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Journal of Solution Chemistry, Vol. 30, No. 5, 2001

Determination of Stoichiometric Dissociation Constants of Benzoic Acid in Aqueous Sodium or Potassium Chloride Solutions at 25◦ C1 Jaakko I. Partanen,2 * Pekka M. Juusola2 and Pentti O. Minkkinen2 Received ; revised February 27, 2001 Equations were determined for the calculation of the stoichiometric (molality scale) dissociation constant K m of benzoic acid in dilute aqueous NaCl and KCl solutions at 25◦ C from the thermodynamic dissociation constant K a of this acid and from the ionic strength Im of the solution. The salt alone determines mostly the ionic strength of the solutions considered in this study and the equations for K m were based on the single-ion activity coefficient equations of the H¨uckel type. The existing literature data obtained by conductance measurements and by electromotive force (EMF) measurements on Harned cells were first used to revise the thermodynamic value of the dissociation constant of benzoic acid. A value of K a = (6.326 ± 0.005) × 10−5 was obtained from the most precise conductivity set [Brockman and Kilpatrick] and this value is supported within their precisions by the less precise conductivity set of Dippy and Williams and by the EMF data set measured by Jones and Parton with quinhydrone electrodes. The new data measured by potentiometric titrations in a glass electrode cell were then used for the estimation of the parameters of the H¨uckel equations of benzoate ions. The resulting parameters were also tested with the existing literature data measured by cells with and without a liquid junction. The H¨uckel parameters suggested here are close to those determined previously for anions resulting from aromatic and aliphatic carboxylic acids. By means of the calculation method based on the H¨uckel equations, K m can be obtained almost within experimental error at least up to Im of about 0.5 mol-kg−1 for benzoic acid in NaCl and KCl solutions. KEY WORDS: Ionic activity coefficients; stoichiometric dissociation constant; benzoic acid; electrochemical cells; conductivity; glass electrode.

1 An

account of the NaCl results of this work was presented at the 11th Euroanalysis Conference, Lisboa, Portugal, September, 2000. 2 Department of Chemical Technology, Lappeenranta University of Technology, P.O. Box 20, FIN53851 Lappeenranta, Finland.

443 C 2001 Plenum Publishing Corporation 0095-9782/01/0500-0443$19.50/0 °

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1. INTRODUCTION In this study, we continue the studies(1−12) dealing with the thermodynamics of weak acid solutions. The purpose of these studies is to determine the stoichiometric dissociation constants (for example, the molality scale dissociation constants, K m ) for solutions of different weak acids as reliably as possible from potentiometric titration data and, in addition, from data measured on galvanic cells without liquid junctions. The cells used in the latter technique contain usually a hydrogen electrode and a silver–silver chloride electrode in a solution of the weak acid considered; these cells are often called Harned cells. The K m values for weak acids can usually be determined more accurately from Harned cell data than from potentiometric titration data, but the determination by the latter technique is experimentally much easier. In the previous studies,(6,8−11) it is shown that these two experimental techniques give identical K m values for formic acid,(10) acetic acid,(11) and propionic acid(8) in salt solutions as a function of the ionic strength Im . In practical problems of solution chemistry, the stoichiometric dissociation constants have a great importance. If it was possible to determine K m for a weak acid solution, it would also be possible to calculate directly the molalities (or other composition variables) for the species existing in the weak acid solution. It is well known that the K m for weak acids is, in very dilute solutions, dependent at constant temperature only on the ionic strength of the solution. In more concentrated solutions, it is also dependent on the other composition variables of the solution. In, for example, analytical and biochemical applications, weak acid solutions contain usually a large amount, compared to the amount of weak acid, of an inert salt (e.g., NaCl or KCl), to keep the ionic strength of the solution constant despite the dissociation of the acid. It has been previously shown that in such solutions K m for the weak acid is dependent only on the molality of the salt (or on Im ; see, for example, the K m,1 results for glycine in Fig. 1A of Ref. 7). For most charge types of the dissociation reactions, the ionic-strength dependence of K m is strong. It has been previously observed, however, that this dependence follows in many cases accurately simple equations for ionic activity coefficients. The other purpose of our research in this field is to determine equations for the calculation of K m for different weak acids in salt solutions at 25◦ C. In the previous studies, the H¨uckel and Pitzer equations have been used for single-ion activity coefficients. In the present study, the H¨uckel equations were used for the analysis of thermodynamic data for benzoic acid (C6 H5 COOH = BenzOH) in NaCl and KCl solutions. Pitzer equations were not considered in this study because Pitzer parameters are missing for the component salts of sodium and potassium benzoate in these solutions, and these parameters (in addition to other relevant parameters) cannot be accurately estimated from the potentiometric titration data only. Usually the binary Pitzer parameters for different salts have been determined in the literature from osmotic coefficients obtained from isopiestic data.

