Estimation of activity coefficients at different temperatures by using the mean spherical approximation

Estimation of activity coefficients at different temperatures by using the mean spherical approximation

www.elsevier.nl/locate/jelechem Journal of Electroanalytical Chemistry 480 (2000) 9 – 17 Estimation of activity coefficients at different temperature...

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www.elsevier.nl/locate/jelechem Journal of Electroanalytical Chemistry 480 (2000) 9 – 17

Estimation of activity coefficients at different temperatures by using the mean spherical approximation G. Lo´pez-Pe´rez *, D. Gonza´lez-Arjona, M. Molero Department of Physical Chemistry, Uni6ersity of Se6ille, E-41071, Se6ille, Spain Received 23 July 1999; received in revised form 26 October 1999; accepted 29 October 1999

Abstract A method of estimating activity coefficients for a variety of electrolytes at different temperatures is presented. The MSA approximation is used to calculate the activity coefficients from experimental data available in the literature. This strategy provides suitable results within a wide range of temperatures, electrolyte stoichiometries and concentrations of investigated solutions. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Activity coefficients; Temperature influence; MSA theory; Aqueous electrolytes

1. Introduction Electrolyte solutions exhibit considerable deviations from ideal behaviour. This fact is specially remarkable at high solute concentrations and/or temperatures. It is usual to find many practical applications under these conditions. Thus, the knowledge of the thermodynamic properties of solutions is indispensable for practical and theoretical purposes. Pitzer’s theory has been widely used for the interpolation of activity and osmotic coefficients because of its high accuracy, but the functional relationships to consider the temperature or mixing effects are complex [1,2]. In fact, the activity coefficients for a single electrolyte at a fixed temperature undergo the influence of different kinds of interactions, i.e. electrostatic, short range and even triple ion interactions. To consider the influence of the temperature, Pitzer et al. [1] used a set of equations with a maximum of 19 adjustable parameters in order to reproduce the experimental data at different temperatures. On the other hand, the mean spherical approximation (MSA) theory [3 – 5] is showing up as an excellent tool for the description of electrolyte solutions in the primitive model [6 – 8]. Unfortunately, the primitive model of electrolytes gives a reasonable representation * Corresponding author. Fax: +34-95-455-7174.

of real solutions only up to approximately 1 or 2 mol l − 1. Nevertheless, the MSA theory has been also applied to solve an extension of the primitive model considering the change of the ionic sizes and/or solution permittivity with the electrolyte concentration at fixed temperature [9–20]. It is remarkable that only Simonin et al. [13] recognised, in 1996, the existence of extra terms in the formulation of the MSA theory when the ionic sizes and/or solution permittivity show concentration dependence. Some important characteristics of the MSA make it a very attractive theory to describe the electrolyte solution behaviour. It has an analytical solution and uses a low number of parameters (usually one per anion/cation couple) and, in addition, these parameters have a microscopic meaning. The temperature effect on the electrolyte thermodynamics has rarely been analysed using MSA [21,22]. Moreover, in the applied model the solution permittivity is fixed to the pure solvent value. Unfortunately, the extra contribution to the activity and osmotic coefficients from the concentration dependence of the ionic sizes, pointed out by Simonin et al. [13], was not considered. A new method to estimate activity coefficients of single electrolytes at different temperatures using the MSA theory is presented. In this treatment, all the solutions were considered as strong electrolytes, and

0022-0728/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 2 2 - 0 7 2 8 ( 9 9 ) 0 0 4 3 8 - 6

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accordingly ionic associations were not considered explicitly. The results show excellent behaviour when the modified primitive model is solved with the aid of the MSA theory considering the temperature effect. Besides, the parameters obtained have physical meaning and the temperature effect is explained with simple mathematical relationships.

2. Theory The electrolyte solution is modelled in the framework of the primitive model, but incorporating two extra features: the cationic radius (r + ) and the solution permittivity (o) depend on the salt concentration [12–20]. These additions incorporated in the model are based on the experimental data observation providing a better representation of the real situation in electrolyte solutions. The cationic solvated radius is expected to decrease with the salt concentration due to the loss of free solvent molecules; so a lower permittivity of the solution can be also presumed. Although there are recent measurements of o for some aqueous solutions with high accuracy [23], they are not available for any chosen salt, concentration or temperature range. These data have been used by Tikanen and Fawcett, to estimate activity coefficients taking into account ionic association [17,20] and to analyse the medium effects in the homogeneous kinetics reactions [19] with the MSA. However, due to the scarce data available, in our paper, the o dependence on salt concentration has been treated as a parameter. These values have been obtained by fitting experimental activity coefficients. In the primitive model, the mean ionic activity coefficients in the molar concentration scale (y) can be split into the electrostatic and hard sphere contributions: D ln y= D ln y el + D ln y HS

