Chemical Modeling of Aqueous Systems II - American Chemical Society

Chapter 27

Numerical Modeling of Platinum Eh Measurements by Using Heterogeneous Electron-Transfer Kinetics 1

2

J. Houston Kempton , Ralph D. Lindberg , and Donald D. Runnells 1

PTI

2

Environmental Services, 1260 Baseline Road, Suite 102, Boulder, CO 80302 Department of Geological Sciences, University of Colorado, Boulder, CO 80309-0250

Downloaded by PURDUE UNIV on December 6, 2016 | http://pubs.acs.org Publication Date: December 7, 1990 | doi: 10.1021/bk-1990-0416.ch027

2

This research evaluates the measurement of the "master" Eh of solutions in terms of heterogeneous electron-transfer kinetics between aqueous species and the surface of a polished platinum electrode. A preliminary model is proposed in which the electrode/solution interface is assumed to behave as a fixed-value capacitor, and the rate of equilibration depends on the net current at the interface. Heterogeneous kinetics at bright platinum in 0.1 m KCl were measured for the redox couples Fe(III)/Fe(II), Fe(CN)6 /Fe(CN) 4-, Se(VI)/Se(IV), and As(V)/As(III). Of the couples considered, only Fe(III)/Fe(II) at pH 3 and Fe(CN) 3-/Fe(CN) 4- at pH 6.0 were capable of imposing a Nernstian potential on the platinum electrode. 3-

6

6

6

Potentiometry is an electrochemical technique in which the electrical potential of an "inert" electrode is measured against that of a reference electrode while both are immersed in an aqueous solution. A problem in potentiometry is that the measured potential may be slow to achieve a steady value. This is especially common in attempts to measure the Eh of solutions that are poorly poised, as is the case with most natural waters, and it is not uncommon for measured redox potentials to drift for many hours (1,2)· The long equilibration times, together with published reports of large discrepancies between platinum Eh values and actual solution compositions (2), have led to a great deal of un certainty and skepticism about the use of Eh measurements. This paper attempts to model and define the conditions under which platinum Eh measurements are likely to reflect the true electrical potential of aqueous solutions. The double layer at the surface of the electrode is modeled as a fixed capacitor (C^), and the rate at which an electrode equilibrates with a solution (i.e. the rate at which C^j is charged) is assumed to be proportional to the electrical current at this interface. The current across the electrode/solution interface can be calculated from classical electrochemical theory, in which the current is linearly proportional to the concentration and electrontransfer rate constant of the aqueous species, and is exponentially proportional to the potential across the interface. The phenomena involved in potentiometric measurement are shown graphically in Figure 1. This is a composite of three linear sweep voltammograms in three different solutions: 10" m ferrocyanide, 10' m ferricyanide, and a mixture 10~ m in each. In Figure 1, a potential has been applied across the electrode/solution interface and the resulting current measured. In potentiometry, a passive electrode is allowed to drift under the resulting current until its potential equilibrates with the solution, at which time the net current is zero. This potential is called the "rest potential" and is equal to the system Eh for a solution which is at homogeneous equilibrium. Figure 1 shows that the point of zero net current is in fact composed of positive and negative components (so called "partial currents"). At the rest potential, the value of the cathodic current, which equals the absolute value of the anodic 4

4

4

0097-6156/90/0416-0339$06.00/0 ο 1990 American Chemical Society

Melchior and Bassett; Chemical Modeling of Aqueous Systems II ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

CHEMICAL MODELING OF AQUEOUS SYSTEMS II

340

current, is defined as the exchange current. Higher exchange currents result in more rapid equilibration and more stable potential. For a solution not in homogeneous equilibrium, the rest potential will be a "mixed potential" (Figure 7.21 in Reference 4), and will he between the electrochemical potentials of the most oxidized and the most reduced couples present. The actual value of a mixed potential will depend on the heterogeneous rate constants for the transfer of electrons between the various aqueous couples and the surface of the electrode. In this study, heterogeneous electron-transfer kinetics were measured for the following: Se(VI)/Se(IV), As(V)/As(III), Fe(CN) 7Fe(CN) -, and Fe(III)/Fe(II). All experiments were done at pH 6.0 with the exception of the iron couple, which was done at pH 3.0. Using electron-transfer kinetic constants, aqueous diffusion coefficients, aqueous concentrations, starting potentials, and a constant double-layer capacitance model, values for the change of EMF as a function of time for a platinum electrode were calculated numerically. The result of this simulation was then compared to the observed potentiometric response for a solution of the same concentration. 3


