197
RATECONSTANT FOR THE DISSOCIATION OF ACETICACID
Application of the Potentiostatic Method. Determination of the Rate Constant for the Dissociation of Acetic Acid* by Ronald R. Schroeder2 and Irving Shain Department of Chemistry, University of Wisconsin, Madison, Wisconsin 63706 (Received August 1, 1968)
The rate constant for the dissociation of acetic acid in aqueous solutions was determined by conducting potentiostatic investigations on several acetate-acetic acid solutions of low bufferingcapacity. The current which was measured resulted from the discharge of the hydrogen ions which were produced by the acid dissociation. kr
CHaCOOH
CH3COO-
+ H+;
H+ B_ l/ZHz
Using existing theories for preceding reaction mechanisms, an apparent rate constant was determined for eac h of five acetic acid-acetate anion concentrations. The values of the apparent rate constant were found t o depend on the thickness of the reaction layer in which the net chemical reaction occurs, relative to the thickness of the electrical double layer in which the reaction is influenced by the potentials and the electric field. A graphical correction procedure was developed which involved plotting the product of the apparent rate constant times the reaction layer thickness against the reaction layer thickness. From the slope of this plot, it was possible to evaluate the characteristics of that portion of the reaction occurring outside of the double layer. Using this correction procedure, a value of the rate constant of ka = 1.58 X lo8 sec-l was obtained.
When the reactant a t an electrode is obtained from a chemical reaction in the bulk of the solution, the current is a function of the rate of the chemical reaction. Early work on kinetic systems of this type involved polarographic studies. ~ - 5Subsequently, the other electrochemical methods were also applied in the investigation of such reactions. These included the galvanostatic the potentiostatic method,* and many others.99’0 Among the early studies were investigations of weak acid systems where an easily reduced acid molecule is produced in the solution by the recombination of the less easily reducible acid anion with hydrogen ions. Several acids were studied and their dissociation and recombination rate constants were reported.’l!l2 However, rates obtained electrochemically were often in conflict with the earlier theories of Debyelaand OnsagerI4 which had established upper limits on attainable recombination rates. These limiting theories were confirmed when nonelectrochemical methods were applied to fast reaction studies arid the general validity of electrochemically measured rates was questioned. Possible sources of error in electrochemical measurements were discussed by Delahay and Vielstich6 and by Strehlow.16 They were thought to arise from the inadequacies of the diffusion equations for describing the mass transfer in fast kinetic systems and from the effect of the electric field of the double layer on the reaction rates (second Wein effect). The first of these problems was resolved, in part, by considering the effect of the electrical double layer on mass transfer.I6 The second problem, the effect of the
electric field on rate constants, was considered by Nurnberg,l” who reported a study of fast reactions in which the second Wein effect was taken into account. I n other work,18 he also corrected his measured values for mass transfer effects in the double layer and thus (1) From the Ph.D. tht *s of R. R. Schroeder, University of Wisconsin, 1967; presented in part at the 155th National Meeting of the American Chemical Society, San Francisco, Calif., April 1968. (2) National Institutes of Health Predoctoral Fellow, 1966. (3) R. Brdicka and K. Wiesner, Collect. Czech. Chem. Commun., 12, 138 (1947). (4) J. Koutecky and R. Brdioka, ibid., 12, 337 (1947). (5) J. Koutecky. ibid., 18, 597 (1953); 19, 857 (1954). (6) P. Delahrty and W. Vielstich, J . Amer. Chem. SOC.,77, 4955 (1955). (7),H. Gerischer and M. Krause, 2. P h y s . Chem. (Frankfurt am Main), 10, 264 (1967). (8) P. Delahay and S. Oka, J . Amer, Chem. Soc., 82, 329 (1960). (9) P.Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers, New York, N. Y., 1954. (10) P.Delahay, “Advances in Electrochemistry and Electrochemical Engineering,” Vol. I, P. Delahay, Ed., Interscience Publishers, New York, N. Y., 1961,p 233. (11) R. Brdicka, “Advances in Polarography,” Vol. 11, I. 8. Longmuir, Ed., Pergamon Press, New York, N. Y., 1960,p 655. (12) H.Strehlow in “Technique of Organic Chemistry,” Vol. VIII, A. Weissberger, Ed., Interscience Publishers, New York, N. Y., 1963,p 799. (13) P. Debye, Trans. Electrochem. SOC.,82, 262 (1942). (14) L. Onsager, J . Chem. Phys., 2 , 599 (1934). (15) H.Strehlow, 2. Elektrochem., 64,45 (1960). (16) L. Gierst in “Transactions of the Symposium on Electrode Processes, Philadelphia, 1959,”E. Yeager, Ed., John Wiley & Sons, Inc., New York, N. Y., 1961. (17) H. W. Narnberg, Discussions Faraday Soc., 136 (1965). (18) H. W. Narnberg in “Polarography 1964,” G. J. Hills, Ed., Macmillan and Coo,Ltd., London, 1966,p 149. Volume 78,Number 1 Januarv 1969
198 obtained rate constant values which were independent of electrode effects. The study of weak acid systems has been one of the principal applications of electrochemical methods in the area of fast reactions. However, the methods used in early studies required that the product of the chemical reaction be electroactive, and thus the recombination rates of common nonreducible acids, such as acetic acid, could not be determined. The dissociation of these acids produces hydrogen ion which does react at the electrode, bdt its polarographic wave is v e y drawn out and appears a t potentials where most supporting electrolytes are also reduced. Ruetschilg used an indirect method which involved a material (azobenzene) whose reduction requires hydrogen ion. Solution conditions were adjusted such that the rate of the dissociation of the acid limited the current. The value obtained for the rate constant for the dissociation of acetic acid in ethanol-water was kd = 2.1 X IO5 sec-l, uncorrected for specific electrode effects. Using the same chemical system in chronopotentiometry, Delahay and Vielstich6 obtained k d = 2.9 X lo5 sec-l, also uncorrected for electrode effects. In the first studies carried out directly on the dissociation of nonreducible organic acids, Nurnbergl7,18,20 used lithium salts as the indifferent electrolyte, thereby decreasing the interference between the reduction current for hydrogen ion and the residual current. Although significant blank corrections were still necessary, it was possible to make measurements using high-level faradaic rectification, apply additional corrections for electrode effects, and obtain a value of k d = 1.39 X lo6 sec-l for acetic acid. Other values of kd for acetic acid ranging from 5 X lo6 to 3 X lo6 sec-l have been obtainedZ1using various relaxation methods. Only one application of the potentiostatic technique to the kinetic study of acids has been reported. Delahay and Oka8 used the azobenzene system to measure the dissociation rate of monochloroacetic acid in an ethanol-water mixture. A fast-rise potentiostat was used, and data were obtained a few milliseconds after application of the potential. The theories of Koutecky , ~ A/latsuda22were applied, and Brdicka, * K o ~ t e c k y and and a value of h d = 1.77 X lo6 sec-l (without doublelayer corrections) was reported. In this work, the potentiostatic method was used to study the dissociation rate of acetic acid in aqueous solution by determining the current-time characteristics of hydrogen ion discharge from a series of acetate buffer solutions. Interference from the residual current was essentially eliminated by using tetramethylammonium chloride as the indifferent electrolyte. This work was done to evaluate the capabilities of a specially constructed potentiostat-cell apparatu~2~ and to demonstrate the applicability of the potentiostatic method to the study of an important kinetic system involving fast reactions. Although the acetic acid system does not The Journal of Physical Chemistry
RONALD R. SCHROEDER AND IRVING SHAIN involve rate constants quite fast enough to test the full capabilities of the instrumentation, the measurements reported here are one or more orders of magnitude faster than previous measurements of reasonable reliability using the potentiostatic method. Correction for double-layer effects on the rate constants was made, and the results are in reasonable agreement with data obtained by other methods.
