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Measurement and correlation of ion activity coefficients in aqueous solutions of mixed electrolyte with a common ion
HIIIDPU[
EOUglglllA
ELSEVIER
Fluid Phase Equilibria 121 (1996)253-265
Measurement and correlation of ion activity coefficients in aqueous solutions of mixed electrolyte with a common ion M . K . K h o s h k b a r c h i , J.H. V e r a * Department of Chemical Engineering, McGill University Montreal. Quebec tI3A 2A7. Canada
Received 15 May 1995; accepted 28 January 1996
Abstract Ion selective electrodes have been used to measure the activity coefficients at 298.2 K of individual ions in aqueous solutions of mixtures of NaC1 and KCI up to 4 molal total chloride ion concentration, and of NaC1 and NaBr up to 4 molal total sodium ion concentration. The experimental results show that the activity coefficient of an ion in the presence of different counterions depends on the nature and concentration of all counterions present in the solution. The results also show that the activity coefficients of the cations in an aqueous mixture are different from those of the anions. These effects can be considered as a corollary of the ion-ion and ion-solvent interactions. A modified form of the Pitzer's model, coupled with a mixing rule which distinguishes between the activity coefficients of the anion and the cation, has been used to correlate the experimental results. Keywords: Activity of ions; Electrolyte solutions; Mixed electrolytes; Ions
1. I n t r o d u c t i o n The knowledge o f the activity coefficients of individual ions is important for the design of equilibrium processes containing electrolyte solutions and in particular for processes involving ion-exchange. The need o f experimental values for the activity o f individual ions for the study of biological systems, has long been noticed in the literature (Eisenman, 1967). However, the lack of reliable experimental methods for the measurement of the activity coefficients of individual ions has been a major barrier. These systems are usually complex in nature and contain more than one electrolyte. Although the measurement of mean ionic activity coefficients of electrolytes in aqueous solutions of mixed salts has been the subject o f many studies (Zemaitis et al., 1986), limited data for the activity coefficients of the individual ionic species are available in the literature (Khoshkbarchi and Vera, 1996). Most of the complexity in measuring activity coefficients of the individual ionic
* Corresponding author. 0378-3812/96/$15.00 © 1996ElsevierScienceB.V. All rights reserved. PII S0378-3812(96)03026-9
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M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
species in electrolyte solutions comes from the interrelationship between the liquid junction potential and the ionic activity coefficients. In fact, as discussed by Taylor (1927), Gibbs (1928), Maclnnes et al. (1938) and Harned and Owen (1958), the measurement of the activity coefficient of ions cannot be done on the basis of exact thermodynamics only and some approximations need to be made. Maclnnes (1961) used a rigorous non-equilibrium thermodynamics method for the determination of the liquid junction potential. Bates (1965) calculated the liquid junction potential, and consequently the activity of the hydrogen ion, using a modified Henderson equation. Shatkay and Lerman (1969) reported measurements of the activities of sodium and chloride ions in a NaC1 solution using a sodium ion selective electrode and an AgC1/Ag electrode, each against a calomel reference electrode. Bates et al. (1970), used some of the data of Shatkay and Lerman and a modified version of the hydration model of Robinson (Robinson and Stokes, 1959) to model the activity coefficients of ions. Komar and Kaftanov (1974) suggested a method for estimation of the activity coefficient of chloride ions in potassium chloride solution from values of the mean ionic activity coefficient. Milazzo et al. (1975) proposed a procedure to obtain the activity coefficient of ions and showed that reliable data can be obtained, by using appropriate experimental techniques and instruments to reduce to an acceptable level the effect of the approximations required. Mokhov et al. (1977) reported values of the activity coefficients of sodium and chloride ions in sodium chloride aqueous solutions at different temperatures which give a poor reproduction of the mean ionic activity coefficient of sodium chloride. In a recent study (Khoshkbarchi and Vera, 1996) we have developed a new method to measure the activity coefficients of the ions in solutions of single electrolytes using ion selective electrodes. The method is based on the definition of a dimensionless electrode potential, and it uses the dimensionless electrode potential measured at low concentrations, to provide a reference level from which the ion activity coefficients can be calculated at higher ionic concentrations. The measurements were independently done for the activity coefficients of both anion and cation in aqueous solutions of single electrolytes. The mean ionic activity coefficients, calculated using the independently measured ionic activity coefficients were in good agreement with values reported in literature which were obtained using isopiestic method. This agreement is a definite proof of the reliability of the new method for measuring the activity coefficients of the individual ions in aqueous single salt solutions. In this study, we have extended the application of the new method to measure the activity coefficients of ions in aqueous solutions of mixed electrolytes with a common ion. We report here results of the measurement of the activity coefficients of chloride ion in aqueous solutions of NaCI and KC1 and of the activity coefficients of sodium ion in aqueous solutions of NaCI and NaBr. When ion selective electrodes are employed to measure the activity coefficients of ions in a mixture of two salts, different ions of the same charge usually interfere with the response of the electrodes. Thus, this first study of mixed salts, is restricted to the accurate measurement of the activity coefficients of the common ion of two salts, in aqueous solutions containing different counterions. The activity coefficients of the counterions can then be calculated using the mean ionic activity coefficients of both salts present in the solution. The mean ionic activity coefficients for these systems, have been obtained by the isopiestic method and are available in the literature (Robinson, 1961). An electrochemical cell, with ion selective electrodes (ISE) and an AgC1/Ag, KCI (4M, saturated with AgC1) single junction reference electrode (S J), has been used. In the experiments, the values of potential difference of the cells; (ISE)lsalt(1)+ salt(2)lKCl, AgC1/Ag (SJ) were measured. These values were then used to obtain the activity coefficients of the single ions in solution using the technique developed by Khoshkbarchi and Vera (1996).
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
255
In our previous work (Khoshkbarchi and Vera, 1996) we proposed a two-parameter correlation based on Pitzer's model (Pitzer, 1980) for the activity coefficients of the individual ions. In this work we have extended the application of this model to correlate the experimental data of the ions activity coefficients in aqueous solutions containing two electrolytes with a common ion.