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Activity coefficients are related to the osmotic coefficients via the Gibbs–Duhem equation. BenzOH is the simplest aromatic carboxylic acid. The thermodynamic value of the dissociation constant K a of BenzOH at 25◦ C has played the central role in the determination of the parameters for the well-known linear free-energy relationship called the Hammett equation [see e.g. Hammett(13) ]. This equation covers the thermodynamic and kinetic data for meta- and para-substituted benzene derivatives. Because of this application, the determination of the dissociation constant of BenzOH has been and is still especially interesting. In thermodynamic handbooks of electrolyte solutions [see e.g. Robinson and Stokes(14) ], K a values have been tabulated for many weak acids. The values in these tables have been most often determined by either the conductometric method or by the method based on electromotive force (EMF) data measured by galvanic cells without a liquid junction (often by Harned cells). For BenzOH in the tables of Robinson and Stokes, a value of K a = 6.295 × 10−5 is given and it was determined by the EMF method by Jones and Parton,(15) see below. Also, the values obtained by other investigations are reported by Robinson and Stokes.(14) Spectrophotometrically, Robinson and Biggs(16) have obtained a value of 6.27 × 10−5 , and this value is supported by Read.(17) On the basis of earlier literature, Matsui et al.(18) suggested a value of 6.25 × 10−5 , and this value is also supported by the glass electrode results of Travers et al.(19) who determined a confidence interval of (6.25 ± 0.07) × 10−5 for K a . The most recent K a value for BenzOH was presented by Kettler et al.(20) on the basis of hydrogen electrode measurements on cells containing a liquid junction (see below), and this value is (6.22 ± 0.09) × 10−5 where the uncertainty represents three times the standard deviation. This paper(20) contains also a review of the K a values recommended in the literature for this acid at 25◦ C. According to these results, benzoic acid is stronger than acetic acid for which K a = 1.758 × 10−5 ,(3,21) but not as strong as formic acid for which K a = 1.82 × 10−4 .(10,22) Stoichiometric dissociation constants have been presently often determined with glass electrodes. For amino acids, these works have been reviewed by IUPAC (see, e.g., Kiss et al.(23) ). These reviews revealed the absence of systematic studies on the influence of Im on K m . For other weak acids, the systematic studies are also rare and the existing studies in this field are usually concentrated on the acids used as pH buffer substances. Results of the systematic glass electrode works have been reviewed by Daniele et al.(24) and Sastre de Vicente.(25) The research group of Sastre de Vicente has also recently measured BenzOH with glass electrodes in aqueous KNO3 solutions,(26) and these data will be considered below. In the present study, potentiometric titrations were carried out in a glass electrode cell containing aqueous mixtures of BenzOH and NaCl or KCl at 25◦ C and the result of these titrations used to determine experimental K m values. It is not easy to solve K m from titration data. Calibrations, asymmetry potentials, and

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liquid junction potentials are difficult problems associated with the determination of K m by this experimental technique, and these problems are not, at the moment, entirely solved. In this study, a recently developed calculation method is used to overcome these problems (see Ref. 10). K m for BenzOH at different experimental salt molalities was calculated by this method. Activity parameters for the H¨uckel equation were then estimated from the K m values obtained by this method. The estimated H¨uckel parameters were compared here to those suggested previously(6,10) for formic and acetic acids (which have almost the same values) on the basis of the literature data measured by the EMF method. The parameters obtained by our glass electrode cell are close to those suggested previously for these two acids (see below). On the basis of the resulting parameters, it is possible to recommend equations for K m of BenzOH in NaCl and KCl solutions at least up to Im of 0.5 mol-kg−1 . The recommended K m values are tabulated below at rounded ionic strengths.

2. EXPERIMENTAL Potentiometric BenzOH titrations were carried out in aqueous NaCl and KCl solutions at 25◦ C. The two series of salt solutions [the NaCl and KCl (pro analysi, Riedel-de Ha¨en) series] were prepared in RO-filtered water (Millipore), and the concentrations in these series were as follows: 0.080, 0.160, 0.240, 0.320, 0.400, 0.500, and 0.700 M. In addition, a 0.02017 M BenzOH (pro analysi, Merck) solution, a 0.200 M NaOH (Fixanal, Riedel–de Ha¨en) solution, and a 0.200 M KOH (Titrisol, Merck) solution were prepared. The solutions titrated were prepared by mixing a volume of 10.00 cm3 of the BenzOH solution, 100.0 cm3 of a salt solution, and 25.00 cm3 of water (see Ref. 2). The NaCl solutions were titrated by using the NaOH titrant and the KCl solutions by using the KOH titrant. During the titrations, the EMF was measured by means of an N62 combination electrode and a CG841 pH meter, both manufactured by Schottger¨ate. The resolution of the meter was 0.1 mV. The titrant was added in increments of 0.050 cm3 by a Dosimat (Metrohm). Standard buffer solutions for which at 25◦ C pH = 4.005 and pH = 6.865(27) were used to check the stability of the measuring system between titrations. The pH meter reproduced usually the same reading within 0.2 mV in these buffer solution tests. To check further the stability of measuring system, a titration of formic acid (ForOH) was carried out before the BenzOH titration in each salt solution at the same conditions. From the results of these ForOH titrations, the K m values [= K m (ForOH, obsd.)] were determined by the method used for BenzOH and described below. The resulting values are shown in Table I (NaCl solutions) and Table II (KCl solutions), where they can be compared to those [= K m (ForOH, recd.)] calculated by the equations recommended in Ref. (10) for K m of this acid in NaCl and KCl solutions. The good agreement between the observed and