Theory Boundary Value Problem. The general boundary value problem for a system in which a chemical reaction precedes a charge transfer has been considered several t i r n e ~ . ~For the acetic acid system, the reactions are
5 kf
CHBCOOH
Hf
+ CH3COO-
H+ “‘Il/zHz
(1) (11)
For the general case, the rate constant, kb’, for the recombination reaction would be second order, and this would make it impossible to obtain an analytical solution to the boundary value problem. However, in potentiostatic experiments it is feasible to maintain the concentration of the acetate ions constant and large compared with the concentrations of the other species. Under these circumstances a pseudo-first-order rate constant can be defined and used in the derivation kb
=
kb‘C*A-
(1)
where C*A- is the concentration of acetate ions in the bulk of the solution. Even with this assumption, however, a completely rigorous solution to the boundary value problem can be obtained only if it is assumed that all diffusion coefficients are equal. The result is6,8*9 ik
= (%FAc*EA
d~m/t) x exp(X2) erfc (XI
(2)
where h =
l/st
(3)
and
(4) Here, ifiis the kinetic current, A is the electrode area, t is the time, CH+ and C H A are the concentrations of hydrogen ions and acetic acid, respectively, C*WAis the bulk concentration of acetic acid, DHAis the dif(19) P. Ruetschi, 2. Phys. Chem. (Frankfurt am Main), 5, 323 (1965). (20) H.W. NOrnberg and G. C. Barker, Natuvwissenschaften, 15, 191 (1964). (21) E. F. Caldin, “Fast Reactions in Solution,” John Wiley & Sons, Inc., New York, N. Y.,1964. (22) H.Matsuda, J. Amer. Chem. SOC.,8 2 , 331 (1960). (23) R. R. Schroeder, Ph.D. Thesis, University of Wisconsin, 1967.
RATECONSTANT FOR
THE
fusion coefficient of the acetic acid, and the other terms have their usual significance. The equilibrium constant, K, which is dependent on the concentration of acetate ion, is related to the acid dissociation constant for acetic acid by Kd = KC*A-. For a system such as acetic acid, eq 2 cannot be applied directly, since the diffusion coefficient of hydrogen ion differs greatly from that of the other materials involved in the reaction. If the differences in the diffusion coefficients are taken into account, an approximate solution can be obtained6t22which is of exactly the same form as eq 2, except that in this case =
+
(D/DHA)dDH+/DHAKd(kf
kb)t
(5)
and
This can be simplified further, since for most experimental situations where the potentiostatic method would be applied (including the acetic acid system , 4/x the concentrations approach their bulk values C’. This, in effect, defines the thickness of the diffuse part of the double layer. When specific ion adsorption is present, the compact layer will contain ionic charges in accordance with the adsorption isotherm for the species involved. I n general, the potentials and charge population in the diffuse layer will be lowered. The values for the parameter describing the diffuse layer can still be det’ermined from the Gouy-Chapman theory but a new capacitance must be considered when calculating $H. Nevertheless, eq 11 remains valid for calculating 1/x, and although #H is changed, the forms of eq 14 and 15 remain the same. For most cases the distance limit for the influence of the potentials remains x = 4/K. Reaction Layer. The reaction layer is the small region around the electrode where net chemical reaction is assumed to occur and where the chemical system is not a t equilibrium, The thickness p of the reaction layer can be estimated for the acetic acid system as p
= dDH+/kb
(16)
and it is considered that all hydrogen ions produced between the plane defined by p and the electrode surface will react a t the electrode surface and produce faradaic current. The size of the reaction layer cannot be calculated without prior knowledge of kb’, the rate constant for recombination. However, for many acids, this recombination rate is nearly diffusion controlled and a value of 5 X 1Olo (mol/l.) -I sec-l can be assumed for the acetic acid case. Under these circumstances, for D E + o lo;* cm2/sec and CA = 1 M , p will be approximately 4.5 A. Thus, for large anion concentrations p can be of the same magnitude as 4/x, the effective double-layer thickness, in which case all the chemical reaction will occur in a region influenced by the double-layer potentials and the electric field. On the other hand, at a lower anion concentration but the same ionic strength, p becomes much larger than 4/x. For example, for CA- = 0.01 M , p will be about 45 A, and much of the chemical reaction will occur in a field-free region of the solution. Correction for Double-Layer Effects. When a chemical rate constant is calculated froin electrochemical data neglecting double-layer effects, the result will be an apparent rate constant, k,, which will differ from the true chemical rate constant, lct, since the region in which the chemical reaction occurs is not homogeneous. The inhomogeneity introduced by the double layer will be reflected by one or more specific effects which may include alteration of the ionic concentration3 in (25) P. Delahay, “Double Layer and Electrode Kinetics,” Interscience Publishers, New York, N. Y . , 1965.