(1)

In this work the MSA has been used to estimate the electrostatic contribution and the hard sphere contribution has been evaluated using the Carnahan – Starling (CS) approximation [24,25]. The dependence of r + and o with the salt concentration introduces extra terms in the activity coefficient: D ln y el = D ln y MSA = D ln y MSA +D ln y MSA +D ln y MSA 0 r o

(2)

and CS D ln y HS = D ln y CS = D ln y CS 0 +D ln y r

(3)

where 0 refers to the quantities in the pure primitive model, r to the terms due to the change of r + with the concentration, and o to the terms that take into account the change in o with the concentration.

Expressions for each term in Eq. (2) and Eq. (3) are given in the literature [13,14]. Conversion from the McMillan–Mayer (MM) to the Lewis–Randall (LR) scale has been performed as given in recent papers [14]: ln y LR = ln y MM − csn¯ sf MM

(4)

where n¯ s is the molar partial volume and cs the molar concentration of the salt respectively, and f MM is the osmotic coefficient in the MM scale, calculated using the MSA theory. Conversion from molar to molal scale has been also performed in the following way: ln g= ln y+ ln

cs r0m

(5)

where g is the mean ionic activity coefficient in the molal scale, m the molal concentration of the salt, and r0 the pure solvent density. For practical purposes, solution densities, taken from Ref. [26], have been calculated as a function of the molar concentration and temperature with the following expression: rs(T) = r0(T)+ d1(T)cs + d2(T)c 3/2 s

(6)

where at a given temperature, r0(T) is the pure water density, rs(T) is the solution density at a fixed salt concentration (cs) and di (T) are coefficients that have temperature dependence. As the starting point in this work, the variation of the cationic radius and the solution permittivity were considered as a function of salt concentration. The radii of the anions were considered constant. The radii of Cl− (1.81 A, ), Br− (1.95 A, ) and OH− (1.76 A, ) were fixed to the crystallographic values taken from Nightingale [27]. This assumption for anionic radii has been used already in the literature [9–11,13–16,20,22] and is supported by experimental evidence for the chloride anion [28]. The following procedure was used to fit the radii of 2− ClO− anions. The Na+ radius was evalu4 and CO3 ated from the fit of the activity coefficient of NaCl solutions. Next, the Na+ radius so obtained, was fixed and used to find the best fit for the activity coefficients of NaClO4 and Na2CO3 solutions at 298 K, modifying the anion radii. A similar procedure has been described for the estimation of the size parameters using the MSA [20]. The values obtained for the size of these anions , for CO23 − . were 1.98 A, for ClO− 4 and 2.00 A A simple linear relationship has been used to allow for the salt concentration effect on the cation radius: r+ = r+ + bcs +

(7)

where r is the hydrated cation radius at infinite dilution and b is a parameter. Although, this equation does not have a fundamental theoretical basis, it has been used successfully in the literature to compare

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experimental values with data calculated by theory [10,13–16]. Non-linear equations have been also employed [21,22], when the dependence of the solution permittivity with the electrolyte concentration is not taken into account in the model. Theoretical [29–38] and experimental [29,39–42] points of view have been used to consider the influence of the electrolyte concentration on the solution permittivity. From statistical mechanics theories of a mixture of hard spheres ions and dipoles [30 – 33], the use of a solute-dependent dielectric permittivity as an improvement of the primitive model has been justified. For moderate concentrations (up to 1 mol l − 1) a linear relationship can be used [29,36,40]. For more concentrated solutions the observed behaviour is non-linear and, at least, an additional term is necessary to fit the experimental measurements [39,40]. The following equation has been suggested by Friedman [35] and it will be used in the present work: o=

o0 (1+acs)

(8)

where a is a parameter. o0 is the solution permittivity at infinite dilution whose value was taken from the literature [43] for pure water. Eq. (8) is able to describe the experimental behaviour up to about 6 mol l − 1 and it has been used previously by Simonin et al. [13 – 16]. The main advantage of this equation is that only one parameter is required, if it is compared with other non-linear relationships [17 – 20,23]. Nevertheless, a polynomial dependence can be obtained easily by serial expansion of Eq. (8). , b and a) are temperature deThe parameters (r + pendent, and the following simple relationships:

Fig. 1. Influence of the three adjustable parameters, r + , b and a, in the logarithm of activity coefficients for LiCl at 298 K. Symbols are experimental data, upper dashed line is r + = 2.55 A, and lower r+ =2.45 A, , upper medium dashed line is b = − 0.060 A, mol − 1 l and lower is b = − 0.010 A, mol − 1 l, upper long dashed line is a = +0.1162 mol − 1 l and lower long dashed line is a= 0.0662 mol − 1 l.

r+ = r+ ,298 + r1t

11

(9)

b= b0 + b1t+ b2t 2

(10)

a= a0 + a1t+ a2t

(11)

2

can be used to describe such an influence. In the Eqs. (9–11), t= T− 298 and r + ,298, b0 and a0 refer to the values of these parameters at 298 K. Consequently, for each single electrolyte solution at a fixed temperature, two parameters have to be fitted, b and a, whereas r + is a characteristic value of the cation considered.

3. Results A reasonable effort was made to consider the most available data for activity coefficients of single electrolytes at temperatures other than 298 K. The data sets for each single salt at different temperatures have been selected for the same author and reference. Furthermore, all temperature data sets have been clipped to the maximum common concentration value. With these conditions, the following single salts were analysed in a temperature range of about 273–350 K, and for the maximum molal concentration indicated between the parenthesis: HCl (4.0 mol kg − 1) [44], LiCl (6.0 mol kg − 1) [45], NaCl (4.0 mol kg − 1) [44], KCl (4.0 mol kg − 1) [46], CsCl (6.0 mol kg − 1) [45], NaOH (17.0 mol kg − 1) [47], HBr (5.5 mol kg − 1) [48], NaBr (4.0 mol kg − 1) [49], KBr (1.2 mol kg − 1) [50], CaCl2 (0.1 mol kg − 1) [51], SrCl2 (0.3 mol kg − 1) [52], BaCl2 (1.5 mol kg − 1) [53], Na2CO3 (1.5 mol kg − 1) [54] and HClO4 (6.0 mol kg − 1) [55]. Small deviations of the parameters from those published previously [15] for some salts at 298 K are due to the different range of concentrations used in the fitting process. A total of 70 experimental data sets have been studied. The influence of the three parameters on the logarithm of activity coefficients is shown in Fig. 1 for LiCl at 298 K. The lines are calculated fixing the values of two parameters and varying the other one in a common range: r + = 2.59 0.05 A, , b= − 0.03509 0.025 A, −1 mol l and a= 0.09129 0.025 mol − 1 l. As can be noted, b and a have a great influence, especially at high salt concentrations. Accordingly, parameters b and a will be very sensitive towards the maximum concentration range. Once a salt is selected, a preliminary approach to obtain the adjustable parameters r + , b and a were carried out by minimising the squared deviations of ln g. Nevertheless, the r + values obtained for salts with different anions and a common cation were either not constant or even negative. If the goal of the paper were exclusively a procedure of data fitting, there would be no objections to these

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Fig. 2. Influence of the temperature on r + for all sodium salts studied in this work. ( — ) Linear regression.

values. But we wanted to obtain some microscopic insight from the fitted parameters, so the results were should be constant for a not satisfactory. Therefore, r + specific cation at a fixed temperature and independent of the accompanying anion. It should depend only on the solvent-cation interaction at infinite dilution. Accordingly, the optimisation strategy was changed as follows: for a specific salt, an initial fixed value of r + was selected and the optimisation was performed only with b and a as parameters. This procedure was re peated for different r + values. The selected parameter set (r + , b and a) was taken as that which gave the minimum value of the percent absolute average relative deviation, AARD(%): AARD(%)=



100 N ln gexp(k) − ln g LR(k) % ABS N k=1 ln gexp(k)



(12)