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Dissociation Constants of Benzoic Acid Table I.

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Results of Titrations of Benzoic Acid in NaCl Solutions by Using the Cells of Type II at Different Ionic Strengths (Im ) with a Base Solution at 25◦ C (Series BNC)

Im /(mol-kg−1 )

0.0595

[104 K m (ForOH,obsd.)]a [104 K m (ForOH,recd.)]b [104 K m (obsd.)]c [104 K m ]d [104 n t /mol]e [E 0 /mV] f Symbol

0.1191

0.1789

0.2388

0.2988

0.3741

0.5251

2.75 3.03 3.21 3.35 3.36 3.54 3.58 2.74 3.02 3.19 3.30 3.37 3.43 3.47 0.996 1.063 1.134 1.161 1.201 1.239 1.293 0.963 1.073 1.144 1.194 1.231 1.264 1.302 1.992 2.008 2.003 2.011 2.006 2.007 2.011 380.15 380.29 380.75 380.77 381.26 381.75 382.52 BNC1 BNC2 BNC3 BNC4 BNC5 BNC6 BNC7

a The stoichiometric dissociation constant determined for formic acid from the data of the corresponding

formic acid titration that preceded the benzoic acid titration. stoichiometric dissociation constant of formic acid calculated by an equation recommended in Ref. (10). c The stoichiometric dissociation constant of benzoic acid determined from the titration data by the method described in the text. d The stoichiometric dissociation constant of benzoic acid calculated by Eq. (4) with the recommended K a and activity parameters. e The optimized amount of benzoic acid. f The value of parameter E used in the calculation of the EMF errors for Fig. 3. It was determined 0 by requiring that the sum of all errors in the data set is zero (i.e., by Eqs. 19, 22 and 23). b The

recommended K m values for formic acid, in all cases (the errors are always less than ±0.11 × 10−4 ), guarantees the quality of the new data for BenzOH. 3. RESULTS In the H¨uckel method, the following equation is generally used for the activity coefficient γ of ion i on the molality scale: ln γi = −α(Im )1/2 /[1 + Bi (Im )1/2 ] + bi,MCl Im Table II.

Results of Titrations of Benzoic Acid in KCl Solutions by Using the Cells of Type II at Different Ionic Strengths (Im ) with a Base Solution at 25◦ C (Series BKC)a

Im /(mol-kg−1 ) 104 K m (ForOH,obsd.) 104 K m (ForOH,recd.) 104 K m (obsd.) 104 K m 104 n t /mol E 0 /mV Symbol a See

(1)

footnotes to Table I.

0.0595

0.1193

0.1792

0.2393

0.2997

0.3755

0.5280

2.71 2.99 3.16 3.24 3.34 3.32 3.37 2.73 3.00 3.16 3.25 3.32 3.36 3.36 0.950 1.083 1.130 1.169 1.199 1.211 1.245 0.957 1.060 1.123 1.166 1.195 1.218 1.235 2.008 2.006 2.007 2.009 2.012 2.009 2.010 380.74 380.38 379.89 379.56 379.48 379.80 380.22 BKC1 BKC2 BKC3 BKC4 BKC5 BKC6 BKC7

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Table III. Ion Parameters for the H¨uckel Equation (Eq. 1) in NaCl and KCl Solutions at 25◦ C and Results of the Regression Analysis Obtained by Eq. (11) Ion 105 K a a pK a e 105 K a,obsd. (NaCl) f pK a,obsd. (NaCl) f 105 K a,obsd. (KCl) f pK a,obsd. (KCl) f B/(mol-kg−1 )−1/2 bNaCl bKCl bNaCl (obsd.)m bKCl (obsd.)m

H+

Cl−

HCOO− 18.2b 3.740

1.25g 0.238g 0.178h

1.25g 0.238g 0.178h

1.4h 0.189k 0.308k

CH3 COO−

C6 H5 COO−

1.758c 4.755

6.33d 4.199 6.23 ± 0.06 4.206 ± 0.004 6.36 ± 0.05 4.197 ± 0.003 1.25 j 0.09l 0.25l 0.09 ± 0.03 0.25 ± 0.03

