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Journal of Solution Chemistry [josc]

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Style file version Nov. 19th, 1999

C 2002) Journal of Solution Chemistry, Vol. 31, No. 1, January 2002 (°

Conductivity Studies of Dilute Aqueous Solutions of Oxalic Acid and Neutral Oxalates of Sodium, Potassium, Cesium, and Ammonium from 5 to 35◦ C M. Beˇster-Rogaˇc,1 M. Tomˇsiˇc,1 J. Barthel,2∗ R. Neueder,2 and A. Apelblat3 Received August 7, 2001 Conductivity measurements of oxalic acid and neutral oxalates (disodium oxalate, dipotassium oxalate, dicesium, and diammonium oxalate) were performed on dilute aqueous solutions, c < 3 × 10−3 mol-dm−3 , from 5 to 35◦ C. These data and those available from the literature were analyzed in terms of dissociation steps of oxalic acid, the Onsager conductivity equation for neutral oxalates, the Quint–Viallard conductivity equation for the acid, and the Debye–Hückel equation for activity coefficients, to give the limiting equivalent conductances of bioxalate anion λ∞ (HC2 O− 4 ) and oxalate anion λ∞ (1/2C2 O2− 4 ) and the corresponding dissociation constants K 1 and K 2 . KEY WORDS: Electrolyte conductivity; electrolyte solutions; ion conductivities; aqueous solutions; oxalic acid; alkali metal oxalates; ammonium oxalate.

1. INTRODUCTION Oxalic (ethanedioic) acid, because of its capacity to form stable complexes with various metal ions, is of considerable importance in many industrial, geological, and biological systems.(1,2) The continuous and widespread interest in the properties of aqueous solutions of oxalic acid and oxalates started early by considering the solubility products of oxalates(3–7) and the dissociation constants of the acid determined by the conductometric,(8–11) potentiometric, and other methods.(2,7,12–22) The dissociation constants of oxalic acid are also available in tables of electrochemical data.(23–28) Recently, the literature data for the first and second dissociation steps of the acid were reexamined by Kettler et al.(2) 1 Faculty of Chemistry and Chemical Technology, University of Ljubljana, SI-1000 Ljubljana, Slovenia. 2 Institut

für Physikalische und Theoretische Chemie, Universität Regensburg, D-93040 Regensburg, Germany. E-mail: [email protected] 3 Department of Chemical Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel. 1 C 2002 Plenum Publishing Corporation 0095-9782/02/0100-0001/0 °

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Beˇster-Rogaˇc, Tomˇsiˇc, Barthel, Neueder, and Apelblat

In contrast to other dibasic carboxylic acids, conductances of oxalic acid, at least at 298.15 K, were repeatedly measured by Ostwald (1889),(8) Collie and Tickle (1899),(29) who also determined conductances of dimethylpyrone oxalate, Drucker (1920)(9) (quoted in Ref. 14,15) and the most extensive measurements by Darken (1941).(10) Only in the last investigation, the second dissociation constant of oxalic acid, is taken into account by the author, but even then the contribution emanating from this step was treated as a small correction to the determined conductances. The Darken conductances of oxalic acid were analyzed by Lee and Wheaton (1977)(30) using the Quint–Viallard conductivity equation. The conductances of oxalic acid at high temperatures in the 298.15–433.15 K range were reported by Maksimova and Yushkevich (1966).(31) Disodium oxalate conductances at 298.15 K were determined by Walden (1891),(32) Lorenz and Scheuermann (1920),(33) Darken (1941)(10) and Taft and Welch (1952)(34) (also at 273.15 and 323.15 K). Noyes and Johnston (1909)(35) reported conductances of dipotassium oxalate in the 273.15–439.15 K temperature range and those of magnesium oxalate at 291.15 K by Kohlrausch and Mylius (1904).(36) With an exception of the Darken work,(10) other investigations are rather of limited value and of an unequal quality of reported results. In the tables of the equivalent limiting conductances, the conductances of bioxalate anion λ∞ (HC2 O− 4 ) are completely omitted when those of oxalate anions are mentioned twice, in the Robinson and Stokes 2 −1 and in the Milazzo book,(37) book,(26) λ∞ (1/2C2 O2− 4 ) = 74.15 S-cm mol 2− ∞ 2 −1 λ (1/2C2 O4 ) = 24.1 S-cm mol , but probably this is a typographic error. For convenience, use of the unit of molar or equivalent conductance, S-cm2 mol−1 , is repressed in the text that follows. Systematic and precise measurements of electrical conductivities of dilute aqueous solutions of oxalic acid and a number of neutral oxalates in the 278.15– 308.15 K temperature range were performed in this work. From determined conductivities, the limiting equivalent conductances of oxalate and bioxalate anions and the dissociation constants of oxalic acid were derived as a function of temperature and compared with the literature values. In the optimization procedure [utilized previously for tartaric acid and tartrates(38–40) ] the law of mass action (activities were calculated using the Debye–Hückel equation for activity coefficients), the Onsager conductivity equation for neutral oxalates, and the Quint–Viallard conductivity equation for oxalic acid were applied to give a reliable set of derived parameters.

2. EXPERIMENTAL 2.1. Materials The Fluka products diammonium oxalate monohydrate (NH4 )2 C2 O4 · H2 O (microselect, >99.5%), dicesium oxalate Cs2 C2 O4 (pure, >97%), dipotassium oxalate monohydrate K2 C2 O4 · H2 O (microselect, >99%), disodium oxalate

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Conductivity of Oxalic Acid and Oxalates


