Thermodynamically Traceable Calorimetric Results for Dilute Aqueous Potassium Chloride Solutions at Temperatures from 273.15 to 373.15 K. Part 2. The Quantities Associated with the Partial Molar Heat Capacity

In previous articles (Partanen et al. J. Chem. Eng. Data 2019, 64, 16−33 and Partanenet al. J. Chem. Eng. Data 2019, 64, 2519−2535), we presented traceable and transparent two-parameter Hückel equations (with parameters B and b1) for the activity coefficients of the salt and for the osmotic coefficients of water in aqueous KCl solutions in the temperature range of 273.15−383.15 K. The latter article is the first part (Part 1) of the calorimetric study. We showed in these articles that our equations for these solutions explain within experimental error the literature data on almost all thermodynamic quantities including the partial molar enthalpies at least up to a molality of 0.2 mol·kg−1 and up to 373 K. In this model, parameter B is regarded as a constant but parameter b1 has a quadratic temperature dependence. No calorimetric data were needed in the parameter estimation. In the second part (Part 2) of the calorimetric study, now, the results obtained for the heat capacity quantities of KCl (aq.) are considered. We show here that all heat capacity literature available for KCl solutions at least up to 0.5 mol·kg−1 is possible to explain within experimental error using exactly the same Hückel equations as those considered previously in our studies for dilute KCl solutions from 273 to 373 K. Because of the success of the used model, we supplement the existing thermodynamic tables with new values for the relative apparent and partial molar heat capacities for KCl solutions. It is likely that the new tables contain the most reliable values available for these heat capacity quantities. ■ INTRODUCTION Due to its central role in biological and industrial systems, potassium chloride is without question one of the most important electrolytes on earth. Its thermodynamic properties in aqueous solutions, such as activity coefficients, osmotic coefficients, heats of dilution and solution, and molar heat capacities of components, have been therefore thoroughly determined in various temperatures, pressures, and concentrations (see the review of Archer). Subsequently, these measurements have been interpreted using different kinds of models that enable the calculation of these properties in regions where measurements are not yet carried out. The most important of these models has been proposed by Pitzer. This Pitzer model also serves as the basis of the multiparameter equations of Archer. For practical treatment, however, these equations are often overly complicated, containing numerous fitting parameters connected to the three independent variables (i.e., temperature, pressure, and concentration). In light of our recent NaCl studies, and our other previous research, the behavior of dilute electrolyte solutions can often be accurately described by a much simpler Hückel-type equation requiring only a few of estimated parameters. To date, heat capacity data of pure aqueous solutions of several uniunivalent electrolytes have been considered in the reviews of Parker, Hepler and Hovey, and Criss and Millero. All of these reviews usually contain only results at the thermodynamic reference temperature of 298.15 K. Heat capacities in solutions of other valence-type electrolytes have been treated in refs 7 and 9 at this temperature. Additionally, the heat capacities of KCl solutions have been studied in the review of Pabalan and Pitzer at various temperatures. Both studies by Criss and Millero utilized Pitzer formalism in the interpretation of their experimental data. In ref 5 and in Part 1 of this study, we demonstrated that the Hückel equation with the two parameters, denoted B and b1, is the most accurate equation to date for predicting the thermodynamic properties of dilute KCl (aq.) at various temperatures. In the present part (Part 2), we have observed that our previous value for parameter B and the quadratic equation with respect to temperature for parameter b1 also apply very well to almost all heat capacity literature available for dilute Received: April 29, 2019 Accepted: August 1, 2019 Published: August 14, 2019 Article pubs.acs.org/jced Cite This: J. Chem. Eng. Data 2019, 64, 3971−3982 © 2019 American Chemical Society 3971 DOI: 10.1021/acs.jced.9b00373 J. Chem. Eng. Data 2019, 64, 3971−3982 This is an open access article published under a Creative Commons Attribution (CC-BY) License, which permits unrestricted use, distribution and reproduction in any medium, provided the author