J . Phys. Chem. 1985,89, 2605-2612 Ui(0) = u(fi,o) vi(0) = u ( f , O ) , i = 1, 2,
fact that this Jacobian is banded with half-bandwidth equal to 3. More details of this algorithm can be found in ref 10 and 11. For the actual computations, a uniform grid of meshwidth hi = 0.01 is employed, resulting in a nonlinear system of 102 ordinary differential equations. Although a nonuniform grid could have easily been used, the uniform grid allows for a second-order local discretization error in the spatial direction. In the time-step integration, the order of the method is optimally between first and fifth, and the stepsize is adjusted to guarantee a relative error or 10-4.
(A4a)
..., NPTS
2605
(A4b)
Unfortunately, standard Adams-Moulton-Bashforth predictor-corrector methods or Rung-Kutta methods cannot be used to solve eq A3-A4 because the stiffness of this system will require time steps which are intolerably too small. Instead, a stiff integrator by Hindmarsh,Io which contains backward differentiation operators of order 1 to 5 , is employed. This integrator uses the Jacobian of the right-hand side of (A3) to form a pseudeNewton's method to converge the corrector equation. It makes use of the
Registry No. PPS, 25212-74-2; AsF,, 7784-35-2; AsF,, 7784-36-3.
(10) A. C. Hindmarsh, Lawrence Livermore Laboratory Report UCID30059, Rev. 1, March, 1975.
(1 1) R. F. Sincovec and N. K. Madsen, ACM Trans. Math Software, 1(3), 232 (Sept 1975).
Recombination of H+ and OH- In Pure Liquid Water Wesley C. Natzlet and C. Bradley Moore* Department of Chemistry, University of California, and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720 (Received: April 25, 1984; In Final Form: February 20, 1985)
Relaxation times for the H20+ H+ + OH- equilibrium are measured as a function of temperature between 0 and 48 O C and in HzO-DzOmixtures at 25 1 OC. Photoionization is produced by a short laser pulse. The measurements test the recombination mechanism proposed by Eigen involving a rate-limiting diffusive encounter between free H+ and OH-. The relaxation times range between 233 and 14 p . The corresponding recombination distance is found to be 5.8 i 0.5 A, independent of temperature and isotopic composition; the result is consistent with Eigen's four-moleculeion-pair intermediate. Encounter rates, diffusive separation rates, overall ionization rates, and the ion pair equilibrium constant are calculated from the 5.8 8, recombination distance for a range of temperatures and isotopic compositions.
*
Introduction Proton transfers occur both in biologically important enzyme reactions and in industrially important acid- and base-catalyzed reactions. Water is a common, albeit complex, solvent for many of these reactions. For some involving methylamine^,'-^ the solvent water molecules play a special, intimate role as intermediaries in the proton transfer. The simplest reaction where this occurs is the liquid water a~toprotolysis~
HzO
perimental apparatus, Figure 2, consisted of (1) a liquid system used to contain, transport, purify, and monitor the water, (2) an optical excitation source, and (3) electronics used to monitor and record the time-dependent conductivity. An electrical pulse opened the gate of a triggered differential amplifier, started a transient recorder (digitizer), and, after a delay, fired the laser. The laser light generated excess ion concentrations by exciting vibrational overtones of the water flowing between the electrodes of a conductivity cell. The transient conductivity was recorded by a signal-averaging computer. The liquid system removed impurity ions by continuously circulating the water in
2H+ + OHkR
The autoprotolysis reaction is one of the fastest reactions in solution.s A technique for following fast reaction kinetics by measuring equilibrium relaxation rates following a small perturbation was developed mainly by Eigen and c o - w o r k e r ~ . ~ , ~ Several methods have been used to perturb the autoprotolysis equilibrium of HZO7-lsand of D20.8 Here we use the method originated by Goodall and co-workers;l2 vibrational activation of the dissociative ionization reaction by single-photon excitation of the &H or &D stretch overtones of liquid water produces excess hydronium and hydroxide ions.lz-ls The concentration of the excess ions during the ensuing recombination is proportional to one component of the transient conductivity, (3) in Figure 1. Ion recombination rates were measured as a function of temperature and isotopic composition. The constant ion recombination distance derived from the data by using the diffusion-limited rate model of Eigen is a physically reasonable result.