1.6i 0.189i 0.308h

a

The thermodynamic dissociation constant for the corresponding acid. Determined by Prue and Read (Ref. 22) from EMF data. c Determined from the conductivity data measured by MacInnes and Shedlovsky (Ref. 21); see also Ref. (3). d Determined in this study from the literature data obtained by the conductance and EMF methods. e pK = −lg K . a a f An estimate of K of benzoic acid, see also footnote e. Determined from the titration data and the a standard deviation, which is also given. (The results of titration BNC1 were omitted from the regression analysis of BNC series.) g Determined from Harned cell data (Ref. 1). h Determined from Harned cell data (Ref. 6). i Determined from Harned cell data (Ref. 3). j Suggested in this study. This value was determined from Harned cell data for hydrogen phthalate and phthalate ions (Ref. 4). k Determined from Harned cell data (Ref. 6) and checked by potentiometric data (Ref. 10). l Recommended in this study. m b for benzoate ions determined from the titration data; the standard deviation is also given (the results of titration BNC1 were omitted from the regression analysis of BNC series). b

where Im is the ionic strength on the molality scale and α is the Debye–H¨uckel parameter equal to 1.17444 (mol-kg−1 )−1/2 [see Archer and Wang(28) ]. Bi and bi,MCl are the parameters that are dependent on ion i, and bi,MCl is, in addition, dependent on the salt MCl present in the system. In Table III are given the parameter values, which have been determined previously for this equation and are considered here. Theoretically, single-ion activity coefficients (like values calculated by this equation) are always hypothetical because there is no direct method to measure them (see discussions, e.g., in Ref. 5). In the present study, as earlier, we use equations for the single-ion activity coefficients to calculate K m values for a weak acid (BenzOH) in salt solutions. This K m is without question an experimentally obtainable quantity. The interest is focused in these studies on the

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simple equations for ionic activity coefficients and not on the other use of these theoretically complicated quantities. The thermodynamic dissociation constant K a for BenzOH is given by K a = γH γA m H m A /(γHA m HA ) = (γH γA /γHA )K m

(2)

where H refers to protons, HA to benzoic acid molecules, and A to benzoate ions. For neutral species HA, it is assumed in the H¨uckel method that γHA = 1 (the validity of this assumption has been discussed previously; see Ref. 6). The stoichiometric dissociation constant K m in Eq. (2) is defined by K m = m H m A /m HA

(3)

The following equation can be presented for K m of BenzOH in aqueous NaCl or KCl solutions: ln K m,MCl = ln K a + α(Im )1/2 {1/[1 + BH (Im )1/2 ] + 1/[1 + BA (Im )1/2 ]} −(bH,MCl + bA,MCl )Im

(4)

where MCl refers to either NaCl or KCl. Here, the values of BH = 1.25 (molkg−1 )−1/2 , bH,NaCl = 0.238, and bH,KCl = 0.178 were used in this equation (see Table III). The thermodynamic value K a in this equation is now redetermined from the conductance data measured by Brockman and Kilparick,(29) Saxton and Meier,(30) Dippy and Williams,(31) Jeffery and Vogel(32) and Strong et al.(33) and from the EMF data measured by Jones and Parton(15) with quinhydrone electrode cells. The preliminary calculations revealed that a value of BA = 1.25 (mol-kg−1 )−1/2 can be used in this equation. This value was previously(4) suggested for the two aromatic anions resulting from ortho-phthalic acid. The parameters bBenzO,NaCl and bBenzO,KCl for this equation are determined in the present study from the new potentiometric titration data. The revision of the thermodynamic dissociation constant of benzoic acid is here based on the conductance and EMF data taken from the literature. Some details of the conductance data used in this determination are shown in Table IV. These data are treated in the following way: For each point of the data sets presented in this table, a value of K c (the stoichiometric dissociation constant on the concentration scale) was iteratively calculated from the reported concentrations and experimental molar conductivities 3m . K c is defined by K c = cH cA /cHA

(5)

where the concentrations are expressed in mol-dm−3 . In these calculations, the following relationship(21) was used: −1

(λH + λA )/(S-cm2 -mol ) = 30m − x1 (ci )1/2 + x2 ci [1 − x3 (ci )1/2 ]

(6)

where x3 has a theoretical value (= 0.2275) and λ refers to the molar ionic conductivities and ci to the ionic concentration (i.e., ci = cH = cA ). The parameters

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Partanen, Juusola, and Minkkinen List of the Conductivity Data Sets Used to Determine the Thermodynamic Dissociation Constant of Benzoic Acid at 25◦ C

Symbola

Ref.

