1216
A. K. R. UNNI, L. ELIAS,AXD H. I. SCHIFF
with the hydrogen bond of reaction (S). Calculations indicate that the contribution of the species ]-COOca+, AHN=I varies from approximately 0.2 per mole of albumin a t p H 7 to approximately 0.003 a t pH 10 for kea, = 4 and may be ignored. Similarly, the Ca++N=,) and -COOH, species ] -COOH, N F ~ are present in concentrations small enough to be ignored above pH 6.5. With these restrictions, ijCa becomes nP(co0- . , , Ca++ . . . N) and, with the assumption that eH = e A
1
Equation 15, with the values for k ~ %k , ~ eH,~ n,, and J C A ~ the same as those used for curve 4, and with (kc&)&= 20, yields curve 5. As eH is allowed to increase ( k C a ) & must also be increased until the curve calculated from equation 15 moves out of a fit with the experimental data as eH approaches 100 and (kCa)ch approaches 70. Equations 10 and 15 represent two
Vol. 67
extremes in treating the uncertain localized electrostatic effect of the pair. Yet in both cases the chelate is the important binding site for calcium ion with the value for the association constant varying between those found from studies on model compounds.
Discussion The peculiar behavior of serum albumin rests as much in its failure to bind ions as predicted as in its generalized affinity for ions. In the acid region serum albumin binds fewer anions than are ca1c~lated.l~In the alkaline region serum albumin binds fewer calcium ions than are predicted. The anion in serum albumin available for competition with other anions in solution is the carboxylate ion. The cation in serum albumin available for competition with calcium in solution is the ammonium ion. The specific pairing and hydrogen bond fornution proposed for calcium and chloride binding represents a consistent treatment for the binding of anions arid cations to serum albumin in the pH range from 2 to 10.5.
THE CONDUCTIVITIES OF SOME QUATERSARY AMMONIUM CHLORIDES AiVD BROMIDES IN NITROXETHANE AT 25’ BY A. K. R. UNN, L. ELIAS, AJXD H. I. SCHIFF Department of Chemistry, McGill University, Montreal, Canada Received November 9, 1962 The conductance of Me4YCl, RlelNBr, Et4KCl, EtaXBr, Pr4NC1, Pr4NBr, Bu4NCl, and BudNBr in nitromethane has been determined over the concentration range 0.0001 to 0.01 N . Considerable attention has been given to the purity of the solvent and salts. Limiting conductances were determined by extrapolation of h-4; plots. Although this method usually yields values which are slightly high, other methods were found not to be suitable to these electrolytes. The values obtained for Ab were consistent with the Kohlrausch rule of independent ionic mobilities within experimental error. The deviations from the Onsager equation increased in the order BudNX < PrdNX < E t 8 X < MeJNX in accordance uTith the decrease in ionic size, as indicated by the ho values. However, the chlorides appeared t o be more associated than the bromides even though the AOvalues indicated C1- to be slightly larger than Br- in solution. The Robinson-Stokes equation was found not to be applicable to these electrolytes.
Introduction Conductance data precise to within a few parts in 10,000 have been recorded for aqueous and alcoholic solutions.1-6 Such data have been important both in the testing and, indirectly, in the development of conductance theory. There are a t present, however, virtually no conductance data of comparable caliber available for non-hydroxylic media. In the best work reported to date for these solvents, the great care taken with the electrical measurements has not, in general, been matched with similar care in the preparation and testing of the solvent and solute. The present investigation was undertaken to provide accurate data for nitromethane solutions of some quaternary ammonium halides over the concentration range 0.01 to 0.0001 *V, a t 25’. The following paper reports ionic conductances for these solutions. Kitromethane was chosen as the sokent, since it has a dielectric constant in the “intermediate” range where (1) T. Shedlovsky, J . Am. Chem. Soc., 6 4 , 1411 (1932). (2) B. B. Owen and H. Zeldes, J . Chem. Phys., 18, 1083 (1950). ( 3 ) G. C. Benson und A. R. Gordon, ibid., 18, 470 (1945). (4) J. P. Butler, H. I. Schiff, and A. R . Gordon, ibid., 19, 762 (1951). (5) R. E. Jervis, D. R. Muir, and A. R. Gordon, J . Am. Chem. Sac., 75, 2855 (1953). (6) J. R. Graham, G. 9. Kell, and A. R. Gordon, ibid., 79, 2352 (1957).
