Anal. Chem. 1997, 69, 431-438
Rate Measurements by the Pulsed-Accelerated-Flow Method Conrad P. Bowers, Kimber D. Fogelman, Julius C. Nagy, Thomas Y. Ridley,† Yi Lai Wang, Sam W. Evetts, and Dale W. Margerum*
Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
A pulsed-accelerated-flow spectrophotometer with UVvisible capability is described that permits measurement of pseudo-first-order rate constants as large as 500 000 s-1 (t1/2 ) 1.4 µs). Chemical rate processes are resolved from physical mixing rate processes by variation of flow velocities under conditions of turbulent flow. Two mixing processes are found in the mixing/observation tube. One mixing rate constant, valid for the full length of the tube, is directly proportional to the flow velocity. The other mixing behavior, proportional to the square of the flow velocity, is found only in the immediate vicinity of the 10 inlet reactant streams that collide with one another in the middle of the observation tube. Contributions from the latter mixing become more important for very fast reactions that take place close to the inlet jets. These mixing models and improved signal/noise permit the measurement of rate constants for very fast reactions. Applications of the PAF method to electron-transfer, proton-transfer, hydrolysis, and non-metal redox reactions are reported for pseudo-first-order and second-order reactions.
kmix
kr
A + B 98 (A‚B)mix 98 P
(2)
magnitude of the mixing rate constant under conditions of turbulent flow down a tube depends on the flow velocity (ν) (eq 3), because the greater the flow velocity the greater the degree
kmix ) kmν
(3)
of turbulence and the better the mixing.4,5 The reciprocal of kapp is the sum of the reciprocals of kmix and kr so that variation of v permits resolution of km (mixing proportionality constant) and kr (eq 4).2
1 1 1 1 + ) kapp km ν kr
(4)
constant kmix, and the reaction rate constant, kr (eq 2). The
Previous publications1-3 from this laboratory have described instruments that used the method of integrating observation4 and continuous flow for short time intervals (i.e., pulsed flow). The PAF-I instrument1 used a pulsed flow, where measurements were taken from velocity plateaus and separate pushes were needed for each velocity tested. First-order rate constants as large as 5000 s-1 could be resolved from data taken in a series of pushes at different velocities. The PAF-II instrument2 used accelerated flow for a short time period, and measurements were taken at 250 different flow velocities in one push. With this instrument, first-order rate constants as large as 12 000 s-1 could be measured and the volume of solution required was much less than when multiple pushes were required. The PAF-III instrument3 used a twin-path mixing/observation cell, and measurements of rate constants as large as 124 000 s-1 were possible. All three instruments used a visible light source, a single monochromator, and optical filters to eliminate stray light. The PAF-IV instrument, described in the present work, continues to use the highly successful twin-path cell design,3 but other design changes significantly enhance its capabilities. These enhancements permit data to be obtained that provide a better understanding of the mixing behavior for very fast reactions and an extension of the range of measureable rate constants. Ultraviolet as well as visible sources are used and stray light is minimized by use of a double monochromator. Larger optical
† Deceased. (1) Owens, G. D.; Taylor, R. W.; Ridley, T. Y.; Margerum, D. W. Anal. Chem. 1980, 51, 130-137. (2) Jacobs, S. A.; Nemeth, M. T.; Kramer, G. W.; Ridley, T. Y.; Margerum, D. W. Anal. Chem. 1984, 56, 1058-1065.
(3) Nemeth, M. T.; Fogelman, K. D.; Ridley, T. Y.; Margerum, D. W. Anal. Chem. 1987, 59, 283-291. (4) Gerischer, H.; Heim, W. Z. Phys. Chem. (Munich) 1965, 46, 345-353. (5) Gerisher, H.; Heim, W. Ber. Bunsen-Ges. Phys. Chem. 1967, 71, 10401046.