6

4

6

THEORY

The following brief discussion reviews electrochemical topics which bear upon the numerical model of electrode drift. For additional details the reader is referred to the comprehensive text by Bard and Faulkner © · An electrochemical cell is a type of electrical circuit. As such, it may be modeled with an electrical analog circuit. The potentiometric cell can be considered to be an electrical potential applied to a capacitor and a resistor in series. The capacitor represents the interface between the electrode and the solution, the applied potential is the solution Eh, and the resistor represents the heterogeneous kinetics of the aqueous redox species. The term "heterogeneous kinetics" denotes electron transfer between different phases, in this case aqueous species and the noble-metal electrode. The time required for the capacitor to equilibrate with the applied potential depends on the size of the capacitor and the electrical current. A model predicting electrode response with time must therefore consider the following: (1) the double-layer capacitance, (2) the concentration of electroactive species at the electrode surface (which in turn is affected by the diffusion coefficients), (3) the values of the formal potentials (E°), (4) the heterogeneous rate constants of the redox species (with respect to the electrode material and electrolyte composition), and (5) the electrical potential of the electrode itself. The capacitance of the double layer, C^, may be described as follows: =Q/E

CQU

(1) 1

where: C$ Q Ε

= capacitance (coul volt" , or farads) = charge in coulombs = electrical potential (volts).

Equation 1 is a simplification because the double-layer capacitance has been shown to be a function of potential, solution composition, electrode material, electrode pretreatment, concentration of adsorbed species, and duration of contact between solution and electrode (6,2). The potential of the electrode at time t is: E(t) = E o+ Γ t i(t)dt JO Cdl t=

where: E(t) E _0 i(t) C^i t t

= potential of the electrode attimet = potential of electrode at t=0 = electrical current attimet (amps) = capacitance (farads) =time(sec).


(2)

27. KEMPTON ET AL.

Numerical Modeling ofPfotinum Eh Measurements

The current due to electrochemical reactions can be described by the current overpotential equation (Equation. 2.5 in Reference 2). For a single redox couple (such as Fe(II)/Fe(III)) this is: 0

i = nFAkOfCo^^Expii-ocnF/RTJÎE-E ')) - Cr

0,

(o>t)

Exp(((l-α)nF/RT)(E-E ))}

(3)

z

where: A = area of electrode (cm ) i = current (coul sec"*) η = electrons in reaction (coul mol" ^ F = Faraday's constant (96,492 coul equiv* k° = heterogeneous rate constant (cm sec' ) R = gas constant (8.314 joule mol" Κ" ) E = potential of electrode (volts) E° = formal potential of the couple (volts) α = electron-transfer coefficient (unitless) C-°(o,t) concentration of oxidized species at electrode surface (mol cm" ) C (o,'t) concentration of reduced species at electrode surface (mol cm" ) 1

1

1


t

=

r

3

=

3

The area of the electrode, A , in Equation 3 was measured in this study by means of chronocoulometry (S) in an aqueous ferricyanide solutions. In the case of multiple redox reactions, Equation 3 can be extended to include the total current by summing up the partial currents for each redox couple. Equation 3 requires the concentration of each electroactive species at the electrode surface. In a vigorously stirred solution, the concentration at the electrode surface can be assumed to be equal to the concentration in the bulk solution and the potential at any time can be calculated by numerically integrating Equation 3 overtimeand substituting it into Equation 2. A more rigorous interpretation of Equation 3 for a quiescent solution can be done by considering the net diffusion of electroactive species due to electrochemical reactions at the electrode. A general solution to the diffusion equation with a planar source is given by Nicholson and Shain (2) and has been expanded on by Imbeaux and Saveant (10): Co

(o,t) =

Cr

(o,t) =

C o

C r

1

2

1

* " IcftVnFADo / - IaftynFADo / *

+

1 2

1

IaftViiFAD,. / + IcftynFAD,. /

(4)

2

2

3

where: Co = bulk concentration of oxidized form (mol cm"- ) Cr* = bulk concentration of reduced form (mol cm" ) D = oxidized-form diffusion coefficient (cm sec" ) D = reduced-form diffusion coefficient (cm sec" ) Ic(t) and Ia(t) (in couls sec / ) , defined below. 3

2

1

2

1

Q

r

_1

2

Ic(t) and Ia(t) above are defined by Imbeaux and Saveant Qfi)

(7)

where: ν =

variable of integration (sec) other symbols as defined before.