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Na2 C2 O4 (microselect, >99.5%), and oxalic acid H2 C2 O4 (p.a., >99%) were used without further purification. Cs2 C2 O4 and Na2 C2 O4 were dried in vacuum at 100◦ C and stored under dry nitrogen. The amount of water in monohydrates was checked by thermogravimetric analysis (Mettler Toledo TGA/SDTA 851e). The values of 1.04 ± 0.01 H2 O for diammonium and 0.98 ± 0.01 H2 O for dipotassium salt were found. The stock solutions were prepared by weighing pure salt and water. The concentration of the stock solution of the oxalic acid was determined with an accuracy of 0.2% by pH titration with standard sodium hydroxide solution. Demineralized water was distilled in a quartz bidistillation apparatus (Destamat Bi18E, Heraeus). The final product with specific conductance of less than 6 × 10−7 Ä−1 cm−1 was distilled into a flask permitting storage and transfer of the solvent into the measuring cell under an atmosphere of nitrogen. 2.2. Conductivity Measurement Measurements were carried out with a measuring assembly consisting of a cold bath (Lauda WK 1400), a precision thermostat (Lauda UB 40J) with immersed conductance cell, and a highly precise conductance bridge (Leeds & Northrup). The thermostat can be set to each temperature of a temperature program with a reproducibility of less than 0.003 K. The temperature in the precision thermostat bath was additionally checked with calibrated Pt100 (MPMI 1004/300 Merz) in connection with a HP 3458A multimeter. The conductivities of solutions were determined with the help of a threeelectrode measuring cell, described elsewhere.(41,42) The cell was calibrated with aqueous potassium chloride solutions.(43) At the beginning of every measuring cycle, the cell was filled with a weighed amount of water. After measurement of the water conductivity at all temperatures of the program, weighed amounts of a stock solution were added using a gas-tight syringe. After every addition the temperature program was executed. From the weights and the corresponding solution densities d the molar concentrations c (mol-dm−3 ) were determined, c(T ) = m˜ × d(T ). A linear change of ˜ where m˜ is d with increasing salt content was assumed, d(T ) = d◦ (T ) + D × m, the molonity of the electrolyte (moles of electrolyte per kilogram of solution) and d◦ (T ) is the solvent density at temperature T , given in Table I. The temperatureindependent density coefficients D (given in Table II) were obtained from density measurements on solutions and pure water using a Paar vibrating-tube densimeter and the method of Kratky et al.(44) 3. RESULTS AND DISCUSSION 3.1. Limiting Equivalent Conductances of Oxalate Anions The results of conductivity measurements with neutral alkali metal oxalates (Na2 C2 O4 , K2 C2 O4 , and Cs2 C2 O4 ) and two samples of diammonium oxalate,

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Table I.

Densities, Viscosities, and Dielectric Constants of Pure Water and Limiting Equivalent Conductances of Ions in Water a

T

d◦b

103 × ηc

εd

278.15 283.15 288.15 293.15 298.15 303.15 308.15

0.99997 0.99970 0.99910 0.99821 0.99705 0.99565 0.99404

1.5192 1.3069 1.1382 1.0020 0.8903 0.7975 0.7195

85.897 83.945 82.039 80.176 78.358 76.581 74.846

∞ + e λ∞ (Na+ )e λ∞ (K+ )e λ∞ (Cs+ ) λ∞ (NH+ 4 ) λ (H )

30.30 34.88 39.72 44.81 50.15 55.72 61.53

46.72 53.03 59.61 66.44 73.50 80.76 88.20

50.03 56.43 f 63.16 69.99 f 77.26 84.51 f 92.10

45.94 f 52.14 f 58.73 f 65.72 f 73.55 80.79 f 88.72

250.02 275.55 300.74 325.52 349.85 373.66 396.90

T , K; d◦ , kg-dm−3 ; η, Pa-s; λ∞ , S-cm2 -mol−1 . (45). c Ref. (46). d Ref. (47). e Ref. (48). f Values calculated from equations λ∞ (Cs+ ) = 73.432 − 1.4276T + 4.8298 × 10−3 T 2 λ∞ (NH+ 4 ) = 5646.6123 + 5.80785T − 1282.1463 In T (based on the data in Ref. 26).

a Units: b Ref.

and

(NH4 )2 C2 O4 , are presented in Table II; the comparison with the literature results at 298.15 K is given in Figs. 1 and 2. In the case of disodium oxalate, the conductances of Lorenz and Scheuermann,(33) Darken,(10) and this study are in a reasonable agreement, but those of Taft and Welch(34) and Walden(32) [original and shifted because old values of conductivities should be multiplied by the factor 1.066(49) ] differ considerably (Fig. 1). As can be seen in Fig. 2, the Noyes and Johnston measurements with dipotassium oxalate(35) have only an indicative character. If neutral oxalates are considered as strong electrolytes, it is expected that in dilute solutions their conductances will follow the limiting slope, which for 1:2 electrolytes is given by the Onsager conductivity equation in the form(47) ½

¾ 132.852 √ 5.602 × 106 q3∞ e 3e = − I √ − (εT )3/2 (1 + q) η(εT )1/2 ¡ ¢¤ £ 2 λ∞ (Me+ ) + λ∞ 1/2C2 O2− 4 q= £ ¡ ¢¤ 3 2λ∞ (Me+ ) + λ∞ 1/2C2 O2− 4 3∞ e

(1a) (1b)

where all symbols have usual meaning, I = 3c is the ionic strength and λ∞ (Me+ ) denotes the limiting equivalent conductances of cations (Table I). The optimization procedure to satisfy Eqs. (1) together with the Kohlrausch law of the independent migration of ions at infinite dilution ¡ ¢ ∞ + ∞ 3∞ 1/2C2 O2− e = λ (Me ) + λ 4

(2)

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Table II. Experimental Molar Conductances of Disodium Oxalate, Dipotassium Oxalate, Dicesium Oxalate, and Diammonium Oxalate from 278.15 to 308.15 Ka 3 × m˜