and source are cited. Dow nloa ded via AA LTO UN IV o n Se ptem ber 19, 201 9 at 19: 08:2 5 (U TC) . See http s://p ubs .acs .org /sha ring guid elin es f or o ptio ns o n ho w to leg itim atel y sh are pub lish ed a rticl es. KCl solutions within 273−373 K. Following the notation of Part 1, this parametrization has been designated PI. In ref 12, we have suggested an alternative parametrization for more concentrated KCl solutions. In this model, which we call in ref 11 and here parametrization PII, the same value was used for B, but the temperature dependence of parameter b1 was determined mainly from the electrochemical data for KCl solutions. The used amalgam cell data of Harned and Cook cover the temperature range 273.15−313.15 K and extend up to a molality of 4.0 mol·kg−1. Because these electrochemical data are not as accurate as the data used for parametrization PI, the resulting model is more utilitarian than the one based on PI. However, in refs 11 and 12, it was observed that PII applies quite well up to 343 K also in less dilute solutions and, therefore, we continue the use of PII in the present study. ■ THEORY In many previous contributions (see, for example, refs 3−5, 11, 12, and 14−17, and the complete list of our studies in this area is in ref 11), it has been demonstrated that Hückel equations


■ INTRODUCTION
Due to its central role in biological and industrial systems, potassium chloride is without question one of the most important electrolytes on earth. Its thermodynamic properties in aqueous solutions, such as activity coefficients, osmotic coefficients, heats of dilution and solution, and molar heat capacities of components, have been therefore thoroughly determined in various temperatures, pressures, and concentrations (see the review of Archer 1 ). Subsequently, these measurements have been interpreted using different kinds of models that enable the calculation of these properties in regions where measurements are not yet carried out. The most important of these models has been proposed by Pitzer. 2 This Pitzer model also serves as the basis of the multiparameter equations of Archer. 1 For practical treatment, however, these equations are often overly complicated, containing numerous fitting parameters connected to the three independent variables (i.e., temperature, pressure, and concentration). In light of our recent NaCl studies, 3−5 and our other previous research, the behavior of dilute electrolyte solutions can often be accurately described by a much simpler Huckel-type equation requiring only a few of estimated parameters.
To date, heat capacity data of pure aqueous solutions of several uniunivalent electrolytes have been considered in the reviews of Parker, 6 Hepler and Hovey, 7 and Criss and Millero. 8 All of these reviews usually contain only results at the thermodynamic reference temperature of 298.15 K. Heat capacities in solutions of other valence-type electrolytes have been treated in refs 7 and 9 at this temperature. Additionally, the heat capacities of KCl solutions have been studied in the review of Pabalan and Pitzer 10 at various temperatures. Both studies by Criss and Millero 8,9 utilized Pitzer formalism 2 in the interpretation of their experimental data.
In ref 5 and in Part 1 of this study, 11 we demonstrated that the Huckel equation with the two parameters, denoted B and b 1 , is the most accurate equation to date for predicting the thermodynamic properties of dilute KCl (aq.) at various temperatures. In the present part (Part 2), we have observed that our previous value for parameter B and the quadratic equation with respect to temperature for parameter b 1 also apply very well to almost all heat capacity literature available for dilute KCl solutions within 273−373 K. Following the notation of Part 1, this parametrization has been designated PI.
In ref 12, we have suggested an alternative parametrization for more concentrated KCl solutions. In this model, which we call in ref 11 and here parametrization PII, the same value was used for B, but the temperature dependence of parameter b 1 was determined mainly from the electrochemical data for KCl solutions. The used amalgam cell data of Harned and Cook 13 cover the temperature range 273.15−313.15 K and extend up to a molality of 4.0 mol·kg −1 . Because these electrochemical data are not as accurate as the data used for parametrization PI, the resulting model is more utilitarian than the one based on PI. However, in refs 11 and 12, it was observed that PII applies quite well up to 343 K also in less dilute solutions and, therefore, we continue the use of PII in the present study.