Equations 4 and 5 represent the change in concentrations due to consumption and production of reactants and products at the electrode. The difference in signs in Equations 4 and 5 is due the sign


341

342


convention for currents: cathodic reactions have positive current, so the convolution integral for the cathodic current, Ic(t), is positive, and visa versa. The relationship between the time and potential is then calculated by numerically integrating the current Q D , Equation 3, until diffusional control of the dissolved species causes the potential to approach a constant value. Details of the procedure can be found in Kempton (12).


E X P E R I M E N T A L METHODS

Electrochemical experiments in this study were performed with a BAS-100 Electrochemical Analyzer (Bioanalytical Systems Inc., Lafayette, Indiana). For potentiometric experiments in calm solutions, platinum disk electrodes were used in the BAS-100 cell stand. For hydrodynamic experiments a Pine Instruments Inc. rotating-disk platinum electrode was interfaced with the BAS-100. Electrode pretreatment is critical for consistent results, so all electrodes were prepared in an identical manner before each experiment. The platinum surface was polished on a felt pad with 0.05 micron alumina and water. The surface was then rinsed with deionized water, wiped dry with a tissue, and the experiment begun as quickly as possible. All experiments were run in 0.1 m reagent grade KC1 without further purification. The pH was adjusted in the experimental cell with small additions of reagent-grade 1.0 m HC1 or 1.0 m NaOH. Except where specifically noted, oxygen was removed by bubbling the cell for 15 to 60 minutes, depending on the volume of the cell, with high-purity nitrogen. Polarographic experiments in this laboratory have shown that solutions purged using this approach contain less than 20 ppb dissolved oxygen. All experiments were thermostatically controlled at 25.0 ± 0.5 °C. Reference electrode were calibrated daily so that potentials could be related to the standard hydrogen electrode (SHE). In all experiments, rates were determined in terms of formal potentials and molal concentrations.

KINETIC M E A S U R E M E N T S

A rotating platinum disk electrode was used for gathering data for the calculation of the kinetic constants. Tafel plots (2) were developed (Figure 2), from which the heterogeneous rate constants, k°, and the electron-transfer coefficients, a, were determined. In all determinations of kinetic parameters, a blank voltammogram, run under identical conditions, was subtracted from the data to remove current due to background reactions or charging of the double layer. The rate of reaction was too slow to produce measurable current (i^) for arsenic and selenium species within the stability range of water at platinum. For those two elements an upper limit was estimated for the value of the rate constant by solving the current overpotential equation (Equation 3) for k° with the assumption of α = 0.5 and i ^ = 4x10"^ amps c m ' . This value of i ^ was chosen empirically from examination of the data; it represents the lowest current that could clearly be distinguished from background currents with the instruments used. The true value of k ° must therefore be equal to or less than the value that is calculated from the minimum limiting current. 2

E L E C T R O D E A R E A A N D DIFFUSION COEFFICIENTS

Diffusion coefficients were taken from the literature or measured using chronocoulometry (£). The area of the electrode, A, was measured using FeiCN)^ " in a solution of 0.004 m K^FeiCN)^ in 0.1 m KC1, for which the diffusion coefficient is given as 0.762 ± 0.01 χ 10* c m sec" (8). It was found from six measurements to be 0.242 c m + 0.005; this value was then used to determine diffusion coefficients for species of interest (from the integrated Cottrell equation, Equation 5.9.1 in Reference 5)· 3

5

2

1

2



27.

KEMPTONETAL.

343 Numerical Modeling of Platinum Eh Measurements

Figure 1. Composite of three independent, experimentally determined, linear sweep voltammograms. One for 0.0001 m K F e ( C î % one for 0.0001 m K Fe(CN)6» and one for a 1:1 mixture of 0.001 m of each. Temperature = 298 ± 1 K. 3

3

-7.0

-13.0 -0.30 -0.20

-0.10

Ο

1.10

0.20

0.30

OVER POTENTIAL (volts) Figure 2. Tafel plots of oxidation and reduction of iron at pH 3 in 0.1 m KC1 determined at a rotating platinum disk electrode. Currents are extrapolated values for an infinite rate of rotation. Temperature = 298 ± 1 K.


