The use of glass electrodes for the determination of formation constants—II

T&ma, Vol 32, No. 6, pp. 483489, 1985 Printed m Great Bntam. All rights reserved

CopyrIght

0039-9140/85 $3.00 + 0.00 Q 1985 Pergamon Press Ltd

THE USE OF GLASS ELECTRODES FOR THE DETERMINATION OF FORMATION CONSTANTS-II SIMULATION

OF TITRATION

DATA

PETER M. MAY*, KEVIN MURRAY and DAVID R. WILLIAMS Department of Applied Chemistry, UWIST, Cardlff, Wales (Received

16 November

1984. Accepted

II January

1985)

Summary-Methods for simulating titration data, includmg various types of corrections for changes in activity, hquid-junction potential and ion-selectivity of electrodes are described. These form the basis of a new library of computer programs, called ESTA (Equilibnum Simulation for Titration Analysis). They permit the range of titration conditions employed in the determination of formation constants to be usefully extended. Simulations have been performed to illustrate the extent to which the effects mentioned above are manifest in many titrations of practical importance.

SYMBOLS

R = the gas constant

T: = calculated total concentration of component I T: = real (analytical) concentration of component i [X,] = free concentration of component i [C,] = concentration of complex j [S,] = concentration of species i (component or complex) {S,} = activity of species i (component or complex) {S:}, {SF}= activity of species i in the test solution and bridge solution, respectively y, = activity coefficient of species i /I, = thermodynamic formation constant of complex j r,, = stoichiometric coefficient of component i in complex j NC = number of components appeanng in complexes NI = total number of components (including inert ions, such as from background electrolyte, not appearing in complexes) NJ = number of complexes NB = number of burettes CT = initial concentration of component i in titration vessel CL = concentration of component i in solution in burette m ~1,= volume added from burette tn V” = initial volume in vessel Ek = electrode potential k Ei = electrode response intercept EU = liquid-junction potential E”‘= contribution of sodium ions to electrode potential s, = electrode response slope C, = empirical constant for H in liquid-junction term Co, = empirical constant for OH in liquid-junction term Kkr = selectivity coefficient of component i 2, = charge on species i

*Author for correspondence. 483

T = absolute temperature F = Faraday constant I = ionic strength 2: = ionic conductance of species i m the test solution n! = iomc conductance of species i in the bridge solution 6 = ionic size parameter A, B = parameters m the Debye-Htickel equation

The procedures developed by Sill& and his colleagues over twenty years ago’ are still widely accepted as the most accurate way of determining formation constants potentiometrically. Experimental conditions were specifically chosen to minimize the adverse effects of various assumptions that had to be made in the analysis of their data. In particular, very high concentrations of supporting background electrolytes were employed to obviate such difficulties as the characterization of activity and liquid-junction potential changes.* However, for many common applications, it is inappropriate to measure formation constants under such conditions. For instance, many biological solutions such as natural waters (e.g., lakes, rivers and soil solutions) and physiological fluids (e.g., plant xylem sap and blood plasma) have much lower ionic strengths, so that the relevant measurements ought to be performed in concentrations of background electrolyte less than, say, 0.2M. Under these conditions, the assumption of constant ionic strength throughout a titration is less valid and can lead to significant errors. Moreover, many systems have formation constants that require measurements to be made at pH values at which it is known that the effects of liquid-junction potentials and alkaline errors can be of a worrying magnitude.

PETER M. MAY et

484

Although a multitude of computer programs have been developed for the analysis of potentiometric titration data, only a few approach the problem in a generalized way. Of these, SCOGS3 and, especially, MINIQUAD have gained widespread popularity. However, neither of these accommodates variations in ionic strength, changes in liquid-junction potential or imperfect ion-selectivity of electrodes. Some programs, such as LETAGROP,5 do permit simulation of changes in liquid-junction potential, albeit in a restricted way, but none allows corrections to be made for all three kinds of effect. Each of these effects is well known and has been extensively studied in its own right. This paper describes how corrections for each may be applied in generalized computer programs for the analysis of potentiometric titration data. Although it is not possible to simulate electrode behaviour entirely satisfactorily for all practical circumstances, the proposed approach satisfies two paramount objectives, namely to improve the interpretation of certain observed data and to provide mathematical forms for the corrections that allow them to be simplified, or even neglected, if that is thought to be appropriate. In this way, the useful range of titration conditions can be extended, the necessity for corrections can be investigated and different forms of the equations used for correction can be readily compared. THEORY

Mass -balance

The conditions of mass-balance are imposed in the standard way by equating calculated total concentrations with real (analytical) concentrations: T:=T;,

i=l,...,NC

(1)

where

“=I

J==I c:v”+

E, = E;,.,’ + s,log[X,]

(3)

(Eyp>iy,.