can be used to predict accurately the mean activity coefficient of salt (γ) and the osmotic coefficient (ϕ) of water in aqueous solutions of many single electrolytes at least up to an ionic strength (I m ) of 1 mol·kg −1 . In eqs 1 and 2, m is the molality, m°i s its unit (i.e., m°= 1 mol·kg −1 ), z + and z − are the charge numbers of the cation and anion, respectively, and B and b 1 are the electrolyte-dependent parameters. The values of the Debye−Huckel parameter α at 101.325 kPa and at various temperatures are given in Table 1 of ref 5, and they have been  taken from ref 18. For a 1:1 electrolyte like KCl, |z + z − | is 1 and I m is the same as the molality. The excess Gibbs energy of solution (ΔG ex ) on the molality scale in pure electrolyte solutions is related to the activity and osmotic coefficients by where T is the temperature in the Kelvin scale and R the universal gas constant. In all of the energy quantities below, the molality scale is used and the mass of water is considered to be 1 kg. The symbol of the apparent molar enthalpy of salt is H app,2 (subscript 2 is associated with the salt), and its definition is given in the following equation: where H is enthalpy of the system, H m,1 * is the molar enthalpy of pure water (symbol 1), H 1 * is enthalpy of the mass of 1 kg water (= w 1 ), M 1 is the molar mass of water (= 0.018015 kg·mol −1 ), and in the last term "kg" is omitted from the unit for simplicity (as now generally) because the molality scale is always used. The relative apparent molar enthalpy (ΔH app ) of salt in these solutions is connected to the excess Gibbs energy by the subsequent thermodynamic identity where H m,2 ∞ is the partial molar enthalpy of the salt at infinite dilution. The apparent molar heat capacity of the salt at a constant pressure is C app,2 , and it is denoted here simply as C app . It is defined analogously to the enthalpy quantity by equation where C is the heat capacity of the system, C m,1 * is the molar heat capacity of pure water, and C 1 * is the heat capacity of the mass of 1 kg water. The relative apparent molar heat capacity (ΔC app ) of the salt is associated with the relative apparent molar enthalpy (ΔH app ) of the salt by equation where C m,2 ∞ is the partial molar heat capacity of the salt at infinite dilution. For a certain molality, the partial molar heat capacity of the salt is related to the apparent quantity by Finally, the relative partial molar heat capacity of the salt (ΔC m,2 ) is defined by equation where T 0 = 273.15 K, and it was presented in ref 5. In Part 1, we have concluded recently 11 that parameterization PI does not apply well to the existing enthalpy data in less dilute KCl solutions in the range 298−303 K. As in Part 1, therefore, we employ here an alternative quadratic parametrization for b 1 that has the form As mentioned in Introduction, this equation was determined in ref 12 mainly from the amalgam cell data, 13 and it was observed to apply quite well up to 343.15 K also in more concentrated solutions. Like before, 11 the calculations associated with eq 11 are referred here to as those of parametrization PII.

■ RESULTS AND DISCUSSION
Calculations of the Relative Apparent Molar Heat Capacities. Equation 10 (or parametrization PI) was tested in Part 1 11 and here using all high-quality calorimetric data that are available for dilute KCl solutions up to 373 K. As described in ref 4 for the NaCl case, the following simple strategy gave accurate and traceable results for the interpretation of the complicated heat capacity data existing in the literature for dilute KCl (aq.) at various temperatures at least up to 0.2 mol·kg −1 : Using eq 3 Table 2. Parameter Values for eq 14 (i.e., for the Dependence of Relative Apparent Molar Heat Capacity on the Molality) for KCl Solutions Obtained by Using Parametrization PI (eq 10)   together with eq 1 for the activity coefficient and eq 2 for the osmotic coefficient, first, the excess Gibbs energy of solution was calculated at temperatures from 273.15 to 353.15 K in intervals of 5 K for rounded values of molalities. Then, these Gibbs energies were fitted to a quadratic equation of the following type For all employed molalities, the resulting values of parameters u, v, and w can be found in Table 1 of Part 1, 11 as well as the standard deviations about the regression line. The relative  apparent molar heat capacity in eq 7 was obtained from the parameters of eq 12 by app app 0 For 298.15 K, the resulting relative values are reported in Table 1 of the present part. Next, the relationship between the apparent heat capacities and molalities was determined by fitting the relative C app values by using the equation where the theoretical Debye−Huckel value was accepted for the coefficient of the square root term (i.e., for α C ) at each temperature. These theoretical values are given in ref 18, and they together with the estimated values of α C,1 , α C,2 , and α C,3 for parametrization PI are presented in Table 2. The corresponding fitted PII parameter values for eq 14 were analogously determined from the values of Table 2 in Part 1 11 and they are here presented in Table 3. Theoretically, parameter α C,1 in Tables 2 and 3 should be zero, but it is considered here as one of the three fitting parameters in this mainly empirical equation.