344


CAPACITANCE OF T H E D O U B L E LAYER

Cyclic voltammetry was used to measure the capacitance of the interface as a function of potential. The current is the product of the scan rate (volt sec' ) times capacitance (coul volt" ) (Equation 1.2.15 in Reference 5). The instantaneous capacitance (in farads, or coul volt" ) at any potential is the current divided by the scan rate. Figure 3 is the current versus potential signal obtained from a cyclic voltammogram in 0.1 m KC1. A slope is superimposed on the charging current, probably the result of IR drop across the solution, and was corrected by using the average of the positive and the negative scan currents for calculating the capacitance. 1

1

1


EXPERIMENTAL RESULTS

Figure 4 shows the calculated apparent capacitances versus potential (as Eh) from experiments using cyclic voltammetry. Measurements at high scan rates are more useful for determining capacitance because the background signal is smaller. Based on the determinations at scan rates of 50, 100, and 200 mV sec' in Figure 4, a constant capacitance of 0.85 χ 10" farad c m ' was chosen for the modeling, which compares satisfactorily with 0.34 χ 10" farad cm" given by Formaro and Trasatti (7) for platinum in 1.0 m perchloric acid. The values and sources of diffusion data used in this study are given in Table I. The values of heterogeneous electron transfer rate constants are given in Table II. Rate constants were not measured above pH 3.0 for the iron couple due to the low solubility of ferric hydroxide. A review of published exchange current densities for iron(14) produced an average k ° of 0.03 + 0.05 cm sec' (n=5) for Fe(III)/Fe(II) in aqueous HC1, and 0.09 ± 0.02 cm sec' (n=6) for Fe(CN) ^ /Fe(CN)5 " in aqueous KC1. These are in reasonable agreement with the measured values in Table II. As explained earlier, for selenium and arsenic, no electrochemical reaction was detected between the upper two valence states, so the limiting maximum rate constants were approximated based on the minimum detectable currents. Stumm and Morgan (4) state that reliable potential measurements require exchange currents above 10" amp c m ' , which they correlate to 10" m Fe(III) and Fe(II), but they note that it is difficult to measure a meaningful Eh in the Fe(III)/Fe(II) system at concentrations below about 10 m. In this study, minimum exchange currents required to obtain meaningful Eh values were determined empirically by noting the lowest concentration of a redox couple, equal molal in reduced and oxidized species, which would produce the approximate thermodynamic Eh at platinum in 10 minutes. In this experiment, concentration, equilibrium potential, and heterogeneous kinetics are known, so exchange current can be calculated from Equation 3. Minimum exchange currents were determined for equimolal solutions of Fe(III)/Fe(II) at pH 3.0 and Fe(CN) '/Fe(CN) ' at pH 6.0. For the Fe(IH)/ Fe(II) couple a stable Nernstian response at a platinum electrode first occurred at about 3x10"^ m each in Fe(II) and Fe(III), corresponding to an exchange current of approximately 1x10"^ amp cm' . In the case of Fe(CN)53-/Fe(CN)5 " at pH 6 a reliable response was first observed at concentrations above about 7xl0"6 m, corresponding to an exchange current of approximately 8x10"^ amp cm" . Both of these are above the minimum value of 10" amp given by with Stumm and Morgan (14). 1

4

4

2

2

1

1

6

4

7

2

6

3

6

4

6

2

4

2

7

NUMERICAL M O D E L

The computer code for the electrode equilibration model (EHDRIFT) was written in PASCAL for use on a microcomputer. The program calculates the rest potential which is the EMF value where the currents sum to zero. If the system is in homogeneous equilibrium the rest potential will represent the system Eh. The numerical algorithm uses Eulers method (1_1) to integrate Equation 2, which involves recalculation of the aqueous concentrations (Equations 4 and 5) at each time step. A full listing of the source code can be found in Kempton (12). The numerical simulations of the drift of the electrode in a stirred solution, the integration of Equation 3, where concentrations at the electrode surface are assumed to be constant, were numerically sound and matched analytical solutions. The inclusion of diffusion terms (Equations 4 and 5),


27. KEMPTON ET AL.

Numerical Modeling ofPhtinum Eh Measurements 345

à.

1 ^

i

ι

1

ι

ι

ι ^χλ

ι


-0.440

Figure 3. Cyclic voltammogram of 0.1 m KC1 at pH 6.0. Platinum working electrode with saturated calomel reference electrode. Approximately constant capacitance between -0.40 and +0.20 volts. Temperature = 298+ 1 K.