2. M e t h o d o f r e d u c t i o n o f e x p e r i m e n t a l data
In this work, we have used a method proposed previously (Khoshkbarchi and Vera, 1996) to measure the activity coefficients of single ions in a solution containing two strong electrolytes with a common ion. The activity coefficient % of an ionic species i, at a molality m k in an aqueous solution, is related to the potential E k of the corresponding ion selective electrode, with respect to a reference electrode, by the general form of the Nernst equation, written as: E k = (E,s E - ERE ) + EjA + S ln(mk% )
(1)
where E;s E is a combination of the standard potential, the junction potential and the asymmetry potential of the ion selective electrode, ERE is the reference electrode electrochemical potential and Ej. k is the junction potential. For generality we use, the term S is in Eq. (1), to represent the slope of E k versus In (mk%), instead of using R T / z ; F which assumes an ideal Nernstian behavior. The slope S in Eq. (1), is a positive number for cations and a negative number for anions. Since it is not possible to measure experimentally the values of S and Ei°k the new method uses the dimensionless ion selective electrode potential, ~k, which eliminates the need to evaluate S and E;°k. The dimensionless ion selective electrode potential is defined by: (U~. - ej.~.) - ( E , - e g . , )
{k =
(2) (E 2 -
Ej,2) -
( E l - E j , I)
where the points 1 and 2 represent two arbitrary fixed salt concentrations. The value of the junction potential E j, k, at a fixed composition, can be estimated by the Henderson equation, as modified by Bates (1965). In this equation, the ionic mobilities of the two end solutions are considered to be equal to the limiting equivalent conductivities of the ions in the original Henderson equation. The modified Henderson equation, with limiting conductivities, is written as (Bates, 1965): RT
EJ'k--
X
[Y'.c+;k+-Y'.c h _ ] , - [ £ c + k + [~c+h+lZ_[+~c_h_[z_l]k-[~c+h+]Z+[-~c
Xln [Zc+x+l =_1 + :Cc_x_l z_ I],
~ c h_]l h_[Z_[];
(3)
where h is the limiting equivalent ionic conductance, and c is the concentration of ions in equivalent per litre. In this equation, the end solutions are represented by the subscripts k and / and the cation and anion by plus and minus signs, respectively. The values of the limiting equivalent ionic
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
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conductances used in this work are those given by Robinson and Stokes (1959). Combining Eqs. (1) and (2), gives: ~k = o~ + [3 ln(mk',h, )
(4)
where ~ and [3 are constants which depend only on the two reference points chosen to define the dimensionless potential. It should be noted that the definition of the dimensionless ion selective electrode potential relies on the assumptions that both S and (Els E - ERE), in Eq. (1), are constant over the concentration range studied. The validity of these assumptions has been discussed elsewhere (Khoshkbarchi and Vera, 1996). In order to evaluate the parameters ~ and [3 in Eq. (4), the range of ion concentrations is divided into two parts, a low ion concentrations (lower than 0.01 m), and a high ion concentration region. At low ion concentrations, the activity coefficient of an ion can be expressed by the Debye-Hiackel equation, and the dimensionless electrode potential at low concentrations, ~cc, is written as: ~LC= C~+ [3[ln(mk) + In('yD-n)]
(5)
For the Debye-Hiackel contribution we use the form: ln~/D-n=
- A z~v/-[ 1 + V~-
(6)
where the Debye-Hiickel constant A, at 298.2 K, is equal to 1.1762 kg 1/2 mole 1/2 and zi is charge of the ionic species i in the solution. In Eq. (6), the ionic strength ! is defined as:
I = ½~(mz2)i
(7)
At higher concentrations the activity coefficient is considered to be the product of a contribution of a Debye-Hiackel activity coefficient term and a residual term written as: ~ = o~ + [3[ln(m~) + ln(3,~ - n ) + ln(3'~)]
(8)
The constants ~ and [3 of the Eq. (5) can be evaluated by a linear fitting of a series of e.m.f. measurements at various ion concentrations lower than 0.01 m. By extending Eq. (8) to higher salt concentrations and subtracting the value of ~ c obtained from Eq. (5), the residual activity coefficient of the ionic species in the solution is calculated as: ln(-,/ff) =
[3
(9)
Once the residual activity coefficient of an ion i is known at a molality m~, its activity coefficient can be calculated from: In(',//) = ln('y,? - n ) + In("//R)
(10)
In principle the method can be directly applied for the measurement of the activity coefficients of ions in aqueous solutions of mixed electrolytes as described below.
M.K. Khoshkbarchi, J.H. Veru / Fluid Phase Equilibria 121 (1996) 253-265
257
3. Materials and methods Sodium chloride of 99.99% purity was obtained from Aldrich (Milwaukee, WI), potassium chloride and sodium bromide 99.9% purity were obtained from A & C American Chemicals LTD. (Montreal, Que.). All three salts were oven-dried for 72 h prior to use. During the drying period, the salts were taken out of the oven after 48 h, 56 h and 72 h and after cooling them in a vacuum desiccator, they were weighed. After 48 h, no change in their weights was observed. A sodium ion selective electrode glass body model 13-620-500, a chloride ion selective electrode polymer body model 13-620-518 and the single-junction reference electrode model 13-620-46 were obtained from Fisher Scientific (Montreal, Que.). The single junction electrode was preferred over a double junction electrode due to the need to minimize and be able to calculate the junction potentials. An Orion p H / I S E meter (MA, USA) model EA 920 with a resolution of ___0.1 mV, was used to monitor the e.m.f, measurements. During the experiments the solutions were stirred continuously and the temperature was kept constant at 298.2 + 0.1 K using a thermostatic bath. The electrodes were conditioned prior to the measurement according to the manufacturer procedure. The experiments were done by measuring the e.m.f, of the ion selective electrodes against a single junction reference electrode in a glass jacketed beaker containing 500 ml of solution. All the instruments were grounded prior to and during the experiments. All the solutions were prepared based on molality and the water was also weighed. The compositions of the initial solutions were accurate within + 0.01 wt.%. The readings of the potentiometer were made only when the drift was less than 0.1 mV. The experiments were started with the lowest concentration of mixed salts and the concentration was increased by addition of solid salts. The error on the molality of the chloride and of the potassium caused by the outgoing flow of the internal solution of the reference electrode was minimized by choosing a large volume of solution, 500 ml, and by performing the experiments at lower concentrations first. The rate of the outgoing flow of the solution from the reference electrode is less than 0.07 ml h -1 and, since the low concentration experiments were performed first, the maximum error in molality of the dilute solutions is estimated to be less than 1%. For more concentrated solutions the maximum error in molality is estimated to be less than 0.1%. All the experiments were replicated three times for the chloride ion and two times for the sodium ion and the data reported are the average of the replicas. Sample variances were obtained from the replicas for each point and a pooled standard deviation was calculated using these values. The 95% confidence interval in % was calculated as +0.018 for sodium ion in NaC1 + NaBr solution and -+0.013 for chloride ion in KC1 + NaC1 solution. In all experiments we used deionized water with the conductivity of less than 0.8 ixS c m - l , prepared by passing the distilled water through ion exchange columns type Easy pure RF, Compact Ultrapure Water System, Barnstead Thermoline.