201
RATECONSTANT FOR THE DISSOCIATION OF ACETICACID the double layer from their bulk values (eq 15), enhancement of the dissociation rate by the electric fieldI4 (eq 14), alteration of the rate and equilibrium expressions by dielectric saturation,26and acceleration or inhibition of mass transfer of ionic material by the potential and the electric field.lB None of these effects has been considered in the derivation of the boundary value problem in which it was assumed that kf, K, and CAwere all constant, independent of distance and time. A rigorous derivation in which these parameters are considered as functions of distance would be difficult. An alternate approach is to estimate the magnitude of these effects on the apparent kinetic parameter, (kfK),. The apparent kinetic parameter as determined from the experimental data represents a mean value of the actual parameter within the reaction layer. Since within this layer the individual parameters, kf and K (=Kd/CA-), are functions of distance, this mean value is given by
or considering the kinetic parameter as a single function
At distances greater than 4/x (the effective doublelayer thickness), the kinetic parameter assumes its bulk value which is independent of distance and, if the experiment is carried out in a system where p is large compared to the dimensions of the double Iayer-ie,, p > 4/x-the integral can be separated
Rearranging the limits of integration
(Ed)(l-14’x
dx dz) (21) C*Aand collecting terms and simplifying, one obtains
The remaining integral could be evaluated if kf(z), Kd(z), and CA-(Z)were known; if p were not greater than 4 / ~ ,this integral would have to be evaluated to obtain the bulk parameters. However, it suffices here to note that this term represents the difference between the average kinetic parameter within the double layer and the kinetic parameter in the bulk of the solu-
tion and contains all the specific effects introduced in the experiment. Thus, representing this integral term as (kfKd/CA-)dl,eq 22 becomes
Dividing each term by K~/C*A-gives &a
=
&f
+
(kf)dl
(24)
For a series of experiments performed at constant ionic strength, (ki)al will be constant, and if p is varied by employing several different anion concentrations, several different values of k , will be obtained. Provided that p > 4/x, a plot of the values of p k a US. p will provide a value of kf from the slope. Since p cannot be calculated before kf is known, it suffices to assume that kb and D H + are constant and, in accord with eq 17, to plot k a / c us. l / c , again obtaining kf from the slope.
Experimental Section One of the major aspects of this work was the development of instrumentation capable of charging the double layer very rapidly, so that reliable measurements of the famdaic current could be made at times as short as 10 psec. A detailed discypion of this portion of the work will be presented elsewhere.2e Instrumentation. Cell. The cell used for potentiostatic experiments was designed to eliminate undesired impedances (lead inductance, contact resistance, etc.) and was arranged so that electrode placement was reproducible. The working electrode was a stationary mercury drop supported on a J-shaped electrode, the counter electrode was a platinum wire immersed in the test solution, and the reference electrode was a saturated calomel electrode which contacted the test solution through a salt bridge and a Luggin capillary. For potentiostatic and stationary electrode polarography experiments, a microburet-type electrode (Brinkmann Instruments) was used to supply mercury drops. For all experiments the solution temperature was determined using a thermometer mounted in the cell lid and immersed in the solution. Room temperature was changed as a means of controlling the cell temperature, and all experiments were carried out at solution temperatures of 25 A 0.5’. Potentiostat and Signal Sources. For potentiostatic experiments a circuit similar to that shown in Figure 15a of ref 27 was used. Positive feedback was used to correct for the uncompensated cell resistance and was obtained by overcompensating the iR drop in a small resistor placed in series with the working electrode. (26) In preparation; see also ref 23. The principles involved have been discussed by G . L. Booman and W. B. Holbrook, Anal. Chem., 35, 1793 (1963); 37, 795 (1965); the subject has been reviewed by R.R.Schroeder and I. Shain, Chem. Instrum., in press. (27) W. M.Schwarz and I. Shain, Anal. Chem., 35, 1770 (1963). Volume 73, Number 1 January 1066
202
RONALD R. SCHROEDER AND IRVING SHAIN