other than the simple coulombic forces of the DebyeHiickel treatment should become important but where a good deal of dissociation will still be expected. Also, comparison of ionic behavior in nitromethane with that in methanol should be instructive, since the solvents have similar dielectric constants but differ chemically. The quaternary ammonium halides were chosen as solutes for reasons of solubility and because they form a homologous series useful for comparison purposes. The salts studied were tetramethyl-, tetraethyl-, tetrapropyl-, and tetrabutylammonium chlorides and bromides. Experimental The conductance measurements were made by the d.c. method using B modification7 of the cell described by Gordon and coworkers.* Briefly, the method consists of passing a known current through the solution and measuring the potential drop across a portion of the solution by means of two probe electrodes; the conductance is then calculated from Ohm’s law. In the present work, the probes were silver-silver chloride (or bromide) electrodes immersed in aqueous chloride (or bromide) solution. Contact with the nitromethane solution was made through the liquid junction formed between the water-nitromethane solutions in a Type A probe chamber.’ The electrodes could not be im(7) L. Elias and H. I. Schiff,J . Phys. Chem., 60, 595 (1956). (8) H. E. Gunning and A . R. Gordon, J . Chem. Phys., 10, 126 (1942),
,June, 1963
CONDUCTIVITIES OF
QUATERNARY AMMONIUM C H L O R I D E S ,4ND BROMIDES
121.7
mersed directly in the more concentrated nitromethane solutions of halides owing t o a fairly rapid dissolution of the silver halide coating. However, below concentrations of about 5 X 10-4 N the rate of dissolution was sufficiently low that the original Gordon cell could be used. I n these cases, comparative tests further verified the previous finding that the conductivities measured by the liquid-junction-probe method were identical wibh those measured with the Gordon cell. The Jones and Bradshaw 0.01 demal standardg was used to obtain the cell constant. The mean of six determinationsgielded a value of 0.1 6461 If: O.OOOOO6 cm.-lfor the cell with liquid junction probes; the same cell with Gordon probes had a cell factor 3.02% lower. The capacity of the cell was about 150 cc. Solvent conductances were measured with a small a.c. cell and bridge circuit to within 3%. The d.c. measurements were made in the manner described by Gunning and Gordon,* with the constant-current circuit modified to accommodate a 6557 rather than 1B4 pentode. Difficulty encountered with the electrical measurements in hot, humid weather was effectively overcome by packing silica1 gel in the glass tubes surrounding the probe leads. The measurements were made after allowing the cell to stand one hour in an oil bath set a t 25.000 =k 0.005O. The bath thermometer was periodically calibrated at the Thermometry Division of the National Research Council in Ottawa. Two calibrations taken one year apart showed a variation of 0.003'. All solutions were prepared gravimetrically, the more dilute ones being prepared by dilution of different stock solutions. Sufficient amounts of materials were used t o ensure an accuracy of better than 0.01% in the weighings. I n the preparation and transferring of the solutions, precautions were taken to ensure a minimum exposure of the materials to air. These precautions included the use of a drybox for handling the more hygroscopic salts and the use of purified nitrogen under pressure for drying Dome of the glassware as well as transferring of solvents and solutions. Complete details of the handling technique may be found e1sewhere.l0J1 The nitromethane densities were determined with a pycnometer of the type described by Shedlovsky and Brown.Iz Salt densities used in correcting weights to vacuum were determined in a 25-cc. specific gravity bottle by displacement of ligroin. The pycnometers were thermostated in a water bath held a t 25.00 f 0.01". Nitrometham--The method of purification was ementially that devised in this Laboratory by Tinkl* and Bulani14: distillation at reduced pressure followed by fractional crystallization. Nitromethane was obtained from Commercial Solvents Ltd. as a clear, pale yellow liquid. A batch of 5 1. was distilled in a column 3.6 cm. in diameter and 300 cm. in height packed with glass helices. The distillation was carried out at 4-6 cm. pressure, which was regulated with a Cartesian manostat. The reflux ratio was adjusted to 10: 1 using a conventional magnetic take off a t the head of the column. First and last fractions of about 1.5 1. were rejected, a middle cut of 2 liters being retained. This distillate was fractionally crystallized by fitting the receiver flask with a smaller side-flask and placing the assembly in a wooden box packed with Dry Ice. The nitromethane was allowed to freeze while the box and its contents were agitated on a mechanical shaker. When all but about 200 cc. of the nitromethane was frozen, which required 4-6 hours, the crystallization was stopped and the unfrozen liquid containing the impurities was poured over into the side-flask. After the nitromethane came to room temperature, the side-flask was removed and the receiver flask stoppered. To obtain a clear, solid mass of frozen nitromethane, it was usually necessary to seed the liquid with a small crystal from a preliminary freezing. Once the solvent had been purified as above and used for conductance measurements, it could be recovered in a sufficiently pure state for further use by distillation alone. Nitromethane contains higher-boiling and lower-boiling impurities, both of which lower the density. The densities of three successive 500-cc. fractions taken at the top of the still were 1.124, 1.130, and 1.131 g./cc. at %5", while successive fractions taken a t the bottom were 1.120, 1.119, and 1.116