Erst

= E; + s,logy, + E,U.

Others’-” assume that yk is constant and that Ep is a linear function of the free proton and hydroxide ion concentrations. The electrode equation is thus Ek = .Y’

+ +Jog[x,l +

dX,I + Kv~m/[X~l(8)

where EP,” = Ei + s,logy,. Electrode selectivity

None of the above programs accommodates the well-known effects of interfering ions. It is possible to extend the useful range of detection by several orders of magnitude if these effects can be taken into account. However, the simple Nikolskii” equation (9), KY = &log({XJ + &,{XJ)

often invoked to describe how the electrode response is ultimately dominated by the interfering ion, is unsatisfactory for this purpose. This is because it describes a very sharp transition in response whereas, in fact, the change is generally much more gradualI As a result, even if equation (9) is used in an entirely empirical way, corrections in the region that is most important, i.e., when the effect is first manifest, can never be more than a very small fraction of the effect itself. At least in the case of glass electrodes, the behaviour observed in the presence of one univalent interfering ion is better described by the Eisenman equation:‘) (10)

where tl and Kkr are empirical parameters. Although a corresponding general multi-ion equation has not been formulated, I4 this has been done in the case of the Nikolskii equation, leading to ELS= s,log({X,} + 1 KL,{X,)--k”i

Electrode equation

In general, the equation relating the potential of an electrode to the activity of the electrode ion can be written in the form (5)

where Ei is the electrode response intercept. For the purpose of formation-constant determination, EL’ is usually given by E;s = s,log{x,} = s,logy, + Sklog&]

(9)

(4)

“=I

E,=E;+E;‘+E,U

(7)

where

EL’ = s,log({X,}“” + K,,{X,}““)

V” + F v, m=l

I-,=

assume that both the liquidSome program@ junction term, E,)‘, and the activity coefficient of the electrode ion, yk, are constant. The resulting electrode equation is

y c;v, m-1

T; =

(2)

al.

(6)

(11)

where the charges z, have the same sign as zI.” For the purpose of titration-data simulation, we therefore propose that an equation of the following form should be used: Eks = s,log[{X,}“” + c (Kk,{X,}rk’--,)“z]cr

(12)

This has the advantage that equations (6), (10) and (11) can all be accommodated by making suitable choices for u and the Kk, values.

Simulation of titration data Liquid-junction

potential

It is common for the liquid-junction coefficients of equation (8) to be obtained empirically. An extension which reduces to this linear form at low concentrations of acid is that derived from the Henderson equationI by Biedermann and Sillen.’ It has the form Ei” = --(RT/F)

In (1 + d[X,]/Z)

(13)

where I is the concentration of the background univalent electrolyte. This equation has been used to correct for liquid-junction potential changes in the calibration of glass electrodes.” The Henderson equation itself, equation (14), predicts potentials across junctions of different univalent electrolytes at constant ionic strength, with reasonable success.” It has also been shown to be useful in calculation of potentials across constant-ionic-medium junctions through potentiometric titrations.”

Much less is known about the applicability of the Henderson equation to salt bridges that have higher concentrations than that of the test solution, alsome typical values though Bates *’ has tabulated predicted for bridges containing saturated potassium chloride solution. Whilst such bridges clearly violate one of the principle tenets of the Henderson equation (namely that activity coefficients remain constant across the junction), they do serve to reduce the size of the junction potential, thus making relative errors in its characterization less significant. However, it remains to be seen whether the equation can be used effectively to simulate changes in the liquid-junction potential when high-concentration salt bridges are involved. Although equation (14) predicts absolute liquidJunction potentials, these can be observed experimentally only as a difference in potentials. It is conventional to have this difference increasingly manifest towards the extremes of pH. This property can be preserved by definmg Ey as follows. ELJ = k

