Single Ion Activities in Multicomponent Systems J. V. Leyendekkers Division of Fisheries and Oceanography, CSZRO,Sydney 2230, Australia A method is presented for predicting single ion activity coefficients in multicomponent electrolyte systems, based on a simple ion interaction theory. As an illustration of the method, the activity coefficients of the fluoride ion in the aqueous systems NaCI-NaF and KCI-NaF are predicted and compared with the values derived from potentiometric measurements. The agreement is encouraging and suggests that the method will be useful for interpreting the response of ion-selective electrodes, especially when they are used as reference electrodes. The residual liquid junction potentials of the “Thalamid” and calomel reference electrodes in the above systems, over the ionic strength range 0.1-2.0, were calculated and found to be approximately linear in 1. COMMERCIAL AVAILABILITY of a large number of ion-selective electrodes has led to renewed interest in the concept of the single ion activity (1-8). The basic problems associated with the measurement of ionic activities using these types of electrodes have been clearly outlined by Bates and Alfenaar (5). Even though an electrode might respond precisely to changes in the activity of a specific ion, the values of the activities can only be derived from relative experimental values, on the basis of certain assumptions. Apart from experimental accuracy, the closeness of the derived values t o the true values will depend on the accuracy of prediction of the reference activity and on the accuracy of prediction of the changes in the various contributing potentials (e.g., the potential of the reference electrode). The first restraint essentially involves the problem of estimating single ion activity coefficients. Values for these coefficients can be obtained by splitting the experimental mean activity coefficient according to some convention. There are several conventions in use but the assumptions on which they are based are generally valid only for dilute solutions (less than ca. 0 . 1 ~ so ) that a reassessment is necessary for higher concentrations. This has already been made (5). An alternative method of estimating the coefficients (necessary in the absence of suitable experimental data) is to use an equation with a reasonable theoretical basis. For dilute solutions the Debye-Huckel (DH) equation has been widely used, the various parameters being derived empirically (9, 10) (the DH equation has also been used in a conventional manner to split the mean activity coefficient). Recently, Bates er a/. (7) have developed an equation based (1) R . M. Carrels, in “Glass Electrodes for Hydrogen and Other Cations,” G. Eisenman, Ed., Marcel Dekker, New York, N.Y., 1967, Chap. 13. (2) A. Shatkay, Biophys. J., 8,912 (1968). (3) A. Shatkay, Anal. Biochem., 29, 311 (1969). (4) A. Shatkay, and A. Lerman, ANAL.CHEM., 41, 514 (1969). (5) Roger G. Bates and Marinus Alfenaar, in “Ion Selective Electrodes,” R. A. Durst, Ed., National Bureau of Standards Special Publication No. 314, 1969, Chap. 6. (6) A. Shatkay, Elecrrochim. Acta., 15, 1759 (1970). (7) Roger G. Bates, Bert R. Staples, and R. A. Robinson, ANAL. CHEM.,42, 867 (1970). (8) G. W . Neff, ibid., p 1579. (9) Jacob Kielland, J . Amer. Chem. SOC.,59, 1675 (1937). (10) James N. Butler, “Ionic Equilibrium,” Addison-Wesley, Reading, Mass., 1964.