The results in these two tables and also in the corresponding tables of ref 4 confirm that the values of α C,1 are always close to zero. The relative apparent molar heat capacities predicted using eq 14 for 298.15 K with parametrization PI (see Table 2) are indirectly provided in Table 1 as errors compared to the ΔC app values given in this table. The agreement between these two values is always good. Table 1 gives, in addition, the real values of the apparent heat capacity (i.e., the values for quantity C app ). These values have been based on the relative values calculated using PI and the value of C m,2 Plot of e C,app (eq 15), the deviation between the suggested apparent molar heat capacity for KCl solutions and that predicted using parametrizations PI or PII of the present study (see text) at 298.15 K as a function of molality m. The suggested values are experimental in graphs A and B except for the sets of Parker 6 and Pabalan and Pitzer. 10 The experimental data sets are introduced in Table 4 where also the used C m,2 ∞ values are given. In the treatment of the sets from refs 6 and 10, the values given in the original papers are used for C m,2 ∞ . These values are −114.6 and −114.0 J·K −1 · mol −1 , respectively. Symbols for graph A where only parametrization PI was used: •, FLD25 (for the notation, see Table 4 Tables 2 and 3 to each other at each temperature, we observe that at temperatures above 298.15 K, eq 14 yields slightly better heat capacities in conjunction with parametrization PI than with PII.
Tests of the New Relative Apparent and Partial Molar Heat Capacities against the Literature Values. The data sets of experimental C app values given in the literature and used in the present study at various temperatures are summarized in Table 4. In addition to these measured results, Parker 6 Table 4 and the smoothed values from refs 6 and 10 were mainly used here to test parametrization PI. Only the results from ref 6 in less dilute solutions were used to test both PI and PII. Because relative heat capacities are not usually reported for the experimental data sets, the C m,2 ∞ values needed in the present comparison were most often determined from the reported C app values by requiring that the appropriate sum of the errors obtained using the tested parametrization is zero at each temperature. The resulting C m,2 ∞ values are also collected in Table  4. Details of the calculations are given in the footnotes to the table.
The test results obtained at 298.15 K are illustrated in three graphs of Figure 1 where the error plots for the various data sets have been drawn. In this figure as in all following figures, these sets are abbreviated as described in Table 4: For single author articles, the acronym contains the first three letters of the author's surname. For articles with two authors, the acronym contains the first two letters of their surnames. Finally, for papers with more than two authors, only the initial letters of the surnames are included. The number in the acronym represents set's temperature in Celsius. The errors for the plots have been calculated from where the predicted values are obtained from PI except in the case of Parker's less dilute solution points 6 where both P1 and P2 were used, as mentioned. Graph A of Figure 1 gives the errors for sets FLD25, OloI25, RaRo25, TaLa25, SLL25, HeGr25, and PCJO-LW25 together with the dilute solution points from Parker. 6 Data points are included in this graph for the different sets up to 1.7 mol·kg −1 . All data support parametrization PI within experimental error up to 1.0 mol·kg −1 except those from the old set of RaRo25 where the error plot shows a clear trend. These old data are not updated here in any way, but it seems probable that the slight revision in the used constants has no influence on the observed trend. Additionally, set HeGr25 seems less precise than the other sets.