EL" !,

_

EL-h r(

485

bridge, i.e., one containing electrolyte ions of the same type and concentration as the background medium, Ep = 0 and its size increases with the concentration of the bridge electrolyte. The advantage of equation (14) for the purpose of simulating titration data is that its general form permits further investigation of liquid-junction-potential phenomena. yet, by an appropriate choice of the ionic conductivities for each ion, it can be used to accommodate the simpler equation (13) when that is applicable. For example, with univalent electrolytes and a constant-ionic-strength junction of the type I(AB) II [X,I(HB),I(AB) equation

(14) reduces

to equation

d = (I.,, - &)I(&

(13), with

+ &).

There is evidence that liquid-junction potentials calculated from the Henderson equation are generally overestimated, especially for high ionic-strength for any given backgroundbridges. 2’.‘2 However, electrolyte and salt-bridge concentrations, there is often only one ion that has an appreciable effect on changes in the liquid-junction potential.‘3 Hence, for junctions involving H +, AH can be empirically adjusted, either at the level of equation (13) or at that of equation (14). Actirit_y coeficients For the calculation of activity coefficients the most common extension to the Debye-Hiickel formula has the form

(15) where c is an empirical constant. Guggenheim24 simplified this by choosing h = l/B and, under this condition, DaviesZs proposed that c = 0.3Az2. The latter proposal is the IUPAC recommended equation for formation-constant work’” and is widely used in geochemical equilibrium calculations.27.2* Others have used Kielland’s29 values for d and empirically determined values for c.~‘,~’ Equation (16) seems a particularly sensible choice to incorporate in a computer program since the parameters d and c for each ion can be chosen such that all of the approaches above can be accommodated. The constants A and B can be calculated within the program from A = 1.8249 x lO”/D’, B = 50.293/D

(15)

where E:“J is the liquid junction potential [calculated from equation (14)] between the bridge solution and a solution containing the bridge ions only (i.e., at pH 7) at the reference ionic strength (i.e., that at which the electrode constants Ef”““’ and q are applicable). This results in a calculated Ek’ that tends to a constant value (usually small or zero) as the pH tends to 7. Note that with a “constant ionic strength”

where D = (3686.2~I”Although the values chosen these formulae adequate for perature range

135.15Z-)“2.

coefficients depend on the particular for the dielectric constant of water, yield results which are more than the present purpose over the temO-100”.

PETER M. MAY el al.

486

At low concentrations of background electrolyte, variations in ionic strength occur during a titration, owing to changes in complexation. By using equation (16), corrections should be made to all the formation constants at each titration point so that they refer to the calculated ionic strength at the point. Corrections of this kind over a large range of ionic strength (say, from I = 0.0 to 0.2) may not be adequate in precise potentiometric work; however, in the majority of cases, variations in ionic strength, although significant, are su~~iently restricted to ensure that corrections with equation (16) are wholly saiisfactory. Conversion into thermodynamic constants Equations (2), (12) and (14) require “thermodynamic” values of the parameters & E,“, K,,, 2; and A::, i.e., those that are defined with respect to a standard state based on reactions occurring at infinite dilution in water at zero ionic strength. However, it is established practice to work with conditional constants (I/?,, ‘Ei. ‘Kkr. ‘A:, ‘2.:) which refer to some other ionic strength (I). Such constants, therefore, need to be converted into the corresponding thermodynamic values in the following way:

parameter (such as u,, & C;, C$,, or 7’:) can be calculated. Consequently, the procedures for solution of the m.b.e. falls into two categories. In the first, these equations are solved for NC free concentrations without any reference to the electrode equation. The electrode potential can subsequently be obtained by substitution into equations (S), (12) and (15). In the second category, it is necessary to solve the electrode equation to obtain the free electrode-ion concentration, [x,J, (given an observed potential, &), before solution of the m.b.e. Solving the electrode equation for [x,) requires a knowledge of ail the free concentrations so it is necessary to implement an iterative solution of this equation and the m.b.e. A Newton-Raphson procedure is generally the most efficient way to solve the m.b.e. The corresponding set of linear equations has the form AS=&