on the mole fraction statistics (MFS) hydration theory. This provides a means of estimating single ion activity coefficients at higher concentrations without the need for mean activity data, provided reliable estimates of the hydration parameters can be made. Although the hydration model used might not be the best, as indicated by recent criticism (11, 12), the derived hydration equation at least represents a more realistic basis for splitting the mean activity coefficient at higher concentrations than that provided by “dilute solution” conventions. The second of the restraints on the accuracy of “experimental” activities involves the difficulty of predicting changes in the response of the reference electrode with precision. For cells with liquid junction (LJ), the residual LJ potential, 4Ej (in pA units), is usually calculated from one of the classical equations, although these do not strictly apply to the “indefinite” type junctions common to most commercial reference electrodes. In addition, for precise evaluation of AEl some estimate must be made of the activities of the “junction” ions (13). If an ion-specific electrode is used as reference, the evaluation of the residual potential will again involve some estimate of ion activities. Obviously, these restraints will be reduced according t o the success with which we can predict ion activity coefficients accurately. The problem is a cyclic one so that an iterative type approach is required. In this paper, as an extension t o the work of Bates et al. (7), equations are derived for predicting single ion activity coefficients in mixed electrolyte solutions. These equations are used to predict the residual potentials of an ion-selective electrode used as reference and the predicted values are compared with those estimated from experiment. The 4Ei values of reference electrodes with liquid junctions are also calculated from experiment and plotted as a function of the ionic strength. THEORY
Multicomponent Systems. A consideration of two-electrolyte systems should be sufficient to allow extension to higher component systems. This is indicated by the free energy equation for a general mixture of electrolytes derived by Reilly and Wood on the basis of Friedman’s theory (14). If only pairwise interaction terms are considered, this equation reduces to a form analogous to that derived on the basis of Guggenheim’s treatment (15-17). (11) J. Rasaiah and H. L. Friedman, J . Chem. Phys., 72, 3352 (1968). (12) Kurt von Schwabe, J . Chim. Phys. Physicochim. Bioi., Special No., October, 205 (1969). (13) Arthur K. Covington, in “Ion Selective Electrodes,” R. A. Durst, Ed., National Bureau of Standards Special Publication No. 314, 1969, Chap. 4. (14) P. J. Reilly and R. H . Wood, J . Phys. Chem., 73, 4292 (1969). (15) E. A. Guggenheim, Tram. Faraday SOC.,62, 3446 (1966). (16) H. S . Harned and R. A. Robinson, The International Encyclopedia of Physical Chemistry and Chemical Physics, Topic 15, R. A. Robinson, Ed., Vol. 2 “Multicomponent Electrolyte Solutions,” Pergamon, London, 1968. (17) J. V . Leyendekkers, J. Phys. Chem., 75,946 (1971).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
1835
Table I. Comparison of Experimental and Calculated Harned Coeftlcientw
System A C HC1-LiC1
I, mol/kg
Calcd
ILXAC
Exptl
0.005
0.5
0.0037
0.007 0.007 LiC1-CsCl
1.0
0.0055
0.0030 0.0050
2.0
0.0110
0.0100
System A C KC1-LiCl
6H-Lib
ILXAC
I,
mol/kg
Calcd
Exptl
2.0 3.0
0.066 0.078 0.104
0.064 0.075 0.096
8K-Li
0.035 0.050 0.056
CsCl-LiCl
6Li-ca
-0.095 -0.108 -0.119 -0.127 NaC1-CsC1
2.0 3.0 4.0 5.0
0.229 0.354 0.492 0.632
4.0
Gce-Li
0.222 0.357 0.508 0.670
as in column
2.0 3.0 4.0 5.0
1
-0.030 -0.016
-0.032 -0.033 -0.032
-0.005
-0.035
-0.011 -0.028 -0.028 -0.028 -0.035
-0.017 -0.021 -0.016 -0.012 -0.015
-0.040
CsC1-NaCl
6Ns-Ca
-0.021
0.5 1 .o
-0,026 -0,036 -0.038 -0.041 HC1-KC1
2.0 3.0
5.0
0.031 0.054 0.102 0.144 0.232
6Ca-Ns
0.028 0.047 0.088 0.129 0.219
0.5 1.0 2.0
as in column 1
4.0 5.0
HC1-CsC1
6H-X
0.012 0.016 0.007 -0.016 NaC1-BaC12
0.5
0.030
1.0 2.0
0.056
4.0
0.276
0.114
6H-Ca
0.031 0.056 0.114 0.264
-0.04 -0.053
-0.032 NaC1-CaC12
1.o 2.0 3.0
0.106 0.206 0.267
0.100
0.198 0.219
6Na-Ca
SN~-B~
-0.028 -0.015 -0.014 CaC12-SrC12
1.0 3.0 5.0
0.011 0.030 0.083
-0.132 -0.036 -0.031
0.007 0.040 0.095
0.008
0.011
0.1 0.7 1.0 3.0 6.0
-0.001
0.007
-0.005 -0.005
-0.004 -0.004
-0.015
-0.006 0.008
-0.009
8Ca-8r
0.001 0.007 0.3 0.002 0.016 -0.001 3.0 0.016 0.037 -0.002 6.0 0.035 The references for experimental data are listed in (17) and (21).
Cationxation interaction kg/mol.