(105 K a )b

Nc

103 Im /(mol-kg−1 )d

(30m )e

x1e

x2e

BroKilKCl SaxMeiNaCl DipWilNaCl JefVogNaClg SBRRPKClh

29 30 31 32 33

6.326 ± 0.002 6.302 ± 0.004 6.338 ± 0.009 6.432 ± 0.006 6.346 ± 0.003

25 10 17 17 8

0.05–0.67 0.095–0.53 0.15–0.69 0.09–0.65 0.05–0.26

382.10 381.96 381.96 f 381.96 f 382.10i

146.72 146.81 146.81 f 146.81 f 146.72i

193.5 174.8 174.8 f 174.8 f 193.5i

a In

the symbol, the initials of the names of the authors are followed by the salt whose conductance was used in the determination of the parameters of Eq. (6). b The thermodynamic dissociation constant calculated from the results shown in Fig. 1, and the standard deviation is also shown. c Number of determinations used in the calculation of K . a d The ionic strength range used in the calculation of K . a e A parameter for Eq. (6); the value is given in the paper that the data have been taken from. f Determined by Saxton and Meier (Ref. 30). g The point where c = 0.0000521 mol-dm−3 was omitted. t h The rest points of Strong et al. (Ref. 33) (N = 8), where I = 0.000358−0.001215 mol-kg−1 m give a value of K a = (7.26 ± 0.03) × 10−5 (K a decreases gradually as a function of Im ) and this value does not agree with the other values. i Determined by Brockman and Kilpatrick (Ref. 29).

used in these calculations are given in Table IV. From a resulting K c value, K m [see Eq. (3)] was calculated by K m = K c /(ρ − MHA ct )

(7)

where ρ is the density of the solution, MHA the molar mass of BenzOH, and ct the total concentration of this acid. The solutions considered in these calculations are so dilute that the following assumption can be made: ρ = ρ(H2 O) = 0.9970 kgdm−3 . The K m values that were obtained from the experimental conductivity data were then converted into the K a estimates by Eq. (2). For the activity coefficients, Eq. (1) was used without the term containing parameter bi . This can be done for the solutions used in conductance measurements because the concentration of ions in these solutions is always small. The K a values obtained in this way are shown in Fig. 1, where they are presented as a function of the ionic strength. From the estimates shown in this figure, the mean values shown in Table IV were obtained for K a for the different data sets. Jones and Parton(15) have measured electromotive forces E on cells of the following type: Pt(s) | quinhydrone(s) | HA(aq, m a ), NaA(aq, m b ), NaCl(aq, m s ) | AgCl(s) | Ag(s)

(I)

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Fig. 1. Estimate of the thermodynamic dissociation constant, K a , of benzoic acid at 25◦ C as a function of the ionic strength Im . The estimates were calculated from the conductivity data shown in Table IV as described in the text. Symbols of the different sets are shown in the figure; see also Table IV.

In the six solutions studied by Jones and Parton, m a varies from 0.0045 to 0.019 mol-kg−1 and the molalities of NaA and NaCl are, in every point, very close to this molality. The electromotive force E of cells of type I is given by E = E Ho -C − (RT /F) ln (γH γCl m H m Cl )

(8)

where Cl refers to chloride ions, E Ho -C is the standard EMF (a value of −0.47699 V was used; see Jones and Parton(15) and Hovorka and Dearing(34) ). The data in this set were used in the determination of the thermodynamic dissociation constant K a as follows: The observed K m value was first calculated from each experimental point of Jones and Parton by equations ¡ ¢ (9) ln m H = E Ho -C − E F/(RT ) − ln (γH γCl m Cl ) K m = m H (m b + m H )/(m a − m H )

(10)

In these calculations, it was assumed that the observed K m depends in these NaCl solutions only on the ionic strength of the solution, as suggested by Eq. (4). In each K m determination, Eq. (1) was used for the activity coefficient of H+ and Cl− ions

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and the relevant parameters for this equation are given in Table III. These parameter values apply to HCl in NaCl or KCl solutions and are also very probably valid for HCl in dilute aqueous mixtures of different weak-acid species and NaCl or KCl (see Ref. 6). Iterative calculations were needed in this determination because the dissociation of BenzOH have an influence on the ionic strength. The following experimental K m values were obtained by these calculations: 103 Im /(mol-kg−1 )

9.0097

19.283

24.264

29.716

33.548

38.078

105 K m

7.621

8.313

8.566

8.677

8.863

8.869

The thermodynamic dissociation constant K a were determined from these K m values by ln K m,MCl − α(Im )1/2 {1/[1 + BH (Im )1/2 ] + 1/[1 + BA (Im )1/2 ]} = y = ln K a − (bH,MCl + bA,MCl )Im

(11)