A “too-fast-to-measure” barrier has long prevented the study of many important chemical reactions in solution. Stopped-flow measurements are generally limited to reactions with half-lives greater than 2-5 ms. Many reactions cannot be maintained under the equilibrium conditions that are necessary for relaxation methods. Photoexcitation methods are often not possible or appropriate. When solutions must be mixed to initiate a reaction, the time required for the physical process of mixing the reactants restricts the ability to observe rapid chemical processes. The pulsed-accelerated-flow (PAF) method1-3 overcomes this problem by measuring the progress of chemical reactions while mixing takes place under conditions of turbulent flow. It is a continuousflow method in which variation of the solution flow velocity in a mixing/observation cell permits the mixing rate constant and the chemical rate constant to be resolved.2,3 The apparent rate constant (kapp) for a rapid reaction between A and B to give products (eq 1) depends upon the mixing rate kapp
A + B 98 P
S0003-2700(96)00599-9 CCC: $14.00
(1)
© 1997 American Chemical Society
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997 431
throughput combined with an electronic low-pass filter improves the signal-to-noise ratio (S/N) significantly. This permits the measurement of large rate constants with smaller signal changes than were possible with previous instruments. A programmable digital positioning motor is used to drive the push syringes so that accelerated flow or decelerated flow velocities can be selected. Extensive software permits data acquisition, calculation, and display on a personal computer. Three sets of electron-transfer reactions are used to calibrate the measured rate constants from 8000 to 500 000 s-1. Applications of PAF-IV to studies of proton-transfer, non-metal redox, and non-metal hydrolysis reactions are outlined for systems that are observed in the ultraviolet as well as visible region, where pseudofirst-order rate constants over a range of 1400-240 000 s-1 and second-order rate constants as large as 2 × 108 M-1 s-1 are measured. EXPERIMENTAL SECTION Instrument Design. Figure 1 is a block diagram of the PAFIV instrument. The syringe ram unit is mounted vertically on a sturdy frame that is isolated (except for flexible tubing used for solution delivery) from an optical table that holds the rest of the instrument. Isolation of the optical bench from the syringe ram eliminates vibrational problems. Vertical mounting of the push system makes it easier to remove bubbles from the syringes and the instrument takes up less floor space. The screw jack rotation is controlled by a programmable positioner (PROPOS) in order to generate the desired accelerated or decelerated push by the syringe ram unit. The mixing/observation cell is positioned inside a thermostated cell holder that also contains two distributor blocks (Kel-F) to divide the two reactant streams from syringes into the 10 streams that go to the manifold of the mixing/observation cell. The optical system has both deuterium and tungsten-halogen sources and utilizes a double monochromator to provide radiation from 200 to 800 nm with minimal levels of stray light. Photovoltaic detectors are used and for the visible light source; a feedback design helps to reduce drift in the light intensity. The detected signal passes through a low-pass filter before the A/D converter in the host computer. We have designed computer programs (in C language) for data acquisition as a function of flow velocities (Capture program) and for off-line analysis of the data to give rate constants (Pcalc program) from the PAF data. Flow System. Two drive syringe barrels (68.6 mm in length) were each made from a 31.8 mm diameter Kel-F rod to give an internal diameter of 14.3 mm. Brass rods with Teflon plungers and two 013 Viton o-rings are used to push solutions from the syringes. The o-rings are lubricated with a minimum amount of silicone or lithium grease to allow ease of movement. The syringe assembly is thermostated. Teflon distribution valves (three-port, L-plug, Hamilton HVX) interconnect the solution reservoirs, the drive syringes, and the leads to the distributor blocks and waste receptacle. The valves are held in place with a restraining device in order to prevent their vertical displacement and leakage during higher velocity pushes, when the back pressure of the system is high. Tefzel tubing (3.2 mm o.d. × 1.6 mm i.d., Upchurch Scientific) with Tefzel flangeless connectors (1/4-28 thread) are used to connect the push system to the distributor blocks, mixing/ observation cell, joiner block, and receiving syringe. The cell holder consists of a 210 mm diameter cylinder made of 25.4 mm thick aluminum with a 15.9 mm thick lids. Depressions in the 432 Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
Figure 1. Schematic of PAF-IV. Key: AD, analog-to-digital converter; AFILT, analog filter for intensity signal; BS, beam splitter; C, counter; CH, thermostated cell holder; DB, distributor blocks; E, displacement encoder, EF, encoder filter; HLS, home and limit switches; J, joiner; L, lenses; MIXCELL, mixing/observation cell; MONOCHR, monochromator; MT, motor; PC, computer; PROPOS, programmable positioner; REF DET, reference detector; RS, receiving syringe; SA, servoamplifier (motor power supply); SCREW JACK, converts angular velocity to linear; SIG DET, signal detector; SRC, light source; SYR BLK, syringe block (thermostated); SYRRAM, syringe ram; T, Tefzel (or Teflon) tubing; TACH, tachometer (velocity); UVPS, ultraviolet power supply; VFB, visible feedback controller; VISPS, visible power supply; VLV, valves.