3.5 χ Ι Ο " , 4

+ 2 0 0 mV/s Ε ο •ο

3 Οχ Ι Ο "

4

2.5 χ Ι Ο "

4

2.0 χ Ι Ο "

i - Ο 50 mV/s x

1

0

0

m

V

/

s

Δ

α

20 mV/s

1 0

m

V

/

4

•

UJ

Ο

1.5χ Ι Ο " Ο < û_

• •

s

Ι.ΟχΙΟ"

Δ

-fX * '

Δ

χ ο
Fe(II) Fe(II)->Fe(III) 3

4

6

4

*

3

a could not be measured by this method; assumed to be 0.5 Replicate determinations and published studies suggest that the precision of these k ° determinations is probably not better than 75 to 100%.


27. KEMPTON ET AL.

Numerical Modeling of Platinum Eh Measurements 347

however, introduced the following numerical limitations: 1) initial time steps on the order of 1x10"° seconds were required to maintain convergence, 2) the use of convolution integrals (Equations 6 and 7) meant that each new step required calculations from all previous steps, resulting in excessive computational time, and 3) increasing the time steps resulted in numerical instability. The net result was that the simulations of potential drift in which diffusion was considered would not converge if the electrode started more than about 200 mVfromequilibrium. Figure 5 shows the modeled and observed changes in potential at platinum over time for 1.3x10"^ m Fe(III) along with 1.3x10"^ m Fe(II) chloride solution at pH 3.0. Aqueous diffusion was included in this simulations. The data in Figure 5 indicate that the Eh-drift model predicts equilibration at a rate which is much faster than is observed. Comparable results were obtained for 1x10"^ m potassium ferrocyanide and ferricyanide at pH 6 in a 0.1 m KC1 solution. The discrepancy between the predicted and observed electrode response suggests that the proposed model omits some key, but unidentified, reactions. It is likely that the same non-modeled reactions which prevent Nernstian response below an exchange current of about 1x10'^ amp cm" are responsible for the poor fit in Figure 5. The most likely reactions involve the platinum electrode, adsorbed oxygen, and water (15. 16). The present model does not include corrosion reactions involving the platinum electrode. Nevertheless, the model is useful because it illustrates the relationship between heterogeneous kinetics and the measurement of Eh, and it confirms the fact that other surface phenomena occur on a platinum electrode which significantly limit the applicability of potentiometry.


2

RESULTS FOR INDIVIDUAL REDOX COUPLES

From the kinetic constants reported in Table II, it is possible to determine which redox couples have the ability to yield meaningful Eh values, as follows: Fe(IID/Fe(ID. A pH 3.0 chloride solution equimolal in the two iron species was found to yield a Nernstian response down to 6x10"^ m total iron, corresponding to an exchange current of about 1x10"^ amp c m ' . It appears that At pH 6.0 Fe(III) is too insoluble to influence a platinum electrode. Potentiometric and kinetic experiments with platinum in Fe(II) solutions at pH 6.0 indicated the formation of a passivating layer, in agreement with Doyle (17). 2

Fe(CN^5^"/Fe(CN)5^". Equimolal solutions of ferrocyanide and ferricyanide at pH 6.0 were found to impart Nernstian responses on platinum down to about 7x10*^ m in each valence. This corresponds to an exchange current of 8x10"^ amp cm" . 2

As(VVAs(IID. Heterogeneous kinetics of the arsenic couple were immeasurably slow at platinum, and it should not be possible to achieve the exchange current required for a Nernstian response on platinum. Conversely, a measured Eh should not reveal anything about the aqueous speciation of As(V) and As(III). Se(VD/Se(IV). As with arsenic, the heterogeneous kinetic constants of oxidized selenium species at platinum are too small to yield a meaningful potentiometric response. Potentiometric measurements in a lxl0"3 m solution of Se(VI)/Se(IV) at pH 6, as expected, showed no statistical deviation from blank 0.1 m KC1 samples. CONCLUSIONS

Aqueous species with fast electron-transfer kinetics at the surface of a platinum electrode are able to produce Nernstian potentials. However, a theoretical effort to model the rate of equilibration between platinum and a redox solution was only partially successful, probably due to reactions not included in the model. The shape of the theoretical Eh-time drift curve matched the observed shape for iron-bearing solutions, but the modeled rate of approach to equilibrium was much faster than the observed rate.

American Chemical Society Library 1 1 5 5 16ih St., N.W Melchior and Bassett; Chemical Modeling of.Aqueous Systems II ACS Symposium Series; American Chemical Washington. DSL Society: 20036 Washington, DC, 1990.