278.15

0.22890 0.48832 0.75604 1.0184 1.2432 1.4726 1.7227 1.9968 2.4216

144.05 142.53 141.24 140.20 139.38 138.62 137.85 137.06 136.00

0.25381 0.49316 0.71175 1.0136 1.2690 1.5706 1.8705 2.1615 2.4518 2.7172

177.03 174.76 173.45 171.96 170.93 169.84 168.88 168.03 167.28 166.55

0.16487 0.32635 0.48668 0.63671 0.82080 1.0046 1.2433 1.4766 1.7171 1.9650

185.81 183.67 182.18 181.10 179.80 178.78 177.40 176.65 175.74 174.87

0.24923 0.50662 0.76038 1.0102 1.2638 1.5106 1.7439 1.9993 2.2350 2.4895

175.86 173.65 171.96 170.58 168.06 167.11 166.51 165.38 165.03 164.39

103

283.15

288.15

293.15

298.15

Disodium oxalate, D = 0.1107 165.91 189.17 213.54 238.95 164.37 187.39 211.54 236.80 162.76 185.56 209.61 234.57 161.56 184.17 207.89 232.70 160.65 183.15 206.67 231.29 159.77 182.13 205.60 230.05 158.85 181.05 204.37 228.69 157.95 180.02 203.17 227.40 156.78 178.73 201.58 225.58 Dipotassium oxalate, D = 0.1339 202.62 229.32 257.16 286.51 200.06 226.47 254.23 282.78 198.53 224.89 252.16 280.50 196.89 222.86 249.91 277.96 195.62 221.39 248.25 276.15 194.34 219.96 246.61 274.29 193.24 218.68 245.16 272.68 192.24 217.56 243.74 271.29 191.36 216.53 242.76 270.00 190.53 215.68 241.76 268.85 Dicesium oxalate, D = 0.2913 211.82 239.11 267.43 296.69 209.45 236.49 264.39 293.37 207.72 234.51 262.23 291.06 206.55 233.06 260.63 289.21 205.03 231.39 258.81 287.18 203.91 230.07 257.34 285.48 202.37 228.47 255.54 283.74 201.39 227.24 254.22 282.00 200.34 226.14 252.84 280.51 199.43 225.00 251.59 279.11 Diammonium oxalate (I), D = 0.06244 201.62 228.63 256.76 286.06 199.05 225.73 253.50 282.37 197.10 223.49 250.92 279.38 195.49 221.61 248.74 277.31 192.68 219.16 246.05 273.92 191.86 217.44 244.11 271.65 190.81 216.29 242.77 270.28 189.71 215.11 241.50 268.87 189.13 214.31 240.61 267.86 188.35 213.50 239.65 266.84

303.15

308.15

265.41 263.01 260.69 258.46 256.99 255.58 254.05 252.62 250.53

293.02 290.46 287.64 285.29 283.60 281.92 280.30 278.56 276.34

315.65 311.91 309.53 306.81 304.81 302.80 301.01 299.45 297.99 296.74

346.23 342.10 339.57 336.60 334.35 332.06 330.18 328.47 326.83 325.36

327.08 323.46 320.73 318.71 316.48 314.66 312.46 310.75 309.08 307.50

357.80 354.27 351.24 348.98 346.76 344.32 342.05 340.34 338.50 336.71

316.18 312.01 309.16 306.09 302.67 300.21 298.80 297.08 296.22 294.90

347.35 343.05 339.34 335.11 332.78 329.65 328.11 325.91 325.29 323.74

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Beˇster-Rogaˇc, Tomˇsiˇc, Barthel, Neueder, and Apelblat Table II. Continued 3 × m˜

278.15

0.21247 0.41451 0.64076 0.86093 1.0839 1.3024 1.5253 1.7513 1.9719 2.3645

176.01 174.34 172.82 170.89 169.48 168.69 167.87 167.13 166.42 165.27

103

a Units:

283.15

288.15

293.15

298.15

303.15

308.15

Diammonium oxalate (II), D = 0.06244 201.66 228.57 256.46 285.48 199.92 226.62 254.31 283.10 198.04 224.51 252.05 280.55 195.83 221.98 249.20 278.14 194.27 220.41 247.96 275.88 193.48 219.48 246.25 274.17 192.39 218.07 244.87 272.57 191.53 217.10 243.72 271.14 190.71 216.13 242.62 270.08 189.34 214.57 240.85 268.15

315.44 312.95 309.94 307.47 304.79 302.85 301.24 299.78 298.39 296.27

346.74 343.51 340.18 337.79 334.59 332.50 330.85 329.13 327.51 325.30

˜ mol-kg−1 ; T , K; 3, S-cm2 -mol−1 ; D, kg2 -dm−3 -mol−1 . m,

as applied to experimental conductances from Table II permits the determination of the limiting equivalent conductances of oxalate anions λ∞ (1/2C2 O2− 4 ) (Table III). Since the limiting equivalent conductances of potassium ion, λ∞ (K+ ) = 73.50 and ammonium ion, λ∞ (NH+ 4 ) = 73.55 are practically the same, in very dilute

Fig. 1. Equivalent conductivities of disodium oxalate at 298.15 K as a function of the square root of concentration c. •, This work; ◦, Ref. (10); N, Ref. (33); ¤, Ref. (34); ¥, Ref. (32); 4, Ref. (32).

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Fig. 2. Equivalent conductivities of dipotassium oxalate at 298.15 K as a function of the square root of concentration c. ¤, This work; ¥, Ref. (35).

solutions it is expected that 3e (K2 C2 O4 ) and 3e ((NH4 )2 C2 O4 ) will be very close to what is actually observed (Fig. 3). However, the conductances of two samples of ammonium oxalate (contrary to other oxalates) unexpectedly differ somewhat and these differences exceed the limits of anticipated error in our experiments. At this moment, there is no explanation for the observed differences, but it is worthwhile to mention that λ∞ (1/2C2 O2− 4 ) derived from conductances of ammonium oxalate (from both sets of data) is slightly, but clearly, lower than that obtained from alkali metal oxalates (Table III). In the system, there is a possibility of formation of Table III.

Limiting Equivalent Conductances of Oxalate Anion λ∞ (1/2C2 O2− 4 ) as a Function of Temperaturea λ∞ (1/2C2 O2− 4 )

T

Na2 C2 O4

K2 C2 O4

Cs2 C2 O4

(NH4 )2 C2 O4 (I)

(NH4 )2 C2 O4 (II)

278.15 283.15 288.15 293.15 298.15 303.15 308.15

44.50 ± 0.30 51.39 ± 0.38 58.68 ± 0.44 66.33 ± 0.50 74.30 ± 0.58 82.61 ± 0.67 91.20 ± 0.71

44.33 ± 0.18 51.27 ± 0.21 58.55 ± 0.26 66.16 ± 0.28 74.15 ± 0.33 82.30 ± 0.46 90.80 ± 0.53

44.45 ± 0.20 51.38 ± 0.19 58.69 ± 0.20 66.31 ± 0.19 74.22 ± 0.17 82.40 ± 0.19 90.75 ± 0.17

43.96 ± 0.33 51.02 ± 0.34 58.35 ± 0.34 65.83 ± 0.37 73.04 ± 0.44 81.36 ± 0.45 89.43 ± 0.50

44.13 ± 0.17 51.17 ± 0.17 58.49 ± 0.17 65.97 ± 0.15 73.19 ± 0.14 81.46 ± 0.18 89.56 ± 0.17

a Units:

T , K; λ∞ , S-cm2 -mol−1 .