4. Results and discussion Figs. 1 and 2 show the dimensionless potentials ~ of sodium and chloride ions, in equimolal aqueous solutions of NaC1 + NaBr and NaC1 + KC1, respectively, as a function of natural logarithm of the total molality of the salts present in solution. In all cases, the values of ~ were calculated using molality 0.5 and 1.0 as reference points for Eq. (2). The straight lines drawn over the experimental points at low concentrations in Figs. 1 and 2 correspond to Eq. (5) which is only valid in the dilute
258
M.K. Khoshkbarchi. J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265 8
4 2
o
-
.
e e
-4 "6-6
I
I
I
-4
-2
0
In (tuNa,) Fig. 1. Effect of concentration on the deviation from linear behavior of the dimensionless potential of sodium ion in an equimolal NaC1 + NaBr solution. • : Dimensionless potential of sodium ion; - - : Linear behavior at low concentrations, Eq. (5).
region of the electrolyte. These lines are tangent to the ~ vs. In (m) curves at low concentrations and deviate from the experimental values at higher concentrations. The deviation of the experimental points from the straight line given by Eq. (5) is caused by the term In (~/R) in Eq. (8). Thus, the difference between the linear behaviour of ~ cc, from Eq. (5), and the experimental points gives the value of In (~R), as indicated by Eq. (9). This procedure is applied, independently, for the data collected at each ratio of the two salts considered in the study. For the NaCI + KCI system we studied the ratios of salts 1/3, 1 / 2 and 2/3. For the NaC1 + NaBr system we studied the ratios 1/3 and 1/2. The measurements for the activity
6
4
2
-~CI"
0
-2
I
4-6
..4
I
I
-2
0
ln(mo. ) Fig. 2. Effect of concentration on the deviation from linear behavior of the dimensionless potential of chloride ion in an equimolal N a C I + K C 1 solution. • : Dimensionless potential of sodium ion; ~ : Linear behavior at low concentrations, Eq. (5).
M.K. Khoshkbarchi, J.H. Vera /Fluid Phase Equilibria 121 (1996) 253-265
259
1.1 1 0.9 0.8 0.7
~¢c r 0.6 0.5
0'40
la0,5
1
15.
2
2.5
Fig. 3. Activitycoefficientsof the chloride ions in solutions of NaCI+ KC1 at various chloride molalities. I1: X Na*= 1; +: XN~+= 1/3; " Xya+= 0.5; -~: XNa+= 2/3; "k: XN~+= 0;. - - : Correlation of experimental data using Eq. (12). coefficients of chloride ions were done in aqueous solutions of NaCI + KCI and those of sodium ions were done in aqueous solutions of NaC1 + NaBr. Fig. 3 shows the activity coefficients of the chloride ions at various total molalities of the salts present in the solution and various ratios of the two counterions of the salts. As shown in Fig. 3, the activity coefficients of the chloride ions in solutions with different ratios of XNa ÷= mNa+/(mK.+ tuNa+) show the same trend, as the molality of chloride ions increases and for a fixed chloride molality, the activity coefficients of the chloride ions increase as the ratio XNa+ increases. This indicates that the activity coefficients of anions, in a solution containing different cations, depend on the nature and concentrations of the cations as a result from the ion-ion and ion-solvent interactions. These results are in agreement with our previous study and show the shortcomings of models which consider the activity coefficients of the anions to be independent of the nature and number of cations present in the solution. Fig. 3 also shows that at high dilution, as suggested by Eqs. (5) and (9), the effect of counterions on the activity coefficients of the common ion of two salts disappears. Fig. 4 shows the activity coefficients of the potassium ions in the solution of NaC1 + KCI. These values have been calculated using the measured values of the activity coefficients of the chloride ion, and values of the mean ionic activity coefficients of each salt in the mixtures of NaC1 + KCi obtained from literature (Robinson, 1961) by means of the exact relation: l
3'_+= (W++~/~) "++v-
(11)
As it is shown in Fig. 4 the activity coefficients of potassium ions in solutions containing NaCl + KCl also depend on the nature and concentrations of the other ions in solution. From this figure it can also be seen that the activity coefficients of the potassium ions in solutions with different ratios XNa+ over the entire range of chloride ions concentrations show the same trend and, for a fixed chloride molality, they increase as the ratio Xna. increases. In principle, the values of the activity coefficients of Na + and K + ions in NaC1 + KCi solutions can also be determined experimentally by direct measurements and use of the same method to reduce the data. As a check, we have also performed
260
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
0.9 0.8
~K ÷ 0.7 0.6 0.5
.
_
"0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Fig. 4. Activity coefficients of the potassium ions in solutions of NaCI + KCI at various chloride molalities. • : X k+ = Xk+= 1/3; 0 : Xk+ =0.5; "I': Xk, = 2/3.
1; m:
these measurements and calculated the mean ionic activity coefficients o f the KCI in the aqueous mixture of KCI + NaC1 using Eq. (1 1). These results for the mean ionic activity coefficients are compared with the values obtained by an isopiestic method (Robinson, 1961) in Table 1. Notably the agreement improves as the relative concentration o f potassium increases due to the decrease in the interference of sodium ions. Although the difference between both values is not large, we prefer to trust the measurements of the mean ionic activity coefficient obtained by the isopiestic method and back calculate the activity coefficients of the potassium and sodium ions with Eq. (11). We have taken this decision considering that in the measurements of the e.m.f, values with ion selective electrodes there is interference between the sodium and potassium ions and thus our measurements for these ions are less reliable than the back calculated values. In Fig. 4 we have used as abscissa the square root of the molality o f chloride ions. This due to the fact that the use o f the molality o f potassium, will result in truncated curves at lower values o f X i. At each chloride molality, the molality of the cation i can be directly obtained as m K + = mc~ X~:+. Fig. 5 shows the activity coefficients of the sodium ions at different molalities in solutions with
Table 1 Comparison of the reported and measured values of the mean ionic activity coefficients of KC1 in the mixtures of KCI + NaCI m 0.5 1.0 2.0 3.0 4.0
XK+ = 0.25
XK+ = 0.5
XK+ = 0.75
Reported a
Measured
Reported a
Measured
Reporteda
Measured
0.656 0.614 0.589 0.593 0.609
0.678 0.629 0.604 0.622 0.633
0.654 0.610 0.578 0.582 0.601
0.675 0.628 0.599 0.596 0.592
0.651 0.607 0.578 0.577 0.589
0.670 0.611 0.580 0.583 0.618
a Robinson (1961).
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
261
1.8
1.6 1.4 ,/
1.2
"~Na ÷ l
0.8
0.(
,
/
~
I
I
0.5
l
1.5
2
2.5
F i g . 5. A c t i v i t y c o e f f i c i e n t s o f the s o d i u m i o n s in s o l u t i o n s o f NaC1 + N a B r at v a r i o u s s o d i u m m o l a l i t i e s . * : X o X c v = 0.5; ,A-: X c l - = 1; C o r r e l a t i o n o f e x p e r i m e n t a l d a t a u s i n g Eq. (12); . . . . Xc~
XcL- = 0; -
-
:
Xc~
= 0; • '
= 0.5; - • - :
= 1.
different ratios X o - = mc~ / ( m B r - + mc~-). As shown in this figure, the activity coefficients of the sodium ions in solutions with different ratios Xc~ over the entire range of sodium ions concentrations show the same trend and they all pass through a minimum. It is clear from Fig. 5 that due to the liquid junction potential, the results obtained with the sodium ion selective electrode have less certainty than those obtained for chloride with the chloride ion selective electrode. The curves of activity coefficients for the sodium ion pass close to each other and within the experimental error they sometimes cross each other. Since the curves for Xc~ = 0 and Xc~ = 1 are close to each other and they actually cross at around a chloride molality of 1.8, we can safely conclude that at intermediate values of X o - , a linear interpolation between the values at Xc~-= 0 and Xc~-= 1 gives a good approximation.