Experimental Section The transient conductivity of pure water was measured as a function of time following an optical excitation pulse. The exPresent address: University of Chicago, James Franck Institute, 5640 S. Ellis Ave., Chicago, IL 60637.
0022-3654/85/2089-2605$01.50/0
(1) E. Grunwald and A. Y. Ku, J . Am. Chem. SOC.,90, 517 (1968). ( 2 ) E. Grunwald and E. K. Ralph, 111, Acc. Chem. Res., 4, 107 (1971). (3) E. Grunwald and D. Eustace, 'Proton-Transfer Reactions", E. Caldin and V. Gold, Eds., Chapman and Hall, London, 1975, p 103. (4) F. H. Stillinger, "Theoretical Chemistry; Advances and Perspectives", Vol. 3, H. Eyring and D. Henderson, Eds., Academic Press, New York, 1978, p 216. ( 5 ) M. Eigen, Angew. Chem. Int. Edit. Engl., 3, 1 (1964). (6) M. Eigen, W. Kruse, G. Maas, and L. De Maeyer, Prog. React. Kinet., 2, 285 (1964). (7) M. Eigen and L. De Maeyer, Z. Elecktrochem., 59, 986 (1955). ( 8 ) G. Ertl and H. Gerischer, Z. Elektrochem, 66, 560 (1962). (9) M. Eigen and L. De Maeyer, Proc. Roy. SOC.London,Ser. A , 247, 505 (1958). (10) G. Britre and F. Gaspard, J . Chim. Phys., 64, 1071 (1967). (11) G. C. Barker, P. Fowles, D. C. Sammon, and B. Stringer, Trans. Faraday SOC.,66, 1498 (1970). (12) B. Knight, D. M. Goodall, and R. C. Greenhow, J . Chem. SOC., Faraday Trans. 2 , 1 5 , 841 (1979). (13) J. F. Holzwarth, "Techniques and Applications of Fast Reactions in Solution", W. J. Gettins and E. Wyn-Jones, Eds., Reidel, Boston, 1979, pp 47-59. (14) W. C. Natzle, C. B. Moore, D. M. Goodall, W. Frisch, and J. F. Holzwarth, J . Phys. Chem., 85, 2882 (1981). (15) J. J. Bannister, J. Gormally, J. F. Holzwarth, and T. A. King, Chem. Br., 20, 227 (1984).
0 1985 American Chemical Society
2606
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985 1.2,
I
I
I
(2)
Figure 1. Transient conductivity in HzO showing (1) the prelaser background conductivity, (2) the peak conductivity immediately following excitation, (3) the relaxation rate, and (4)the base line increase after relaxation. The magnitude of the transient is as small as 3 X IOd times the level of background conductivity, (1). The signal shown here has a peak magnitude about lo4 times the background conductivity. Curve is for HzO at 19.8 O C with Y = 9936 cm-'. The time scale is 400 gs/division. LIQUID
TRIGGERED
+D I F F E R E N T I A L AMPLIFIER
1 PULSE GENERATOR
i
LOW PASS FILTER
1 DIGITIZER
1 COMPUTER AVERAGER
Figure 2. Apparatus for measuring transient conductivity changes induced by photodissociative ionization of water.
