J.C.S. Faraday I, 1980,76,2510-2518Apparent Molal Volumes of some Highly ChargedElectrolytes in WaterBY FRANCESCO MALATESTA" AND ROBERTO ZAMBONIIstituto di Chimica Analitica ed Elettrochimica dell' Universiti,Via Risorgimento 35, 56100 Pisa, ItalyReceived 5th December, 1979The apparent molal volumes of Mg,Fe(CN),, Sr2Fe(CN)6 and K,Co(CN), were measured at 25'C bypyknometric and dilatometric methods. The 2 :4 salts appreciably deviate from the theoretical limitinglaw in a direction which is opposite, in the more dilute solutions, to that predicted by Debye-Huckeltheory (DH). For K,Co(CN), no appreciable deviations from the theoretical slope are seen below0.01 mol dm-3 and the deviations are in the same direction as DH predicts when the concentration isincreased. Such behaviour is similar to that of Ca,Fe(CN), and K,Fe(CN),.The results are analysed by means of the Mayer theory and by numerical integration of the Poisson-Boltzmann equation.Both treatments can justify, in terms of electrostatic interaction, volumetric behav-iour similar to those of Mg2Fe(CN)6 and Sr2Fe(CN)6 for the 2:4 salts and to that of K,CO(CN)6 for the1 : 3 salts.An enhanced, qualitative discrepancy from the predictions of Debye-Huckeltheory (DH) occurs for the apparent molal volumes (4") of some strong multivalentelectrolytes.'-6 According to DH, the 4v slopes should usually be less than thetheoretical limiting slope (DHLL) at real concentrations, and approach DHLL asthe concentration decreases. In Nd(N03)3,2 K4Fe(CN)6,3'4 Ca2Fe(CN)2 and someother salts,'*5 the slope of 4v becomes greater than DHLL at low concentrationand then increases as the concentration is further decreased. The limiting slope isnot approached in the experimental range of concentrations, and it is difficult toextrapolate the molal volume at infinite dilution ( V O ) .Deviations from DH, when observed in aqueous dilute solutions of 1:1 electro-lytes, are reliably assumed as evidence of non-electrostatic interactions.3 7 However,the positive deviations of highly charged electrolytes'-' (and the similar deviationsobserved in apparent relative molal enthalpies, 4L8) may perhaps be justified interms of fundamentally electrostatic interactions. In aqueous solutions of highlycharged electrolytes' (as well as in non-aqueous solutions of weakly charged elec-trolytes") the linearisation of the Poisson-Boltzmann exponential term is math-ematically unjustified at ionic strengths ( I ) which are much lower than in aqueoussolutions of 1:1 salts, and DH gives a misleading view of the electrostatic behav-iour.Indelli and De Santis' found that Mayer theory"?l2 was able to justifypositive deviations from DHLL in dilute solutions of 1 : 3 and 1 :4 salts. They usedthe so-called DHLL + B213 approximation of Mayer theory, i.e. the same physicalmodel of the Debye-Huckel theory (primitive model). DHLL + B2 and a numeri-cal integration of the Poisson-Boltzmann equation, IPBE93l 0,14-1 were also ableto explain the positive deviations in 4L of highly charged electrolyte^.'^.^^ (In spiteof the well known inconsistencies of the Poisson-Boltzmann equation,' IPBEagrees with the experimental results better than does DHLL + B2.16917)We have therefore studied other, multivalent electrolytes.In this paper, theresults obtained for the 2:4 salts Mg,Fe(CN)' and Sr2Fe(CN)', and for the 1:3 salt251F . MALATESTA AND R . ZAMBONI 251 1K,Co(CN),, are shown. The volumetric behaviour of other highly charged electro-lytes [in particular Ca2Fe(CN)z and K,Fe(CN);] are reexamined for comparison.EXPERIMENTALThe apparatus was the same as that used p r e v i ~ u s l y ~ . ~ and did not show any difference incalibration. The experimental technique was also the same as in previous ~ o r k . ~ , ~MgFe(CN)6 and Sr,Fe(CN), were prepared by ion exchange on a Dowex 50 wx8 resin in thehydrogen form, so as to give H4Fe(CN)6, which was immediately gathered in MgO or SrC03.Both salts were recrystallised several times from water + ethanol mixtures and then three timesfrom conductivity water.K,Co(CN), was obtained following the Benedetti-Pickler method"and purified by iterated crystallisations. All salts were air-dried for 1 week and