5. Modelling The experimental values of the activity coefficients of the individual ions in mixed electrolyte solutions are correlated by extending the application of the modified form of the Pitzer's model for ions in single electrolyte solutions proposed by Khoshkbarchi and Vera (1996). The equation used here has the form: 3 2
In ~/i-
1 + P/~-x + Bi 1 + P / ~ x + Ciln 1 + pI X
where A x is the usual Debye-Hi~ckel constant which in mole fraction basis, at 298.2 K, is equal to 8.766. As before we used here p -- 9 and I x is defined as: I
I, = 7Y',xizi
2
(13)
262
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
where x i is the mole fraction of ion i in the mixture. In Eq. (12) B i and C i are the two parameters for ion i whose values are calculated from the following mixing rules: n i = ~., X l Bit~l l
14)
Ci :
15)
and EXIC
° i,l
t
where the summation runs over all counterions present in the solution. For a molality m t of counterion 1, its fraction in a mixture of 1:1 electrolytes is defined as: ml
XI
=
16)
)--, mk k
In Eqs (14) and (15) the terms C°i.t and BiO.tare the values of the respective parameter for ion i in the presence of counterions only. In all cases studied in this work, we were able to reproduce the experimental results without introducing adjustable parameters. The relation for the mean ionic activity coefficient of the electrolyte obtained by combining Eqs. (11) and (12) takes the form: 3
ln~/±=
l+p/~
+B+_ l + p / ~ ' x + C ± l n
(
l+pl~
(17)
with E Ui Bi i
B±=
EU i
(18)
i
and
E Ui C i C _t_=
i E
ui
(19)
i
where the summation runs over all ions present in the solution. The solid lines in Figs. 3 and 5 show the results obtained with Eq. (17). The parameters for single electrolyte solutions are those reported previously (Khoshkbarchi and Vera, 1996). No new adjustable parameters have been used.
6. Conclusions A new method for the reduction of the experimental data proposed by Khoshkbarchi and Vera (1996) has been applied to measure the activity coefficients of individual ions in the solutions of mixed electrolytes with a common ion. The activity coefficients of the common ion of two salts was
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
263
measured and the activity coefficients of the two counter ions were calculated using the values of the mean ionic activity coefficient. In this way we have eliminated the effect of interference of ions with the same charge in the e.m.f, measurements. The results obtained indicate that the activity coefficients of the cation and the anion of an electrolyte in aqueous solution are different one from the other. The activity coefficient of an ion in a solution of mixture of two 1:1 electrolytes with a common ion depends on the nature and concentration of other ions present in the solution. We have used an extension of a modified form of the Pitzer's model for ions in single electrolyte solutions (Khoshkbarchi and Vera, 1996) we correlated the experimental values of the activity coefficients of the individual ions in mixed electrolyte solutions. This model considers the effect of nature and concentration of different ions present in the solution on the activity coefficients of each ion in the solution. The general for of model needs the binary parameters for aqueous solutions of each single salt and a interaction parameter for the aqueous solutions of mixed salts. For the systems studied here, the interaction parameter ktm was set to zero and the model used only the parameters obtained previously from aqueous solutions of single electrolytes. Thus, the model can be used for prediction of systems formed by mixed salts with a common ion.
7. List of symbols A Ax B C c
E F I Ix m R S T x Zi
Debye-Hi~ckel constant molality base Debye-Hi~ckel constant mole fraction base Adjustable parameter Adjustable parameter Concentration of ions in equivalent per litre Potential of electrochemical cell Faraday number Ionic strength on a molality basis Ionic strength on a mole fraction basis Molality Universal gas constant Slope of electrode potential Absolute temperature Mole fraction Charge of species i
7.1. Greek letters
h
Constant of the dimensionless electrode potential versus logarithm of the activity of an ion at low concentrations Slope of the dimensionless electrode potential versus logarithm of the activity of an ion at low concentrations Activity coefficient Limiting equivalent ionic conductance
264 oi q'i
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
Stoichiometric number of ions i Dimensionless electrode potential Transference number of species i
7.2. Subscripts i ISE J k 1 RE +
Species i Ion selective electrode Junction Level of concentration Level of concentration Reference electrode Cation Anion Mean ionic
7.3. Superscripts O
D-H LC R
Reference state Debye-Hi~ckel contribution Low concentration Contribution of residual activity coefficient
Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research of Council of Canada for financial support.
References R.G. Bates, Determination of pH Theory and Practice, 2nd edn., Wiley, New York, 1965. Bates, R.G., Staples, B.R. and Robinson, R.A., 1970. Anal. Chem., 42: 867. G. Eisenman, Glass Electrodes for Hydrogen and Other Cations, Marcel Dekker, New York, 1967. J.W. Gibbs, Collected Works, Vol. 1, Longmans Green, New York, 1928. H.S. Harned and B.B. Owen, The Physical Chemistry of Electrolyte Solutions, 3rd edn., Reinhold, New York, 1958. Komar, N.P. and Kaftanov, A.Z., 1974. Russ. J. Phys. Chem., 48: 246. Khoshkbarchi, M.K. and Vera, J.H., 1996. AICHE J., 42: 249. Maclnnes, D.A., Belcher, D. and Shedlovsky, T., 1938. J. Am. Chem. Soc., 60: 1094. D.A. Maclnnes, The Principles of Electrochemistry, Dover, New York, 1961. Milazzo, G., Bonciocat, N. and Borda, M., 1975. Electrochim. Acta, 21: 349. Mokhov, V.M., Bagdasarova, I.P., Kekeliya, A.G. and Lavrelashvilig, L.V., 1977. Russ. J. Phys. Chem., 51: 1406. Pitzer, K., 1980. J. Am. Chem. Soc., 102: 2902.
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R.A. Robinson and R.H. Stokes, Electrolyte Solutions, 2nd edn., Butterworths, Boston, MA, 1959. Robinson, R.A., 1961, J. Phys. Chem., 65: 662. Shatkay, A. and Lerman, A., 1969. Anal. Chem., 41: 514. Taylor, P.B., 1927. J. Phys. Chem., 31: 1478. J.F. Zemaitis, D.M. Clark, M. Rafal and N.C. Scrivner, Handbook of Aqueous Electrolyte Thermodynamics, A1CHE, New York, 1986.