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Fig. 3. Equivalent conductivities at 298.15 K as a function of the square root of concentration c. •, Dipotassium oxalate; ◦, diammonium oxalate (I); M, diammonium oxalate (II).

undissociated molecules of ammonia and hydrolysis products, which may reduce the conductivity of solutions. If the hydrolysis process is time-dependent, then some kind of the age effect will explain the differences in the ammonium oxalate samples. However, without additional and detailed potentiometric studies, this is only a plausible guess. Thus, taking into consideration a somewhat uncertain situation with ammonium oxalates, in calculations of the average values of λ∞ (1/2C2 O2− 4 ), only alkali metal oxalates were included (Table IV). At 298.15 K, it is possible to compare our value λ∞ (1/2C2 O2− 4 ) = 74.27 ± 0.08 with those estimated from sodium oxalates by Lorenz and Scheuermann(33) Table IV.

Limiting Equivalent Conductances of Bioxalate λ∞ (HC2 O− 4 ) and Oxalate a Anions λ∞ (1/2C2 O2− ) and Walden Products 4

T

λ∞ (HC2 O− 4)

ηλ∞ (HC2 O− 4)

λ∞ (1/2C2 O2− 4 )

ηλ∞ (1/2C2 O2− 4 )

278.15 283.15 288.15 293.15 298.15 303.15 308.15

26.61 ± 0.036 30.47 ± 0.38 34.46 ± 0.41 38.89 ± 0.39 43.06 ± 0.39 47.76 ± 0.41 52.91 ± 0.43

0.04043 0.03982 0.03922 0.03897 0.03834 0.03809 0.03806

44.43 ± 0.09 51.35 ± 0.07 58.64 ± 0.08 66.27 ± 0.09 74.27 ± 0.08 82.44 ± 0.16 90.91 ± 0.25

0.06750 0.06711 0.06674 0.06640 0.06612 0.06575 0.06541

a Units:

T , K; λ∞ , S-cm2 -mol−1 ; ηλ∞ , S-cm2 -mol−1 -Pa s.

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(50) λ∞ (1/2C2 O2− ] λ∞ (1/2C2 O2− 4 ) = 77.35–79.10 [quoted by Gonick 4 ) = 75.6, which is based on the Bredig compilation from 1894;(51) the recalculated Walden(32) value is λ∞ (1/2C2 O2− 4 ) = 72.7 (calculations were performed for 1:1 strong electrolytes and not 1:2 electrolytes, because in the last case the result is far from correct value). The modern results of Darken(10) λ∞ (1/2C2 O2− 4 ) = 74.10 and Taft ) = 74.10 are practically the same as our value. The and Welch(34) λ∞ (1/2C2 O2− 4 (35) ) = 74.0 was derived by Noyes and Johnston from same result λ∞ (1/2C2 O2− 4 the potassium oxalate conductances. All these values were derived in the liter√ ature using the Kohlrausch equation (3 against c with c → 0), but the conductances did not satisfy the Onsager slope. It is interesting to note that Lee and Wheaton,(30) using the Quint–Viallard conductivity equation, reexamined the Darken conductances(10) to obtain λ∞ (1/2C2 O2− 4 ) = 73.79 − 74.15. The limiting equivalent conductances of oxalate anion and the corresponding Walden products λ∞ (T )η(T ), as a function of temperature, are presented in Table IV. Brummer and Hills,(52) applying the Eyring theory of transition state to movement of individual ions, derived

µ

∂ ln λ∞ ∂T

¶ p

6=

1Hλ 2 = − 2 RT 3

µ

∂ ln d◦ ∂T

¶ (3) p 6=

If the partial molar enthalpy of activation associated with the process, 1Hλ , is independent of temperature, then the integral form of Eq. (3) is 6= ¢ ¡ 1Hλ + const. ln λ∞ d◦2/3 = − RT

(4)

The temperature dependence of limiting conductances, as expected from Eq. (4), is a straight line, which is actually observed in Fig. 4. For the oxalate anion, we have ¤ £ 2031.5 2/3 = 11.112 − ln λ∞ (1/2C2 O2− 4 )d◦ T

(5)

6=

where 1Hλ = 16.89 kJ/mol and density of pure water d◦ is given in Table I. 3.2. Limiting Equivalent Conductances of Bioxalate Anions The properties of oxalic acid, H2 C2 O4 , in dilute aqueous solutions are expressed in terms of the dissociation equilibria H2 C2 O4 → H+ + HC2 O− 4; 2− + HC2 O− 4 → H + C2 O4 ;

K 1 (T ) K 2 (T )

(6a) (6b)

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Fig. 4. Values of ln(λ∞ d◦ ) as a function of 1/T . •, Oxalate anion; ¥, bioxalate anion; line and dashed line calculated using Eq. (4). 2/3

The equilibrium constants of these reactions are £ ¤ [H+ ] HC2 O− c(α1 + 2α2 )α1 4 F1 = F1 K1 = [H2 C2 O4 ] 1−α £ ¤ [H+ ] C2 O2− c(α1 + 2α2 )α2 4 ¤ F2 = F2 K2 = £ α1 HC2 O− 4

(7a) (7b)

They are expressed in terms of the total degree of dissociation, α = α1 + α2 and the partial degrees α1 and α2 , which are assigned to the primary and the secondary steps of dissociation, and c is the total (analytical) concentration of oxalic acid. F1 and F2 denote the quotients of the activity coefficients F1 =

f H+ f HC2 O−4 f H2 C2 O4

;

