Points of zero charge in the presence of specific adsorption

Points of Zero Charge in the Presence of Specific Adsorption J. L Y K L E M A Laboratory for Physical and Colloid Chemistry, Agricultural University, De Dreijen 6, 6703 BC Wageningen, The Netherlands

Received July 19, 1983; accepted October 6, 1983 In the absence of specific adsorption, the point of zero charge (pzc) of a colloid or suspension can be obtained by measuring the relative surface charge as a function of the concentration of the chargedetermining ions at various concentrations of the indifferent electrolyte. These curves show a common intersection point (cip) which may be identified with the pzc. In the presence of electrolytes containing specifically adsorbing ions, also cip's are observed but they do not coincide with the pzc. The physical meaning ofcip's is that at such points the Esin-Markov coefficientis zero, indicating equal compensation of the surface charge on a molecular basis by cations and anions. For the establishment of cip's in the presence of specific adsorption, the surface charge must be measured in a mixture of the specifically adsorbing electrolyte and an indifferent one. A thermodynamic theory is developed to obtain from this surface charge and from information on cip's the complete composition of the double layer at any composition of the bulk. Examples are worked out for an insoluble oxide in a mixture of a specifically adsorbing (2-1) and an indifferent (1-1) electrolyte. Points o f zero charge (pzc) are i m p o r t a n t characteristics o f all amphoteric colloids a n d suspensions. T h e y serve as reference points for double-layer parameters, such as the surface charge ao and the free energy a n d their values reflect the chemical nature o f the interface. Hence it is m a n d a t o r y to have reliable sets o f pzc values at one's disposal for various compositions o f the solution. However, the determination o f pzc values is n o t always straightforward. Apart f r o m the familiar chemical requirements (such as ensuring uttermost cleanliness o f the surface) there are methodical problems (different m e t h o d s sometimes measure different zero points) a n d issues o f principle: points o f zero charge are t h e r m o d y n a m i c a l l y inoperable quantities so that each procedure o f m e a s u r e m e n t is, either or not tacitly, based on some m o d e l assumption. These problems are c o m p o u n d e d if the system contains specifically adsorbing ions, in which case the all but a u t o m a t i c application o f standard procedures has led to grossly erratic results.

One o f the established techniques to measure a pzc is (potentiometric) titration with so-called potential-determining ions at various electrolyte concentrations G. fftitration curves exhibit a c o m m o n intersection point (cip) this is identified as the pzc. As we shall show, this procedure is justified only if the electrolyte is indifferent. With specifically adsorbing ions, c o m m o n intersection points can be observed, but they are not points o f zero charge. W e shall show which interpretational steps are consciously or unwittingly m a d e to identify a cip with a pzc a n d on that basis offer an alternative interpretation for the tip in the presence o f specific adsorption. Before doing so, it will appear necessary to revisit some seemingly obvious double-layer parameter definitions. The following discussion will be focused on surface c h a r g e - p H titrations for insoluble oxides, but the analyses are mutatis m u t a n d i s valid for other systems, e.g., for silver halides if pAg is substituted for pH. The isoelectric point (iep) is an entirely different notion. Typically, due to specific ad109 0021-9797/84 $3.00

Journal of Colloid and Interface Science, Vol. 99, No. 1, May 1984

Copyright © 1984 by Academic Press, Inc. All tights of reproduction in any form reserved.

110

J. LYKLEMA

sorption it moves in a direction opposite to that of the pzc. In this paper, iep's will not be discussed. ABSENCE OF SPECIFIC ADSORPTION The procedure of determining the pzc runs follows. The surface charge is defined through ~0 - F(P~+ - Po.-) [1] as

which is equivalent with O"0 ~ F(rHN03

[2]

-- F K O H )

because analytically only the adsorption of electroneutral species is measurable, solution and double layer as a whole being electroneutral. The indifferent electrolyte is KNO3. The sol or suspension is equilibrated at a certain, say high, pH, and low c,. (point 1 in Fig. 1) and then titrated till low pH (point 2). An absolute value cannot be assigned to this curve because there is no way to measure ao at the start of the titration. At point 2, electrolyte is added. The pH increases and a0 becomes more positive (downward in Fig. 1, till point 3). Now the titration is resumed in the inverse direction (till point 4), again electrolyte is

