciprocal depen-
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987 0.10
0.02
f
1
'
'
'
4
0
v.
8
12
m/s
Flgure 7. Plots of M,,,((A - A - ) / ( A,, - A -)) for fast reactions: A, 3.16 X lo-' M W(CN),& 6.01 X lo4 M IrCIt- in 6.5 M H,S04, kY = 2.67 X lo4 s-'; B, 6.02X lo4 M W(CN),& 1.20 X lo-' M IrCI, in 0.5 M H2S04,k , = 5.14 X lo4 s-'; C, lo-' M [H2P04-]T,pH ;=7, thymol blue; D, lo-* M [KHPIT, pH ~ 3 ,thymol blue.
+
+
+
+
dence (eq 3 and 4) can be used for the entire range of k, values. However, a simpler data treatment is advantageous when kapp is greater than 4000 s-l. Under this condition, e-y > lo6 s-l (19). Curves C and D of Figure 7 show reaction data for thymol M, pH 7) and KHP blue (pKa 8.9) with phosphate M, pH 3), respectively. In these two cases, M plots have negative slopes, which indicates that the reaction is too fast to resolve. For these curves, signal changes with velocity are attributed solely to mixing effects. Mixing. When k, exceeds lo6 s-l, ulbk,