348


0.680

TIME

(sec)

Figure 5. Numerical simulation and observed Eh drift curve for platinum electrode in 1.3X10"** m FeCl and 1.3xl0" m FeCl in 0.1 m KC1 adjusted to pH 3 with HC1. Diffusion included in numerical simulation. Temperature = 298 ± 1 K. 5

2

3


27. KEMPTON ET AL.

Numerical Modeling of Platinum Eh Measurements

349

In terms of natural environments, quantitative application of platinum Eh measurements to equilibrium chemical modeling is extremely limited. Of the four systems studied here, only the Fe(III)/Fe(II) couple in acidic solutions is common in natural waters and is capable of yielding a Nernstian response. Considering the limitations of Eh measurement and the common occurrence of homogeneous disequilibrium Q), it would seem prudent to limit the use of Eh to either specific reactions or to qualitative descriptions when discussing natural systems. It has been suggested (IS, 1& 2Q) that the measurement of dissolved gases such as 0 , H S, NH4, CH4, and H2 may be a preferable method for characterizing the redox status of natural waters. 2

2


ACKNOWLEDGMENTS

Research grants from the Basalt Waste Isolation Project of Rockwell International Corporation (Hanford, Washington) and from the Electric Power Research Institute (Contract 8000-16, Dr. Ishwar Murarka) are gratefully acknowledged. Dr. Karl Koval, Department of Chemistry, University of Colorado, gave generously of his time and patience. LITERATURE CITED

1. Back, W.; Barnes, I. In Equipment for field measurement of electrochemical potentials: U.S. Geol. Survey Research 1961, 1961, p. C366-C368. 2. Runnells, D.D.; Lindberg, R.D.; Kempton, J.K. Scientific Basis for Nuclear Waste Management X, Material Res. Soc. Sympos. Proc. 84, J.K Bates; W.B. Seefeldt, Eds., 1987, p. 723-733. 3. Lindberg, R.D.; Runnells, D.D. Science. 1984, 225, 925-927. 4. Stumm, W.; Morgan, J.J. Aquatic Chemistry. 2nd ed. Wiley-Interscience: New York, 1981; pp. 490-495. 5. Bard. A.J.: Faulkner. L.R. Electrochemical Methods: Fundamentals and Applications, Wiley and Sons: New York, 1980; Chapter 1, 3, and 4. 6. Bockris, J. O.; Reddy, A.K.N. Modern Elecotrochemistry, Vol. II, Plenum Press; New York, 1970; pp. 718-790. 7. Formaro, L.; Trasatti, S. Electrochimica Acta, 1967, 12, 1457-1469. 8. Sawyer, D.T.; Roberts, J.L. Experimental Electrochemistry for Chemists, John Wiley and Sons; New York, 1974, p 77. 9. Nicholson, R.S.; Shain, I. Anal. Chem. 1964,36,706-723. 10. Imbeaux, J.C.; Saveant, J.M. Jour. Electroanal. Chem. and Interfac. Electrochem. 1973, 44, 169187. 11. Maron, J.M. Numerical Analysis: A Practical Approach; Macmillan Inc.: New York, 1982; pp. 377-386. 12. Kempton, J.H. Ms. Thesis, Univ. of Colorado, Boulder, 1987. 13. Heusler, Κ. E. In Encyclopedia of Electrochemistry of the Elements, Bard, A. J., Ed.; Marcel Dekker, Inc.: New York, 1982; pp. 228-381. 14. Morris, J.C.; Stumm, W. In Equilibrium Concepts in Natural Water Systems, Stumm, W., Ed.; Adv. in Chem. Series No. 67, American Chemical Society: Washington, DC, 1967; pp. 270-285. 15. Hoare, J.P. In The Electrochemistry of Oxygen; Interscience: New York, 1968; pp. 13-46. 16. Whitfield, M. Limnol. and Oceanogr; 1974, 19, 857-865. 17. Doyle, R.W.S. Amer. Jour. Sci., 1968, 266, 840-859. 18. Berner, R.A. Jour. Sed. Petrol., 1981, 51, 359-365. 19. Lindberg, R.D. Ph.D. Dissertation, Univ. of Colo., Boulder, 1983. 20. Stumm, W. Schweiz. Z. Hydrol., 1984, 46, 291-295. RECEIVED July 20, 1989


Chemical Modeling of Aqueous Systems II - American Chemical Society

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