~
For simplicity only those systems will be considered here where interactions other than pairwise are negligible. Consider a mixture of electrolytes A (cation 1, anion 2) and C (cation 3, anion 4) in water, w , at constant ionic strength. The excess free energy due to the pairwise interactions is given by (16)
follows, since this term and the Debye-Huckel term are common to the single electrolyte and the mixture, that log
=
YI/YI'
- ~ C [ ( ~ V ~ / Z I Z ZVA)BIZ
( ~ v ~ / z ~ z ~ v-c (2~3/~3~4~c)6131 )BI~ (2)
where 2.3038 = B and 2.3036 = 8, the zero superscript identifies single electrolyte quantities. Similarly,
where B and 8 are parameters characteristic of the subscripted pair of ions and are taken as dependent on the total ionic strength, I , but independent of solution composition at a fixed I. B relates t o anion-cation pairs, 8 to cation-cation or anion-anion pairs. The interactions between ions of the same species (e.g. are neglected. n represents the number of moles, W the weight of the solvent. The partial derivative with respect to nl is
all)
lOgY2/yza = -IC[(~V~/ZIZ~VA)BIZ ( ~ v ~ / z ~ z ~ v-c () ~B vz ~~ / z ~ z ~ v (3) c)~x] Assuming that Harned's rule is applicable, the mean molal activity coefficient of electrolyte A is given by log Ya/Y.iO where
(YAC
=
--(YAC~C
(4)
is the Harned coefficient.
So that, from Equations 2 and 3 (YAC
= (~VIV~/Z~ZZ -V A ~ ) B I ~
( ~ v ~ v ~ / z ~ z ~ v-A (v~cv) ~B vI ~ / z ~ z ~ v-A v c ) B z ~ (2vlv3/z3z4vAvc)~13- (2V2V4/Z3Z4VAVC)624 ( 5 ) since (with the usual valence and charge symbols)
When IC= 0, Equation 1 reduces to the first term which represents the pairwise ion interaction relevant to cation 1 in the single electrolyte solution at an ionic strength I . It 1836
In terms of thermodynamics the Equations 2 and 3 are not operationally significant. However, we can assume that they represent a reasonable (albeit concenrionaf) splitting of aAC into its ionic components a1and (YZ given by 01
=
(YZ =
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
( ~ v ~ / z I z ~ vA )( B~ ~v ~ / z ~ z ~ v-c (2~3/~3zavc)613 )BI~ (6) (2~l/zizzva)B~ - (2~3/~3~4~c)B23 - ( 2 ~ 4 / ~ 3 ~ 4 ~ ~(7) )624
Table 11. Comparison of Experimental and Calculated Values of the Change in the Activity Coefficient of the Calcium Ion Due to the Presence of Sodium Ion in an Aqueous Chloride System” 1,
mol/kg
BCa-C1, kg/mol
B’NS-CI, kg/mol
0.750 0.750 0.740 0.739 0.412 0 ,340d
0.100 0.100 0. loo 0.136 0.084 0.099d
0.03 0.05 0,075 0.10 0.70
OICaCI*(NaCI), kg/mol -0.108 -0.080 -0.072 -0.047 -0.011
log (YCalY
log (YCl/Y OC1) Exptl Calcdc
Oca)
Exptl
Calcdb
0.030 0.035 0.036 0.034 0.035
0.019 0.027 0.038 0.036 0.097 0.046
-0.011 -0.012 -0.010 -0.010 -0,007
-0.005 -0.007
-0.011 -0.011 -0.037 -0.010
Calcium chloride at trace concentration.
* Equation 9. c
Equation 10. Value of interaction coefficient derived on the basis of DHG-Equation
Equations 5 , 6, and 7, together with data on the mean molal activity coefficients of single electrolytes, and the Harned coefficients for two-electrolyte common ion mixtures, form the basis for predicting the interactions in multicomponent systems. When ion association occurs, use can be made of the relationship proposed by Davies (18), log
YA’
=
log
YA
+
(1/vA)
- R))”‘] (8) ionic activity coefficient and (1 - R )
log
[ v i ” ’ v ~ Y 2 / ( V i R ) Y 1(V~ vi(1
where yA’i s the mean represents the fraction of the cations that form ion pairs. This equation should enable a better approximation of the B terms t o be made. As regards the range over which these equations can be expected t o hold, the following points are noted. Values of B can be derived from single-electrolyte data (19) while estimates of 6 depend as well on data for the common ion mixtures. It is not possible, as yet, t o ensure that the values assigned t o the various types of interactions are the correct ones. However, any large discrepancies should show up in comparisons of calculated and experimental values of the CYAC terms. Table I shows the results for a number of common ion mixtures, the maximum change in log YA for a given ionic strength being listed. The B. values given by Pitzer and Brewer (20) were used for the B terms. The 6 terms for the charge-symmetric systems were calculated from the relationship (aAC
ffCA)
-(4VlV3/Z1Z2VAVC)613
=
15a.