stored inhermetic containers.In order to check compositions and water content, salts were analysed, so as to obtain theirequivalent weights with respect to at least one of the components. K&O(CN)6 was tested forCo by an electrogravimetric method.20,21 We found Co = 17.75 f 0.05% (theoretical value forthe anhydrous salt, 17.73%), which gives an apparent molecular weight of 332.0 & 0.9 (theoreti-cal, 332.35).In addition the salt was tested once for its Co(CN):- content: H,Co(CN), wasobtained by ion exchange, gathered in NaOH solution and then back titrated with HC1. Theresult was consistent with a molecular weight of 332.7.Mg,Fe(CN), and Sr,Fe(CN), were analysed by a similar method4 for H,Fe(CN), ; apparentmolecular weights of 475.5 k 0.8 [theoretical, 476.76 for Mg,Fe(CN), - 12H20] and 619.8 & 0.9[theoretical, 621.39 for Sr2Fe(CN)6 * 13H20] were obtained. Mg,Fe(CN), was also tested forMg by EDTA titration: an apparent molecular weight of 477.0 & 0.9 was found.we assumed that the salts had the exact compositions K,CO(CN),,Mg,Fe(CN), - 12H20 and Sr,Fe(CN), * 13H20.Slight differences in water content could beresponsible for the systematic errors in 4; (z0.9cm3 rno1-l for the Sr salt, for instance,resulting from the difference between theoretical molecular weight and apparent molecularweight obtained by the ion-exchange-acidimetric method).In calculatingCALCULATIONSAll calculations were performed using the same value for the distance of closestapproach, 4, between cation and cation, cation and anion and anion and anion.4DHLL + B2 formulae for the apparent molal volumes have already been de-The IPBE calculations for apparent molal volumes are similar to thosefor 4,,,16717 except that the numerical differentiation of the activity coefficients iscarried out against pressure ( p ) instead of temperature (7').The value of the dielec-tric constant of water and of its derivative with respect to pressure were taken fromOwen.22 The water compressibility coefficients, p, are those of Kell and W h a l l e ~ . ~ ~Of course, both DHLL + B2 and IPBE (as well as DH) cannot directly providethe 4" values, but 4; quantities which differ from q5v in the V" term (ie.,4; = theoretical 4v - I/" value).RESULTS AND DISCUSSIONThe experimental results are reported in tables 1 and 2. K,CO(CN)6 behaves in asimilar way to K3Fe(CN)6.4 Mg2Fe(CN)6 and Sr,Fe(CN), display very similarbehaviour to Ca2Fe(CN),.' At low concentrations (below z 3 x lo-, mol dm-,)the 2:4 salts show striking positive deviations from the limiting slope (DHLL)2512 APPARENT MOLAL VOLUMES OF STRONG ELECTROLYTESTABLE 1 .-DENSITIES AND APPARENT MOLAL VOLUMES OF CONCENTRATED SOLUTIONSsolution c/mol dm-3 a di5 4" cm3 mol-'0.3 1 5050.092950.171730.421 10.214030.214050.1 I7740.21 3290.0975 11.0996891.0673631 .0279631.0330791.021 9271.0721701 .0356491.014911.041 74661.6,58.9555.0152.025 1 .0549.61154.50151.92149.79a Based on the assumption of the exact compositions Sr,Fe(CN)6. 13H20,Mg2Fe(CN)6-12H20, and K3Co(CN), for solid salts.bFor water at 25 C we took di5 =0.997 07TABLE 2.-4., FROM DILATOMETRIC MEASUREMENTS. NUMBERS IN BRACKETS IDENTIFY THE START-ING SOLUTION IN TABLE 1.(1) 2.806(1) 2.341(2) 1.906(2) 1.590(3) o.8277(1) 0.7392( I ) 0.7392(3) 0.6907(1) 0.3400(2) 0.291 2(1)O.l6l7(2) 0.1099(2) 0.0495(3) 0.0335(1 ) 0.0324(2) 0.0220(3) 0.021 5(3) 0.021 551.3050.9050.2449.8448.2748.0,47.7946.9446.0045.32"44.9643.842.540.841.440.639.040.0~~(1) 3.05 1(1) 1.906(1) 1.590(2) 1.529(3) 1.049(3) 0.8749(1) 0.5022(3) 0.241 7(1) 0.1099(3) 0.0869(3) 0.0577(2) 0.025( I ) 0.031 3(1) 0.0220(1) 0.0220~~45.7044.6544.2044.1t3.4743.0241.8940.3838.93K336.036.535.537.635.2~~ ~(1) 3.751(1) 3.130(2) 1.899(2) 1.899(2) 1.585(3) 0.86g3(2) 0.6027(1)0.2162(1) 0.9882(2) 0.2407(3)0.1100(2)(1) 0.053(3) 0.050(1) 0.0974(2) 0.0493(1) 0.043 3~~147.76147.48146.85146.S4146.6,145.80145.95145.40144.7144.4144.6143.7144.2143.5144.5143.s144.3~ ~~"Erratic value.An anomalous, progressive descent of the level in the capillary of the dilat-ometer was observed, which did not correlate with temperature (a bubble of air'I4).unlike the 1:3 salts. This cannot be justified in terms of DH theory,* but it agreescomparison, the positive deviations in $v of Mg2Fe(CN),, Sr2Fe(CN)6 andwith previous results for the & values of multiply charged electrolyte^.