In graph A, however, the KCl set of FLD25 requires some extra considerations in the same way as in ref 4 the NaCl sets measured by Perron et al. 29 30 this group observed that a small systematic difference exists between the heat capacities per unit volume obtained by using the commercial instruments and this prototype. Through various tests, the group concluded that commercial instruments give more reliable values after normalizing to the data for aqueous NaCl solutions and after determination of the numerical correction factor specific to each instrument. In ref Plot of e C,app (eq 15), the deviation between the suggested apparent molar heat capacity for KCl solutions and that predicted using parametrization PI of the present study (see text) at temperatures below 323 K as a function of molality m. The suggested values are experimental, and the data sets are introduced in Table 4. Symbols for graph A: •, TaLa5 (for the notation, see Table 4); o, HeGr15; ▼, RSSA30; △, HeGr35; ■ , RSSA40; □ , HeGr45; ⧫ , TaLa45. Symbols for graph B where the data of Patterson et al. 21 (sets PCJO-LW in Table 4   Plot of e C,app (eq 15), the deviation between the suggested apparent molar heat capacity for KCl solutions and that predicted using parametrization PI of the present study (see text) at high temperatures as a function of molality m. The suggested values are experimental (see Table 4 for the notations) except for the sets of LiBr80, LiBr100, and Pabalan and Pitzer 10 where smoothed values were predicted. Symbols for graph A where only the data of Patterson et al. 21 (sets PCJO-LW in Table 4)   For the parameter representing the partial molar heat capacity at the infinite dilution (i.e., C m,2 ∞ ), the value for this KCl set (according to Table 4) is −111.42 J·K −1 ·mol −1 . Parameter q C,1 is a general parameter, and its value at this temperature is 28.95 J· K −1 ·mol −1 (given, for example, in ref 4). Parameter q C,2 depends on the salt and for KCl its value is (according to ref 30) 8.2 J·K −1 · mol −1 . In the tests of the present study, the heat capacities given in set FLD25 were predicted using eq 16, but they are so close to the original values that no corrections are needed. This is shown in Table 5, where the experimental apparent molar heat capacities from ref 24 and the corrected values obtained using eq 16 are reported.
Graph B displays data for solutions up to 4 mol·kg −1 . Above 1.5 mol·kg −1 , all data deviate from the predicted values and the deviations seem to be consistent. Parametrizations PI and PII are compared in graph C using the smoothed data from Parker up to 3.0 mol·kg −1 . PII is only slightly better than PI in the less dilute solutions, and the difference is close to the scattering of errors between various sets in graph B of this figure. Hence, PII is not Defined by equation ΔC app = C app − C m,2 ∞ , where C app is defined in eq 6 and C m,2 ∞ is the partial molar heat capacity at infinite dilution. b The values have been calculated using parametrization PI (see text). c m°= 1 mol·kg −1 . d Unit is J·K −1 ·mol −1 and the value is given at temperature T expressed in parenthesis as (T − 273.15 K)/K. Defined by equation ΔC app = C app − C m,2 ∞ , where C app is defined in eq 6 and C m,2 ∞ is the partial molar heat capacity at infinite dilution. b The values have been calculated using parametrization PI (see text). c m°= 1 mol·kg −1 . d Unit is J·K −1 ·mol −1 and the value is given at temperature T expressed in parenthesis as (T − 273.15 K)/K.

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Article considered in the present study any further because this parametrization lacks the traceability and transparency of PI.
The heat capacities obtained from dilute KCl solutions at temperatures other than 298 K up to 318 K are considered in graphs A and B of Figure 2. According to graph A, almost all heat capacities up to 2.0 mol·kg −1 can be reproduced within experimental error using parameterization PI. In graph A, the only exceptions are four points in set TaLa5, one point in set TaLa45 (see the caption), and one point in set HeGr45 where m = 0.01 mol·kg −1 . The points at the last-mentioned molality were omitted here as outliers from the treatment of the sets given by Hess and Gramkee 22 at temperatures other than 308.15 and 318.15 K (see Table 4). In graph B where the sets from Patterson et al. 21 are considered, only one point at 278.15 K that is close to 0.5 mol·kg −1 shows a substantial error. This error is about 12 J· K −1 ·mol −1 . All other absolute errors are much smaller than this value. Thus, these data support parametrization PI at all temperatures from 273.15 to 318.15 K. However, it should be noted that Patterson et al. 21 measured their sets at a pressure of 350 kPa, but no corrections were used here to treat these data because this pressure is quite close to the standard pressure of 101.325 kPa.
In chemical thermodynamics, heat capacities in solutions are often expressed by means of the partial molar heat capacities of the solvent and solute. The solute values can be easily obtained with eq 8 from the polynomials presented in eq 14 together with the parameter values given in Table 2. Parametrization PI from 273.15 to 313.15 K was also tested against the relative partial molar heat capacities reported by Harned and Cook 13 on the basis of their amalgam cell measurements at various temperatures in this range using intervals of 5 K. In these tests, the reported ΔC m,2 value (definition is shown in eq 9) is compared to the predicted one. The resulting error is defined by and it is presented as a function of molality m. These errors are shown in Figure 3. Up to 1.0 mol·kg −1 , all of these data agree well with the predicted values obtained using PI. At temperatures below 288.15 K, this agreement also continues above this molality limit. At 278.15 K, the good agreement extends up to 4 mol·kg −1 with all of the absolute errors being smaller than 15 J· K −1 ·mol −1 (these errors are outside the molality scale of Figure 3 and cannot be seen in this figure). One possible interpretation

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Article for this result is that the calorimetric data from sets TaLa5 and PCJO-LW5 (considered in graphs A and B of Figure 2, respectively) at molalities close to or higher than 0.5 mol·kg −1 are not entirely reliable. This is probably true despite the amalgam cell measurements not being the most accurate method to obtain thermodynamic data.