F2 =

f H+ f C2 O2− 4 f HC2 O−4

(8a,b)

The activity coefficients of individual ions f i in dilute solutions can be approximated by the Debye–Hückel equation √ z i2 A I (9) log f i = − √ 1 + ai B I where z i are the charges of ions, ai are the ionic size parameters, A and B are constants depending on temperature, and the dielectric constant of water ε.(26) I is the ionic strength, which is equal to I = c(α1 + 3α2 ). The activity coefficient

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of undissociated oxalic acid is assumed to be unity. If the equilibrium constants and activity coefficients are known, the degrees of dissociation can be successively evaluated for every concentration c by an iterative solution of two quadratic equations   s µ ¶ µ ¶2 K1 1 4K 1 (1 − α2 )  K1 − + 2α2 + + 2α2 + (10a) α1 =  2 cF1 cF1 cF1 s ( ) 1 8K 2 α1 2 −α1 + α1 + α1 = 4 cF2

(10b)

Because of the scattering of results, the dissociation constants of oxalic acid, which is a moderately strong acid, were determined many times.(2,7–22) Kettler et al.(2) reexamined these constants and proposed, for the first dissociation step, the expression for “best” values of K 1 35399.5 log K 1 (T ) = −626.41 − 0.097611 × T + T 2.17087 × 106 + 98.2742 × ln T (11) T2 In his expression for K 2 , the last term is obviously incorrect. The Kettler et al. values of K 1 were used in our calculations, whereas K 2 was determined here, together with λ∞ (HC2 O− 4 ), from the conductivity data of oxalic acid. Experimental conductances of oxalic acid as a function of concentration and temperature are presented in Table V and at 298.15 K in Table VI. In Fig. 5 are compared our values at 298.15 K with those reported in the literature (the old results were shifted, as mentioned above). As can be seen, the agreement is very satisfactory, but only our conductances cover the dilute solutions region where behavior of oxalic acid as a weak electrolyte is clearly observed. Following the procedure established in our previous works for similar systems,(38–40,53–56) the measured molar conductances 3 were considered as the sum of the ionic contributions n X |z i |ci λi κ (12) 3= = c c i=1 −

where κ is the specific conductance, λi and ci are the conductivities and concentrations of individual ions present in the solution. The ionic conductances λi in Eq. (12) are expressed by √ 3/2 (13) λi = λ∞ i − Si I + E i I ln I + J1i I − J2i I where the coefficients are available from the Quint–Viallard theory.(57–60) The coefficients Si and E i depend on the viscosity and dielectric constant of water and

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Table V. Experimental Molar Conductances of Oxalic Acid from 278.15 to308.15 K, D = 0.0451a 103 × m˜

3exp

3calc

3exp

3calc

3exp

3calc

0.08706 0.17586 0.25912 0.34243 0.51678 0.67792 0.84064 1.0070 1.2436 1.5514

278.15 K 380.47 380.43 343.46 342.72 326.46 325.91 315.83 315.57 302.60 302.77 295.54 295.69 290.49 290.70 286.55 286.86 282.39 282.69 278.27 278.58

283.15 K 419.76 419.83 379.10 378.23 360.29 359.74 348.67 348.39 334.45 334.35 326.41 326.59 320.92 321.12 316.59 316.92 312.00 312.34 307.32 307.82

288.15 K 456.55 457.51 413.02 412.34 392.96 392.37 380.72 380.18 365.13 365.06 356.71 356.73 350.80 350.86 346.12 346.33 341.16 341.40 336.04 336.52

0.08706 0.17586 0.25912 0.34243 0.51678 0.67792 0.84064 1.0070 1.2436 1.5514

293.15 K 495.51 495.72 447.72 446.87 425.95 425.35 412.74 412.20 395.78 395.97 386.83 387.01 380.49 380.68 375.43 375.80 370.10 370.47 364.85 365.19

303.15 K 565.38 566.44 511.98 511.26 487.78 487.24 473.09 472.63 454.50 454.66 444.65 444.73 437.62 437.71 432.11 432.26 426.11 426.29 420.22 420.32

308.15 K 597.59 598.67 541.91 541.12 516.90 516.27 501.76 501.22 482.62 482.72 472.40 472.49 465.16 465.24 459.41 459.61 453.11 453.39 446.94 447.15

a Units:

˜ mol-kg−1 ; T , K; 3, S-cm2 -mol−1 ; D, kg2 -dm−3 -mol−1 . m,

on the equivalent conductance at infinite dilution of ion i, λ∞ i , when J1i and J2i depend also on the distance of closest ion approach ai . The coefficients are rather complex functions of parameters and they are explicitly given in Refs. (39,53,56). In the case of oxalic acid, Eq. (12) takes the form (14a) 3(H2 C2 O4 ) = α1 31 + 2α2 32 £ ¡ ¢¤ ¢¤ 32 = λ(H+ ) + λ 1/2C2 O2− (14b,c) 31 = λ(H+ ) + λ HC2 O− 4 ; 4 £

¡

where expressions for individual ions are given by equations of type Eq. (13). The experimental set of (3, c) data, when represented by the dissociation model, require choice and preselection of parameters. At constant temperature T , the values of dielectric constant and viscosity of pure water, ε and η, and the limiting equivalent conductances of λ∞ (H+ ) (Table I) and λ∞ (1/2C2 O2− 4 ) (Table IV) are needed. The first dissociation constant K 1 was taken from the Kettler et al. work(2) (Eq. 11). Thus, the calculated quantities 3i , αi , f i , and I depend only on the limiting conductances of the organic anions λ∞ (HC2 O− 4 ), the equilibrium constants K 2 , and the distances ai . The ionic size parameters are unknown for organic anions

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Table VI. Degrees of Dissociation α1 and α2 , Contributions to the Conductance from the Primary and Secondary Dissociation Steps, 31 and 32 , and Experimental and Calculated Molar Conductances 3exp and 3calc of Oxalic Acid at 298.15 Ka 103 × c 0.00868 0.17535 0.25835 0.34143 0.51526 0.67594 0.83819 1.0040 1.2401 1.5469