(17)

where A,m= F=;;Ti’li. m=l

I.‘.,

NC: i=l

,.a.* NC.

s,, = AX;, = shift to mth unknown = .u;+’ -.x2. h, = T:(F) - T$?‘),

‘I, is defined as in equation (4). Expressing equation (12) in terms of concentrations, and rearranging, yields EiS = qlog’y,

Hence, the supplied value of ‘I$ and the ionselectivity coefficients are converted into thermodynamic values as follows. El = ‘E; - s, log ‘vk K,, = ‘Kd~&,)l”~‘~) The supplied ionic conductivities to be used in the calculation of EkJ are treated similarly:

a;= ‘Ayy:;

a;=‘a;,$;

Whether Debye-Htickel corrections are to be invoked or not, only conditional constants thus need to be supplied by the program user. In either case, the same values (say, for electrode potential) are calculated for any point in the titration which actually has the reference ionic strength. So&ion qf mass-balance equations The NC mass-balance equations (m.b.e.), equation (11, can, in principle, be solved for any NC unknowns as long as all remaining parameters are known. In particular, there are NC free concentrations that can be determined. On the other hand, if one of the free concentrations has been measured experimentally, the remaining free concentrations and one other

and z” =

,,th

iteration

estimate of the unknowns.

The equations can be set up in terms of the absolute values of unknowns except for the free concentrations, [XI], and formation constants, p,, for which natural logarithms are more convenient. The equations (17) can then rapidly be solved by forward and backward substitution using the Gout factors for A (obtained by using partial pivoting).j2 Scaling of the shift vector Z is only required when particularly large shifts are predicted or if 6’.b increases. If a failure in the Newton-Raphson procedure occurs in those tasks for which the only unknowns are free concentrations, the slower but more robust secant method13 can he used to solve the equations. Initial estimates It is well known that the efficiency of the Newton-Raphson method is considerably improved if good initial estimates can be obtained for the unknowns. Here, there are two classes of unknowns to consider. The first consists of the free concentrations of the components. Unless such concentrations can be obtained from electrode equations, they are very difficult to estimate. Secondly, there are those unknowns which are experimental parameters such as total titrand and titrant concentrations and titration volumes. Formation constants can also be included in this class. Generally it is possible to estimate reasonable initial values. At the first and second points, initial estimates for the unknowns of the second kind can readily be

Simulation of titration data

487

Table 1. Formation constants used in

Table 2. Initial volumes (ml), concen-

the simuiations~6

trations (M), electrode parameters (mV) and sekctivitv narameters used

Species ~_-.I__. I. OH 2. HCys 3. H,Cys 4. H,Cys 5 ZnCysz 6. ZnHCys 7. ZnHCys? 8. Zn,Cys, 9. Zn,HCys, 10. Zn,H$Jys,

fog B - 13.380 .-10.110 18.078 20.050 17.905 14.604 24.114 42.278 48.313 54.082

provided from the experimental values. If measured electrode potentials are available, initial estimates of the corresponding free electrode-ion concentrations can readily be calculated from the simple electrode equation (7). At the first titration point, estimates of the other free concentrations can be obtained from a proportional formula of the kind used in COMICS34 and ECCLES.IS Initial estimates for these concentrations at the second point can then be taken as the values obtained for the solution of the equations at the first point. Initial estimates of all parameters at the third and each subsequent point of a titration are best obtained by linear extrapolation of the solution from the previous two points. This can, in principle, be based on the change in either the observed electrode potentials or the titration volumes. Experience has shown that, when they are availabLe, the potentials are the more effective, It is found that the number of Newton-Raphson iterations required to solve the m.b.e. at each point can be reduced by as much as 50% by this technique, compared with one which uses the solution of the previous point for the initial estimate. As storage and extrapolation of two previous solutions mvolves considerably less work than the application of a number of Newton-Raphson iterations, a correspondingly marked decrease in execution time is obtained. Quadratic extrapolation has also been investigated but proved to be less satisfactory, in part because the extra calculation over three points does not always lead to a decrease in the number of Newton-Raphson iterations required to solve the m.b.e.