g./cc. One crystallization of a rather impure sample raised the density from 1.129 to 1.130, the den& of the discarded portion being 1.128 g./cc. The conductance of samples taken a t the top or bottom of the still was the same, and is therefore no criterion for purity. After repeated distillations and crystallizations, the density of the solvent was 1.13124 i. 0.00004 g./cc. a t 2 5 O , and its specific conductance 0.5-1 X ohm-' crn.-l. The density found here compares favorably with that reported by Hartley,16 although his conductance value is higher by a factor of 10, undoubtedly due to his distillation procedure a t atmospheric pressure and consequent decomposition. The purity of the nitromethane was determined absolutely by the freezing-curve method of Rossini.le About 100 cc. of nitromethane was placed in a vacuum-jacketed test-tube immersed in an acetone-Dry Ice slurry. The liquid was agitated by a vertically operated mechanical stirrer at 160 strokes/min. while dry nitrogen wati blown in to exclude air moisture. The cooling rate was less than 2'/min. Temperatures were read off a Beckmann thermometer a t recorded time intervals until about 1/2 of the nitromethane was frozen. From a geometrical analysis of a time-temperature plot, the molar purity was found to be 99.96%. The particular sample of solvent used in the purity determination had a density of 1.13120 g./cc. and was certainly not the best product used for the conductance work. Nevertheless, conductivity data obtained with it were indistinguishable from data obtained using nitromethane of higher density. On the other hand, use of water-free solvent of density less than 1.130 g./cc. gave erroneous data. It is interesting to note that Dreisbach and Martin'? found the purity of a nitromethane sample of density 1.12346 g./cc. to be 99.45%. Titration of the purified solvent with Karl Fischer reagent18 showed the water content to be less than 0.001% by weight. The titration proceduIe was standardized with sodium acetate trihydrate in methanol and checked against nitromethane samples to which small amounts of water had been added. Corrections were made for the small water content of the solvent by measuring the effect of water on the conductivities of nitromethane solutions. Conductance measurements were made on 0.0005,0.002, and 0.007 N Pr4NBr solutions to which known amounts of water up to 0.04% by weight were added, the additions being checked by Fischer titration. After allowing for the slight change in A due to the small change in concentration, it was found that the increase in A for a given electrolyte concentration was linear in the amount of water added. One per cent by weight of water caused an increase of 7.3,9.6, and 15.0%, respectively, in the conductances of 0.007, 0.002, and 0.0005 N solutions. A correction, amounting at most to a few hundredths of a A unit, was accordingly applied to the results. The water content of a nitromethane solution left standing for three hours in the cell with liquid junction probes increased by less than 20%, indicating no serious contamination of the cell solution by the probe solution. Quaternary Ammonium Halides.-The purity of the salts was estimated on the basis of the halide content as determined by gravimetric analysiei. The analytical procedure was checked against "Conductivity grade" KCl and KBr; the accuracy of the analysis is believed to be better than 0.03%. After recrystallization from solution, the salts were dried in vacuo in an "Abderhalden" type of drying pistol, with a trap immersed in Dry Ice-acetone serving as the desiccant. The part of the drying tube holding the salt was heated by the vapors of a boiling liquid, the drying temperature being determined by the choice of liquid used. The effect of drying temperature on salt purity was the same for all the salts studied: a t lower temperatures the purity improved as the temperature was increased, remained constant within a temperature range at higher temperatures, then finally deteriorated at still higher temperatures. Temperatures below 110" were not high enough for drying Pr4NCl. I n this low temperature region the length of drying time had only a slight effect on improving the purity; for example, a sample of Et4NBr dried a t 80" for 12 and 60 hours, respectively, analyzed as 99.55 and 99.57%, which are identical within the limits of analytical precision. Dried a t 160" for 12 hours, the salt analyzed 99.97'3,. A t higher tem-
(9) G. Jones and 13. B. Bradshaw, J . A m . Chenv.. Soc., SS, 1780 (1933). (10) L. Elias, Ph.D. Thesis, MoGill University, 1956. (11) A. K.R. Unni, Ph.D. Thesis, MoGill University, 1958. (12) T. Shedlovsky and A. S.Brown, J . A m . Chem. Soc., S6, 1066 (1934). (13) R. R. Tink, Ph.D. Thesis, MoGill University, 1953. (14) W, Bulani, Ph.D. Thesis, MoGill University, 1954.