EOUglglllA
ELSEVIER
Fluid Phase Equilibria 121 (1996)253-265
Measurement and correlation of ion activity coefficients in aqueous solutions of mixed electrolyte with a common ion M . K . K h o s h k b a r c h i , J.H. V e r a * Department of Chemical Engineering, McGill University Montreal. Quebec tI3A 2A7. Canada
Received 15 May 1995; accepted 28 January 1996
Abstract Ion selective electrodes have been used to measure the activity coefficients at 298.2 K of individual ions in aqueous solutions of mixtures of NaC1 and KCI up to 4 molal total chloride ion concentration, and of NaC1 and NaBr up to 4 molal total sodium ion concentration. The experimental results show that the activity coefficient of an ion in the presence of different counterions depends on the nature and concentration of all counterions present in the solution. The results also show that the activity coefficients of the cations in an aqueous mixture are different from those of the anions. These effects can be considered as a corollary of the ion-ion and ion-solvent interactions. A modified form of the Pitzer's model, coupled with a mixing rule which distinguishes between the activity coefficients of the anion and the cation, has been used to correlate the experimental results. Keywords: Activity of ions; Electrolyte solutions; Mixed electrolytes; Ions
1. I n t r o d u c t i o n The knowledge o f the activity coefficients of individual ions is important for the design of equilibrium processes containing electrolyte solutions and in particular for processes involving ion-exchange. The need o f experimental values for the activity o f individual ions for the study of biological systems, has long been noticed in the literature (Eisenman, 1967). However, the lack of reliable experimental methods for the measurement of the activity coefficients of individual ions has been a major barrier. These systems are usually complex in nature and contain more than one electrolyte. Although the measurement of mean ionic activity coefficients of electrolytes in aqueous solutions of mixed salts has been the subject o f many studies (Zemaitis et al., 1986), limited data for the activity coefficients of the individual ionic species are available in the literature (Khoshkbarchi and Vera, 1996). Most of the complexity in measuring activity coefficients of the individual ionic
* Corresponding author. 0378-3812/96/$15.00 © 1996ElsevierScienceB.V. All rights reserved. PII S0378-3812(96)03026-9
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M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
species in electrolyte solutions comes from the interrelationship between the liquid junction potential and the ionic activity coefficients. In fact, as discussed by Taylor (1927), Gibbs (1928), Maclnnes et al. (1938) and Harned and Owen (1958), the measurement of the activity coefficient of ions cannot be done on the basis of exact thermodynamics only and some approximations need to be made. Maclnnes (1961) used a rigorous non-equilibrium thermodynamics method for the determination of the liquid junction potential. Bates (1965) calculated the liquid junction potential, and consequently the activity of the hydrogen ion, using a modified Henderson equation. Shatkay and Lerman (1969) reported measurements of the activities of sodium and chloride ions in a NaC1 solution using a sodium ion selective electrode and an AgC1/Ag electrode, each against a calomel reference electrode. Bates et al. (1970), used some of the data of Shatkay and Lerman and a modified version of the hydration model of Robinson (Robinson and Stokes, 1959) to model the activity coefficients of ions. Komar and Kaftanov (1974) suggested a method for estimation of the activity coefficient of chloride ions in potassium chloride solution from values of the mean ionic activity coefficient. Milazzo et al. (1975) proposed a procedure to obtain the activity coefficient of ions and showed that reliable data can be obtained, by using appropriate experimental techniques and instruments to reduce to an acceptable level the effect of the approximations required. Mokhov et al. (1977) reported values of the activity coefficients of sodium and chloride ions in sodium chloride aqueous solutions at different temperatures which give a poor reproduction of the mean ionic activity coefficient of sodium chloride. In a recent study (Khoshkbarchi and Vera, 1996) we have developed a new method to measure the activity coefficients of the ions in solutions of single electrolytes using ion selective electrodes. The method is based on the definition of a dimensionless electrode potential, and it uses the dimensionless electrode potential measured at low concentrations, to provide a reference level from which the ion activity coefficients can be calculated at higher ionic concentrations. The measurements were independently done for the activity coefficients of both anion and cation in aqueous solutions of single electrolytes. The mean ionic activity coefficients, calculated using the independently measured ionic activity coefficients were in good agreement with values reported in literature which were obtained using isopiestic method. This agreement is a definite proof of the reliability of the new method for measuring the activity coefficients of the individual ions in aqueous single salt solutions. In this study, we have extended the application of the new method to measure the activity coefficients of ions in aqueous solutions of mixed electrolytes with a common ion. We report here results of the measurement of the activity coefficients of chloride ion in aqueous solutions of NaCI and KC1 and of the activity coefficients of sodium ion in aqueous solutions of NaCI and NaBr. When ion selective electrodes are employed to measure the activity coefficients of ions in a mixture of two salts, different ions of the same charge usually interfere with the response of the electrodes. Thus, this first study of mixed salts, is restricted to the accurate measurement of the activity coefficients of the common ion of two salts, in aqueous solutions containing different counterions. The activity coefficients of the counterions can then be calculated using the mean ionic activity coefficients of both salts present in the solution. The mean ionic activity coefficients for these systems, have been obtained by the isopiestic method and are available in the literature (Robinson, 1961). An electrochemical cell, with ion selective electrodes (ISE) and an AgC1/Ag, KCI (4M, saturated with AgC1) single junction reference electrode (S J), has been used. In the experiments, the values of potential difference of the cells; (ISE)lsalt(1)+ salt(2)lKCl, AgC1/Ag (SJ) were measured. These values were then used to obtain the activity coefficients of the single ions in solution using the technique developed by Khoshkbarchi and Vera (1996).
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255
In our previous work (Khoshkbarchi and Vera, 1996) we proposed a two-parameter correlation based on Pitzer's model (Pitzer, 1980) for the activity coefficients of the individual ions. In this work we have extended the application of this model to correlate the experimental data of the ions activity coefficients in aqueous solutions containing two electrolytes with a common ion.