%

6

pH °

added (till point 5) and this is followed by acid titration till point 6, and so on. If a cip develops, this is identified as the pzc rendering the set of curves absolute value. The underlying reasoning is as follows. If a0 is negative, it becomes more negative at given pH if electrolyte is added, because then the surface charge is better screened by the more abundant cations. Likewise, if ao would have been positive, it would become more positive upon electrolyte addition, now because of screening by NO~ ions. Only if the surface is uncharged has electrolyte addition no effect, hence the cip is the pzc. The reasoning leads to the correct answer but contains a missing link. Strictly speaking, a pzc cannot be measured, because one cannot measure PH+ and Fon- (or, for that matter, PHNO3 and FKOU) independently. Not ao is measured, but the way in which a0 is compensated. If K ÷ and NO~- ions contribute to an equal extent to the compensation (i.e., each by 50%), it is inferred from the lack of preference for these ions that the surface bears no charge. This equal compensation rule fits in the familiar double-layer picture: if ao --~ 0 and no specific adsorption occurs, the double layer is purely diffuse and a0 is for 50% compensated by an excess of counterions and for 50% by a deficit of coions. For electrolytes of other valence type the fractional compensations are different because of the unequal screening power of ions of different valency. However, because of the physical picture we shall also in this case speak of equal compensation. A thermodynamic way of restating the issue in a more general fashion is in terms of the Esin-Markov coefficient ft. This quantity was originally introduced to account for certain anomalies in the shift ofpzc's on mercury due to electrolyte addition (1). For oxides, fl is defined as

pH

fl = (6pH/6 In a±)~o, FIG. 1. Possible set o f ao(pH) curves for a n insoluble oxide. The c o n c e n t r a t i o n o f the indifferent electrolyte is indicated. In this case cip = pzc. Journal of Colloid and Interface Science. Vol. 99, No. l, May 1984

[3]

where a± is the mean activity of the electrolyte. For silver iodide pH is again substituted by

POINTS OF ZERO CHARGE pAg. The coefficient /3 is a measure of the horizontal distance between the curves of Fig. 1. It can be proven by thermodynamic arguments (2) that generally for a z+z_ electrolyte 1

~ ~-

Z+

__

z+ -

z_

bff_

Z+Z_

a+_

1 + z+ + z_ ( r a + ]

z_

z+z_

\6aola±

[4]

where a_ and or+ are the charges attributed by anions and cations, respectively. As in the cip /3 = 0, it is concluded that at that point in a (1-1) electrolyte ~O'O!a+- "~

~ff0

a~. =

- - 1/2

[5]

which is a more formal way to describe the equal contributions of cations and anions to the compensation o f the surface charge. If the electrolyte is asymmetrical, the compensation at 13= 0 remains 50-50 with respect to the number of ions, but chargewise it becomes asymmetrical. For instance, for Me(NO3)e

a~

\ ~°0 ] a±

It may be noted that for the above analysis it was not necessary to introduce the ambiguous notion "potential determining ion," which we shall avoid in this paper. On the other hand, it had to be agreed that the surface charge is entirely and completely given by [ 1] or [2], i.e., H + and O H - ions are identified as the surface ions. If for a given oxide, for a number of (indifferent) electrolytes at various G the same cip is found, it may be safely assumed that this tip is a property of the uncontaminated surface and then it is justified to identify this point as the pzc in absence of specific adsorption. Following Pyman et al. (3) we shall call this the pristine p z c (Pyman et al. define their double-layer parameters in an unusual way, but their criterion for "pristine" agrees de facto with ours).