(15) C. P. Wright, D. M. Murray-Rust, and H. Hartley, J . Chem. S o c . , 199 (1931). (16) F. Rossini, J . Res. Natl. Bur. Std., SS, 355 (1945). (17) R. R . Dreisbaoh and R. A. Martin, Ind. Eng. Chem., 41,287.5 (1949). (18) K. Fisoher, 2. angew. Chem., 48, 394 (1935).
A. IC. R. UNNI, L. ELIAS,AND H, I. SCHIFF
1218
peratures, 160' in the case of Pr4NC1 and 180' in the case of EtaNBr, the salts underwent some decomposition, as indicated by a strong amine smell and decrease in purity; a t these temperatures the length of drying time had a marked effect on the purity. The range of drying temperature best suited for each salt was determined by such trials. Salts prepared by alternate procedures and dried a t the suitable temperature gave identical conductance results. All were used within a week or two after purification. The densities of nitromethane solutions of the salt3 could be adequately expressed by p =
1.13124
+ b&
where iiz is the concentration in moles/1000 g. of solution and b is a constant for each salt given below. Me4NCl and Me4NBr were Eastman Kodak products recrystallized from methanol and twice precipitated from methanol by addition of ether. They were dried at 140' for 24 hours, and each analyzed as 99.98% halide. For Me4NCl, Halt density = 1.16 g./cc. and b = 0.017; for Me4NBr,density = 1.58 g./cc. and b = 0.057. EtaNC1, Pr4NC1, and Bu4NC1 were prepared by the metathesis of the corresponding iodides by AgCl in methanol, the conversion being checked by the starch test for iodide. After filtration through a sintered-glass funnel, the solutions were evaporated to dryness under vacuum. The ethyl salt was recrystallized from chloroform and precipitated from chloroform with ether, three times; it was dried at 140' for 24 hours; analysis 99.97%, density 1.37 g./cc., b = 0. The propyl salt was thrice precipitated from acetone with ether and dried a t 132'; analysis 99.96%, density 1.10 g./cc., b = -0.019. The butyl salt, which was extremely hygroscopic, was given a preliminary drying a t room temperature for 12 hours, then thrice precipitated from acetone with ether and dried at 56"; analysis The qua99.98%, m.p. 75", density 1.1 g./cc., b = -0.053. ternary ammonium iodides used in the methateses were prepared by refluxing the alkyl iodide with the trialkyl amine in methanol for about 24 hours. The unreacted reagents were removed by extracting the solution with ligroin, after which the salts were recovered by evaporating the methanol layer to dryness under vacuum. The iodides so obtained were recrystallized in acetone and precipitated from acetone with ligroin. Et4NBr was an Eastman Kodak product precipitated from chloroform with ether and from methanol with ether. I t was dried a t 160'; analysis 99.S'l%, density 1.37 g./cc., b = 0.010. PrJVBr was prepared and purified in a similar manner t o that described for the iodides, using n-propyl bromide in place of the iodide; it was dried a t 132', analysis 99.97%, density 1.19 g./cc., b = 0. Bu4NBr was prepared as suggested by Sadek.lQ Tri-n-butylamine and n-butyl bromide were refluxed in methanol for 30 honrs. Water was added and the unreacted reagents extracted with ligroin, The water layer was cooled to a slush and dried in vacuo over PzO6. One batch of the crude product was recrystallized six times from bensene-ligroin mixtures, a second batch was precipitated three times from acetone with ether. The salt was dried a t 80-100"; analysis 99.98% m.p. 118.5", density 1.13 g./cc., b = -0.010. An alternative preparation in which Bu4NOH was neutralized with anhydrous HBr in methanol yielded a product of the same purity; the conductance data obtained using this sample agreed with the data derived using the first preparation within experimental precision.