a closed loop through a thermostated ion-exchange column immedidtely preceding the conductivity cell. A . Liquid System. This section describes the components, the preparation, and the operation of the closed loop water circulation system. The components of the H 2 0 system consist of a water reservoir, a pump, an ion-exchange column with constant temperature jacket, a thermocouple, and a conductivity cell. They were constructed of quartz, Pyrex, fluorinated ethylene-propylene (FEP Teflon), poly(tetrafluoroethy1ene) (PTFE Teflon), and silicone rubber. A 225-mL Pyrex bulb was used as a water reservoir during system fill and evacuation/degassing. It supplied a Cole Parmer Masterflex pump. The pump used a Model 7545-00 drive unit with a 7017 peristaltic pump head using 641 1-47 silicone tubing. The tubing was 0.25 in. i.d. and 0.379 in. 0.d. An 18 cm piece of I / & SGA Scientific no. R8428-14 corrugated, flexible Teflon tubing was connected to the pump outlet. The ion-exchaxige column was a 78 cm length, of 8 mm o.d., 7 mm i.d., T F E Elastofluor tubing partially filled with Rohm & Haas Amberlite MB-1 mixed-bed ion-exchange resin. Trace amounts of amine and styrene decomposition products were leached from the resin by soaking in high-purity water for several days. Resin (10-14 mL) was then poured into the column in several small portions to minimize separation of cationic and anionic forms of the resin caused by a higher settling rat: for the denser cationic form. Column supports were made of 0.21-mm-mesh Teflon Spectramesh grid no. 146464 from Spectrum Medical Industriek The grid was stretched over a few-millimeter length of PTFE tubing and then inserted in the column. Larger diameter Vycor glass and polyethylene columns with glass wool column supports were
Natzle and Moore tested with satisfactory results. Bio Rad AG501-X8 ion-exchange resin columns were not used because they produced H 2 0 with twice the conductivity of the column described. The column was temperature regulated by circulating methanol in an outer jacket. The absolute water temperature was measured with a copperconstantan thermocouple against a water-ice reference. The conductivity cell was machined from a in. X 13/, in. X 6 in. telfon block with an inner chamber in. X in. X 4 in. Inner walls were smooth and edges perpendicular to the flow were rounded to minimize water turbulence. The bare platinum electrodes were made of 0.05 1-in. diameter platinum wire bent and flattened into a staple with 2.7 mm X 9 mm electrode faces. Electrode pairs were located 21/2-in. and 3 in. downstream from the cell inlet to allow inlet turbulence to damp out before reaching in. X 1 in. X 6 in. the electrodes. Quartz windows ' / 8 in. or were sealed with Viton O-rings. A 1/16-in.thick protective rubber pad was sandwiched between the window and a 1/8-in. X 6 in. X 13/4 in. stainless steel retaining plate which screws into the teflon cell. The Teflon, quartz, Pyrex, and silicone rubber parts of the system must be cleaned to prevent contamination of the high-purity circulating water. The Teflon cell body, Elastofluor fittings, and Spectramesh column supports were cleaned by soaking in aqua regia and then rinsing in water for about 5 h. The PTFE tubing and flexible FEP tubing were assembled with the clean Elastofluor fittings and filled with concentrated nitric acid for about 5 h. The system was then rinsed and filled with water for 4 h. Quartz windows were cleaned with methanol then soaked in an aqueous 2% sodium hydroxide-1% sodium EDTA solution to remove adsorbed ions. The pyrex water reservoir and thermocouple well were boiled for several minutes in dilute nitric acid. Electrodes were cleaned before assembly by immersion in 1 part of nitric acid to 3 parts of hydrochloric acid followed by passage of 1 A of current for 2 min through the positively charged electrode. A platinum wire served as the negative electrode. Silicone tubing for the peristaltic pump was cleaned by rinsing with methanol followed by immersion in boiling water which was allowed to cool to room temperature. Feedstock water for the circulating