α1 0.68487 0.79893 0.84291 0.87137 0.90171 0.91683 0.92594 0.93175 0.93650 0.93924

0.4883 0.9768 1.9530 3.9060 7.8125 15.625 31.250

0.89821 0.93097 0.93971 0.92662 0.89071 0.83030 0.74686

0.9768 1.9530 3.9060 7.8125 15.625 31.250

0.93097 0.93971 0.92662 0.89071 0.83030 0.74686

2.3886 2.8225 5.2032 5.3939 7.2036 7.9151 9.4687 10.347 12.656 14.443 14.946 19.744 24.897 27.433 34.196 34.294

0.93816 0.93548 0.91466 0.91287 0.89617 0.88981 0.87644 0.86921 0.85128 0.83839 0.83491 0.80448 0.77644 0.76405 0.73462 0.73422

α2

31

0.31369 267.87 0.20103 310.87 0.15256 328.89 0.12364 339.33 0.08932 351.04 0.07150 356.53 0.05973 359.73 0.05126 361.66 0.04279 363.09 0.03536 363.67 Ostwaldb 0.09327 349.75 0.05247 361.41 0.02889 363.31 0.01581 356.36 0.00869 340.32 0.00483 314.85 0.00271 280.89 Collie and Ticklec 0.05247 361.41 0.02888 363.31 0.01581 356.36 0.00869 340.32 0.00483 314.85 0.00271 280.89 Darkend 0.02425 362.20 0.02097 360.71 0.01233 350.86 0.01196 350.06 0.00932 342.69 0.00860 339.93 0.00737 334.20 0.00684 331.13 0.00577 323.58 0.00516 318.20 0.00501 316.76 0.00397 304.27 0.00327 292.77 0.00302 287.75 0.00251 275.87 0.00250 275.71

32

3exp

3calc

264.25 168.97 128.03 103.63 74.69 59.68 49.78 42.66 35.54 29.31

531.25 480.53 457.41 443.43 425.56 416.17 409.44 404.08 398.43 392.85

532.12 479.83 456.92 442.95 425.73 416.22 409.50 404.31 398.63 392.99

78.02 43.68 23.89 12.96 7.05 3.87 2.14

435.99 408.29 388.02 368.84 345.38 318.73 284.62

427.77 405.08 387.20 369.32 347.38 318.72 283.03

43.68 23.89 12.96 7.05 3.87 2.14

408.00 388.00 369.00 345.00 319.00 285.00

405.08 387.20 369.32 347.38 318.72 283.03

20.01 17.27 10.07 9.75 7.57 6.97 5.96 5.52 4.64 4.14 4.02 3.16 2.60 2.39 1.98 1.98

379.95 376.16 358.74 357.50 348.02 344.78 338.10 334.73 326.50 320.75 319.27 306.26 294.71 289.72 277.94 277.81

382.21 377.98 360.93 359.81 350.26 346.90 340.16 336.65 328.22 322.34 320.78 307.43 295.37 290.14 277.85 277.69

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Beˇster-Rogaˇc, Tomˇsiˇc, Barthel, Neueder, and Apelblat Table VI. Continued 103

×c

41.509 42.151 49.356 50.183 54.131 58.225 65.919

α1

α2

0.70744 0.70524 0.68233 0.67989 0.66868 0.65780 0.63910

0.00214 0.00211 0.00186 0.00183 0.00172 0.00162 0.00146

31 Darkend 264.95 264.07 254.91 253.93 249.45 245.11 237.66

32 1.68 1.66 1.46 1.44 1.35 1.27 1.14

3exp

3calc

267.27 266.41 257.50 256.57 252.26 248.07 240.93

266.64 265.73 256.36 255.36 250.80 246.38 238.80

c, mol-dm−3 ; 3, S-cm2 -mol−1 . (8). c Ref. (29). d Ref. (10).

a Units: b Ref.

and, therefore, should be prescribed. Fortunately, it was observed that, based on the Quint–Viallard conductivity equation, in a wide range of ai values, the final results are very weakly dependent on the closest approach distances. Rather arbitrarily, but in accordance with the values suggested by Kielland(61) and Harris,(62) ˚ a(C2 O2− ) = 5.0 A, ˚ a(HC2 O− ) = 3.0 A, ˚ the following distances a(H+ ) = 9.0 A, 4 4 2− − + + ˚ ˚ a(H /C2 O4 ) = 7.0 A, and a(H /HC2 O4 ) = 6.0 A were used for calculation of the individual activity coefficients f i , Eq. (9), and in the conductivity equations, Eq. (13). It was assumed that the ionic size parameters are independent of

Fig. 5. Equivalent conductivities of oxalic acid at 298.15 K as a function of the square root of concentration c. •, This work; ◦, shifted values of Ref. (8); ♦, Ref. (29); ¤, Ref. (10).

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temperature. After choosing ai , for a known set of (3, c) data, the optimization problem for oxalic acid is, therefore, reduced to 3 = f [c, K 2 , λ∞ (HC2 O− 4 )]. The calculation procedure starts by first solving the chemical problem, Eqs. (7–9), by consecutive evaluation of αi , f i , and I for every given concentration c, starting with initial values α2 = 0 and f i = 1, until repetition does not change αi , f i , and I . These values were introduced to Eqs. (14) to obtain the limiting conductances of bioxalate anion λ∞ (HC2 O− 4 ) and K 2 , (Eqs. 10), by the search for the best agreement between the measured and calculated conductances. The final results of calculations, together with experimental conductances in the 278.15 − 308.15 K temperature range, are presented in Table V. As can be observed, there is an excellent agreement between 3exp and 3calc values. At 298.15 K, more detailed data for oxalic acid, also including the degrees of dissociation α1 and α2 , the contributions to the molecular conductance 3 from the primary and secondary dissociation steps, 31 and 32 , Eqs. (14), are reported in Table VI. In addition, using the set of the here derived parameters, the literature conductances of Ostwald,(8) Collie and Tickle,(29) and Darken(10) were recalculated and compared with experimental results. Once again, a very nice agreement can be noted with the Darken conductances (even for not so dilute solutions) that support the limiting equivalent conductances determined in this work. The same is observed with the historical measurements from 1889 of Ostwald(8) and Collie and Tickle.(29) The values of λ∞ (HC2 O− 4 ) and corresponding Walden products are given in Table IV and in Fig. 4. The limiting equivalent conductance of bioxalate anion as a function of temperature can be expressed as £ ¤ 2/3 ln λ∞ (1/2HC2 O− = 10.267 − 1940.3/T (15) 4 )d◦ 6=