2. M e t h o d o f r e d u c t i o n o f e x p e r i m e n t a l data
In this work, we have used a method proposed previously (Khoshkbarchi and Vera, 1996) to measure the activity coefficients of single ions in a solution containing two strong electrolytes with a common ion. The activity coefficient % of an ionic species i, at a molality m k in an aqueous solution, is related to the potential E k of the corresponding ion selective electrode, with respect to a reference electrode, by the general form of the Nernst equation, written as: E k = (E,s E - ERE ) + EjA + S ln(mk% )
(1)
where E;s E is a combination of the standard potential, the junction potential and the asymmetry potential of the ion selective electrode, ERE is the reference electrode electrochemical potential and Ej. k is the junction potential. For generality we use, the term S is in Eq. (1), to represent the slope of E k versus In (mk%), instead of using R T / z ; F which assumes an ideal Nernstian behavior. The slope S in Eq. (1), is a positive number for cations and a negative number for anions. Since it is not possible to measure experimentally the values of S and Ei°k the new method uses the dimensionless ion selective electrode potential, ~k, which eliminates the need to evaluate S and E;°k. The dimensionless ion selective electrode potential is defined by: (U~. - ej.~.) - ( E , - e g . , )
{k =
(2) (E 2 -
Ej,2) -
( E l - E j , I)
where the points 1 and 2 represent two arbitrary fixed salt concentrations. The value of the junction potential E j, k, at a fixed composition, can be estimated by the Henderson equation, as modified by Bates (1965). In this equation, the ionic mobilities of the two end solutions are considered to be equal to the limiting equivalent conductivities of the ions in the original Henderson equation. The modified Henderson equation, with limiting conductivities, is written as (Bates, 1965): RT
EJ'k--
X
[Y'.c+;k+-Y'.c h _ ] , - [ £ c + k + [~c+h+lZ_[+~c_h_[z_l]k-[~c+h+]Z+[-~c
Xln [Zc+x+l =_1 + :Cc_x_l z_ I],
~ c h_]l h_[Z_[];
(3)
where h is the limiting equivalent ionic conductance, and c is the concentration of ions in equivalent per litre. In this equation, the end solutions are represented by the subscripts k and / and the cation and anion by plus and minus signs, respectively. The values of the limiting equivalent ionic
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
256
conductances used in this work are those given by Robinson and Stokes (1959). Combining Eqs. (1) and (2), gives: ~k = o~ + [3 ln(mk',h, )
(4)
where ~ and [3 are constants which depend only on the two reference points chosen to define the dimensionless potential. It should be noted that the definition of the dimensionless ion selective electrode potential relies on the assumptions that both S and (Els E - ERE), in Eq. (1), are constant over the concentration range studied. The validity of these assumptions has been discussed elsewhere (Khoshkbarchi and Vera, 1996). In order to evaluate the parameters ~ and [3 in Eq. (4), the range of ion concentrations is divided into two parts, a low ion concentrations (lower than 0.01 m), and a high ion concentration region. At low ion concentrations, the activity coefficient of an ion can be expressed by the Debye-Hiackel equation, and the dimensionless electrode potential at low concentrations, ~cc, is written as: ~LC= C~+ [3[ln(mk) + In('yD-n)]
(5)
For the Debye-Hiackel contribution we use the form: ln~/D-n=
- A z~v/-[ 1 + V~-
(6)
where the Debye-Hiickel constant A, at 298.2 K, is equal to 1.1762 kg 1/2 mole 1/2 and zi is charge of the ionic species i in the solution. In Eq. (6), the ionic strength ! is defined as:
I = ½~(mz2)i
(7)
At higher concentrations the activity coefficient is considered to be the product of a contribution of a Debye-Hiackel activity coefficient term and a residual term written as: ~ = o~ + [3[ln(m~) + ln(3,~ - n ) + ln(3'~)]
(8)
The constants ~ and [3 of the Eq. (5) can be evaluated by a linear fitting of a series of e.m.f. measurements at various ion concentrations lower than 0.01 m. By extending Eq. (8) to higher salt concentrations and subtracting the value of ~ c obtained from Eq. (5), the residual activity coefficient of the ionic species in the solution is calculated as: ln(-,/ff) =
[3
(9)
Once the residual activity coefficient of an ion i is known at a molality m~, its activity coefficient can be calculated from: In(',//) = ln('y,? - n ) + In("//R)
(10)
In principle the method can be directly applied for the measurement of the activity coefficients of ions in aqueous solutions of mixed electrolytes as described below.
M.K. Khoshkbarchi, J.H. Veru / Fluid Phase Equilibria 121 (1996) 253-265
257
3. Materials and methods Sodium chloride of 99.99% purity was obtained from Aldrich (Milwaukee, WI), potassium chloride and sodium bromide 99.9% purity were obtained from A & C American Chemicals LTD. (Montreal, Que.). All three salts were oven-dried for 72 h prior to use. During the drying period, the salts were taken out of the oven after 48 h, 56 h and 72 h and after cooling them in a vacuum desiccator, they were weighed. After 48 h, no change in their weights was observed. A sodium ion selective electrode glass body model 13-620-500, a chloride ion selective electrode polymer body model 13-620-518 and the single-junction reference electrode model 13-620-46 were obtained from Fisher Scientific (Montreal, Que.). The single junction electrode was preferred over a double junction electrode due to the need to minimize and be able to calculate the junction potentials. An Orion p H / I S E meter (MA, USA) model EA 920 with a resolution of ___0.1 mV, was used to monitor the e.m.f, measurements. During the experiments the solutions were stirred continuously and the temperature was kept constant at 298.2 + 0.1 K using a thermostatic bath. The electrodes were conditioned prior to the measurement according to the manufacturer procedure. The experiments were done by measuring the e.m.f, of the ion selective electrodes against a single junction reference electrode in a glass jacketed beaker containing 500 ml of solution. All the instruments were grounded prior to and during the experiments. All the solutions were prepared based on molality and the water was also weighed. The compositions of the initial solutions were accurate within + 0.01 wt.%. The readings of the potentiometer were made only when the drift was less than 0.1 mV. The experiments were started with the lowest concentration of mixed salts and the concentration was increased by addition of solid salts. The error on the molality of the chloride and of the potassium caused by the outgoing flow of the internal solution of the reference electrode was minimized by choosing a large volume of solution, 500 ml, and by performing the experiments at lower concentrations first. The rate of the outgoing flow of the solution from the reference electrode is less than 0.07 ml h -1 and, since the low concentration experiments were performed first, the maximum error in molality of the dilute solutions is estimated to be less than 1%. For more concentrated solutions the maximum error in molality is estimated to be less than 0.1%. All the experiments were replicated three times for the chloride ion and two times for the sodium ion and the data reported are the average of the replicas. Sample variances were obtained from the replicas for each point and a pooled standard deviation was calculated using these values. The 95% confidence interval in % was calculated as +0.018 for sodium ion in NaC1 + NaBr solution and -+0.013 for chloride ion in KC1 + NaC1 solution. In all experiments we used deionized water with the conductivity of less than 0.8 ixS c m - l , prepared by passing the distilled water through ion exchange columns type Easy pure RF, Compact Ultrapure Water System, Barnstead Thermoline.