The adder-controller amplifier was a Burr-Brown 1607B operational amplifier driving a Burr-Brown 16348 booster amplifier. The voltage follower was an Analog Devices 102C operational amplier and was mounted as close to the reference electrode as possible. The inverter-compensator used to provide the positive feedback was a Philbrick P45A operational amplifier. Circuit details and operating characteristics will be presented elsewhere.26 The signal used to supply the cell potential was derived from a Hewlett-Packard 214A pulse generator. A Tektronix 545A oscilloscope equipped with a Type W differential comparator plug-in was used for signal monitoring and recording of the current-time curves. The pulse generator used to apply the cell potential signal also functioned as the main timing device for the experimental system. Oscilloscope base line drift problems were overcome by providing a second triggering sequence which initiated the oscilloscope sweep and delayed the main pulse by a time equivalent to one graticule division on the oscilloscope face. Thus, the first time division on the resulting photograph presented a zero current base line for reference. Polarographic experiments were performed employing conventional three-electrode methods. For stationary electrode polarography a circuit similar to Figure 4 of ref 28 was employed. The input signals required to impose linear and triangular potential functions on the cell were obtained from a circuit which converted the time base sweep signal from the oscilloscope into triangular signals of the proper magnitude and sl0pe.23 Reagents. Chemicals. The chemicals used were of the best available quality (reagent grade when possible) and, except for the supporting electrolyte material, were used without further purification. Tetrarnethylammonium chloride, TMACl (Eastman Organic Chemicals), was purified by recrystallization from 95% ethanol. Tetramethylammonium hydroxide, TMAOH (Eastman Organic Chemicals), was titrated with standard hydrochloric acid and an exact weight: equivalent ratio was determined. Two solutions of acetic acid (0.173 and 3.46 F ) were prepared and standardized against a sodium hydroxide solution which was, in turn, standardized using potassium acid phthalate. Triply distilled water was used for the preparation of solutions and the rinsing of all glassware. The nitrogen, which was bubbled through the SOIUtion in the cell to remove oxygen, was of high purity and was further purified by passing it through solutions of vanadous sulfate and sodium hydroxide and finally through a solution of the supporting electrolyte. The mercury used for both stationary and dropping mercury electrodes was purified by rinsing several times alternately with 1M nitric acid, 1 M sodium hydroxide with potassium cyanide added, and triply distilled water. This chemically purified mercury was then The Journal
of
Physical Chemistry
Table I : Concentrations of Acetic Acid, Tetramethylammonium Acetate, Tetramethylammonium Chloride, and Hydrogen Ion for the Test Solutions Solution
CIIA,
CA,
mM
M
CTYACI, M
PH
I I1 111
2.17 4.32 8.65 17.3 34.6
0.025 0.049 0,100 0 I200 0.400
1.475 1.450 1.40 1.30 1.10
5.80 5.81 5.78 5.83 5.78
IV V
distilled at reduced pressure to reinove inert, refractory materials. Solution Preparation. Solutions for the electrochemical investigations were prepared by weighing out an appropriate amount of 10% aqueous TMAOH, titrating this to equivalence with 3.46 and 0.173 F acetic acid, adding a predetermined excess of 0.173 F acetic acid and sufficient solid TMACl for a final ionic strength of 1.5, and finally diluting to the appropriate volume. In all, five different solutions (Table I) were prepared in this manner. Procedure for Electrochemical Experiments. Preparation. The cell ma filled and nitrogen was bubbled through the solution for at least 10 min. Then the capillary leading to the reference electrode was filled by drawing solution from the cell into the capillary. A center compartment of the reference electrode salt bridge was filled with concentrated TMACl solution. Determining Electrode Positions and Recording Conditions. The cell and the potentiostat were matched as closely as possible by analyzing the transfer functions of the components and adjusting the interelectrode spacings to obtain the best possible frequency response without instability.26 Final positioning of the electrodes was done using either blank or test solutions. The pulse generator was triggered and the current-time function was observed on the oscilloscope screen. Then the reference electrode capillary position was adjusted until the most rapid current decay was observed. When optimum conditions had been established on blank solutions, an estimate was made of the shortest time a t which accurate faradaic current measurements could be made, The value of the charging current which could be considered negligible was different for the various solutions employed, but, in general, a current of less than 100 pA introduced negligible error and was usually attained at times between 20 and 40 psec. Recording Current-Time Curves. Several curves were included on the same photograph by vertically spacing the base lines and curves. Generally, three or four curves were obtained on the same electrode with 2-min delays between them. A new mercury (28) W. L. Underkofler and I. S h a h Anal. Chem., 35, 1778 (1963).
203
RATECONBTANT FOR THE DISSOCIATION OF ACETICACID
F 1-
I
54
I7t 14
ti"+----.