where 1Hλ = 16.13 kJ/mol, i.e., the molar enthalpy of activation for bioxalate anion is nearly the same as for the oxalate anion. As pointed out above, due to the large value of K 1 , oxalic acid is a moderately strong acid and in the concentration range considered in this work, c < 1.6 × 10−3 mol-dm−3 , the total degree of dissociation α = α1 + α2 is near unity with substantial conversion of bioxalate anions into oxalate anions (in part, the same is observed in Refs. 8,29). The result is that both contributions to the determined conductance 3, which are coming from the primary and secondary dissociation steps, 31 and 32 , are comparable (Table VI) and this is true at any T . On the other hand, in more concentrated solutions of oxalic acid, the Darken assumption that 32 is a small correction to 3 is justified, but evidently this makes estimation of − ∞ λ∞ (HC2 O− 4 ) rather difficult. Determined by Darken, λ (HC2 O4 ) = 40.15, (30) reexamined his conductances to give the same value [Lee and Wheaton λ∞ (HC2 O− 4 ) = 40.0] is lower and probably less accurate than our result at 298.15 K, λ∞ (HC2 O− 4 ) = 43.06 (Table IV).

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Coefficients of the Quint–Viallard conductivity equation, the thermodynamic functions, and equilibrium constants of the secondary dissociation step K 2 , which were evaluated in this work are presented in Table VII. If we compare K 2 , based on the conductivity data [K 2 = 5.3 × 10−5(10,30) ] and from potentiometric measurements [K 2 = 5.0 − 6.4 × 10−5(2,7,12–28) ] with our result, K 2 = 4.94 × 10−5 at 298.15 K, the determined value is the lowest one, but not far from others. The derived values of K 2 can be correlated by ln K 2 = 192.0912 − 8206.25/T − 30.6252 ln T

(16)

Differentiation of Eq. (16) with respect to temperature yields the molar enthalpies 1H2◦ and the molar entropies 1S2◦ of the secondary dissociation step; they are Table VII. Dissociation Constants, the Standard Changes of the Molar Enthalpy and Entropy, Limiting Conductances, and Coefficients of the Quint–Viallard Conductivity Equation from 278.15 to 308.15 Ka T K 1 × 102 1H1◦ 1S1◦ K 2 × 105 1H2◦ 1S2◦

b

3∞ 1 S11 (H+ ) S12 (HC2 O− 4) E 11 (H+ ) E 12 (HC2 O− 4) J11 (H+ ) J12 (HC2 O− 4) J21 (H+ ) J22 (HC2 O− 4) 3∞ 2 S21 (H+ ) S22 (1/2C2 O2− 4 ) E 21 (H+ ) E 22 (1/2C2 O2− 4 ) J11 (H+ ) J12 (1/2C2 O2− 4 ) J21 (H+ ) J22 (1/2C2 O2− 4 ) a Units: b Taken

278.15

283.15

5.635 −0.13 −24.4 5.70 −2.60 −90.6

5.605 5.550 5.400 5.291 5.135 4.963 −1.25 −2.28 −3.22 −4.09 −4.90 −5.66 −28.4 −32.0 −35.2 −38.2 −40.9 −43.3 5.55 5.31 5.15 4.94 4.66 4.34 −3.87 −5.14 −6.41 −7.69 −8.96 −10.23 −95.1 −99.6 −104.0 −108.3 −112.5 −116.7 31 = [λ(H + ) + λ(HC2 O− )] 4 306.02 335.20 364.40 392.91 421.42 449.81 82.16 91.45 100.97 110.69 120.61 130.71 27.29 31.34 35.70 40.20 45.02 50.14 52.16 57.47 62.87 68.34 73.90 79.55 3.159 3.569 4.064 4.501 5.050 5.699 477.0 527.1 578.0 629.7 682.2 735.4 78.56 89.97 102.55 115.16 129.14 144.37 835.6 923.5 1013.0 1103.8 1196.0 1289.5 144.63 165.96 189.48 213.23 239.54 268.21

276.63 73.11 23.48 46.93 2.763 427.8 67.78 749.2 124.55 294.52 103.01 50.33 152.81 −21.11 1017.4 −73.72 1317.2 −16.51

288.15

293.15

298.15

32 = [λ(H + ) + λ(1/2C2 O2− 4 )] 326.94 359.38 391.79 424.12 115.63 128.56 141.78 155.28 58.68 67.61 77.09 87.15 171.50 190.77 210.65 231.14 −24.84 −28.87 −33.16 −37.69 1142.4 1271.3 1403.9 1540.4 −89.21 −106.54 −125.63 −146.55 1480.1 1648.0 1821.0 1999.1 −22.19 −28.94 −36.66 −45.38

303.15

456.27 169.05 97.75 252.28 −42.46 1680.8 −169.45 2182.4 −55.27

308.15

448.10 183.05 108.88 274.02 −47.53 1824.8 −194.71 2370.6 −66.81

T , K; K 1 , K 2 , mol-dm−3 ; 1H , kJ-mol−1 ; 1S, J-K−1 mol−1 ; 3, λ, S-cm2 -mol−1 . from Ref. (2), Eq. (11).

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presented in Table VII. Similar calculations for the primary dissociation step were performed using Eq. (11).(2)

ACKNOWLEDGMENTS This work was supported by the Ministry of Education, Science and Sport of Slovenia (no. 01030-P-506). The authors are grateful to Romana Cerc Koroˇsec, dipl. chem., for thermogravimetric measurements.