1 11

PRESENCE OF SPECIFIC ADSORPTION Two problems arise when the electrolyte contains specifically adsorbing ions, i.e., ions having an affinity for the surface in excess of the coulombic term. The former is a problem of definition. If in addition to H + and OH-- other ions bind to the surface, imparting charge there is no a priori reason to identify only H + and O H - as the ions contributing to tr0. Indeed, some authors prefer to define a0 as the charge attributed by all chemisorbed species. This definition has some drawbacks. One of them is that it is often not easy to discriminate between nonspecifically, weakly specifically, and strongly specifically adsorbing ions, and another one is that then multidimensional titration curves are needed: ao should now be plotted as a function o f p H and the logarithms of the concentrations of all specifically adsorbing species. In view of the dominant sensitivity of oxides to H + and OH-, and considering that almost exclusively acid-base titrations are done, we decide to maintain definition [1] or [2]. Consequently, specifically adsorbed ions belong by definition to the countercharge, or, more specifically, to the Stern charge. The latter problem is more methodical. It is the screening of charges, already present on the surface, that determines/3. However, specifically adsorbing ions do not only passively screen existing charges, they actively contribute to that charge themselves. Electrolytes containing specifically adsorbing ions are therefore not suitable tools to obtain pzc values. Nevertheless, the pzc in the presence of specific adsorption remains fully defined. In our definition of surface charge it is still the pH where a0 = 0. If the cation adsorbs specifically this pzc is lower than the pristine pzc because cations in the Stern layer favor the adsorption of O H - over that o f H +, so that a lower pH is required to restore the H + - O H adsorption balance. Similarly, specific adsorption of anions leads to pzc's above the pristine value. Journal of Colloid and Interface Science, Vol. 99, No. 1, May 1984

112

j. LYKLEMA

Several authors, carrying out titrations as in Fig. 1 with electrolytes containing specifically adsorbing ions, observed sharp c o m m o n intersection points. For instance, Breeuwsma and Lyklema (4) reported on hematite (aFe203) a cip for Ca(NO3)2 which was about 2 pH units lower than the pristine pzc, whereas for K2SO4 an increase of 1.1 p H units was observed. Similarly, Pyman et al. (3) reported for an amorphous coprecipitate of silica and ferric hydroxide in Na2SO4 a crossover point which was 1.3 units higher than the pzc. The existence of cip's in the presence of specific adsorption appears well established and so is the trend as a function of pH, which happens to be the same as that for the pzc. In fact, this similarity has led Breeuwsma and Lyklema (4) to the (erroneous) conclusion that also in the presence of specific adsorption cip and pzc were identical. That in the presence of specific adsorption the cip ¢ pzc was for the first time recognized by Pyman et al. (3), although the relative complexity of their system was not conducive for a rigorous analysis. The discrepancy was also observed by Huang (5) though in a more primitive fashion (the plots contain amounts of acids or base instead o f ao-values) and this author did not offer an explanation. The feature was independently discovered by Ardizzone et al. (6) and investigated under better defined conditions. The trend is that for specifically adsorbing cations the cip is situated at a positive ao, the more positive the higher the affinity of the cation for the surface. Conversely, in the case o f specific adsorption of an anion, the cip is at negative values of ao. The only way to locate the cip on the troaxis in the presence of specific adsorption is by conducting acid-base fitrations in mixtures of the specifically adsorbing electrolyte and an indifferent one. The latter is needed to obtain a reference to the pristine pzc which may be considered well established. Figure 2 sketches how such titrations in mixed electrolyte look: Consider the case of a heavy metal Me2+; ads0rbir~g specifically on an oxide. First a titration is done in dilute, Journal of Colloid and Interface Science, Vol. 99~ No. 1, May 1984

10-1

oo

i0 -2 10-~0. 4 10"5M Me(NO3) 2 10-3 M KNO3

c ~p

pH°

pH

FiG. 2. Trends observedif an insoluble oxideis titrated in a mixture of Me(NO3)2(concentrationindicated) and dilute KNO3. The Me ion adsorbs specifically.

say 10-3 M, KNO3 to establish the absolute values of ao. The concentration of KNO3 must be high enough to define the ionic strength (conc >> conc of acid and base) but sufficiently low to allow complete displacement of K + from the double layer by Me 2÷ ions at not too high concentration of the latter. If low concentrations o f Me(NO3)2 are added (10 -5 M ) no changes are observed at the positive side (since there NO~ is the counterion and its concentration is hardly increased), but some increase in the negative direction is observed at high pH, because here the Me 2+ ion is the counterion and because o f its specific adsorbability (and to a lesser extent, due to its higher valency) it can already compete with K + if CM~+ < Cry+. Similar trends are observed at higher concentrations of Me(NO3)2 on the understanding that also at low p H the absolute value of Cro starts to rise as soon as the NO~ concentration attributed by Me(NO3)2 is of a magnitude comparable to that stemming from KNO3. As Fig. 2 shows, this leads to a cip which is not identical with the pzc and differs the more from it, the stronger Me 2+ ions adsorb specifically. The figure shows also that the pzc is well established and moves in positive direction (to the left) as predicted above.