end pieces hold Delrin supports, which align the twin-path cell and hold it firmly in place (Figure 2). Mylar shims totalling 0.3 mm are used to give a tight fit in order to prevent cell expansion during a push. The cell holder provides a thermostated enclosure for the twin-path cell. Mixing/Observation Cell. The twin-path cell and holder are shown in Figure 2. The cell (machined from PVC) has 10 inlet jets (0.33 mm diameter) that converge in the center of an observation tube (1.40 mm in diameter and 2.050 cm in length). The flow velocity (v) in the observation tube is accelerated or decelerated as the mixture flows from the center toward the windows. The reaction path is 1.025 cm. Solutions exit near the edge of the windows, which are fused quartz, 19.1 mm diameter and 6.4 mm thick. The outlets from the cell connect to a Kel-F joiner block and through exit tubing to a receiving syringe (Hamilton 1010-TLL gas-tight with Teflon-tipped plunger). The length of the exit tubing can be adjusted to change the back
Figure 3. Velocity profile for decelerated flow. Velocities and acceleration (a) are for solutions in the observation tube. Data are acquired after a 210 ms delay with a ) -39.9 m/s2 for v ) 11.8-3.0 m/s.
Figure 2. Twin-path cell with cell holder. (top) Cross-sectional view parallel to the light path and flow direction in the observation tube: CH, cell holder; CS, cell supports; W, windows: 9, o-rings; A and B, reactants: P, products. (bottom) Cross-sectional view at the cell center perpendicular to the observation tube, where 10 streams of reactants impinge upon one another.
pressure in order to prevent cavitation and to minimize pressuredependent delivery effects. Syringe Ram. The motor, screw jack assembly, and push plate for the syringe plungers are mounted vertically on a 0.47 m × 1.16 m × 25.4 mm aluminum plate, which is on a frame isolated from the rest of the instrument. The ball bearing screw (Beaver Precision) has a lead of 0.400 in./revolution. Digital Positioning System. The motor (Electrocraft Model E26-3) is driven differentially by a pulsed-width modulator servoamplifier/power supply (Electrocraft SA-9030/1). The motor provides a tachometer voltage and TTL signal from an optical encoder (Renco Model R-80) to indicate the angular velocity. The servoamp is controlled by a PROPOS II programmable controller (Electrocraft). The PROPOS and data acquisition computer count pulses from the encoder. There is some noise on the encoder due to the motor and servo-amp. The noise is removed by a Schmitt-triggered NAND gate pair with one input filtered with a simple 1 µs RC circuit. It is not necessary to externally filter the signal to the PROPOS. A requirement for the PAF push is that a known relationship exist between solution velocity and time during periods of data acquisition. Although the PROPOS system permits all types of velocity profiles to be programmed, we have primarily used a profile that gives a linear deceleration of velocity during data acquisition. Figure 3 shows the profile of velocities of solutions within the observation cell during the push time of 0.43 s. In order to avoid oscillations, a two-step approach is used to attain a velocity of 12.5 m/s. Typically, data are acquired as the velocity decreases from 11.8 to 3.0 m/s. The displacement of the drive syringes as a function of time in the data acquisition window is determined by encoder pulses supplied by the motor. These data
are fit to the appropriate law of motion to calculate the solution velocity for each absorbance measurement (Av). The reproducibility (0.7%) allows data to be ensembled when desired and permits point-by-point subtraction of two pushes to be made if necessary for light scattering corrections. After the flow stops, a small portion of solution in the center of the observation cell is not well mixed because of the lack of turbulence at the end of the push where v is less than 3 m/s. In order to obtain accurate static readings for Ao or A∞ measurements (see below), the wellmixed solution in the receiving syringe is pushed back into the observation cell. The PROPOS is programmed to pull back the push plate the minimum distance necessary for accurate static readings. The decelerated profile has several advantages: (1) The volume of solution that passes through the mixing/observation cell prior to the start of data collection ensures that the preceding reaction products are thoroughly flushed from the system. This eliminates the need for intermediate rinses. (2) Any problems with small losses of volatile reactants in the Tefzel tubing are overcome because the deceleration portion of the profile does not begin until the solutions that were originally in the push syringes have passed into the observation cell. (3) Similarly, the temperature control of reactants is better because they have been stored in the thermostated push syringes, rather than in the tubing. Optical System. The visible (50 W tungsten-halogen) and UV (30 W deuterium) sources are contained in an Oriel 7340 universal monochromator illuminator housing. The fan is not used because its vibration affects the visible lamp and the mirror in the housing. A programmable Hewlett-Packard 6267B dc power supply is used for the visible lamp, and an Oriel 6312 deuterium power supply is used for the UV source. The light passes through a dual-grating monochromator (Instruments, SA: DH-10; 200800 nm; 250 nm blaze concave holographic gratings). The visible light throughput of PAF IV is ∼4 times that of the PAF III instrument (with a 100 W bulb and 350 nm band-pass filter),3 which did not have UV capability. The better throughput significantly increases the signal relative to the dark noise. The light is collimated and then focused at the center of the cell using Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