Results and Discussion Values of the equivalent conductance a t round concentrations and limiting conductances are given in Table I. These were obtained from a large scale plot of the Shedlovsky function A,' in a manner similar to that described by Benson and Gordona3 Such a plot was made from conductance measurements of at least 24 different solutions with conceiitratioiis spanning the concentration range given. The precision of the data over the concentration range 0.01 to 0.001 N was better than 0.027?0, but decreased to 0.037?0 a t the highest dilution. The Shedlovsky plots all exhibited pro(19) H. Sadek and R. M. Fuoss, J . A m . Chem. Soc., 72, 301 (1950)
Vol. 67
nounced minima except in the case of the methyl salts, where the curves lay entirely below ~ i o . ~ O A-dc plots of the data were in all cases linear in the range 0.001 to 0.0005 N within experimental precision. With the values D = 36.6721and 17 = 0.627 centipoise15 for nitromethane a t 25O, the Onsager equation becomes A = ha - (0.708&
+ 125.1)dC
The per cent deviations of these linear portions from the theoretical slopes are: Me4YC1, 72; Me4YBr, 55; EtdNC1, 13; Et4NBr, 10; Pri?JCl, 10; Pr4n'Br, 5; Bu4n'C1, 8; Bu4NBr, 4. Except far the methyl salts, the quaternary ammonium halides in nitromethane are all reasonably strong electrolytes. The order of increasing ionic strength is Me4NX < Et4NX < Pr4NX < Bu4SX; this is also the order of iiicreasiizg size of the cation in solution, if one attributes differences in A, to changes in ionic diameters. The behavior is therefore consistent with the smaller ion having the greater tendency to associate. On the other hand, the chlorides appear to be more associated than the corresponding bromides, while, if A, is again taken as a measure of ionic size, the chloride ion is slightly larger than the bromide ion. This behavior is not consistent with the smaller ion having a greater tendency toward association. The inconsistency may be resolved by assuming that the radius of the migrating C1- is larger than that involved in the distance of closest approach with the cation. This assumption implies that the solvent molecules are loosely held by C1- and that the cation is able to penetrate the solvent sheath. Since nitromethane is aprotic and solvation of the anion probably occurs through weak ion-dipole interactions, this interpretation is not unreasonable. It is interesting to note the different behavior of these salts in methanol, for which Hartley has reported conductance data of the tetramethyl- and tetraethylaminoiiium chlorides and In methanol, where anion-solvent effects should be much stronger, C1- is also larger than Br- but the chlorides are less associated than the bromides, in conformance with the more usual type of behavior. The A, values listed in Table I were obtained by a least-squares analysis of the linear portion of the A - 4 2 plots. Although this extrapolation yielded Ao's for which the difference between bromides and chlorides was 0.22 for each cation within 0.01 A unit, the procedure is objectionable on theoretical grounds and leads to Ao's which are too high. It was adopted because none of the other available extrapolation methods proved superior. Extrapolation of the Ao'--c plots was uncertaiii because of the steep curvature at the dilute end, except for Pr4;?JBr and Bu4r\iBr; for these salts, Ao's derived from the Shedlovsky curves were 0.05 unit lower than those obtained from the A-dF plots. Attempts to fit the data to the form12 &' = A0 Bc Ec log c were for the most part unsuccessful.
+ +
( 2 0 ) The Shedlovsky plot for BuaNBr shown in the previous publication t o illustrate the conductance method lies above d o over the entire concentration range. Further work in the diliite range revealed that the curve actually goes through a minimum near c = 0.0005 1~ and approaches Ao from below. (21) A A. Maryott and E. R Smith, Natl. Bur. Standards. Cirs. 514 (1951). (22) G. Unmack, R. Bullook, D. M. Murray-Rust, and H. I-Iartley, Proc. Roy. Soc. (London), 8132,427 (1931).