discrepancies in the cesium chloride and the charge asymmetric systems since it is likely that 6 ~ , - c , # and ~ C , - C , # and even if 6Na--h.a= Gca-ca, the split is not symmetrical because of the charge differences. Interactions other than pairwise might also be significant, especially at higher concentrations. Equations 6 and 7 will be more sensitive to errors in estimating the B parameters than Equation 5 since the corresponding weights are usually higher, but they are expected t o be valid over the same range. Experimental data are needed to confirm this and assess the accuracy. The use of these relationships is now illustrated. CaCkNaCI System. From Equations 5, 6, and 7
+
acn2+ = ~ B c I - N, ( * / ~ ) B c ~ - c I~ C ~ C ~ C I ? ( N ~ C I ) (9) W - =
(‘/~)Bc~-cI - BCI-N~
These coefficients were estimated using the B’ values listed by Pitzer and Brewer (20) and the experimental Harned coefficients (21). There is good agreement, (at Z 6 0.1) with the values derived experimentally (Table II), especially in view of the limitations of the experimental cells. B values derived from Equation 15a (see below) give a better fit at 0.7~1. NaF-NaCI System. Ion association occurs in this system so that Equation 8 should be used in determining the B coefficients from the single electrolyte data, viz, for this 1 :1 system, log YA’
=
log Y A
- log R
The B’ of Pitzer and Brewer is defined by
derived from Equation 5 and its corresponding form for
(4ViVz/Va2ZiZ2)B’i?= (log YA
ffCA.
=
- log ’yDH)/r
+
Similarly, for the 1 :1/2 :1 charge systems 613
(10)
where log ynH = -0.5107z1z21/1/(1 d b a t 25 “Cso that the modified B value, say B’ is given by
+ f f c ~ )- (‘/9)B23
-2(ff~c
In general, the results indicate that for these strong electrolyte mixtures Equation 5 gives a reasonable division of the free energy of mixing up to ionic strengths of 4 or 5. The equivalent of omitting 611 and 6 3 3 in Equation 5 is t o absorb them in the 813 term so that they receive incorrect weights when this term is split t o form Equations 6 and 7, unless they are equivalent. This probably accounts for the (18) C. W. Davies, “Ion Association,” Butterworths, London, 1962. (19) D. R. Rosseinsky and R. J. Hill, J. Electroanal. Chem.,30, App. 7 (1971). (20) G. N. Lewis and M. Randall, “Thermodynamics,” 2nd ed., revised by K . S. Pitzer and L. Brewer, McGraw, New York, N.Y., 1961.
(~V~V~/VA~Z~ZZ)B‘I~ = (log YA - log YDH)/z - (l/r) log R = (4vlV2/~A2z1z2)B‘12 - (I/Z) log R (11) From Equations 5 , 6, and 7 and taking account of ion association o(S%T=
=
B’Na--F - B’Na-ci B ’ N a - ~- (l/Z) log R
f f ~ -= h N a F ( N a C 1 )
- B’xs-ci
(12)
- B’s~-F-I- B’N~--CI -I- (l/Z) log R (13)
Equation 13 is used below in the analysis of the present experimental results. (21) J. V. Leyendekkers and M. Whitfield, J . Phys. Ckem., 75, 957 (1971).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
1837
NaF-KCI System. This is a more complicated system since there is no common ion. However, the necessary data can be derived from the common-ion systems NaF-NaC1 and NaC1-KCI. From Equation 5 C Y N ~ F ( N ~ C I )=
=
'/ZBNa-Cl
+
= O!N~F(N~CI)
-
'/Z~F-CI
- '/28Na-K B'N~-F- ' / ~ B N ~ - c-I '/zB'K-F 'd8Na-K -
( Y N ~ c ~ ( K c ~= )
oNaF(Kc1)
'/~B'N~-F - '/zBN~-cIl/~~C-CI
there remains a good possibility of remedying this situation, There might, for example, be some simple relationship between the two parameters (23). In the meantime, the calculation of B from mean activity coefficient data introduces a conventional quality to Equation 15 analogous to that inherent in the equations for the mixed electrolyte solutions. The appropriate values of the B terms are calculated from log YA
'/28F-C1
f '/~B'NB-F-
where pt a,,'
~N~CI(XCI)
=
'/zB'K-F- '/zBN~-cI -k '/ZBK-CI
From Equations 6 and 7 C Y N ~ +=
(YF-
+
=
hNaCl(KC1)
2aNaF(NaC1)
=
2aNaCl(KCl)
=
log 71
+
( ~ 1 ' u1a2'
+
+ Bizdl(a1'vl
f Uz'v2)]
(vA/zvl) log
YA
+
If complexing occurs, then BIZ'should replace Blz where
+ B'N*-F- (111) log R +
2aNaF(NaCI)
0.328a,' and
+ B N ~ - c-~ B'F-K B'K-ci
aF-
=
+ p & d E + ~VIYZBIZI/VA~ZIZ~
Bnd&AQi'az')/[VA
~BN*-cI
which become, in terms of B' CYNS+
-AzizzdI/(l