Quantity y can be calculated from each experimental K m value and, therefore, Eq. (11) represents an equation of a straight line y vs. Im . By means of this method, the following results were obtained from these K m values: K a = 6.3 × 10−5 (pK a = 4.199 ± 0.005) and bA,NaCl + bH,NaCl = 0.5 ± 0.4, where the standard deviations are also given for the estimated quantities. The estimate obtained for K a in this regression analysis agrees well with the results of conductivity data presented in Table IV, but is not so precise as those. The data set BroKilKCl seems to be the most reliable of the sets used in this study in the determination of K a for BenzOH at 25◦ C. According to Table IV, the following confidence interval at a significance level of 0.95 can be presented for K a on the basis of this data set: K a = (6.326 ± 0.005) × 10−5 . From the other conductivity sets in Table IV, set DipWilNaCl supports this value within its precision. The following confidence interval is obtained from this set: K a = (6.34 ± 0.02) × 10−5 . The points included in the determination from set SBRRPKCl give the following interval: K a = (6.346 ± 0.007) × 10−5 , which is rather close to the interval obtained from the data of set BroKilKCl. The same is also true for the interval obtained from set SaxMeiNaCl [i.e., that of K a = (6.302 ± 0.008) × 10−5 ]. Only the conductivity data of set JefVogNaCl in Table IV gives for K a a value [= (6.432 ± 0.013) × 10−5 ] that is considerably different from the other values. The EMF data set of Jones and Parton gives the following confidence interval [see above, K a = (6.3 ± 0.2) × 10−5 ], and this large interval also supports, within its precision, the confidence interval obtained from set BroKilKCl. According to these results, a value of 6.33 × 10−5 will be here recommended for K a of benzoic acid at 25◦ C.

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Activity parameters bNaCl and bKCl for Eq. (1) for benzoate ions are determined from the potentiometric titration data measured for this study. The cell used can be expressed by the following symbol: Reference electrode | KCl(aq, c = 3.0 mol-dm−3 ) ||HA(aq, m t − m b ), MA(aq, m b ), MCl(aq, m s ) | glass electrode

(II)

The experimental K m values obtained by the present calculations are shown in Tables I and II. These values were calculated as follows. A glass electrode parameter (E0 , the exact definition is given below), the dissociation constant K m , and the amount of the acid (nt ) in the titration vessel were simultaneously estimated from each titration data set. The following equation is valid for electromotive forces measured on the glass electrode cell of type II in any weak acid solution: E = E o + (RT /F) ln aH

(12)

o

where aH is the activity of protons and E is a term that includes contributions of the reference electrode, liquid junction, standard-glass electrode, and asymmetry potentials [see, e.g., May et al.(35) ]. It is assumed in all present titrations that this term remained constant during the titration. The three parameters were estimated by means of the following equation: E = E o + (RT /F) ln γH + (RT /F) ln m H = E 0 + (RT /F) ln m H

(13)

where E 0 = E o + (RT /F) ln γH is the glass electrode parameter that must be estimated, and it is also constant during each titration at a constant ionic strength [see Eq. (1)]. m H is calculated for each titration point by m 2H + (K m + m b )m H + K m (m b − m t ) = 0

(14)

where m b is the molality of base, NaOH or KOH, in the solution titrated and m b = cb V /w1 , where cb is the concentration, V is the volume of the base solution added in the titration, and w1 is the mass of water in the solution titrated. m t in Eq. (14) is the total molality of BenzOH in the solution titrated and m t = n t /w1 , where n t is the amount of this substance. K m and the glass electrode parameter E 0 were calculated for each data set studied, containing N points, by equations X (E i − E pred,i ) = 0 (15) ln xi = (E i − E 0 )F/(RT )

(16)

K m,i = xi (xi + m b,i )/[(n t /w1,i ) − m b,i − xi ] µX ¶ Km = K m,i /N

(17)

m H,i = {[(K m + m b,i )2 + 4[(n t /w1,i ) − m b,i ]K m ]1/2

(18)

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− (K m + m b,i )}/2 E pred,i = E 0 + (RT /F) ln m H,i

(19) (20)

where x i is the experimental molality of H+ ions in point i. In addition, the amount (n t ) of BenzOH was optimized by requiring that the following square sum, S(E),

Fig. 2. Flow chart for the calculation of K m , n t , and E 0 from the potentiometric titration data (m o = 1 mol-kg−1 , for the other symbols; see text and Eqs. 15–21).