433
f/3 and f/7 lenses (Oriel), respectively. Photovoltaic detectors (EG&G Model HUV-4000 B) are used. Prior to the cell, a 45° quartz beam splitter directs a portion of the light to a reference detector. The feedback circuit controls the power supply to keep the signal constant at this detector; the drift is reduced to less than 0.6 mA/h (at the signal detector). The feedback control cannot be used for the UV lamp, but the UV intensity drift corresponds to less than 0.7 mA/h. Absorbance measurements were calibrated with CrO42- in KOH.6,7 This verified that the length of the observation tube is 2.050 cm. Dilution experiments of absorber in one syringe vs medium in the other were used to test for equal volume delivery under flow conditions. After the necessary adjustments, absorbance vs velocity traces with changes in absorbance of 4000 s-1, then e-Y , 1 and M ) v/kappb. Substitution into eq 4 gives eq 6,
M ) (v/bkr) + (1/bkm)
(6)
where kr can be evaluated without iteration from linear plots of M vs v. Under these conditions, kr ) 1/[b(slope)]. This relationship has been used previously3 to measure reaction rate constants from 4500 to 124 000 s-1; however, when kr g 50 000 s-1, only the higher velocities could be used in the M vs v plots. Four data files are usually required for a determination of a rate constant on the PAF instrument. One file contains data collected for a PAF push of media vs media. This file serves as a reference intensity with which all other intensity data are compared in order to calculate absorbance values. For a measurement of a pseudo-first-order reaction, the next two files contain data collected for a PAF push of the limiting reagent vs medium (Ao1) and excess reagent vs medium (Ao2). These data are used to calculate the initial absorbance of the reactants (Ao ) Ao1 + Ao2). The final data file contains data collected during the reaction push of excess reagent vs limiting reagent. The intensity vs velocity data of this file are used to calculate the Av values. The static measurement following this push gives the intensity data corresponding to the products when the reaction is finished (A∞). Light Scattering Corrections. Ideally, the absorbance data for limiting reagent vs medium or for excess reagent vs medium have no velocity dependence since no reaction occurs. If the excess reagent is concentrated enough to affect the refractive index of its solution, Schlieren scattering will cause an apparent absorbance when the excess reagent is mixed with medium or limiting reagent.2,10 Because the mixing efficiency increases with velocity, this apparent absorbance will decrease as the velocity increases. The reaction absorbance values (Av) can be corrected by eq 7, where reactant 2 is assumed to be in excess, d indicates
M)
Av - (Ado2 - Aso2) - A∞ Aso1 + Aso2 - A∞
(7)
dynamic, and s static values. If the excess reagent absorbs, there may be a change in its absorbance with velocity due to instru(10) Dickson, P. N. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1986.
mental effects. Equation 7 also corrects for any background changes in an absorbing excess reagent. Reagents. Na2IrCl6‚6H2O (Aldrich, 99.9+%, λmax ) 487 nm) was used as received. K4W(CN)8 was prepared by the method of Leipoldt11 and was recrystallized from water and ethanol. Its purity was checked by microanalysis and titration with primary standard (NH4)2Ce(NO3)6 (G. F. Smith) with Fe(phen)32+ as indicator. The W(CN)84- solutions were relatively stable (decomposition was less than 1%/h) in the 0.50 M H2SO4 medium, but were slightly light sensitive12 so the solutions were protected from light. Solutions of Feaq2+ and Ru(bpy)33+ were prepared as described previously.2 Solutions used in the PAF are typically filtered and degassed under aspirator vacuum for 10-20 min. -Log[H+] and pKa Determination of Imidazole. A combination electrode (Sargent-Welch, S-30072-15) was calibrated by means of a HNO3/NaOH titration to give -log[H+] ) p[H+] readings at 25.0 °C and an ionic strength of 0.10 M KNO3. The method of Gran13 was used to determine the end point, and the treatment of Molina et al.14 was used to determine the electrode parameters. The potentiometric data were analyzed with the program SCOGS.15 The pKa at µ ) 0.10 M (KNO3) is 7.03 ( 0.03. pKa Determination of Chlorophenol Red. Absorbance values at 434 and 572 nm (λmax for the acidic and basic forms, respectively) of solutions with varying H+ concentrations were measured on a Perkin-Elmer 320 spectrophotometer. The pKa is 5.92 ( 0.01 (25 °C, µ ) 0.10 M KNO3).
Figure 4. M plots for pseudo-first-order rate constants for IrCl62with excess W(CN)84- where eq 6 applies kr (s-1): (A) 21 400; (B) 44 100; (C) 73 600.