N being the number of the experimental data points. Changes of 90.025 A, in r + shifted the AARD less than 0.4% for all the salts studied, once the b and a parameters were optimised. This type of hierarchic optimisation has been used commonly in the literature [56] for other purposes: i.e. the fitting method for electrode admittances to the Randles’ circuit. The values of r + , for salts with different anions and the same cation at a given temperature, evaluated following this procedure were quite close, the differences being less than 0.02 A, . Furthermore, it is possible to select the same value for r + at a fixed temperature for salts with a common cation. Fig. 2 shows the good linear relationship between r + and the absolute temperature for all sodium salts studied in this work. This optimisation method sacrifices fairly the precision of the interpolation method, investing it in a physical meaning and coherence of the resulting parameters. Tikanen and Fawcett [20] considered the ion association to obtain a good fit of several 1:1 electrolytes,

among them NaCl, LiCl, CsCl, KBr and KCl. In our case, the degree of agreement between the experimental and theoretical activity coefficients at different temperatures can be observed in Fig. 3, without considering ion association as was stated before. It is interesting to point out two special cases. The procedure has allowed us to obtain a fair estimation of the activity coefficient for NaOH (Fig. 3c) and Na2CO3 (Fig. 3d) in a wide range of concentrations. These results are satisfactory in spite of their well-known atypical behaviour as strong electrolytes. With this optimisation strategy the temperature effect in the activity coefficients has been analysed. A relatively high average deviation has been obtained for HBr solutions but this can be explained on the basis of the experimental values in original data sets, as can be noted in Fig. 4.

4. Discussion The results of the calculations are gathered in Tables 1 and 2. One contribution of this work may be that, using a relatively small number of parameters, it is possible to interpolate the activity coefficients of 14 salts in a wide range of concentrations (typically 0–5 mol kg − 1) and temperatures (273–353 K). However the aim of this work is not only to give an efficient fitting method but to obtain a physical insight into the real systems, interpreting the values obtained for the parameters. Before beginning a deeper discussion of each parameter, it should be remarked that for all the salts studied, at any concentration or temperature, the cationic radii (r + ) and solution permittivities (o) are always reasonable values. Moreover, they are within the expected physical range (except Na2CO3 at T] 313 K where r + : 0 at the higher concentrations. This fact may be due to a hydrolysis effect not considered in our treatment). The change of these parameters with concentration and temperature is always smooth. The values of cationic radii at infinite dilution for 298 K are included in Table 1. It has to be noted that the values for H+, Na+ and K+ were obtained from linear regression of several different salts, according to the fact that the radius at infinite dilution should be independent of the accompanying anion. The second column of the table gives the values of r + ,298, and it can be verified easily that they are quite well correlated with the various scales existing in the literature of hydrated radii [27,57,58]. The following sequence can be found: Ca2 + \ Sr2 + \ Ba2 + : H+ : Li+ \ Na+ \ K+ \Cs+ and it agrees with other fits of experimental data to theories based on the primitive model [13–15,20,21]. This sequence is also obtained at any other studied

G. Lo´pez-Pe´rez et al. / Journal of Electroanalytical Chemistry 480 (2000) 9–17 temperature. The dependence of r + with temperature could be described for all cations by a linear relationship, with a positive slope, r1 for all cases. This increase of the cationic radius at infinite dilution with the temperature may be a consequence of the increasing ther-

13

mal agitation, which moves the solvent molecules of the solvation shell a little farther away from the central ion. For monovalent cations the following sequence is observed for r1: H+ B Li+ B Na+ B K+ : Cs+

Fig. 3. Comparison between experimental (symbols) and calculated (lines) activity coefficients for several electrolytes. Temperature values are indicated in each plot. (a) NaCl; (b) NaBr; (c) NaOH; (d) Na2CO3; (e) CsCl and (f) LiCl.

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Fig. 4. Comparison between experimental (symbols) and calculated activity coefficients for HBr at 313 K. Differences at high concentrations are found in all experimental data sets (see Table 2). Residuals=(ln gexp −ln g LR).

This arrangement is in agreement with the sequence of the temperature derivative of the B-coefficient of Jones–Dole [59], which is related directly to the ionic standard partial volumes at infinite dilution, and therefore to the solvated ionic radii. Other evidence of similar behaviour for a comparable parameter can be found in the literature in the work of Krestov [60]. He analysed the influence of the temperature in the crystalline radius through the isobaric temperature coefficient given as a function of the Goldschmidt radius. The same sequence was reported, even with the same order of magnitude. Thus, the greater change in the solvated radius at infinite dilution with temperature, of Cs+ than Li+ does not originate in a bigger change in the solvation shell but in the ionic core. The sequence of divalent cations is the opposite and cannot be rationalised with the above arguments. Nevertheless it must be considered that the range of concentration studied for these divalent salts is significantly shorter. Table 1 Fitted parameters for the individual cationic radius at infinite dilution (r+,298 ) and its temperature coefficient (r1) Cation