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X

(E i − E pred,i )2

(21)

Details of these calculations appear in the flow chart presented in Fig. 2. The activity parameters bA,NaCl and bA,KCl were determined linear regression analysis using Eq. (11). The results obtained by this regression analysis from the K m (obsd.) data presented in Tables I and II are shown in Table III. In this table are also included the activity parameters for salt solutions of formic and acetic acids determined previously from data measured by the Harned cells. Table III shows that the present calculation method gives for benzoate ions in NaCl and KCl solutions the b parameters which are rather close to those of formate and acetate ions (which have the same values). According to this table, the following H¨uckel parameters can be recommended in this study for Eq. (1): BBenzO = 1.25 (mol-kg−1 )−1/2 , bBenzO, NaCl = 0.09 and bBenzO,KCl = 0.25. The regression analysis by Eq. (11) with the data in Tables I and II also gives two estimates for K a of BenzOH, shown in Table III. Both estimates agree well with the thermodynamic value suggested above for this acid (i.e., with K a = 6.33 × 10−5 ). The recommended K m values, calculated by Eq. (4) by means of this K a and the activity parameters recommended in the previous paragraph, are shown in Tables I and II for all benzoic acid solutions considered. 4. DISCUSSION The H¨uckel parameters determined above are tested with the experimental data measured in this study (see Tables I and II). The recommended K m values given in these tables were used in the tests. Parameter E 0 for Eq. (13) was calculated for each data set by Eq. (19) and the following equations: E 0,i = E i − (RT /F) ln m H,i ³X ´ E0 = E 0,i /N

(22) (23)

The resulting values for E 0 are included in Tables I and II. The results of the tests are shown as error plots in graphs A (NaCl) and B (KCl) of Fig. 3, where the EMF error defined by eE = E(obsd.) − E(pred.)

(24)

is presented for each data set as a function of the added base volume. Most of the error plots in these graphs are random and all errors are comparable to the resolution of the pH meter (0.1 mV). Therefore, the data presented in Tables I and II support the calculation method well. The H¨uckel parameters were further tested with the EMF data measured by Jones and Parton on cells of type I (see above) and by Briscoe and Peake(36) on

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Fig. 3. The difference between the observed and predicted electromotive forces, eE in Eq. (24), as a function of the titrant volume in the titrations of benzoic acid by the base (NaOH or KOH) solution in NaCl solutions (A, series BNC) and in KCl solutions (B, BKC). The predicted EMF was calculated by Eqs. (4, 13, and 14) using the thermodynamic dissociation constant of 6.33 × 10−5 , the recommended activity parameters, and the glass electrode parameter E 0 shown in Tables I and II. Symbols of the different sets are shown in the graphs.

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cells of the following type: Pt(s) | H2 (g, f = 101.325 kPa) | HA(aq,m a ), NaA(aq,m b ), NaCl(aq,m s ) | AgCl(s) | Ag(s)

(III)

where f is the fugosity. In these tests, the observed K m values are compared to the predictions of Eq. (4) calculated by means of the recommended parameter values. The observed K m values for the data set of Jones and Parton were shown above and those obtained in the same way from the set of Briscoe and Peake are the following (the point where m a = 0.02338 mol-kg−1 was omitted as an erroneous point): 103 Im /(mol-kg−1 ) 105 K m

24.482

32.117

33.248

33.717

77.429

8.599

8.831

8.949

8.850

9.828

The best value of E Ho -C = 0.2234 V was used in latter calculations [Harned and Ehlers(37) suggested for this cell a value of E Ho -C = 0.22250 V], and the authors of this study emphasized that it was extremely difficult to obtain reproducible results with aqueous benzoic acid solutions in hydrogen electrode cells of type III. The results of the comparison between the observed and predicted K m values are shown in Fig. 4. In this figure, the pK m error defined by e(pK m ) = pK m (obsd.) − pK m (pred.)

(25)

is presented for each experimental point as a function of the ionic strength. Despite the experimental difficulties explained above, the predicted K m values in this graph agree well with the experimental counterparts. The largest absolute pK m error in these tables is less than 0.01. This value can be compared to the conventional pK m error of 0.06 suggested by Albert and Serjeant(38) in their well-known monograph. Therefore, the recommended activity parameters are well supported by the data of Jones and Parton and Briscoe and Peake measured on cells without a liquid junction. Kettler et al.(20) have succeeded in measuring reproducible EMF data by using hydrogen electrodes in BenzOH solutions. They measured concentration cells of the following type: Pt(s) | H2 (g) | HA(aq), NaA(aq), NaCl(aq,m s ) || HCl(aq), NaCl(aq,m s ) | H2 (g) | Pt(s) (IV) The observed potential is related in this cell to the cell composition by the relationship E = −(RT /F) ln (m H,t /m H,r ) − E lj

(26)

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Fig. 4. Difference, e(pK m ) in Eq. (25), between the observed pK m values of benzoic acid and those calculated by the H¨uckel method as a function of the ionic strength (Im ) for the data set of Jones and Parton (Ref. 15) (see cell I) and Briscoe and Peake (Ref. 36) (cell III). The observed pK m values were calculated as described in the text and are shown there and the predicted pK m values were calculated by Eq. (4) with the value of K a = 6.33 × 10−5 and with the recommended activity parameters.