Parametrization PI is tested in the four graphs of Figure 4 against the experimental C app values above 318 K and up to 373 K. All errors in graph A were determined using data from the sets of Patterson et al., 21 and the results support parametrization PI as the agreement is excellent up to 0.2 mol·kg −1 . At various temperatures from 323 to 368 K up to this molality, the errors are highly overlapping in this graph, but all absolute errors are below 5 J·K −1 ·mol −1 . For larger molalities, the absolute errors increase, but the agreement is still good as can be seen in this graph and in graph D (where T = 373.15 K). The results in graph C regarding the temperature range of 323−363 K and especially those in graph D at 373.15 K show the scattering of the errors from one set to another at the same molality. All errors in graphs A and B are less than or only comparable to this scattering. Saluja et al. 25 measured their data sets (sets SLL in Table 4) at a pressure of 600 kPa, but again no corrections were utilized to

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Article modify these data as in the case of Patterson et al. 21 in Figures 2A  and 4A because of the proximity of this pressure to the standard value of 101.325 kPa.
Recommended Values for the Relative Apparent and Partial Molar Heat Capacities of KCl in Aqueous Solutions. The practically main and also final result of the present study is eq 14 with the parameter values presented in Table 2 for the relative apparent molar heat capacity. As described above, we have calculated the parameter values for this table using only eqs 1−3 and 10 with the procedure based on eq 12 and on the thermodynamic equations of eqs 5 and 7. The resulting table for eq 12 is given in Table 1 of ref 11. On the basis of eq 14 and the parameter values in Table 2 of the present study, the relative apparent molar heat capacities for dilute KCl solutions are given at rounded molalities in Table 6 from 273.15 to 298.15 K. Table 7 displays these values in the range 303.15− 328.15 K, Table 8 in the range 333.15−348.15 K, and finally Table 9 in the range 353.15−373.15 K. The corresponding values for the partial heat capacities are collected in Tables 10−13, respectively. The values in all of these tables have been obtained using parametrization PI and are therefore transparent and fully traceable. It has been shown below and previously that the values in the tables are supported by all high-quality experimental data in the literature for dilute KCl solutions.

■ CONCLUSIONS
Based on extensive testing against the existing activity and osmotic coefficient data, 5,12 high-quality enthalpy data, 11 and heat capacity data (present study), we conclude that the measured results for dilute KCl solutions can often be predicted within their experimental uncertainty up to a molality of 1.0 mol· kg −1 in the temperature range 273−373 K with a simple reparametrization of the Huckel equation. At least up to 0.2 mol· kg −1 , completely traceable thermodynamic quantities can be obtained using the new Huckel parametrization in all of these temperatures. These main results for solutions of this salt are fully analogous to those obtained in refs 3 4, and 12 for NaCl solutions. In the present article, new values for the relative apparent and partial molar heat capacities (i.e., for ΔC app and ΔC m,2 , where 2 refers to the solute) at the temperatures from 273 to 373 K in intervals of 5 K are tabulated for dilute KCl solutions at least up to 0.5 mol·kg −1 using our simple model. The tabulated values are congruent with all of the available highquality experimental data and represent the most accurate values suggested for this system to date. This publication supplements the tables of the previous articles, 5,11 where we observed that the two-parameter Huckel equation can be safely used for activity and osmotic coefficients 5 and for the quantities associated with the partial molar enthalpies 11

Notes
The authors declare no competing financial interest. Defined by equation ΔC part = C m,2 − C m,2 ∞ , where C m,2 is the partial molar heat capacity of the salt and C m,2 ∞ is its value at infinite dilution. b The values have been calculated using parametrization PI (see text).