Titration zn/cys

&‘CYS

20.0 0.055

20.0 0.050

0.015 0.15 0.2

0.015 0.014 0.15 0.102 0.2

0.35 0.15

0.35 0.15

400.0 61.54

400.0 61.54

0.125

3.27 4.17 x 10-16

3.27 4.17 x IO-‘6

major objective of this project is to provide a flexible computing tool for investigating phenomena associated with chemical interactions in solution and for their quantitative characterization. Although liquid-junction potential, sodium-ion interference and activity changes have long been individually characterized, they have been widely neglected in the context of poten~ometric titrations. As no computer program for analys~ng titration data has hitherto incorporated all of these corrections, their importance has been difficult to assess and, as a result, has tended to be ignored. To illustrate the extent to which each effect is typically manifest in many titrations of practical importance, two simulations have been performed. These relate to the dete~ination of proton and zinc formation constants of cysteine. 36 The titration conditions and parameters are shown in Tables 1-3. The Davies equation” was used to simulate activity-coefficient changes. Values for the formation constants, ionic conductivities and sodium-eon selectivity parameters were estimated from the literature, unless otherwise indicated. The results of the simulations are shown in Tables 4 and 5. Dist~butions of the complexes during the titrations, as a percentage of total cysteine and total

Table 3. Ionic conductivities RESULTS AND DISCUSSON

Constant-I bndge (I= 0.15)

The ability to apply the more general electrode equation [see equations (5), (12) and (15)J and to correct for activity changes, forms the basis of a generalized computer library called ESTA (Equilibrium Simulation for Titration Analysis) which is currently being developed in this department.* The ~~~_.-.--_.-__-_~ *Interested readers can obtain details of how to acquire copies of these programs by writmg to

PMM.

kidge

Na K

Cl

Test solution

Na K Cl CYs Ii OH

-

Satd. KCI bridge (I = 4.2) -~___

53

-

-83

36 -39

53 -83 -10 375 -200

53 81 -83 -10 125* - 100*

*Empirical values estimated ex~~mentaliy.37

PETERM.

488 Table 4. Simulated and tonic-strength Volume added. ml

-log:H:

0.0 0.4 0.8 1.2 1.6 2.0 24 2.6 2.8 3.2 3.6 4.0 4.4 4.x 5.2 6.0 6.8 7.4 7.8

1.92 2.02 2.14 2.29 2.49 2.78 3 52 6.94 7.49 8 02 8.49 9.15 9.79 10.21 10.58 11.19 11.51 11.65 11.72

liquid-Junction potenttals, changes for a typical titration O<,of total Cys m species 2 3 4 0 0 0 0 0 0 0 7 20 46 71 85 72 50 30 10 5 4 3

40 46 53 61 71 83 96 93 80 54 28 8 1 0 0 0 0 0 0

respectively, are also given. The magnitude of changes in ionic strength that commonly occur in titrations of this kind can be seen, emphasizing the need to correct formation constants to the appropriate ionic strength at each point. Secondly, the effect of sodium ion interference towards higher pH (Es”), even at the concentrations which occur in these titrations, is significant. Moreover. as temperatures and sodium-ion concentrations increase. this effect will increase dramatically.3x It is thus interesting to note that corrections of this kind are also important when higher concentrations of background electrolyte are employed to minimize ionic-strength and liquid-junction potential changes. For example, in zinc

Table

MAY et al. sodium-ion cystemate

Ip.’ , ‘EL’.

60 54 47 39 29 17 4 0 0 0 0 0 0 0 0 0 0 0 0

mV

mV

E”‘. mV

I, M

-5.5 -4.4 -3.3 -2.4 -15 -0.8 -0.1 -0.1 -0.2 -0.4 -0.7 -0.9 - 1.0 - I.1 - 1.1 -05 05 1.1 1.6

0.7 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.7 0.8 0.9 0.9

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.6 0.9 1.2 1.6 2.4 3.0 3.4 3.6

0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.151 0 153 G. 156 0.160 0.163 0.169 0.175 0.180 0.187 0.193 0.196 0.199