REFERENCES 1. The Merck Index, An Encyclopedia of Chemicals, Drugs and Biologicals. 11th edn., S. Budavari, ed. (Merck, Rahway, NJ, 1989). 2. R. M. Kettler, D. A. Palmer, and D. J. Wesolowski, J. Solution Chem. 20, 905 (1991). 3. H. Schäfer, Z. Anorg. Chem. 45, 293 (1905). 4. C. W. Davies, Trans. Faraday Soc. 23, 351 (1927). 5. R. W. Money and C. W. Davies, J. Chem. Soc., p. 2098 (1938). 6. K. J. Pedersen, Trans. Faraday Soc. 35, 277 (1939). 7. A. McAuley and G. H. Nancollas, J. Chem. Soc., p. 2215 (1961). 8. W. Ostwald, Z. Phys. Chem. 3, 241 (1889). 9. C. Drucker, Z. Phys. Chem. 96, 381 (1920). 10. L. S. Darken, J. Amer. Chem. Soc. 63, 1007 (1941). 11. I. N. Maksimova and V. F. Yushkievich, Elektrokhimiya 2, 577 (1966). 12. E. E. Chandler, J. Amer. Chem. Soc. 30, 713 (1908). 13. E. Larson, Z. Anorg. Chem. 140, 292 (1924). 14. R. Gane and C. K. Ingold, J. Chem. Soc., p. 2153 (1931). 15. H. N. Parton and R. C. Gibbons, Trans. Faraday Soc. 35, 542 (1939). 16. H. N. Parton and A. J. C. Nicholson, Trans. Faraday Soc. 35, 546 (1939). 17. H. S. Harned and L. D. Fallon, J. Amer. Chem. Soc. 61, 3111 (1939). 18. J. C. Speakman, J. Chem. Soc., p. 855 (1940). 19. G. D. Pinching and R. G. Bates, J. Res. Natl. Bur. Std. 40, 405 (1948). 20. M. L. Dondon, J. Chim. Phys. 54, 304 (1957). 21. J. J. Christensen, R. M. Izatt, and L. D. Hansen, J. Amer. Chem. Soc. 89, 213 (1967). 22. J. L. Kurz and J. M. Farrar, J. Amer. Chem. Soc. 91, 6057 (1969). 23. R. Parsons, Handbook of Electrochemical Constants (Butterworths, London, 1959). 24. G. Kortüm, W. Vogel, and K. Andrussow, Dissociation Constants of Organic Acids in Aqueous Solution (Butterworths, London, 1961). 25. B. E. Conway, Electrochemical Data (Greenwood Press, Westport, CT, 1969). 26. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd rev. edn. (Butterworths, London, 1970). 27. D. Debos, Electrochemical Data (Elsevier Sci., Amsterdam, 1975). 28. R. K. Freier, Aqueous Solutions, Data for Inorganic and Organic Compounds, Vols. 1 and 2 (Walter de Gruyter, Berlin, 1976, 1978). 29. J. N. Collie and T. Tickle, J. Chem. Soc. (London) 75, 710 (1899). 30. W. Lee and R. Wheaton, J. Chim. Phys. 74, 689 (1977). 31. I. N. Maksimova and V. F. Yushkevich, Elektrokhimiya 2, 577 (1966). 32. P. Walden, Z. Phys. Chem. 8, 433 (1891).

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33. 34. 35. 36. 37.

R. Lorenz and A. Scheuermann, Z. Anorg. Chem. 117, 121 (1921). R. Taft and F. H. Welch, Trans. Kansas Acad. Sci. 55, 223 (1952). A. A. Noyes and J. Johnston, J. Amer. Chem. Soc. 31, 987 (1909). F. Kohlrausch and F. Mylius, Sitzungsber. Akad. Wiss. Berlin, p. 1223 (1904). G. Milazzo, Electrochemistry, Theoretical Principles and Practical Applications (Elsevier Sci., Amsterdam, 1963). M. Bester-Rogac, R. Neueder, J. Barthel, and A. Apelblat, J. Solution Chem. 26, 127 (1997). M. Bester-Rogac, R. Neueder, J. Barthel, and A. Apelblat, J. Solution Chem. 26, 537 (1997). M. Bester-Rogac, R. Neueder, J. Barthel, and A. Apelblat, J. Solution Chem. 27, 299 (1998). R. Wachter and J. Barthel, Ber. Bunsenges. Phys. Chem. 83, 634 (1979). J. Barthel, R. Wachter, and H.-J. Gores, in Modern Aspects of Electrochemistry, B. E. Conway and J. O’M. Bockris, eds., Vol. 13 (Plenum, New York, 1979). J. Barthel, F. Feuerlein, R. Neueder, and R. Wachter, J. Solution Chem. 9, 209 (1980). O. Kratky, H. Leopold, and H. Stabinger, Z. Angew. Phys. 27, 273 (1969). E. F. G. Herington, Pure Appl. Chem. 48, 1 (1976). L. Korson, W. Drost-Hansen, and F. J. Millero, J. Phys. Chem. 73, 34 (1969). B. B. Owen, R. C. Miller, C. E. Miller, and H. L. Cogan, J. Phys. Chem. 65, 2065 (1961). H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, 3rd edn. (Reinhold, New York, 1958). J. F. J. Dippy, Chem. Rev. 25, 151 (1939). E. Gonick, J. Phys. Chem. 50, 291 (1946). G. Bredig, Z. Phys. Chem. 13, 191 (1894). S. B. Brummer and G. J. Hills, Trans. Faraday Soc. 57, 1816 (1961). A. Apelblat and J. Barthel, Z. Naturforsch. 46a, 131 (1991). A. Apelblat and J. Barthel, Z. Naturforsch. 47a, 493 (1992). A. Apelblat, J. Mol. Liquids, 73–74, 49 (1997). E. N. Tsurko, R. Neueder, J. Barthel, and A. Apelblat, J. Solution Chem. 28, 973 (1999). J. Quint, Thesis, University of Clermont-Ferrand, April 1976. J. Quint and A. Viallard, J. Solution Chem. 7, 137 (1978). J. Quint and A. Viallard, J. Solution Chem. 7, 525 (1978). J. Quint and A. Viallard, J. Solution Chem. 7, 533 (1978). J. Kielland, J. Amer. Chem. Soc. 59, 1675 (1937). D. C. Harris, Quantitative Chemical Analysis (Freeman, San Francisco, CA, 1982).

38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.