system came from two sources. One was a Pyrex and quartz pyrolytic still. The water vapor in the still was passed in an oxygen atmosphere through a 900 "C quartz tube pyrolyzer. Trace organic impurities were oxidized to COz and H 2 0 during passage through the pyrolyzer. The candensed HzO was then redistilled in a second stage and stored. An oxygen purge prevented atmospheric C 0 2 from reaching the water. In the second source, water was prefiltered in Barnstead charcoal and ion-exchange resin filters, then distilled in a Corning Mega-Pure still. No difference in conductivity was noticed with water from either source. The measured conductivity in the liquid system was typically 110% of the equilibrium conohm-' ductivity due to hydroxide and hydronium ions (5.5 X at 25 "C). Water was introduced to the system by filling the expansion bulb. Dissolved COz and many included air bubbles must be removed after pouring an ion-exchange column and filling the system with HzO. Gases were removed by evacuation of the system which caused the water to boil, The system was repressurized with argon to exclude atmospheric COz. Several evacuation/repressurization cycles with the water circulating were needed to completely remove dis,solved atmospheric gases. The liquid flow rate was about 2.5 cm3 s-l. Flow rate variations caused by the peristaltic pump can produce low-frequency noise in the conductivity signal. Flow rate changes were minimized by impeding the fluid flow a t the conductivity cell inlet and by allowing the system volume between the pump and the flow restriction to expand and contract in response to pressure pulses from the pump. An 18-cm section of corrugated FEP Teflon tubing and, to some extent, the flexibility of the silicone pump tubing allowed system expansion. The flow restriction was provided by the ion-exchange column. Water temperature was varied from 47.6 to about 0.13 "C. The stability was f0.2 "C over 2 min. At 8.5 "C, the water warmed a maximum 0.2 "C while traveling
Recombination of H+ and OH- in Liquid Water from conductivity cell to thermocouple. The overall uncertainty in the absolute temperature measurement was i 0 . 4 'C. Most tubing, the ion-exchange column, and the conductivity cell were thermally insulated with foam rubber or Armstrong Armaflex. Experiments in DzO and isotopically mixed water required DzO feedstock, deuterated ion-exchange column, and an isotopic composition measurement. Feedstock 99.8% DzO (Bio-Rad catalog no. 710-1003 or Aldrich catalog no. 15,188-2) was pyrolytically distilled. The still was predeuterated by distilling and then discarding about 10 mL of D 2 0 while heating all normally cool regions of the still with a heat gun. After cooling, 200-300 mL of DzO was added to the still and collected after the first stage of distillation. The ion-exchange column was deuterated by preparing the column with HzO, followed by three successive cycles of draining, filling, and mixing with high-purity DzO. Mixing was accomplished by degassing the system a t least six times. Vigorous boiling during degassing assisted mixing. The mole fraction of hydrogen in the final mixture with highest isotopic enrichment was 0.0057 f 0.002. The ionic conductivity was 110% of the theoretical conductivity of equilibrium D30+ and OD- in the mixture. The isotopic composition of the water was determined from samples drawn through a Teflon-coated silicone rubber septum with a syringe. Density measurements with a 3-mL pycnometer were used to determine isotopic enrichment for hydrogen mole fractions greater than 0.10. Liquid-phase infrared spectroscopy was used for smaller hydrogen mole fractions. Measured hydrogen mole fractions have an uncertainty of 0.002 at 0 mole fraction hydrogen increasing to 0.01 for mole fractions of hydrogen of 0.1 and greater. Infrared spectra with 8-cm-' resolution were taken on a Nicolet 7199 FTIR spectrometer using a HgCdTe detector, KBr beamsplitter, and Globar source. The liquid samples were contained in 1-cm-path cylindrical quartz cells. The magnitude of the HOD absorption a t 6030 cm-' increases linearly as hydrogen