4. Results and discussion Figs. 1 and 2 show the dimensionless potentials ~ of sodium and chloride ions, in equimolal aqueous solutions of NaC1 + NaBr and NaC1 + KC1, respectively, as a function of natural logarithm of the total molality of the salts present in solution. In all cases, the values of ~ were calculated using molality 0.5 and 1.0 as reference points for Eq. (2). The straight lines drawn over the experimental points at low concentrations in Figs. 1 and 2 correspond to Eq. (5) which is only valid in the dilute
258
M.K. Khoshkbarchi. J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265 8
4 2
o
-
.
e e
-4 "6-6
I
I
I
-4
-2
0
In (tuNa,) Fig. 1. Effect of concentration on the deviation from linear behavior of the dimensionless potential of sodium ion in an equimolal NaC1 + NaBr solution. • : Dimensionless potential of sodium ion; - - : Linear behavior at low concentrations, Eq. (5).
region of the electrolyte. These lines are tangent to the ~ vs. In (m) curves at low concentrations and deviate from the experimental values at higher concentrations. The deviation of the experimental points from the straight line given by Eq. (5) is caused by the term In (~/R) in Eq. (8). Thus, the difference between the linear behaviour of ~ cc, from Eq. (5), and the experimental points gives the value of In (~R), as indicated by Eq. (9). This procedure is applied, independently, for the data collected at each ratio of the two salts considered in the study. For the NaCI + KCI system we studied the ratios of salts 1/3, 1 / 2 and 2/3. For the NaC1 + NaBr system we studied the ratios 1/3 and 1/2. The measurements for the activity
6
4
2
-~CI"
0
-2
I
4-6
..4
I
I
-2
0
ln(mo. ) Fig. 2. Effect of concentration on the deviation from linear behavior of the dimensionless potential of chloride ion in an equimolal N a C I + K C 1 solution. • : Dimensionless potential of sodium ion; ~ : Linear behavior at low concentrations, Eq. (5).
M.K. Khoshkbarchi, J.H. Vera /Fluid Phase Equilibria 121 (1996) 253-265
259
1.1 1 0.9 0.8 0.7
~¢c r 0.6 0.5
0'40
la0,5
1
15.
2
2.5
Fig. 3. Activitycoefficientsof the chloride ions in solutions of NaCI+ KC1 at various chloride molalities. I1: X Na*= 1; +: XN~+= 1/3; " Xya+= 0.5; -~: XNa+= 2/3; "k: XN~+= 0;. - - : Correlation of experimental data using Eq. (12). coefficients of chloride ions were done in aqueous solutions of NaCI + KCI and those of sodium ions were done in aqueous solutions of NaC1 + NaBr. Fig. 3 shows the activity coefficients of the chloride ions at various total molalities of the salts present in the solution and various ratios of the two counterions of the salts. As shown in Fig. 3, the activity coefficients of the chloride ions in solutions with different ratios of XNa ÷= mNa+/(mK.+ tuNa+) show the same trend, as the molality of chloride ions increases and for a fixed chloride molality, the activity coefficients of the chloride ions increase as the ratio XNa+ increases. This indicates that the activity coefficients of anions, in a solution containing different cations, depend on the nature and concentrations of the cations as a result from the ion-ion and ion-solvent interactions. These results are in agreement with our previous study and show the shortcomings of models which consider the activity coefficients of the anions to be independent of the nature and number of cations present in the solution. Fig. 3 also shows that at high dilution, as suggested by Eqs. (5) and (9), the effect of counterions on the activity coefficients of the common ion of two salts disappears. Fig. 4 shows the activity coefficients of the potassium ions in the solution of NaC1 + KCI. These values have been calculated using the measured values of the activity coefficients of the chloride ion, and values of the mean ionic activity coefficients of each salt in the mixtures of NaC1 + KCi obtained from literature (Robinson, 1961) by means of the exact relation: l
3'_+= (W++~/~) "++v-
(11)
As it is shown in Fig. 4 the activity coefficients of potassium ions in solutions containing NaCl + KCl also depend on the nature and concentrations of the other ions in solution. From this figure it can also be seen that the activity coefficients of the potassium ions in solutions with different ratios XNa+ over the entire range of chloride ions concentrations show the same trend and, for a fixed chloride molality, they increase as the ratio Xna. increases. In principle, the values of the activity coefficients of Na + and K + ions in NaC1 + KCi solutions can also be determined experimentally by direct measurements and use of the same method to reduce the data. As a check, we have also performed
260
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
0.9 0.8
~K ÷ 0.7 0.6 0.5
.
_
"0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Fig. 4. Activity coefficients of the potassium ions in solutions of NaCI + KCI at various chloride molalities. • : X k+ = Xk+= 1/3; 0 : Xk+ =0.5; "I': Xk, = 2/3.
1; m:
these measurements and calculated the mean ionic activity coefficients o f the KCI in the aqueous mixture of KCI + NaC1 using Eq. (1 1). These results for the mean ionic activity coefficients are compared with the values obtained by an isopiestic method (Robinson, 1961) in Table 1. Notably the agreement improves as the relative concentration o f potassium increases due to the decrease in the interference of sodium ions. Although the difference between both values is not large, we prefer to trust the measurements of the mean ionic activity coefficient obtained by the isopiestic method and back calculate the activity coefficients of the potassium and sodium ions with Eq. (11). We have taken this decision considering that in the measurements of the e.m.f, values with ion selective electrodes there is interference between the sodium and potassium ions and thus our measurements for these ions are less reliable than the back calculated values. In Fig. 4 we have used as abscissa the square root of the molality o f chloride ions. This due to the fact that the use o f the molality o f potassium, will result in truncated curves at lower values o f X i. At each chloride molality, the molality of the cation i can be directly obtained as m K + = mc~ X~:+. Fig. 5 shows the activity coefficients of the sodium ions at different molalities in solutions with
Table 1 Comparison of the reported and measured values of the mean ionic activity coefficients of KC1 in the mixtures of KCI + NaCI m 0.5 1.0 2.0 3.0 4.0
XK+ = 0.25
XK+ = 0.5
XK+ = 0.75
Reported a
Measured
Reported a
Measured
Reporteda
Measured
0.656 0.614 0.589 0.593 0.609
0.678 0.629 0.604 0.622 0.633
0.654 0.610 0.578 0.582 0.601
0.675 0.628 0.599 0.596 0.592
0.651 0.607 0.578 0.577 0.589
0.670 0.611 0.580 0.583 0.618
a Robinson (1961).
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
261
1.8
1.6 1.4 ,/
1.2
"~Na ÷ l
0.8
0.(
,
/
~
I
I
0.5
l
1.5
2
2.5
F i g . 5. A c t i v i t y c o e f f i c i e n t s o f the s o d i u m i o n s in s o l u t i o n s o f NaC1 + N a B r at v a r i o u s s o d i u m m o l a l i t i e s . * : X o X c v = 0.5; ,A-: X c l - = 1; C o r r e l a t i o n o f e x p e r i m e n t a l d a t a u s i n g Eq. (12); . . . . Xc~
XcL- = 0; -
-
:
Xc~
= 0; • '
= 0.5; - • - :
= 1.
different ratios X o - = mc~ / ( m B r - + mc~-). As shown in this figure, the activity coefficients of the sodium ions in solutions with different ratios Xc~ over the entire range of sodium ions concentrations show the same trend and they all pass through a minimum. It is clear from Fig. 5 that due to the liquid junction potential, the results obtained with the sodium ion selective electrode have less certainty than those obtained for chloride with the chloride ion selective electrode. The curves of activity coefficients for the sodium ion pass close to each other and within the experimental error they sometimes cross each other. Since the curves for Xc~ = 0 and Xc~ = 1 are close to each other and they actually cross at around a chloride molality of 1.8, we can safely conclude that at intermediate values of X o - , a linear interpolation between the values at Xc~-= 0 and Xc~-= 1 gives a good approximation.