1 13

POINTS OF Z E R O C H A R G E

Figure 3 illustrates this. It may be added that once the cip has been established in sufficiently dilute KNO3, also absolute a0(pH) curves for specifically adsorbing electrolytes in the absence of KNO3 may be drawn. In the present section we shall assume this to be realized. The conclusion is that pzc and cip are both well-defined characteristics that have in common that they differ the more from the pristine pzc the stronger the specific adsorption. They differ, however, with respect to the influence of CMe(NO3)2:the pzc depends on it, whereas the cip is typically an invariant parameter. Automatically the question presents itself as to what is the physical meaning of this cip. As the interpretation of the cip in terms of the Esin-Markov coefficient, given in the previous section, is a thermodynamic argument, pristine pzc ~9

it is of general validity, irrespective of whether or not ion adsorption is specific. The conclusion is therefore that for pure Me(NO3)2 the cip is the equal compensation point, just as it is for pure KNO3. In the example of Fig. 3 this amounts to the applicability of [6] at the cip. That in the case of specific adsorption cip ~ pzc can be understood as follows: at the pzc adsorption of Me 2+ is (apart from the higher valency) favored over that of NOr because of the specific affinity of Me 2+ for the surface. In order to reach a situation where the intrinsic adsorbabilities of cation and anion are identical (so that only the valency difference remains) a positive charge on the surface is needed. This charge should be the more positive the higher the specific affinity of the cation for the surface. Once the equal corn-

!

\ pzc B

cd,.o3,2

co, 72

-

~

~

Cd (NO3) 2 o0=*3.6 [.tC cm -2

b(N03) 2

o o - .4.5

/,

I~C cm -2

~--- Pb (NO3) 2 I

I

I

I

I,

I

-6

-5

-4

-3

-2

-1

log

CMe(N%):~ /M

FIG. 3. Shift of pzc due to specific adsorption of bivalent cations on hematite. Data from Ardizzone et al. (6); CKNO3= 10 -3 M. The arrows indicate the p H o f the cip. The surface charge o f the cip is also given. Journal of Colloid and Interface Science, "Vol.99, No. 1, May t984

114

J. LYKLEMA

pensation point has been reached, further increase of the Me(NO3)2 concentration does not lead to further shifts, i.e., all successive curves at increasing CM~NO~)~must pass through the same point. Similar reasoning applies to double layers in the presence of specifically adsorbing anions. It may be noted that the above interpretation is purely thermodynamical, i.e., no model has been needed. For instance, it has not been necessary to introduce p K values, double-layer potentials, or Stern-layer capacitances. Such models (7, 8) are, however, needed if interpretations of the actual values of the pzc's and cip's are sought. In the following sections we shall now give a double-layer analysis valid for the cip and a thermodynamic description of the ionic components of charge in mixed electrolytes. Using the former as the reference points, the latter provides a complete picture of the double-layer composition if the required a0(pH) titration data are available, without further invoking model assumptions. Elaborations will be given for the (important) special case of mixtures of KNO3 + Me(NO3)2 on oxides.

Above it was noted that in the absence of specific adsorption for ( l - l ) electrolytes the situation of equal compensation (fl = 0, ha+/ ~ro = ~ _ / ~ r 0 = - 0 . 5 ) corresponded with the diffuse double-layer composition near the pzc. This is generally the case for other electrolyte types and we shall now present an analysis for a mixture of KNO3 and Me(NO3)2. As the situation is such that there is no Stern layer and equal compensation, we can use the D H approximation. According to Gouy theory, for the potential gradient we have

Zx

[ 2e 2 ~112 = -T-1-7-1,~I [ ~ n ie-~jy + const] ~/2, \ec0m/ !