mole fraction is increased, Figure 3. To determine isotopic composition, the peak height for a sample from the liquid system was compared to a calibration based on measurements at 0.01 1, 0.032, 0.058, and 0.140 mole fraction hydrogen. The peak height was determined from a base line drawn on the slope of the 6250-cm-' DzO peak. B. Excitation Sources. A Quanta' Ray DCR-1, Nd3+:YAGpumped PDL-1 dye laser was used directly or Stokes shifted in a high-pressure hydrogen or methane Raman cell. The dye laser output was focussed first with a 50-cm lens external to the 1-m Raman cell. Sixty centimeters from this lens an internal 6.7-cm focal length lens produced a second focus to enhance the second Stokes output. The dye laser wavelength was checked with a Beck wavelength reversion spectrometer. Reported wavelengths are accurate to 1 nm. Near-infrared frequencies were calculated from the dye laser frequency and the Raman frequency of the gas in the Raman cell. The desired output wavelength was selected with a quartz prism or Coming 7-56 or Corning 7-69 fdters. Typical transmitted energies were between 1 and 4 mJ. The corresponding light flux at the cell is 8-32 M W cm-* in a spot 1.5 mm2 for the 8-11spulse duration. For some experiments a t 1.06 pm, a Raytheon SS404 Nd3+:YAG laser was used. The maximum fluence of 0.83 J cm-z at 1.06 Mm was used for measurements in DzO.The pulse length was 15 ns, the fluence 0.83 J/cm2, and the beam area 1 mmz. Pulse energies were measured with a Scientech 38-0101 calorimeter. The conductivity cell was initially aligned with a visible beam by tilting the cell until the spot reflected from the rear window did not strike the electrodes. Spurious signals resulted when the laser beam touched the cell electrodes. A mask made from a pair of anodized razor edges 1.4 mm apart, 9 mm in front of the electrodes oriented parallel to the electrode plane eliminated the problem. C. Electronics. This section describes both the important features of the signals and the electronics chosen to record the
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985 2607
u 0) c 0
2
v) 0
n
U
I
I
I
I
6500 6200 5900 5600 Frequency (cm-'1 Figure 3. The infrared spectra of 0.003,0.017, and 0.038 mole fraction hydrogen samples of water are shown by curves A, B, and C, respectively. The peak at 6030 em-' is from HOD. The peak at 6250 em-' is from D20. The pathlength is 1 em. The absolute scales for spectra A, B, and C are 0.0760, 0.0804, and 0.0800 base 10 absorbance units/division, respectively.
useful and eliminate the unwanted portions of the signal. The transient signal magnitude was as small as 3 X 10" times the background conductivity resulting from the equilibrium hydronium and hydroxide ion and from the small concentration of unidentified ions. The equilibrium conductivity had a time varying noise component with a frequency between dc and about 20 H z and a magnitude up to 700 times as large as the signal magnitude. The noise came from flow turbulence in the cell, from bubbles, and possibly from other causes. The static conductivity used to assess the ionic purity was monitored by measuring the voltage drop across a resistor in series with the conductivity cell. The electronics for recording the desired transient signal consisted of several parts.16 The conductivity monitor separated the static conductivity from the transient conductivity and amplified only the transient. An operational amplifier used as a current-to-voltage converter amplified current differences between an illuminated and a dark pair of conductivity electrodes. A triggered differential amplifier then increased the transient voltage and attenuated the low-frequency noise. It sampled the voltage when a trigger pulse arrived and amplified the subsequent voltage difference. The signal was then dc coupled to the 7A16 vertical amplifier of a Tektronix 7912 transient digitizer with a PDP 11/10 Digital Equipment computer averager or to the 7A16P vertical amplifier of a Tektronix 7912 AD transient digitizer with an LSI-11 com(16) W. C. Natzle, Ph.D. Thais, University of Califomia, Berkeley, 1983.