where r and t refer to the reference and test solutions, respectively. According to the authors, the use of matching ionic media minimizes the second term in Eq. (26), reduces the ratio of the activity coefficients for minor ions to unity, and permits use of molal concentrations rather than activities. The liquid junction potential E lj was obtained by the Henderson equation and was always very small. Finally, Kettler et al. (see Table V in that study) suggest the following K m values for BenzOH in NaCl solutions at 25◦ C (these researchers actually report the values for the logarithm of the molality dissociation quotient Q a , which is same as K m , at saturation vapor pressure, but these values can probably be used in this connection without any correction): Im /(mol-kg−1 )

0

0.1

0.5

1.0

3.0

5.0

pK m

4.206

3.991

3.907

3.917

4.131

4.412

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Table V. Stoichiometric Dissociation Constant (K m ) at 25◦ C for Benzoic Acid as a Function of the Ionic Strength (Im ) in Aqueous NaCl and KCl Solutions Im /(mol-kg−1 )a

104 K m (NaCl)

104 K m (KCl)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.30 0.40 0.50

0.633 0.777 0.834 0.876 0.910 0.939 0.964 0.987 1.007 1.026 1.043 1.112 1.163 1.231 1.273 1.297

0.633 0.777 0.832 0.873 0.906 0.934 0.958 0.980 0.999 1.016 1.033 1.096 1.140 1.195 1.223 1.234

aI

m

is the same as m NaCl or m KCl .

The comparison of these values to those obtained by the H¨uckel method is shown in Fig. 5 where the pK m errors defined by Eq. (25) are presented. In the calculation of the predicted values, both the thermodynamic value of K a and the activity parameters recommended in this study and those estimated by us from the data of Kettler et al. [i.e. those of K a = 6.194 × 10−5 (pK a = 4.208) and BA = 1.25 molkg−1 and bA,NaCl = 0.133] were used. It seems important that practically the same value of B for benzoate ions was obtained in these estimations as that recommended above. The recommended activity parameters apply satisfactorily to the data up to a molality of 1.0 mol-kg−1 , and the fitted parameters apply very well to all of these data (the errors can be again compared to the conventional error of pK m , discussed in the previous paragraph). The pK m values determined by Barriada et al.(26) by using potentiometric titrations for BenzOH in aqueous KNO3 solutions were used to test the new H¨uckel parameters for benzoate ions. The parameter values for KCl solutions apply better to these data. The results of the comparison of the experimental values to those calculated by means of the H¨uckel parameters for KCl solutions are shown in Fig. 6 where the pK m errors defined by Eq. (25) are presented. In the calculation of the predicted values for this figure, both the K a value recommended in this study and that fitted by Barriada et al.(26) by using their activity coefficient model (Guggenheim model) were used. It is clear that the latter value applies better to the experimental data, but Fig. 6 shows that the former also applies satisfactorily to these data (compare to the conventional pK m error, discussed above). Thus, the

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Fig. 5. Difference, e(pK m ) in Eq. (25), between the pK m values suggested by Kettler et al. (Ref. 20) on the basis of measurements on cells of type IV for benzoic acid in NaCl solutions and those predicted by the H¨uckel method as a function of the ionic strength (Im ). The predicted pK m values were calculated by Eq. (4) with the K a values of 6.33 × 10−5 or 6.194 × 10−5 and with the activity parameters of BA = 1.25 (mol-kg−1 )−1/2 and bA,NaCl = 0.09 or 0.133, respectively.

H¨uckel parameters for KCl solutions seem to apply in this case also satisfactorily for KNO3 solutions. Table V shows the recommended K m values for benzoic acid in aqueous salt solutions at 25◦ C at rounded ionic strengths. These values apply to the case where NaCl or KCl alone determines the ionic strength of the solution. The K m values were calculated by ln K m,MCl = ln K a + 2α(Im )1/2 {1/[1 + B(Im )1/2 ]} − (bH,MCl + bA,MCl )Im

(27)

where K a = 6.33 × 10−5 , α = 1.17444 (mol-kg−1 )−1/2 , B = 1.25 (mol-kg−1 )−1/2 , bH,NaCl = 0.238, bH,KCl = 0.178, bA,NaCl = 0.09, and bA,KCl = 0.25, where A refers to benzoate ions. It was shown above that these parameter values are supported by potentiometric titration data, by conductance data and by EMF data measured on cells without a liquid junction. Therefore, it seems to us that the K m predictions obtained by Eq. (27) and presented in Table V are reliable.

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Fig. 6. Difference, e(pK m ) in Eq. (25), between the pK m values of determined by Barriada et al. (Ref. 26) for benzoic acid in KNO3 solutions and those predicted by the H¨uckel method as a function of the ionic strength (Im ). The predicted pK m values were calculated by Eq. (4) with the K a values of 6.33 × 10−5 and 5.88 × 10−5 and with the recommended activity parameters for KCl solutions.

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