The values of these B terms will be identical with the corresponding B' values when pt = 1 . The single ion activity coefficient can be calculated directly from
B'N~-F- B N ~ - c-~ ~ N ~ - - K
f B'N~-F BK-c~= B'N~-F- B'F-K - 8 F 4 1 =
=
- 2B'xa-c1 (14)
+ B'NaC1 - B'F-K + (I/[)
+
~ B I ~ ' Z / V A= Z I~ZB~~ Z I / U A Z I Z Z
log R
(1/2UlYZ) log { v l Y l ! J Z Y ~ / ( v l R )[uz Y~
- Ul(1 - R)1'*}
In practice, the second term on the right hand side, multiplied by u1 or v2 can simply be added t o log yl or log y2 calculated from Equations 15 or 16. For a 1 :1 electrolyte, this term is -log R. Figure 1 gives a comparison of some of the ion activity coefficients estimated from Equation 15, with the corresponding values given by Bates et al. (7). The values of ai' were taken from Reference 9, mean molal activity coefficients were taken from Reference 24 or 25. No attempt is made to select the best theoretical approach, As mentioned above, the theory used here was chosen for convenience. However, a few points are noted. The log y1 = - A z l Z d G ( l pldr> ~ U Z B ~ Z I / U (15) ~ Z ~ Z ~ agreement between the two theories for sodium chloride is good up t o an ionic strength about 1 molal. These values where A = 0.5107 mole-I'* kgl/z and p: = 0.328 al' at can therefore be used with some confidence. This is fortunate 25 "C,al' being the effective diameter of the hydrated ion in in view of the dominance of sodium chloride in natural Angstroms, (log yz is obtained by symmetry). The first aqueous systems. The agreement is also good in this range term on the right covers interionic forces of the Debyefor the divalent cations when the more exact form of log Huckel type. These may be represented more accurately by ye' (Equation 15a) is used. It is interesting t o note that the Guggenheim convention gives values very close to those derived from the MFS theory for the divalent ions. Both sets of results illustrate the divergence of values of ycl- in different cation systems-Le., Y C I - ( K C I ) # Y C I - ( I I C I ) where (3/4 lir(c/dom))for 2 (15a) M is any univalent cation except K'. after Glueckauf (22). Where c, m represent molar and molal Specific Ion Electrodes as Reference Electrodes. With concentrations, b or d / d iis approximately equivalent to these equations in hand it is possible to make a n estimate of the response of a n ion specific electrode in mixed electrolyte P. The second term on the right of Equation 15 is taken to solutions. If such a n electrode is used as a reference elecrepresent the averaged ion-solvent plus ion-ion interactions. trode to another electrode specific for a n ion i , say, and the This is equivalent t o assuming that both these interactions estimate is close to the true response, then the measurements can be represented by a power series in I and that the first should yield a quantity which is close to the true value of ai, power is always dominant. An advantage in using Equation 15 is that values of ai', estimated independently of any mean (23) H. S . Harned and B. B. Owen, "The Physical Chemistry of activity data, are readily available for about 130 ions, including Electrolytic Solutions," 3rd ed., Reinhold, New York, N. Y., organic ions (9, IO). Although there is no way of estimating 1958, p 511. the B parameters in a similarly independent way at present, (24) R. Parsons, "Handbook of Electrochemical Constants," Butterworths, London, 1959. (25) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2nd ed, Butterworths, London, 1959. (22) E. Glueckauf, Proc. R o y . Soc. A . . 310, 449 (1969). Equation 14 is used below. Single Electrolyte Solutions. The above analysis illustrates a method for estimating the change in the activity coefficient of a n ion i with solution composition, at a fixed ionic strength, from data on the mean activity coefficient. In order to predict the value of the ionic activity coefficient it is necessary to make a n estimate of log y t o . The values given by Bates et al. (7) for seven unassociated chlorides could be used. However, t o be consistent with the above treatment, it is assumed that for a n electrolyte A dissociating into ions 1 and 2
+
+
1838
+
&di