RESULTS AND DISCUSSION Pseudo-First-Order Calibration Reactions with Rate Constants up to 75 000 s-1. Single-step, irreversible electrontransfer reactions were selected to calibrate the PAF-IV instrument. Three reactions were used (eqs 8-10) where the medium
IrCl62- + Feaq2+ f IrCl63- + Feaq3+
(8)
IrCl62- + W(CN)84- f IrCl63- + W(CN)83-
(9)
Ru(bpy)33+ + Feaq2+ f Ru(bpy)32+ + Feaq3+
(10)
for the Ir(IV) reactions was 0.50 M H2SO4 and the medium for the Ru(bpy)33+ reaction was 1.0 M HClO4. Figure 4 shows plots of M vs v (eq 6) for the IrCl62-/W(CN)84- reaction with three different concentrations of W(CN)84-, where kr values for loss of IrCl62- equal 1/[0.01025(slope)]. The results give kr (s-1) values of 21 400 ( 300 (curve A), 44 100 ( 1500 (curve B), and 73 600 ( 1800 (curve C). The precision is determined from the results for five to six pushes for each set of conditions. In these experiments, W(CN)84- was in 10-fold or greater concentration than the initial IrCl62- in order to give pseudo-first-order rate constants. The initial postmixed IrCl62- concentrations varied from 0.02 to 0.04 mM for these reactions, which were observed at 487 nm ( ) 4075 M-1 cm-1). The Ao values ranged from 0.15 to 0.35 and A∞ = 0. The M plots in Figure 4A are linear for the full velocity range used (3.0-11.5 m/s), but for (B) are linear (11) Leipoldt, J. G.; Bok, L. D. C.; Cilliers, P. J. Z. Anorg. Allg. Chem. 1974, 350-352. (12) Nya, A. E.; Mohan, H. Tetrahedron 1984, 3, 743-747. (13) Rossotti, F. J. C.; Rossotti, H. J. Chem. Educ. 1965, 42, 375-379. (14) Molina, M.; Melios, C.; Tognolli, J. O.; Luchiari, L. C.; Jafelicci, M. J. Electroanal. Chem. 1979, 105, 237-246. (15) Sayce, I. G. Talanta 1968, 15, 1397-1411.
Figure 5. Pseudo-first-order rate constants used to calibrate PAFIV in the range where eq 6 is valid. b, Fe2+ + IrCl62-; 0, Fe2+ + Ru(bpy)33+; O, W(CN)84- + IrCl62-.
only above 4 m/s and for (C) are linear only above 5 m/s. The deviations from linearity become more pronounced as the kr values increase. As discussed below, a different evaluation procedure is recommended when kr values exceed 75 000 s-1. Figure 5 shows plots of the resolved kr values from eq 6 for 13 different concentrations of W(CN)84- between 0.1 and 0.8 mM. The slope gives a second-order rate constant of (9.28 ( 0.06) × 107 M-1 s-1 for the reaction in eq 9. Figure 5 also shows kr values for the reactions in eqs 8 and 10 plotted against the concentration of Feaq2+ (excess reagent). The slopes give second-order rate constants of (3.48 ( 0.05) × 106 M-1 s-1 for the IrCl62-/Feaq2+ reaction and (8.43 ( 0.04) × 105 M-1 s-1 for the Ru(bpy)33+/Feaq2+ reaction. All the second-order rate constants are in excellent agreement with previous evaluations.3 The standard deviations of the resolved rate constants for these three systems are (0.6, 1.4, and 0.5%, respectively. Mixing Behavior in the Twin-Path Cell. Previous work3,16,17 has shown that, for turbulent flow of solutions down a tube, the mixing rate constant is proportional to the flow velocity (eq 3). (16) Gerischer, H.; Holzwarth, J.; Seifert, D.; Strohmaier, L. Ber. Bunsen-Ges. Phys. Chem. 1969, 73, 952-955. (17) Toor, H. L.; Singh, M. Ind. Eng. Chem. Fundam. 1973, 12, 448-451.
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
435
Figure 6. M plots for rate constants fit with eq 11. (A-E) IrCl62- + W(CN)84-: (A) kpred 133 000 s-1, kr (s-1) found 115 000; (B) 245 000, 224 000; (C) 373 000, 360 000; (D) 500 000, 570 000; (E) curve D without the kr term contribution; (F) thymol blue + H2PO4- (0.02 M, -log[H+] 6.47) kr = 2 × 107 s-1, curve fit to eq 12.