2608 The Journal of Physical Chemistry, Vol. 89, No. 12, I985
Natzle and Moore
TABLE I: Experimental Equilibrium Relaxation Time, Recombination Rate, and Recombination Distance for the Diffusion-Controlled Reaction L+ + OL- + LiO
kR, M-I
‘4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
T, O C 0.13 f 0.4 0.4 f 0.4 0.7 f 0.4 1.1 f 0.4 2.1 f 0.8 2.2 f 0.4 3.35 f 0.4 3.7 f 0.9 4.2 f 0.4 6.5 f 0.7 6.7 f 0.4 9.0 f 1.0 10.1 f 0.4 21.1 f 1.2 25.1 f 0.8 29.5 f 0.4 30.4 f 0.3 39.1 f 0.8 42.5 f 0.4 47.6 f 0.4
233 f 14 230 f 26 227 221 f 2 199 f 4 202 i 18 220 f 26 178 f 4 173 f 10 144 f 6 146 f 18 122 f 2 113 f 20 56 f 2 45 f 1.4 35 f 1.4 33 f 1.0 21.5 f 0.6 19.1 f 1.2 14 f 2
6.30 f 0.36 6.2 f 0.7 6.3 6.34 f 0.04 6.70 f 0.12 6.6 f 0.6 5.8 f 0.75 6.97 f 0.14 7.0 f 0.4 7.5 f 0.26 7.4 f 0.9 7.90 f 0.16 8.1 f 1.5 10.3 f 0.2 11.1 f 0.4 11.9 f 0.6 12.3 f 0.4 14.0 f 0.4 14.1 f 0.8 16.5
6.0 f 0.7 5.7 f 1.3 5.77 5.72 f 0.08 6.03 f 0.22 5.8 f 1.0 4.0 f 1.0 5.95 f 0.24 5.8 f 0.7 5.9 f 0.4 5.7 f 1.3 5.8 f 0.22 5.8 f 2.0 5.75 f 0.4 5.74 f 0.35 5.55 f 0.45 5.80 f 0.3 5.58 f 0.26 5.1 f 0.6 6.04
0.0057 f 0.0020 0.0174 f 0.0034 0.038 f 0.005 0.111 f 0.01 0.123 f 0.01 0.153 f 0.01 0.22 f 0.01 0.28 f 0.01 0.32 f 0.01 0.346 f 0.01 0.403 f 0.01 0.406 f 0.01 0.46 f 0.01 0.51 f 0.01 0.60 f 0.01 0.70 f 0.01 0.81 f 0.01 0.90 f 0.01 1
25.4 f 0.5 24.7 f 1.0 24.3 f 0.6 24.5 f 0.5 24.4 f 0.6 24.5 f 0.5 24.2 f 0.6 25.3 f 0.6 24.6 f 0.4 25.3 f 0.5 24.4 f 0.4 24.9 f 0.7 24.5 f 0.5 24.8 f 0.6 25.4 f 0.4 25.4 f 0.4 24.9 f 0.4 24.5 f 0.4 24.1 f 0.6
138 f 80 180 f 36 170 f 36 160 f 32 155 f 12 147 f 8 156 f 28 116 f 10 117 f 4 100 f 14 107 f 2.6 100 f 12 9 0 f 10 89 f 2.2 75 f 1.7 66 f 2.8 59 f 2 53 f 3 47 f 1.7
9.5 f 5.4 7.2 f 1.4 7.9 f 2 7.3 f 1.4 7.66 f 0.6 7.76 f 0.45 6.9 f 1.2 8.2 f 0.7 8.10 f 0.25 8.8 f 1.2 8.15 f 0.2 8.6 f 1.0 9.1 f 1.0 8.67 f 0.2 9.2 f 0.2 9.60 f 0.4 9.86 f 0.3 10.4 f 0.6 11.O f 0.4
8.8 f 10 5.5 f 2.2 6.5 f 2.8 5.4 f 2 5.9 f 0.9 5.9 f 0.7 4.4 f 1.6 5.9 f 1.0 5.74 f 0.34 6.5 f 1.8 5.48 f 0.26 6.0 f 1.4 6.4 f 1.4 5.57 f 0.26 5.70 f 0.26 5.6 f 0.5 5.5 f 0.32 5.7 f 0.7 5.9 f 0.4
XH
7,
puter averager. A low-pass filter preceding the digitizer input filtered out high-frequency noise. A cutoff frequency of 300 kHz was used for signal decay times less than about 75 ps. The cutoff frequency was 60 kHz for longer decay times. An optical isolator on the triggering lines prevented electrical noise from the laser from entering the signal electronics. Signal collection proceeded at the IO-Hz laser repetition rate. Typically, 5 12 shots were averaged, the conductivity cell polarity was reversed, and the same number of additional shots averaged. Subtraction of the two signals eliminated pick-up and multiplied the conductivity signal by two. The subtracted pick-up was about A in magnitude. A step increase of about 1.5 X 5X A with 300 V applied to the conductivity cell produced a conductivity step with signal-to-noise of one after averaging 1024 shots. This step increase corresponds to a 3 X M change (3 ppm) in hydronium and hydroxide ion concentrations at 25 O C . Figure 1 shows a typical transient conductivity signal. Data