=I
,
I
microseconds
2
Figure 2. Polarograms of test solutions; composition as in Table I.
milliseconds Figure 1. Typical current-time curves obtained on test solution I. Solution composition: 0.025 M TMAOAc, 2.17 mM HOAc, 1.475 M TMACl; load resistor, 300 ohms; oscilloscope settings: upper photograph, top: 20 psec/div, 200 mV/div; middle: 50 psec/div, 200 mV/div; bottom: 100 psec/div, 100 mV/div; lower photograph, top: 200 rsec/div, 100 mV/div; middle: 500 psecldiv, 100 mV/div; bottom: 1 msec/div, 100 mV/dive
drop electrode was collected after the third or fourth run. Since three separate photographs could be taken on each film, 9-12 separate current-time curves were recorded on each picture. Typical curves are shown in Figure 1. Time Range. Experiments on each solution were conducted over several time ranges corresponding to oscilloscope time base settings of from 100 psec to 20 msec full scale. The time scales used in the experiments were limited by the onset of convective mass transfer a t about 40-50 msec after the application of the potential step. Observation of the electrode, using a microscope, showed that this stirring was caused by two effects. Upon application of the pulse the mercury drop moved because of surface tension changes, and also because large volumes of hydrogen were liberated. The form of the current-time curves indicates, however, that these processes were rather slow compared to the time scales employed and that the stirring did not affect the currents during the first few milliseconds after the potential change. At times longer than 100 msec, the stirring caused the current to increase slightly and eventually to become constant. The 20-msec time
limit on accurate current measurement made it nearly impossible to observe a diff usion-controlled current which would have made it possible to use eq 10. For each current-time curve, 7-10 points equally spaced on the time axis were selected and their horizontal and vertical coordinates were measured. Initial and Final Potentials. The potentials employed in the potentiostatic experiment were selected from the current-potential data obtained by polarography. I n polarographic experiments with the dme, the waves obtained were, in general, poorly shaped and exhibited maxima and adsorption phenomena €or most of the solution concentrations employed (Figure 2). I n order to determine a suitable final potential for the potentiostatic experiments, stationary electrode polarography had to be used. The stationary electrode polarograms (Figure 3) did not exhibit maxima or other anomalies. The processes which cause the poorly shaped dme polarograms were probably related to the motion of the drop a t these very negative potentials.
0.4'
0.3 .
Figure 3. Stationary electrode polarograms on solution I. Scan rates: curve A, 0.41 V/sec; curve B, 4.10 V/sec; curve C, 41.0 V/sec. Volume '78,Number 1
January 1060
204
RONALDR. SCHROEDER AND IRVING SHAIN
The potential values employed for the potentiostatic work were an initial potential of -1.5 V and a final potential of -2.1 V (vs. sce).
Results and Discussion Computation Procedures. Gruphical Determination of the Rate Constant. The simplest manner in which the current-time data can be used to obtain the parameter X and the kinetic term krK involves a graphical pro-
cedure based on rearranging eq 10, in which i k d i is graphed vs. Vi. In actually applying this method, however, there are several serious limitations. Values of f(X) are obtained point by point, and any scatter in the data results in a substantial uncertainty in the calculated values of the kinetic parameters. A second factor is that experimental data must be available for times at which the current is diffusion controlled. When the limiting values of i k d i are not experimentally obtainable, the problem of determining kiK becomes complex. Theoretical diffusion currents can be calculated (eq 8) but this requires that the diffusion coefficient DHA be known accurately. I n some cases, values of the diffusion coefficient can be obtained from polarographic data, but this approach could not be used in this work, since the polarographic waves were poorly shaped. Thus, an alternate procedure was sought which would not be so sensitive to the scattered data and which would not depend so directly on the value of i d or DHA.