However, initial mixing in the center of the twin-path cell is the result of 10 streams of reactants that impinge upon one another perpendicular to the direction of flow down the observation tube. These inlet jet velocities are 3.6 times larger than the observation tube flow velocities. We propose that the mixing rate constant in the center portion of the cell with head-on collision of multiple jets is proportional to the square of the velocity, kmix ) km′(v′)2, where v′ ) 3.6v and km′ is a proportionality constant. It is difficult to define the exact boundary condition between the two mixing dependencies. However, the center jets are only 0.33 mm in diameter relative to the twin-path cell length of 20.5 mm, so that only 1.6% of the observation cell has appreciable mixing from the inlet jets. For slower reactions, the experimental Av values correspond to reactions that occur in the full length of the observation cell and the mixing behavior within the small path length of the cell center does not make an appreciable contribution. As the reactions become faster, more of the reaction takes place near the cell center and the observed Av values reflect a different dependence of mixing on velocity at the center jets. If kmix depends on v2 in the cell center and on v in the rest of the tube, the result will be an expression that corresponds to eq 11, where c is proportional to 1/km′.
M)
v c 1 + + v bkr bkm
(11)
In practice, we find that the c term is not needed when kr is less than 50 000 s-1 and that it becomes important to evaluate the c term as kr becomes greater than 100 000 s-1, so we have selected 75 000 s-1 as the dividing line between the use of eq 11 vs eq 6. Equation 11 is a semiempirical relationship that is evaluated for each set of conditions by a nonlinear least-squares fit of the data where c, kr, and km are fitted parameters. Deviations in the linearity of M vs v plots become more pronounced as kr values become larger, as shown in Figure 6. Curve F shows the response of the PAF data when the kr value is adjusted to be extremely large (∼2 × 107 s-1) for the reaction 436
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
Figure 7. Pseudo-first-order rate constants greater than 75 000 s-1 evaluated by eq 11 for the W(CN)84- + IrCl62- reaction. kpred, predicted; kr, observed. (Error bars are one std dev.)
between the basic form of an indicator (thymol blue) and a solution containing 0.02 M H2PO4- at p[H+] 6.47. Under these conditions the v/bkr term in eq 11 becomes negligible at all velocities used and the data can be fit to eq 12, where c ) 0.0218
M ) c/v + 1/bkm
(12)
( 0.0006 m/s and km ) 5345 ( 15 m-1. Thus, at infinitely large values of kr, the PAF instrument shows a behavior characteristic of the two regions of mixing. M Plots for kr Values from 75 000 s-1 to 500 000 s-1. Curves A-D in Figure 6 show the experimental M plots for reactions of IrCl62- with increasing concentrations of W(CN)84to give kr values that increase from 115 000 s-1 to 570 000 s-1. The solid lines give the fits to eq 11. Curve E shows the results from the data in curve D when the kr term is dropped. The difference between curves D and E illustrates that the contribution from the kr term is still significant even when the rate constant is as large as 500 000 s-1. The signal-to-noise capabilities of the instrument are very important in regard to the ability to obtain an accurate fit of kr in eq 11. Figure 7 shows the measured kr values plotted against the predicted kr values as the concentration of W(CN)84- is increased. The predicted kr values are calculated on the basis of the value of 9.28 × 107 M-1s-1 for the second-order rate constant evaluated from data in Figure 5 where kr varied from 8000 to 74 000 s-1. These results show that valid kr values can be obtained from eq 11 for 75 000 to 500 000 s-1. The standard deviation for kr values between sets of up to three ensembled pushes is typically (5%, but the precision becomes poorer (up to (9%) at very large kr values. The accuracy of the kr values is generally within 15% of the predicted value. At the highest kr of 500 000, the result is 570 000 or 14% high. The fit of eq 11 to the low-velocity portion of the data (Figure 6) is difficult because of S/N limitations and this may cause the trends above and below the correlation line in Figure 7. As kr increases, the v/bkr term decreases while the c/v term does not change very much. This limits the ability to measure larger kr values. Dependence of km and c on kr. We have not attempted to assign c or km values, but rather permit these constants along with kr to be evaluated by least-squares fit of each set of data.
values between 63 000 and 101 000 s-1, but at these kr values, the c/v term in eq 11 is small compared to the v/bkr term and is determined with less precision. Proton-Transfer Rate Constant for the Reaction between Chlorophenol Red and Imidazole. The ability to use the PAFIV instrument to measure fast proton-transfer reactions is demonstrated in the reactions of chlorophenol red (an indicator with pKa ) 5.92 for HCPR-) with excess imidazole buffer (pKa ) 7.03 for Him+). An acidic solution (p[H+] 4.3) of HCPR- [(1.5-3.8) × 10-5 M] was mixed with an imidazole solution [(2.1-6.7) × 10-4 M] at p[H+] 7.8-8.0, so that the final p[H+] was 7.42 ( 0.05. The observed reaction at 434 nm (eq 13) goes to 96-97%
HCPR- + im h CPR2- + Him+
Figure 8. Dependence of the mixing proportionality constant, km, on the pseudo-first-order rate constants, kr, for the reaction of IrCl62with excess W(CN)84-.