[71

J

where y is the dimensionless potential e4~/kT, Journal of Colloid and Interface Science, Vol. 99, No. 1, May 1984

dy

[ e 2 ~1/2 = -T-[~) (2nl + 6ni)'/2y-~ Ay.

Yxx

[8]

From Gauss' law,

-~ -T-(eeokT)l/Z(2nl + 6n2)l/Zyd,

[9]

where Yd = e4~d]kT and qrd is the potential of the diffuse double layer. The ionic components of charge are defined by fig =

zieni

f;

(e-~y - 1) dx

[101

which can be integrated by changing variables using parameter A defined by [8]. In the D H approximation

DOUBLE-LAYER COMPOSITION IN EQUAL COMPENSATION POINTS

dy

e the elementary charge, k the Boltzmann constant, T the absolute temperature, e the relative dielectric constant of the medium, e0 = 8.854 × 10-12 C V -1 m -1 and nj is the number of ions j per unit volume. For a mixture of KNO3 (salt 1) and Me(NO3)2 (salt 2) HK+ = HI, nMei+ = n 2 , and riNG3- = n+ + 2n2, z_ = 1, z2 = 2, z3 = --1. Substituting this in [7], expanding the exponentials till the quadratic terms, finding the constant from the boundary conditions x = o% y = 0, and dy/ dx = 0, elaboration leads to

aK+ = --(enl/A)yd,

[ I I a]

aM~+ = --(4en2/A)yd,

[ 1 lb]

aNO~ = --(e(nl + 2nz)]A)yd,

[1 lC]

satisfying aK+ + aM¢2++ aNO~+ a0 = 0. From [11 ] the relative contributions o f the various ionic species to the compensating charge can be computed under the conditions for which [ 11 ] has been derived. This double-layer analysis serves two purposes. First, under specific conditions the composition of the double-layer is completely known. This information can be used as a reference point for the ionic components of charge analysis of the following section, which

POINTS OF ZERO CHARGE

gives generally valid expressions for the variations of the three az's with a0 or with pH. Second, Eqs. [ 11 ] can be used to show under which conditions cip's develop. Generally when titration curves intersect, these intersection points depend on composition, meaning that if one of the electrolytes is added, the intersection point moves, both on the pHand a0-axis. From [ 11 ] this can be read because generally the three components depend on n~ and n2. However, if one o f the electrolytes is present in excess, and more of this electrolyte is added, the composition becomes invariant, i.e., all successive curves have to pass through the same point, so that a cip develops. In the case under study for excess KNO3 (n~ >> n2) o-M~2+ "-- 0, aK+ = aNO~ = 0.5 and for excess Me(NOn)2 (nab>n0 aK+ ~ 0 , au~+ = 2aNOn, in agreement with [5] and [6]. T H E R M O D Y N A M I C DESCRIPTION OF THE IONIC COMPONENTS OF CHARGE IN BINARY ELECTROLYTE MIXTURES

A thermodynamic theory for a+ and a_ has been given for the example of silver iodide in (1-1 ) electrolytes (9). If a0 is known as a function ofpAg and c~t it is possible to find (be+~ 6o0) and (6o-_/6a0) so that absolute values of a+ and a_ are obtainable by integration if a reference value is available. This theory applies also to oxides if pAg is replaced by pH. We want now to generalize this theory for the present case of a mixture of a (1-1) electrolyte, say K N O 3 , concentration c~, and a (2-1) electrolyte, say Me(NO3)/, concentration c2. In this case cromust be known as a function of pH, c~, and c2. Such data are not difficult to obtain although to the knowledge of the present author they are not available in sufficient detail yet. In order to avoid unnecessarily cumbersome expressions, the following assumptions are made. The oxide is insoluble, the aqueous solution contains HNO3, KNO3, KOH, and Me(NO3)2 as the added electrolytes. At this instance, there is no need to account explicitly for hydrolysis of the Me 2+ ion, should it occur, because the above four components fully de-