Curve-Fitting Method for Determining the Rate Constant. When current-time data are available over a reasonably wide range of experimental times, a graphical method can be used which relies on the magnitude and time dependence of the kinetic current. For this method, eq 9 can be used to calculate theoretical kinetic current-time curves assuming convenient values of k f K and i d . Three principal regions in the resulting curves are obtained corresponding to the short-time, the midrange, and the long-time limits of f(X). At very short times (small A) the current is constant, while at long times (large A), ik -+ i d . On log-log plots, such as Figure 4, these curves appear as a constant current at short times, then a transition region where the currents gradually decrease, and finally a straight-line region corresponding to a limiting slope of -I/*. However, all three regions can be seen for a single curve only if a very wide range of times is included. Then experimental current-time data are plotted in a similar fashion, and the experimental curves are matched with one of the theoretical current-time curves. The matching can be done by superimposing one graph over the other and moving one in a vertical or horizontal direction until the experimental and calculated curves show the best fit. The matched curves imply that the values of X for the two current-time curves are the same, Le., Xcalcd = Xexpt1. In the matching process, the graphs become shifted relative to each The Journal of Physical Chemistry
I
other on the time scale. The ratio tc&lcd/t@xptl is easily determined by comparing the time scales. Since hoalad/ hexptl = 1, it follows from the definition of X (eq 7) that
where y = DB+/DHA. Then the experimental kinetic parameter can be calculated from the value of ktK which was used to generate the theoretical curve. When this method is used, the current-time data along the entire curve are used simultaneously, and scatter in the data introduces much less error in the calculation of the kinetic parameters. In addition, it is much easier to estimate the limiting value of the current-time data at long times, and only the ratio DE+/DHAneed to be known explicitly. This ratio was taken as equal to the ratio of the limiting equivalent conductances of H+ and CHsCOO-, 8.55.29 Determination of Equilibrium Constant. The equilibrium constant for the dissociation of acetic acid in concentrated solutions of tetramethylammonium chloride has not been reported previously and had to be determined for this work. Acetic acid-tetra,methylammonium acetate buffers of several concentration ratios were prepared and their pH was measured. Also, dilute solutions of hydrochloric acid were made up and tetramethylammonium chloride was added to make the ionic strength 1.5. The pH of these solutions was also measured. The chemical equilibrium constant was then determined from the relation Kd
cA(CH/CHA)
(26)
or Kd
=
(cA/cHA)
(aH/fH)
(27)
where aH and fH are the activity and activity coefficient for hydrogen ion. A value of Kd = 2.28 X 10-6 mol/l. was found. (29) L. Meitea, Ed., “Handbook of Analytical Chemistry,” MoGrawHill Book Go., Inc., New York, N. Y., 1963.
RATECONSTANT FOR
THE
205
DISSOCIATION OF ACETICACID I
Table I1 : Typical Potentiostatic CurrentcTime Data Obtained from Solution IV -100 t, @sea
23 33 43 53 63 73 83 93
fiseoai, mA
8.40 2.11 2.00 1.92 1.73 1.57 1.46 1.34 mseca-
-1
7 2 0 0 pseca-
-600
$3
6,
t,
fisec
mA
rsea
40 60 80 100 120 140 160 180
1.97 1.83 1.63 1.54 1.47 1.40 1.33 1.24
100 150 200 250 300 350 400
m6eca----.
-5
t, mea
a,
1,
mA
mseo
mA
86 186 286 386 486 586 686 786 886
1.58 1.25 1.03 0.899 0.801 0.745 0.697 0.660 0.622
0.50
0.822 0.637 0.548 0.490 0.452 0.418 0.389 0.368 0.350
a
1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50
$8
50
t
p~&'-.r
it mA
2.26 1.54 1.37 1.21 1.09 1.01 0.941 0.876
7 - 1 0msec"-t, 6 msea mA
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00
I\
100
0.621 0.476 0.404 0.357 0.328 0.306 0.288 0.272 0.257
,.'.,,'
100
'
.
"""'
1000
'
'
""'e
t(ped
Figure 5 . Experimental current-time data on log-log plots; solution composition as in Table I.
Table 111: The Apparent Rate Constants and Equilibrium Constants for the Five Solutions ke SOIU-
Full-scale.
CAS
tion
M
IC X 104
I I1 I11 IV
0.025 0.049 0.100 0.200 0.400
9.12 4.56 2.28 1.14
V
Experimental Results I n all, 31 sets of current-time data were obtained on the five solutions listed in Table I. Each set consisted of a series of replicate current-time curves obtained over the time range from 10 psec to 10 msec. Usually, this time range was covered in six groups of currenttime curves: 0-100 psec; 0-200 psec; 0-500 psec, 0-1 msec; 0-5 msec; and 0-10 msec; three or four curves were obtained within each group. Thus, each set of data for a particular solution required the analysis of from 18 to 24 individual current-time curves. In the normal procedure, the three or four replicate current-time curves for a particular time range were averaged, and data were tabulated for seven to ten different values of the time. A typical set of such tabulated current-time data is shown in Table 11. Then each set provided a single composite log-log plot of the current as a function of time. Typical plots (one for each solution) are shown in Figure 5. The plots of the experimental currents correspond closely to the theoretical curves of Figure 4 except at very short times, where the currents are higher than predicted. A major portion of the current at these short times is due to the charging of the electrical double layer. Simultaneous adsorption of tetramethylammonium ions appears to extend the charging current to longer times than encountered for solutions of perchloric acid or lithium salts. An additional contribution to this initial current is caused by the reduction of the free hydrogen ions present a t the start of the experiment. Although these processes were not considered in the boundary value
.