Figure 9. Dependence of the c term in eq 11 on kr for the IrCl62+ W(CN)84- system.
This is important because the mixing behavior depends on the medium, the exact construction and performance of the mixing cell, and the level of concentration of the excess reagent. We have observed experimentally that km increases for a series of reactions where kr increases. The effect is shown in Figure 8 for the IrCl62-/W(CN)84- system, where the km values increase from 2000 to 4300 m-1 as the kr values increase from 9000 to 500 000 s-1. As already discussed, different portions of the observation tube are used for reactions of different speed. One explanation of the change of km is that the degree of turbulence is higher closer to the inlet jets than further down the tube. This would cause km to increase as kr increases. In these experiments, we change kr by increasing the concentration of excess reagent. Higher excess reagent could speed the mixing process near its final stages, where small inhomogeneities are influenced by diffusion of reactants from one area to another. Higher concentrations will assist the rate of diffusion. Figure 9 shows a plot of the value of c (eq 11) vs kr for the IrCl62-/W(CN)84- system. The c term is almost constant and equals 0.018 ( 0.002 m/s for kr values from ∼100 000 s-1 to 500 000 s-1. The values of c vary from 0.013 to 0.015 m/s for kr
(13)
completion, so that the reverse reaction can be neglected. The experimentally observed kr values varied from 124 000 to 242 000 s-1 as the free imidazole concentration increases from 0.15 to 0.50 mM. This corresponds to a rate constant of (3.3 ( 0.1) × 108 M-1 s-1 for the direct proton transfer between the acidic form of the indicator and the basic form of imidazole. In this reaction mixture, even faster acid/base neutralization reactions (eqs 14 and 15) precede the indicator reactions. The
H3O+ + OH- h 2H2O
(14)
H3O- + im h Him+ + H2O
(15)
rate constant for the reaction in eq 14 is 1.3 × 1011 M-1 s-1 18 and the value for the reaction in eq 15 is 1.5 × 1010 M-1 s-1.19 These reactions quickly remove excess acid as the indicator solution mixes with the imidazole solution. The rates of proton-transfer reactions between weak acids and bases, where the protons are on oxygen or nitrogen donor groups, depend in part on the driving force of the reaction (∆pKa ) 1.11 in this case) and often approach the diffusion-controlled limit of 7 × 109 M-1 s-1 when ∆pKa > 3.18 Our experimental value for eq 13 is smaller than the value of (9 ( 3) × 108 M-1 s-1 reported from temperature-jump relaxation experiments.19 Other Applications of the PAF-IV Instrument to the Measurement of Fast Reactions. The PAF-IV instrument has been utilized with many different chemical systems to observe reactions over a wide range of wavelengths and rate constants (Table 1).20-33 First-order rate constants (kr) have been measured (18) Eigen, M. Angew. Chem., Int. Ed. Engl. 1964, 3, 1-72. (19) Eigen, M.; Hammes, G. G.; Kustin, K. J. Am. Chem. Soc. 1960, 82, 34823483. (20) Nagy, J. C.; Kumar, K.; Margerum, D. W. Inorg. Chem. 1988, 27, 27732780. (21) Troy, R. C.; Margerum, D. W. Inorg. Chem. 1991, 30, 3538-3543. (22) Wang, Y. L.; Nagy, J. C.; Margerum, D. W. J. Am. Chem. Soc. 1989, 111, 7838-7844. (23) Troy, R. C.; Kelley, M. D.; Nagy, J. C.; Margerum, D. W. Inorg. Chem. 1991, 30, 4838-4845. (24) Fogelman, K. D.; Walker, D. M.; Margerum, D. W. Inorg. Chem. 1989, 28, 986-993. (25) Yiin, B. S.; Margerum, D. W. Inorg. Chem. 1990, 29, 1559-1564. (26) Yiin, B. S.; Margerum, D. W. Inorg. Chem. 1990, 29, 1942-1948. (27) Scheper, W. M.; Margerum, D. W. Inorg. Chem. 1992, 31, 5466-5473. (28) Gerritsen, C. M.; Margerum, D. W. Inorg. Chem. 1990, 29, 2757-2762. (29) Gerritsen, C. M.; Gazda, M.; Margerum, D. W. Inorg. Chem. 1993, 32, 5739-5748. (30) Gazda, M.; Margerum, D. W. Inorg. Chem. 1994, 33, 118-123.