5. Modelling The experimental values of the activity coefficients of the individual ions in mixed electrolyte solutions are correlated by extending the application of the modified form of the Pitzer's model for ions in single electrolyte solutions proposed by Khoshkbarchi and Vera (1996). The equation used here has the form: 3 2
In ~/i-
1 + P/~-x + Bi 1 + P / ~ x + Ciln 1 + pI X
where A x is the usual Debye-Hi~ckel constant which in mole fraction basis, at 298.2 K, is equal to 8.766. As before we used here p -- 9 and I x is defined as: I
I, = 7Y',xizi
2
(13)
262
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
where x i is the mole fraction of ion i in the mixture. In Eq. (12) B i and C i are the two parameters for ion i whose values are calculated from the following mixing rules: n i = ~., X l Bit~l l
14)
Ci :
15)
and EXIC
° i,l
t
where the summation runs over all counterions present in the solution. For a molality m t of counterion 1, its fraction in a mixture of 1:1 electrolytes is defined as: ml
XI
=
16)
)--, mk k
In Eqs (14) and (15) the terms C°i.t and BiO.tare the values of the respective parameter for ion i in the presence of counterions only. In all cases studied in this work, we were able to reproduce the experimental results without introducing adjustable parameters. The relation for the mean ionic activity coefficient of the electrolyte obtained by combining Eqs. (11) and (12) takes the form: 3
ln~/±=
l+p/~
+B+_ l + p / ~ ' x + C ± l n
(
l+pl~
(17)
with E Ui Bi i
B±=
EU i
(18)
i
and
E Ui C i C _t_=
i E
ui
(19)
i
where the summation runs over all ions present in the solution. The solid lines in Figs. 3 and 5 show the results obtained with Eq. (17). The parameters for single electrolyte solutions are those reported previously (Khoshkbarchi and Vera, 1996). No new adjustable parameters have been used.
6. Conclusions A new method for the reduction of the experimental data proposed by Khoshkbarchi and Vera (1996) has been applied to measure the activity coefficients of individual ions in the solutions of mixed electrolytes with a common ion. The activity coefficients of the common ion of two salts was
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263
measured and the activity coefficients of the two counter ions were calculated using the values of the mean ionic activity coefficient. In this way we have eliminated the effect of interference of ions with the same charge in the e.m.f, measurements. The results obtained indicate that the activity coefficients of the cation and the anion of an electrolyte in aqueous solution are different one from the other. The activity coefficient of an ion in a solution of mixture of two 1:1 electrolytes with a common ion depends on the nature and concentration of other ions present in the solution. We have used an extension of a modified form of the Pitzer's model for ions in single electrolyte solutions (Khoshkbarchi and Vera, 1996) we correlated the experimental values of the activity coefficients of the individual ions in mixed electrolyte solutions. This model considers the effect of nature and concentration of different ions present in the solution on the activity coefficients of each ion in the solution. The general for of model needs the binary parameters for aqueous solutions of each single salt and a interaction parameter for the aqueous solutions of mixed salts. For the systems studied here, the interaction parameter ktm was set to zero and the model used only the parameters obtained previously from aqueous solutions of single electrolytes. Thus, the model can be used for prediction of systems formed by mixed salts with a common ion.
7. List of symbols A Ax B C c
E F I Ix m R S T x Zi
Debye-Hi~ckel constant molality base Debye-Hi~ckel constant mole fraction base Adjustable parameter Adjustable parameter Concentration of ions in equivalent per litre Potential of electrochemical cell Faraday number Ionic strength on a molality basis Ionic strength on a mole fraction basis Molality Universal gas constant Slope of electrode potential Absolute temperature Mole fraction Charge of species i
7.1. Greek letters
h
Constant of the dimensionless electrode potential versus logarithm of the activity of an ion at low concentrations Slope of the dimensionless electrode potential versus logarithm of the activity of an ion at low concentrations Activity coefficient Limiting equivalent ionic conductance
264 oi q'i
M.K. Khoshkbarchi, J.H. Vera / Fluid Phase Equilibria 121 (1996) 253-265
Stoichiometric number of ions i Dimensionless electrode potential Transference number of species i
7.2. Subscripts i ISE J k 1 RE +
Species i Ion selective electrode Junction Level of concentration Level of concentration Reference electrode Cation Anion Mean ionic
7.3. Superscripts O
D-H LC R
Reference state Debye-Hi~ckel contribution Low concentration Contribution of residual activity coefficient
Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research of Council of Canada for financial support.
References R.G. Bates, Determination of pH Theory and Practice, 2nd edn., Wiley, New York, 1965. Bates, R.G., Staples, B.R. and Robinson, R.A., 1970. Anal. Chem., 42: 867. G. Eisenman, Glass Electrodes for Hydrogen and Other Cations, Marcel Dekker, New York, 1967. J.W. Gibbs, Collected Works, Vol. 1, Longmans Green, New York, 1928. H.S. Harned and B.B. Owen, The Physical Chemistry of Electrolyte Solutions, 3rd edn., Reinhold, New York, 1958. Komar, N.P. and Kaftanov, A.Z., 1974. Russ. J. Phys. Chem., 48: 246. Khoshkbarchi, M.K. and Vera, J.H., 1996. AICHE J., 42: 249. Maclnnes, D.A., Belcher, D. and Shedlovsky, T., 1938. J. Am. Chem. Soc., 60: 1094. D.A. Maclnnes, The Principles of Electrochemistry, Dover, New York, 1961. Milazzo, G., Bonciocat, N. and Borda, M., 1975. Electrochim. Acta, 21: 349. Mokhov, V.M., Bagdasarova, I.P., Kekeliya, A.G. and Lavrelashvilig, L.V., 1977. Russ. J. Phys. Chem., 51: 1406. Pitzer, K., 1980. J. Am. Chem. Soc., 102: 2902.
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R.A. Robinson and R.H. Stokes, Electrolyte Solutions, 2nd edn., Butterworths, Boston, MA, 1959. Robinson, R.A., 1961, J. Phys. Chem., 65: 662. Shatkay, A. and Lerman, A., 1969. Anal. Chem., 41: 514. Taylor, P.B., 1927. J. Phys. Chem., 31: 1478. J.F. Zemaitis, D.M. Clark, M. Rafal and N.C. Scrivner, Handbook of Aqueous Electrolyte Thermodynamics, A1CHE, New York, 1986.