1 15

fine the system. Pressure and temperature are constant, c I and c2 are both >>CHNo3and CKon and all mole fractions are ~Xwater. In thermodynamic terms this implies that in the Gibbs equation there is no problem with respect to the location of the Gibbs dividing plane; all surface excesses r i may be identified with the real physical excess of i. The Gibbs equation

d7 = - E ri d~,i

[121

i

contains the above four components, because we set d#w ~ 0. Because of d#HNO3 =dmc~o3 - d/~KoiJthe number of independent variables can be reduced to three. Using [2] and identifying F~÷ a s (FKO H ~- FKNO3 ) and FMe2+as FM
a~+ du~o3

-- 1/20"Me2+ d#Me(YO3)2, [13]

where we have introduced the ionic components of charge aK+ ------FFK+ and ~rM~z+ -~ 2FFM:+. Two useful cross differentiations are

2 \ ~#HNO3/ #KNO3'#Me(NO3)2

_(

6ao

]

,

[14a]

,

[14b]

\ O#Me(NO3)a/~HNO3,UKN03

~ H N O 3 ] #-KNO3'g~Me(NO3)2

=(

b~o )

k ~]~'~KNO3//~HNO3'/XlMe(NO3)2

Here

~#HNO3/#KNO3 ,#Me(NO3)2

\~)["£HNO3Ict,c2

= (because c~, c2 >> C~No~) ~ \ ~ H + 10,c2

2 . 3 0 R T \ ~pH ]~,~ " Journal of Colloidand InterfaceScience, Vol. 99, No. i, May 1984

116

j. LYKLEMA

Hence,

\ ~Me(NO3)2/.UHNO~,~-KNO~

2.30R T \ 6pH 1¢~,~ =(

~r° )

.

[15b]

~K~KNO3/gHNO3,/tMe(NO3)2

(

The RHSs of [15a] and [15b] must be converted into (~o/~ In C2)pH,~ and (~a0/ In c~)v~,~, respectively. This can be done by using the extended chain rule several times. For instance, from [ 15a],

/ ~/'tMe(NO3)2//~HNO3,UKNO3

_#oo)

Because cl and c2 ~ CHNO3,the mean activity coefficient of the HNO3 depends only on cl and c2. The activity coefficients depend in a rather complex fashion on the composition and there are rules to take this into account but for the present purpose we shall ignore this dependence. Constancy of ~t~-~o3 then means c1(cl + 2c2) = const or (Cl + c2) dct = -Cl dc2 and constancy of/~M~(NO3)2implies 2c2 dc~ = -(Cl + 6c2) dc2. Using this, [18a] and [ 18b]

k ~/'tMe(NO3)2//~HNOB,c1

,(,c,)

\ {~CI ]/~FINO3,,°'Me(NO3)2 \ (~b/'Me(NO3)2I/-tHNO~,t~KNO~

[161

~C1 / =(~C21 r~e(NO3)2 #HNO~,#KNO3 \ ~/'tKNO31~I-INO3#tMe(NO3)2 -

R T \cl + 3c2]

[19]

which can be substituted in the RHS of [16] and the corresponding elaborated RHS of [ 15b], respectively. Using these expressions, [ 15a] and [ 15b] can be reworked into 4 . 6 0 R T \ ~pH ]~lX2 _

cl + 3c2 ~,6 In Cl/".NoS2

in which

5Cl ]/XldNO3,#Me(NO3) 2

+ 2(Cl + C2) [ _ _ ~ 0 ~ , cl + 3c2 ~,~iIn C2],HNo~,e~

~ ~C 1]/~HNo3,C2

,(,c2I

[17]

\ ~C2/UrlNO3,ct \ ~C1//zHNO3dzMe(NO3)2

and similarly for the RHS of [15b]. The three thermodynamic potentials are related. The way in which they depend on c~, c2 and cry+ = CnNO~depends on speciation. In the absence of hydrolysis o / ~ o ~ = t~NO3 + R T l n yZ~(Cl, c2) + R T l n Cl(Cl + 2c2),

[18a]

o //'Me(NO3)2 -~ /-~Me(NOD2 + R T In y3(cl, c2) + R T l n c2(Cl + 2c~)2,

[18b]

~tHNO3= t~ONo~+ R T In y2(c~, Cz) + R T I n cn(cl + 2cz).