r+,298 /A,

104 r1/A, K−1

H+ Li+ Na+ K+ Cs+ Ca2+ Sr2+ Ba2+

2.57 2.52 2.01 1.76 1.03 2.90 2.75 2.50

14.7 26.4 94.7 116 103 225 139 105

In our treatment, the anions have been considered as unsolvated, so any change in their radius in solution with the temperature should come from changes in the crystallographic radius. In this work, the anionic radii are fixed, but a more realistic assumption may be to allow that these values were temperature dependant. This possibility will be considered in future work. The interpretation of b from a microscopic point of view is not so direct, because it is related to the change on the solvation shell, due to the effect of salt concentration. Column 2 in Table 2 collects the values at 25°C (t= 0), corresponding to values of b0, accordingly to Eq. (10). They are negative for almost all salts studied. This fact is in agreement with the loss of solvation associated with the increase of concentration, as a consequence of the lower availability of solvent molecules for a given ion in the solution. The exceptions to this behaviour are the values for CsCl, CaCl2 and SrCl2. The highest concentrations studied for these bivalent salts are much smaller (0.08 and 0.3 mol kg − 1, respectively) compared with the rest, so, the fitted values may lose some physical meaning. This narrow concentration range may be also the reason for the high values fitted of b0, b1, b2, a0, a1 and a2, when they are compared with the other entries of Table 2. Nevertheless, at 298 K, there are experimental data in a wider range of concentration for these bivalent salts, and then a negative value of b0 is recovered [14] in the fitting process. The case of CsCl is different, and it has no simple explanation. The anomalous increment of the ionic size with concentration can be also found in the literature [14] for Cs+, Rb+ and alkylammonium salts. This is the general behaviour observed at all the temperatures studied, with some exceptions. The b values are small, so no big changes in r + with cs are observed (generallyB0.5 A, in the whole concentration and temperature range). The temperature dependence of b can be described by means of a smooth quadratic function, Eq. (10), which decreases in nearly all cases when raising the temperature (Fig. 5a). In other words, the dehydration of an ion at a given concentration is smaller at higher temperatures, as a consequence of thermal agitation. From Eq. (10) we have that:

 

#b = b1 + 2b2t #T

(13)

then the values of b1 tabulated in Table 2 give directly the following cross derivative:

    #b # #r + = #T #T #cs

(14)

at 298 K. For the alkaline chlorides, b1 decreases in the sequence: HCl\ LiCl\ NaCl: KCl\ CsCl

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Table 2 Temperature effect on the b and a parameters. A quadratic relationship, Eqs. (10) and (11), with the absolute temperature is assumed. MAARD(%) represents the mean AARD obtained for the different temperatures of each electrolyte in the fitting procedure Salt

b0/ A, mol−1 l

104 b1/ A, mol−1 l K−1

106 b2/ A, mol−1 l K−2

a0/ mol−1 l

104 a1/ mol−1 l K−1

106 a2/ mol−1 l K−2

MAARD/ %

T range/ K

mmax/ mol kg−1

HCl LiCl NaCl KCl CsCl NaOH HBr NaBr KBr HClO4 CaCl2 SrCl2 BaCl2 Na2CO3

−0.044 −0.035 −0.0092 −0.0023 0.027 −0.011 −0.028 −0.018 −0.0044 −0.024 3.65 0.348 −0.132 −0.320

−1.39 −1.98 −6.64 −5.14 −6.12 −7.32 −1.93 −6.28 42.3 −1.91 2250 340 6.48 −5110

2.19 −1.07 3.02 17.4 1.90 18.3 −0.437 13.4 −108 8.89 −17000 −355 282 581

0.076 0.095 0.085 0.075 0.018 0.080 0.091 0.069 0.083 0.077 0.386 0.185 0.0044 0.054

8.32 7.55 6.18 10.7 0.757 9.32 2.40 6.34 17.4 −0.998 370 93.1 19.8 −163

1.82 3.69 15.1 26.1 0.576 28.6 1.26 30.4 −2.78 6.99 −1920 37.5 114 239

1.77 0.42 0.34 0.36 0.36 0.48 11.2 a 1.43 0.42 1.63 0.40 1.16 0.58 2.01

273–323 273–348 273–323 273–313 273–348 273–313 273–333 273–313 283–333 283–313 288–308 283–343 273–318 288–338

4.0 6.0 4.0 4.0 6.0 17.0 5.5 4.0 1.2 6.0 0.1 0.3 1.5 1.5

a

See text for explanation.