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
437
Table 1. Applications of PAF IV to Fast Reaction Kinetics chemical system
species monitored
λ, nm
HOCl + INCl3 + IOBr- + I ICl2-/ICl + H2O, OHIBr2-/IBr + IIBr/IBr2- + H2O, OHHOCl + SO32I3-/I2 + SO32RNCl2 + SO32NCl3 + HSO32-
I3I3OBrICl2I2BrIBr2OClI3RNCl2 NCl3
353 353 329 230 428, 440 253 292 353 310, 320 360
I3-/I2 + S2O32HOCl/OCl- + CNOBr- + CN-
I3OClOBr-
353 292 329
OI- + CN-
OI-
280
Br2 + NH2Cl Br2 + NH2OH HOBr + Br- + H+
Br3 Br3Br2
266 266 390
HOBr + Br- + H+ I2 + NH2OH
Br3I3-
266 353
-
range of kr, s-1 a ref 17 000-142 000 10 000-26 000 5 900-13 000 75 00-150 000 2 100-9 500 4 700-140 000 4 800-107 000 6 300-74 000 4 200-45 000 (1.9-5.8) × 107 M-1 s-1 b 30 000-150 000 3 900-13 000 (5-10) × 107 M-1 s-1 b (6 ( 2) × 107 M-1 s-1 c 11 900-20 000 3 900-60 000 (6.4-19.8) × 107 M-1 s-1 b 8 000-16 000 1 400-10 600
20 20 21 22 23 23 24 25 26 26 27 28 29 29 30 31 32 32 33
a Pseudo-first-order rate constants. The range of values given are used to evaluate dependence on [H+] as well as excess of the second reactant. b Pseudo-second-order rate constants (unequal concentrations). c Second-order rate constant (unequal concentration).
from 1400 to 150 000 s-1 (t1/2 values from 0.5 ms to 4.6 µs). Pseudo-second-order conditions have also been used with rate constants as large as 2 × 108 M-1 s-1 and initial half-lives as small as 3.1 µs.29 Electron-transfer reactions between IrCl62- and W(CN)84- were used to calibrate very rapid reactions under second-order conditions with c/v corrections.29 Rates of hydrolysis of inter-halogens were measured for the first time in studies of IClaq22 and IBraq23 at 230 and 253 nm, respectively. Halogen cation (31) Beckwith, R. C.; Cooper, J. N.; Margerum, D. W. Inorg. Chem. 1994, 33, 5144-5150. (32) Beckwith, R. C.; Wang, T. X.; Margerum, D. W. Inorg. Chem. 1996, 35, 995-1000. (33) Liu, R. M.; McDonald, M. R.; Margerum, D. W. Inorg. Chem. 1995, 34, 6093-6099.
438 Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
(Cl+, Br+, I+)-transfer rates and mechanisms have been determined for many reactions of electrophiles (NCl3, HOCl, OCl-, HOBr, OBr-, Br2, OI-, I2) with nucleophiles (CN-, SO32-, I-, S2O32-, NH2OH). In some instances these reactions are subject to general-acid or general-base catalysis. Halogen cation transfer is so fast that a few rates are limited by prior proton-transfer steps. In other cases, rapid adduct formation is observed as in the formation of I2S2O32- with a second-order rate constant of 7.8 × 109 M-1 s-1.27 The fast reaction capabilities of PAF-IV permitted detailed reaction mechanisms to be worked out for all these reactions. RECOMMENDATIONS AND CONCLUSIONS We have been able to measure first-order rate constants as large as 500 000 s-1 (t1/2 ) 1.4 µs) by additional characterization of the mixing process and by careful optimization of the solution delivery system. In general use, a range of first-order rate constants (kr) from 1000 to 170 000 s-1 is recommended. The larger the kr value to be determined, the larger the value needed for Ao - A∞. The PAF method permits study of irreversible reactions (not possible by relaxation methods); however, it is also possible to measure systems with reversible first-order processes.23 Reactions can also be measured under second-order conditions (equal or unequal concentrations).21,26,29 Initial halflives greater than 3 µs are recommended for second-order reactions. The PAF method moves the frontier of the “too-fastto-measure” barrier for irreversible reactions forward by a factor of 1000. ACKNOWLEDGMENT This work was supported by National Science Foundation Grants CHE-9024291, CHE-8720318, and CHE-8616666. The Ru(bpy)33+/Fe2+ calibration reactions were measured by R. C. Troy. This is part 4 in a series. Previous papers: (1) Anal. Chem. 1980, 51, 130-137; (2) Anal. Chem. 1984, 56, 1058-1065; (3) Anal. Chem. 1987, 59, 283-291. Received for review June 17, 1996. Accepted November 19, 1996.X AC960599E X
Abstract published in Advance ACS Abstracts, January 1, 1997.