L
10
0.57
X 10-5, seo-1
Std dev X 10-6
11.0 10.0 4.8 3.2 1.5
2.0 0.3 0.4 0.1 0.1
No. of deter-
minations
6 7 10
5 3
problem, they did not extend to times long enough to interfere with the analysis of the kinetic data, i.e., times longer than about 10-20 psec. Each of these log-log current-time plots was treated as discussed above, and an experimental value of the apparent rate constant IC, was calculated. The values obtained on each solution were averaged, and these results are summarized in Table 111. Correction for Double-Laver E$ects. With these values of IC, and the appropriate acetate concentrations, eq 24 'was employed to obtain the true rate constant by plotting k a / l / c vs. l/dc (Figure 6). The slope, calculated by the method of least squares, gave a value for kf of 1.58 X lo6sec-l. To test the validity of this approach to the determination of the true value of the rate constant kf, it was
Figure 6. G r a p k a l correction for double-layer effects. units are lc&/d~*( x I o - ~ ) , d l . / m o l x sec-1, vs.
+.
1/6
Volume 73, Number 1 January 1969
206
necessary to determine the value of the reaction layer thickness for each of the five solutions to confirm that the variation of k, with acetate concentration could in fact be accounted for by field effects. This was done by calculating kb from eq 4, assuming D H + = cmz/sec and substituting these values in eq 16. The calculated values of p were 24, 17, 12, 8.5, and 6.0 for solutions I-V, respectively. Since the effective thickness of the diffuse part of the double layer was estimated at 4/x = 10 8, it is apparent that a substantial portion of the chemical reaction must take place in a region of strong potential fields. Thus, the extrapolation technique is valid and the kr determined in this experiment represents a true chemical rate constant. Discussion of Error. The uncertainty in the value of the rate constant can be assessed qualitatively in terms of the experimental errors. The principal sources of error in the experiment are the oscilloscope, uncertainty in the value of the equilibrium constant and the diffusion coefficients, and the method of calculation. The measurement error ( ~ 3 % on each axis of the oscilloscope) was reduced by data averaging, although some scatter in the current-time data remained. I n general, the error introduced by the curve-fitting method is determined by the time range over which experimental data are available. I n this case, sufficient data were available so that the curve-fitting procedure itself introduced only minor error; the major error was the uncertainty introduced by the lack of exact values for DHa(or id). Thus, the uncertainty in the quantities Kd and DHA are the largest sources of error. The error in these terms is not readily calculable, but for the determiaation of Kd, a f12% error could be introduced by read-
T h e Journal of Physical Chemistry
RONALD R, SCHROEDERAND IRVING SHAIN ing errors of 0.05 pH unit. The error in the value of DH+/DHAcannot be assessed, but assuming 20% possible error, an estimated error in kf would be +30%,. The final result would then be a value of kf equal to 1.6 (i0.5)X lo8 sec-l. AlSignQkance of the Double-Layer Correction. though the magnitude of the estimated error precludes any quantitative interpretation of the double-layer effects, the form of the plot in Figure 6 merits comment. For the effects of accelerated dissociation and/or deviation from pseudo-first-order nature for the recombination, an increase in the rate constant with small y would be expected. That is, the last term in eq 24 would be positive, but actually the opposite effect, a decreased value of kr is observed, indicating that the double layer inhibits the net reaction. One possible explanation for this would be the specific adsorption of the tetramethylammonium ion which could cause a decrease in the dielectric constant near the electrode surface and also a significant change in the potentials within the double layer. That is, because of the specific adsorption of cations, the potential in the electrical double layer would be less negative than anticipated from merely considering the cell potential. The last term in eq 24 would be negative, indicating that the kinetic parameter within the double-layer region is smaller than its bulk value. Thus, from the form of the Forrection term in eq 24, a negative intercept in the plot of k f p vs, p is reasonable in the system studied. 3
Acknowledgments. The authors are grateful to the National Science Foundation, which supported this work in part through Grant No. GP-3907, and the U.S. Atomic Energy Commission, which supported this work in part through Contract No. AT(l1-1)-1083.