The influence of temperature in a can also always be described by using a quadratic relationship, Fig. 5b, but here the value increases with the temperature. This would be the expected behaviour; the decrease of the permittivity when the temperature increase agrees well with the theoretical and experimental studies [23]. From Eqs. (8) and (11), we have:

   

#o 1 #o0 cs =o − (a +2a2t) 1 +acs 1 #t o0 #t

n

Also in Fig. 6, the dotted line originating from Eq. (8) clearly proves that the experimental data can be described by this equation.

(15)

The negative values of some entries in Table 2 for a1 and a2 could imply an increase of the permittivity when #o0 is about the temperature becomes greater. But #t − 0.4 K − 1 and the anomalous behaviour can be observed only for Na2CO3 and for concentrations above 4.3 mol kg − 1 for HClO4 at 283 K. For Na2CO3, as was stated before, the hydrolysis effect has not been considered. The remaining salts show the same behaviour observed with the available experimental data, decreasing the solution permittivity with the rise of the temperature. The slope is greater for higher temperatures and lower electrolyte concentrations. It is worthwhile to analyse the optimised values of the solution permittivity, to see if they are within the range and sequence of available experimental data. Fig. 6 shows the influence of the electrolyte concentration in o for several chlorides at 298 K. The optimised values (open symbols) follow the same tendency and have satisfactorily close values to the experimental data (solid symbols). Moreover, both data sets (experimental and optimised) show the same sequence for a fixed electrolyte concentration with a:

 

Li+ BNa+ BK+

Fig. 5. Influence of the temperature on b (a) and a (b) parameters for KCl.

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formed with the remaining experimental data. The parameters so obtained were used to estimate theoretically the values for the omitted temperatures. The mean absolute average relative deviation for the logarithm of the activity coefficient was less than 2%, in both cases.

5. Conclusions

Fig. 6. Solution permittivity as a function of the solution concentration of some chloride salts. () K+; ( ) Na+ and () Li+. Experimental and fitted values are represented by solid and open symbols respectively. Dotted lines are generated using the Eq. (8).

The experimental permittivity has two contributions: static and dynamic [34,36,38]. Only the static part of the solution permittivity should be included in the model. Thus, the differences found between calculated and experimental o data, may be due to the dynamic contribution. Some estimations of the dynamic contribution provides results [38] that agree qualitatively with our findings: the static values are higher than the experimental ones. On the other hand, the scarcity of permittivity measurements of electrolyte solutions preclude the use of the experimental o values in the model. Nevertheless, an attempt has been carried out with the available data for LiCl, NaCl, KCl and KBr at 298 K. The parameters associated with the cationic radius were optimised only for the above mentioned salts. The new r + ,298 values calculated are about 0.4 A, higher than those gathered in Table 1 and the values for KCl and KBr, also maintain the individual cation property. Although the quality of the calculations was worse, the new AARD being twice as large compared with that obtained previously, it looks a very promising pathway when experimental electrolyte solution permittivity measurement at different temperatures are available. From another point of view, and independently of a physical explanation of the optimised parameters, the MSA theory provides a method to interpolate activity and osmotic coefficients within a wide range of temperatures and concentrations. This method uses a relatively low number of parameters, namely six. Pitzer’s treatment [1,2] considers the temperature effect using at least nine adjustable parameters. The interpolation method has been tested for HCl solutions using the following procedure. From the original data sets [44], the data collection for 283 and 313 K were removed. After that, the optimisation was per-

This work shows the use of MSA to solve a modification of the primitive model by considering the concentration and temperature effect. These effects are considered using very simple mathematical relationships. Some of the adjustable parameters can be interpreted from a microscopic point of view that greatly simplifies the analysis. Thus, the MSA theory gives an easy method to consider the temperature effect in thermodynamic excess properties of solutions, providing a simple and accurate way to interpolate data at different temperatures.

Acknowledgements The authors have been supported partially by the Direccio´n General de Investigacio´n Cientı´fica y Te´cnolo´gica (DGICYT) under Grant PB095-0537. The authors are indebted to Professor A.G. Gonza´lez for his helpful comments.

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