[18c]

JournalofColloidandInterfaceScience,Vol.99, No. 1, May 1984

[20a]

2.30R T \ ~pH ]~1,~2 _

c1+6c2 ( 6~0 / 2(cl + 3c2) \6 In cll~r~o~,C2 +

c---2----1( ~ao ~ . Cl + 3C2 ~,6 In C2]ta.HN03,CI

[20b]

The differential quotients in the RHSs of [20] may be converted into the corresponding quotients at constant pH using In c11~.No3,C2

\~ In C11pH,c2

POINTS OF ZERO CHARGE

1 17

where because of [18c] constancy of ~HNO3 (bpH/6 In c2).O,Cl, which may be considered and c2 together implies dcH+ = - d c l , hence the Esin-Markov coefficients 131 and /32 for electrolyte mixtures.

_

6 In CI]..No~,C2

CI~+ "

CONCLUSION

Similarly, 6pH ~ _ -2c2 In c2]~,nNo~,C, CH+ so that eventually

,o0 6 In Cl],nNo~.C2 \6 In CJpn,c2

CH+ I,rpH]c,,~ 2'

6 I n C21~1_1NO3,CI

_(

a0/

\6 In C2]pH,c,

2c2( a0

t2lbl

CH+ \rpH]cl,~ ~"

For other types of electrolyte mixtures, of course expressions differing from [20] and [21 ] are found. The main point we want to make, however, is that the variation of aMe2+ and arc+ with pH (or, for that matter, with a0) is experimentally accessible if titration curves a0(pH, Cl, c2) are available. Using the reference points as obtained in the previous section, [20] can be integrated to give absolute values, after which aNo~follows as a0 -- a•+ -- aMe2+.Hence, a full mapping of the ionic components of charge emerges. Expressions [21 ] contain vertical cross sections and slopes of the titration curves. In crossover points, the former assumes the value zero, and in a cip the value remains zero over a range of the concentration of the independent variable. An alternative approach would have considered horizontal cross sections between the titration curves. In that case 6~rK/6a0and &rMff ha0 would have been expressed in terms of the differential quotients (6pH/6 In cl),o.,2 and

By essentially thermodynamic arguments it is shown that common intersection points, observed in the presence of specifically adsorbing ions, are no points of zero charge but equal compensation points, i.e., such points do have a simple and well-defined physical meaning. Experimentally they can be established only in the presence of a second, not specifically adsorbing electrolyte. A theory is given to obtain the ionic components of charge, i.e., the complete composition of the double layer, from titration curves ao(pH, ct, c2). For this evaluation no model parameters, such as binding constants, pK's, or doublelayer capacitances, are needed and the ambiguous notion "potential-determining ion" need not be introduced. ACKNOWLEDGMENT The author appreciates useful discussions with Dr. A. de Keizer. REFERENCES 1. Esin, O., and Markov, B., Acta Physicochim. URSS 10, 353 (1939). 2. Lyklema, J., J. Electroanal. Chem. 37, 53 (1972). 3. Pyman, M. A. F., Bowden, J. W., and Posner, A. M., Austr. J. Soil Res. 17, 191 (1979). 4. Breeuwsma, A., and Lyklema, J., Discuss. Faraday Soc. 52, 324 (1971). 5. Huang, C. P., in "Adsorption of Inorganies at SolidLiquid Interfaces" (M. A. Anderson and A. J. Rubin, Eds.), Chap. 5, especially Fig. 4. Ann Arbor Sei., 1981. 6. Ardizzone, S., Formaro, L., and Lyklema, J., J. Electroanal. Chem. 133, 147 (1982). 7. Bolt, G. H., and van Riemsdijk, W. H., in "Soil Chemistry" (G. H. Bolt, Ed.), Vol. B. PhysieoChemical Models, Chap. 13, p. 459. Elsevier, Amsterdam/Oxford, New York, 1982. 8. James, R. O., and Parks, G. H., in "Surface and Colloid Science" (E. Matijevi~, Ed.), Plenum, Vol. 12, p. 119. New York/London 1982. 9. Lyklema, J., Trans. Faraday Soc. 59, 189 (1963).

Journal of Colloidandlnterface Science, Vol. 99, No. 1, May 1984