June, 1963
121.9
CONDUCTIVITIES OF QUATERNARY AMMONIUM CHLORIDES ASD’BROMIDES TABLEI EQLJNALENT CONDUCTANCE AT ROUND CONCENTRATIONS
10%
(g.-equiv./l.)
MeaNCl
XeaNBr
EtaNCl
100 70 50 20 10 5 2 1
82.18 87.21 91.51 101.08 106.08 109.58 112.63 114.22 117.62
86.34 90.91 94. ‘77 103.14 107.49 110.61 113.27 114.62 117.83
90.24 93.00 95.29 100.39 103.15 105.20 107.07 108,07 110.37
A?
Equivalent conductance APrrNCl EtpNBr
Similarly, a Fuoss-Shedlovsky treatmentz3 of the data was also of limited use in determining Ao. In the case of the methyl salts, a straight-line relation between AS and c ( A X ~ )existed ~ a t higher concentrations but deviated from linearity a t lower concentrations. The propyl and butyl salts and Et4NBr mere amenable to the treatment only a t concentrations below 5 X N, although even here there was a tendency of the plots to curve upward. The Ao’s obtained by this method were consistently lower by about a tenth of a A-unit than the values in Table I. An analysis of the data by the revised Fuoss-Onsager equation24 is discussed iii a subsequent paper. 25 While some uncertainty exists in the extrapolations, it could by no means account for the discrepancy between A, for Bu4KBr found here and that reported by Miller and F u o s ~ . These ~ ~ workers give a value of 86.80 for this salt in nitromethane a t 2 5 O , which differs by more than 10 A-units from the present value. At least part of this discrepancy can be attributed to the lorn purity of their solvent, as indicated by its low density of 1.1251 g./cc. It may also have been contaminated by PZO, from the distillation procedure used, as found in the investigation of Hartley15 and W a l d e i ~ ~ ~ A comparison of the Walden product Ao7 for the data (23) R. M. Fuoss, J. A m . Chenz. Soc., 57, 488 (1935). T. Shedlovsky, Inst., 226, 739 (1938). (24) R . M. Fuoss, J. A m . Chem. SOC.,81, 2659 (1959). (25) R. L. Kay, S. Blum, and H. I. Schiff, J . Phys. Chem., 67,1223 (1963). (26) R. C. Miller and R. M. Fuoss. J. A m . Chem. Soc., 75, 3076 (1953). (27) P. Walden and E. J . Birr, Z . physrk. Chem., A16S, 263 (1833).
J. Franklzn
90.53 93.29 95.64 100.75 103.54 105.58 107.41 108.36 110.60
83.76 86.18 88.21 92.70 95.17 97.04 98.76 99.70 101.88
Pr4NBr
Bu4NC1
BurNBr
83.95 86.44 88.49 93.10 95 66 97.53 99.21 100.03 102.10
79.98 82.18 84.04 88.14 90.42 92.18 93.86 94.72 96.83
79.97 82.23 84.14 88.39 90.75 92.48 94.14 94.98 97.04
of this research with work in other solvents is shown in Table 11. The agreement between solvents is not too surprising, since all three have similar dielectric constants and are electron donors. TABLEI1 COMPARISON OF Nitromethane = 0.627 cp.)
(7
Me4NBr EtnNCl EtnYBr Pr4NBr Bu4NBr
A,q
-
Aov
c -
0,738 .6!34 .6!33 .640 .608
Nitrobenzene28 1.1811 cp.)
(7 =
Dimethylformamldez8 (7 = 0.796 cp.)
0.738
0.700 0.710 0.659 0.607
The conductance equation of Robinson and Stokes30 combined with the law of mass action was not applicable to the data of this research. No reasonable value of d could be found for which the dissociation constant was invariant with concentration. Acknowledgments.-The authors wish to thank the National Research Council for a grant-in-aid and for the award of Fellowships to L. E. They are also grateful for the award of a Columbo Plan Fellowship to A. K. R. U. (28) C. R. Witschonke and C. A. Kraus, J. A m . Chem. Soc., 69, 2472 (1947). (29) P. G. Sears, E. D. Wilhoit, and L. R. Dawson, J. P h y s . Chem., 59, 373 (1955). (30) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd Edition, Butterworths Scientihc Publications, London, 1959.