3.OM sodium perchlorate at 25 , errors of 1.0 mV may occur at pH values as low as 8. The tables also show the liquid-junction potentials calculated from the Henderson equation for both constant ionic strength (‘I?‘) and saturated potassium chloride (‘EL’) bridges. For titrations at these ionic strengths it is evident that constant-ronicstrength bridges may give rise to significant and somewhat irregular changes in liquid-junction potentials over the whole pH range. This is due to the opposing effects of hydrogen and hydroxide ions on the one hand and the steadily increasing concentration of sodium on the other. The saturated potassium chloride bridges are much less problematical in

5. Stmulated liquid-Junction potentials, sodium-ton strength changes for a typtcal zmc cysteinate

Volume added, ml

-log{Hj

5

0.0 0.4 0.8 1.2 1.6 2.0 2.4 28 3.2 3.6 4.0 4.4 4.8 5.2 6.0 6.8 7.4

2.04 2.17 2.33 2.54 2.88 3.93 4.87 5.07 5.25 5.44 5.68 6.00 6.55 10.70 11.39 1163 11.74

0 0 0 0 0 0 0 0 0 1 1 2 4 6 6 6 6

9, of total Zn in species 6 7 8 9 10 0 0 0 0 0 3 15 15 14 I1 8 5 2 0 0 0 0

0 0 0 0 0 0 1 3 4 5 5 5 3 0 0 0 0

0 0 0 0 0 0 0 0 1 2 6 17 46 72 72 71 71

Interference protonation

0 0 0 0 0 0 0 2 5 10 18 25 18 0 0 0 0

0 0 0 0 0 0 4 14 23 29 28 19 4 0 0 0 0

interference titratton

and

ionic-

‘EU, mV

ZEL’

p.%’

mV’

rni

L M

-2.1 - 1.7 -0.7 0.2 0.9

1.1 1.1 1.1 1.1 1.1 1.1 I 1 1.1 1.0 1.1 0.9 0.9 0.9 0.9 1.0 1.0 1.1

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 1.7 2.7 3.2 3.5

0.164 0.164 0.164 0.163 0.163 0.162 0.160 0.159 0.157 0.157 0.156 0.157 0.160 0.164 0.170 0.175 0.179

15 1.3 1.1 0.8 0.6 0.4 0.2 -0.1 0.1 1.2 2.2 2.8

Simulation of titration data this respect. Thus, a choice must generally be made between the better characterized constant-ionicstrength bridge and the smaller, but not so well characterized, effects of saturated potassium chloride bridges. In either case, more experimental work is probably required to match the values of these parameters to the accuracy of modern research potentiometers. However. it is clear that with saturated potassium chloride bridges, under normal circumstances, errors from changes in the liquid-junction potential are likely to be much less than those arising from other sources.3” They do not, therefore, justify the use of pH buffers4’ in preference to internal calibration of glass electrodes. As is well known, the values of liquid-junction potentials and single-ion activity coefficients calculated as described here do not have any absolute significance. They cannot be measured experimentally and, as a corollary, they cannot, in themselves, be meaningfully interpreted in any data analysis. However, the effects of dzflerences in these quantities can have an experimental reality and thus can validly be taken into account, either theoretically or empirically. Although the approaches developed in this paper allow for large changes in liquid-junction potential, ionic strength and sodium-ion interference, it obviously remains desirable to design experiments which minimize these effects whenever possible. Nevertheless, it is envisaged that programs which apply such corrections will help to establish the extent to which titration conditions can be extended beneficially. This will permit a wider variety of equilibrium systems to be satisfactorily characterized. A major advantage in this respect will lie in the ability to integrate and quantify, simply and rapidly, expected effects in the actual context of each application. This will provide an insight, for the system under investigation, which has hitherto often proved elusive in practice. REFERENCES

I. L. G. Sill&n. Laboratorv Methods m Current Use at the Department of Inorganic Chemistry, KTH. Stockholm, 19.59. 2. G. Biedermann and L. G. SillCn, Ark. Kemi, 1953, 5, 425. 3. I. G. Sayce. Talanta, 1968, 15, 1397. 4. A. Sabatim. A. Vacca and P. Gans, ibid., 1974. 21. 53. 5. L. G. Sill&n, Acra Chem. &and., 1964, 18, 1085. 6. P. M. Mav. D. R. Williams. P. W. Linder and R. G. Torrington, Talanta, 1982, 29, 249. 7. N. Ingri and L. G. Sill&n, Ark. Kemi, 1964, 23, 47. 8. G. Arena, E. Rizzarelli, S. Sammartano and C. Rigano. Talanta, 1979, 26, I.