The experimentally measured quantities are the exponential decay time of the transient conductivity, e.g., segment (3) of Figure 1, the temperature, and the isotopic composition of the water. Results are given in the first three columns of Table I. The first step in the analysis of raw conductivity data was to subtract the final base line conductivity from the digitized conductivity vs. time. The base line was determined from the average conductivity at long times, typically, between the third and the fourth or fifth decade of the exponential decay. Semilog plots of the decay were linear over at least two decades when signalto-noise was high. The conductivity decay is a single exponential. The decay time, T, was determined from a least-squares fit of the high signal-to-noise region of the natural log of the conductivity
P
s-I
X
0,
decay. A small fraction of the first T following laser excitation was rejected to avoid any error caused by the finite electronic rise time, and the last portion was rejected because of a diminishing signal-to-noise in the log plot as the conductivity approached the final base line conductivity. Typically, about 1 T was included in the least-squares fit. The decay times shown in Table I average between 4 and 16 determinations. Exceptions are for H 2 0 at 0.7, 3.35, and 47.6 C which are averages of two, three, and two determinations, respectively. For hydrogen mole fractions other than 1.O, weighted averages are calculated with weighting factors proportional to the square of the peak conductivity perturbation. The decrease in absorbance and in photoionization quantum yield for D20vs. H 2 0leads to a decrease in transient conductivity and therefore signal-to-noise as X, decreases. Uncertainties in Table I are two standard deviations. The temperature “uncertainty” in Table I gives the range of measured temperatures from which T values were averaged. In addition, the conductivity cell may be as much as 0.2 O C further from room temperature than the thermocouple temperature reported. Measurements of T were made for 0.58 fim I X I 1.19 fim. A large fraction of the data was recorded at 1.06 pm. Studies in pure water at and below room temperature were made at many wavelengths. No change in T with X was measurable.
Kinetic Model for Ion Recombination This section describes the reaction model used to analyze the primary data and to calculate the ionic recombination rate constant, the rate constant for diffusive separation from an ion-pair intermediate, the net dissociation rate constant, the ion-pair equilibrium constant, and the ion recombination distance. A . Recombination Model. The mechanism of eq 1 is oversimplified. A more complete mechanism that includes the pos-
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985
Recombination of H+ and OH- in Liquid Water sibility of a finite rate of reaction in an L+-OL- ion pair has been described by Eigen and co-workers.6 The mechanism is
+ OL-
L+
* k
L+...oL-
k2l
&2 L20 k31
7*hort-I Tlon['
+
= [S + (S2- 4 P ) 1 / 2 ] / 2
(3)
[s - (s2- 4P)'/2]/ 2
(4)
=
+
+
+
with S = k*12 k Z 1 k23 k32 and P = k*12k23 k21k32 + k*12k32. The rate k*12= k12(CLt CoL-)where CLt and CoLare the equilibrium L+ and OL- concentrations. If 4P > k 2 1 ) . Thus kR, the ion recombination rate , diffusion rate constant to form constant of eq 1 , is equal to k I 2 the an ion pair. Only a small concentration of L+-OL- is present in the liquid, and the overall equilibrium constant for water dissociation favors neutral water, i.e., kD