489

9. M. Wozniak, J. Canonne and G. Nowogrocki, J. Chem. Sot., Dalton. 1981, 2419. IO. M. Molina. C. Melios. J. 0. Tormolli. L. C. Luchiari and M. Jafelicci. Jr., i. Electroanal. Chem. 1979, 105, 237. II. B. P. Nikolskii, Acta Physicochim. USSR, 1937, 7, 597. 12. B. P. Nicolsky (B. P. Nikolskii), M. M. Schultz, A. A. Belijustin (A.-A. Belyustin) and A. A. Lev, in Glass Electrodes for Hvdronen and Other Cations. G. Eisenman (ed.),“Chapier 67 Dekker, New York, 1961. Biophys. J., 1962, 2, 259. 13. G. Bsenman. 14. Idem. in Glass Electrodes for Hydrogen and Other Catrons. G. Elsenman (ed.), Cbpter 9. Dekker, New York, 1961. Pure Appl. Chem., 1976, 48, 127. 15. G. G. Guilbault, Z. Phys. Chem.. 1907, 59, 118; 1908, 63, 16. P. Henderson, 325. and D. Midgley, Anal. Chim. Acta, 17. H. S. Dunsmore 1972. 61, 115. The Principles of Electrochemistry. 18. D. A. MacInnes, Dover, New York, 1961. 19. G. T. Hefter, Anal. Chem., 1982, 54, 2518. 20. R. G. Bates. CRC Crit. Ret). Anal. Chem.. 1981, 10,247. Trans. Faraday Sot.. 1968, 64, 1059. 21 R. G. Plcknett. 22. D. A. MacInnes and L. G. Langsworth, Cold Spring Harbor Svmposium &ant. B~ol., 1936, 4, 18. 23. C. F. Ba& jr. and k. E. Mesmer, The Hydrolysis of Cations. Wilev. New York. 1976. Phd. Mag.. 1935, 19, 588. 24. E A. Guggenheim. London, 25. C. W. Davies, ion Association, Butterworths, 1961. 26. G. H. Nancollas and M. B. Tomson, Pure Appl. Chem., 1982, 54, 2675. 27. G. Sposito and S. V. Mathgod, GEOCHEM: A computer program for the calculatron of chernrcal equilibria m soil solurrons and other natural water systems. University of California, 1980. D. C. Thorstenson and L. N. Plum28. D.. L. Parkhurst, mer, PHREEOE-A computer prozram for aeochemical analysis, U.‘S: Geological S&vei, Gate; Research Division, Richmond V.A., PB 81-167801, 1980. 29. J. Kielland, J. Am. Chem. Sot., 1937, 59, 1675. 30. P. W. Lmder and K. Murray, Talanta, 1982, 29, 377. K. Kharaka and I. Barnes. SOLMNEQ: 31. Y. Solutron-mineral equdibrium co,-zputations, U.S. Geologlcal Survey, Water Research Division, Washington, D.C., PB-215-899, 1973. 32. J. H. Wllkmson and C. Reinsch. Handbook for Auramatic Computation. Vol. II. Linear Algebra, p. 93. Sprmger-Verlag, Berlm, 197 I. L. G. Sill& and B. Warn33. N. Ingn, W. Kakolowicz, qvist. Talanta. 1969, 14, 1261; corrections. 1970, 15, No. 3, p. xi. 34. D.-D. Perrin and I. G. Sayce, ibid., 1967. 14, 833. 35. P. M. Mav. P. W. Linder and D. R. Williams. J. Chem Sot. Dalton. 1977. 588. 36. G. Berthon. P. M. May and D. R. Williams, ibid., 1978, 1433. 37. C F. Jones, P. M. May, K. Murray and D. R. WIlllams, unpublished work. 38. N. Linnet, pH Measurements m Theor) and Practice, Radiometer. Copenhagen, 1970. 39. P. M. May. Talanta. 1983, 30, 889. 40. H. K. J. Powell and M. C. Taylor, Talanfa, 1983, 30. 885