^*'^^'^ BY* In principle, DH predicts positive deviations from DHLL for high enough, positive values ofd In B/dp. In practice, values of d In B/dp that are too high (of the order of 10' b) are required in order tojustify positive deviations such as those in 2 :4 salts, and they seem unreasonable (furthermore, theshapes of the DH curves calculated in this way are entirely different from experimental ones).For similarconsiderations about $,, see ref. (17)F . MALATESTA A N D R . ZAMBONI 25137-3I - zi 5 0a3-Ca2Fe(CN)6 occur over a wider range of I than the corresponding deviations in #L.(This cannot fully be explained at present; however, other salts behave ~imi1arly.l~In terms of the theories based on the primitive model, it may mean that thedistances of closest approach cannot be considered as independent from T andAlthough the 4" values of the 2:4 salts are not linear functions of I+ below0.4 mol dmP3 ionic strength, their differences, A#", are almost linear when plottedagainst I' (fig. 1). The A#" values were obtained by subtracting the experimentalvalue of 4" for Mg2Fe(CN), and Ca2Fe(CN),* from linearly interpolated values of# v for Sr2Fe(CN)6 (similar results are obtained from graphical interpolation).A 2:4p . I 7 ) .0 . /./- I .---~ ~ ~ o ~ - o - o0 .o 08a1 I 1salt should reasonably simulate another 2:4 salt at low concentration, and one canexpect that the almost linear trends of A$" are maintained up to the I = 0 limit(although fig. 1 does not illustrate this). This suggests a difference in vc of = 5 cm3mol- ' between Sr2Fe(CN)6 and Mg2Fe(CN)6 [1.5 between Ca2Fe(CN)6 andThe V o values of the 2:4 salts cannot be obtained from direct extrapolation ofthe experimental #v curves. An evaluation of the vo values is provided by theadditivity relationships from vJ of weekly charged electrolytes.The values 30.1,36.1 and 36.7 cm3 mol- ', for Mg2Fe(CN)6, Sr2Fe(CN)6 and Ca2Fe(CN)6, respect-ively, are calculafed from Millero's selected data' if one assumes 108S4 (instead ofl-10.083) as the V" values of K,Fe(CN),. The difference, 6 cm3 rnol-I between theI/" values of Sr2Fe(CN), and Mg,Fe(CN),, compares favourably with the valuesuggested by fig. 1 (= 5 cm3 mol-'); the small discrepancy may be due to system-atic errors in $" (water content uncertainties in the salts) or to inaccuracies in theliterature data. The vc values of 30.1 and 36.1 were used to calculate the values of4: for Mg2Fe(CN)6 and Sr,Fe(CN),, respectively, to be compared with theoretical4: (DHLL + B2 and IPBE). A discrepancy of a few cm3 mo1-I from true extrapo-lations cannot change the meaning of such comparisons.Mg 2 F e m 6 1 .* The basic experimental data of ref.( 5 ) were used for Ca2Fe(CN)6.1-82514 APPARENT MOLAL VOLUMES OF STRONG ELECTROLYTESThe evaluation of v' as 36.7 cm3 mol-' for Ca2Fe(CN)6 is far too high, in ouropinion. The 4" curve of this salt5 lies between those of MgzFe(CN)6 andSr,Fe(CN),. A value of 34 cm3 mol-' has been given in a previous paper.5 Fig. 1suggests a smaller value, perhaps 31.6 according to the value of 30.1 forMgzFe(CN)6 [or 32.6 according to 36.1 for SI-~F~(CN)~]. The ionic conventionalvolume of Ca2+ was assumed to be greater than that of Sr2+ at 25"C,' according tothe data of CaC1z,24 a very hygroscopic salt (systematic errors?).According to thepresent results, it would lie between Mg2+ and Sr2+ (in analogy to results reportedat higher temperatures ').0.2 0.4 0.6 0.84: IFIG. 2.-Plots of 4: against J'l for 1 : 3 salts. DHLL + B2 : dashed lines [i = 3,4 and 8 8, for (a), (b) and(c), respectively]. IPBE: full lines d In h/dp = 0 [h = 3 and 4 A for (d) and (e), respectively]; dotted line(f), d In i/dp = p, 6 = 5.5 A. As a comparison, K,Co(CN), (0) and K,Fe(CN)z (0) are given. Dash-and-dot line: DHLL.As for K,CO(CN)~ [and likewise K,Fe(CN),4], there seem to be no problems inthe extrapolation vo. 143.3 cm3 mol-' is obtained on the basis of DHLL C146.4 forK3 Fe(CN), "1.Fig. 2 shows some DHLL + B2 and IPBE curves for the 1:3 salts. This nowconfirms, in terms of IPBE, what had already been found by means of DHLL + B2calculations :6 i.e., positive deviations from DHLL are justified at low concen-trations, for sufficiently small values of ii [the volumetric behavour of Nd(N03)3Zmay also be explained in this way].The greatest positive deviations are expected inthe approximate concentration range 5 x 10-4-5 x 10- mol dm- ; an occasionaland misleading parallelism with DHLL must occur in such cases at higher concen-trations (and may lead to erroneous extrapolation from data obtained at far toohigh ionic strength). Greater ii values lead to negative deviations only, similar tothose the DH theory predicts. The inversion occurs at ii z 4 A (in the assumptionthat d In h/dp = 0) and, in consequence, a slope close enough to DHLL should bemaintained up to high concentrations (0.01 or 0.02mol drnp3).The data forK,Co(CN), do not deviate appreciably from DHLL below 0.01 mol dm-3, and theV" value of 143.3 cm3 mol- is probably accurate; however, minor positive deviF. MALATESTA A N D R. ZAMBONI 2515ations cannot be excluded, and a slightly different I/' value, perhaps 0.5 cm3 rno1-lless, is also possible.Both DHLL + B2 and IPBE predict very large, positive deviations from DHLLat low concentrations for the 2:4 salts (fig. 3), except at unusually high ii values.When reasonable ii values are used, the positive deviations are similar or greater tothe experimental values of Mg,Fe(CN),, Sr,Fe(CN), and Ca,Fe(CN),. Accordingto both treatments, the limiting slope cannot be reached for 2:4 salts within theexperimental range of concentrations.From a quantitative viewpoint, the agree-ment with the experimental data is very feeble in the case of the Mayer theory. TheIPBE curves appear more satisfactory and a remarkable improvement can be15.+I -10m-..5I I0.2 0.4 0.6JIFIG. 3.---Plots of 4; against ,'I for 2:4 salts. DHLL + BZ: dotted curves [i = 7.5 and 6 A for (a) and (b),respectively]. IPBE: dashed lines, d In i/dp = 0 [i = 5 and 4.5 A for (c) and (d), respectively]; full lines,d In &/dp = p [ii = 7 and 6 A for (e) and (f), respectively]. As a comparison, Mg,Fe(CN), (0) andSr,Fe(CN)6(0) are given. Dash-and-dot line : DHLL.obtained when using a positive value of d In iildp, of the same order as the com-pressibility coefficient of water, p.The theoretical meaning of this parameter is ratherdoubtful; however, a value of d In h/dp equal to p also improves the resemblance inthe cases of K3Co(CN), and K,Fe(CN), [fig. 2, curve ( f ) ] . Greater d values arecorrespondingly required, which agree better with the d values to be used in &.I7This agrees with previous results for K4Fe(CN)6.4917 A pressure rise could perhapsmodify the water structure around the hydrated ions (increasing the ion hydration),so as to require a positive value of dlni4/dT in terms of the primitive model.However, the inaccurate shapes obtained with d In d/dp = 0 might come from innerinconsistencies of the IPBE treatment. An analysis of the data in terms of someother treatment of the primitive model, for instance HNC,25 would be helpful.IPBE (and DHLL + B2) may be useful for highly charged electrolytes within theconcentration range where DH works for 1 : 1 electrolytes. By plotting the differ-ences between experimental 4v values and & values calculated from IPBE (o2516 APPARENT MOLAL VOLUMES OF STRONG ELECTROLYTESDHLL + B2) one might identify a convergence towards the I/' value, indepen-dently of the arbitrary choice of ii and d In iildp.This method is not effective for thepresent data for 2:4 salts; it seems impossible to obtain sufficiently precise 4vvalues at such high dilutions as the method requires [different extrapolations areapparently suggested for d In i/dp = 0; slightly better results are obtained ford In G/dp = p, but not so good as to provide a reliable value of T/'.This is shown infig. 4 for Mg,Fe(CN)6]. However, the same method can usefully be applied to other(less charged) multivalent electrolytes, when appreciable positive deviations occur32 1 M AA AAI I0.2 0.4\'IFIG. 4. Plots of 4L - 4: (IPBE) against 4'1 for Mg,Fe(CN)6. Upper section, d In G/dp = 0 [C; = 5 (A), 4.5(0) and 4 A (o)]. Lower section, d In i/dp = p [i = 8.5 (A), 7 (0) and 6 A (O)].(but not so high as in the 2:4 salts). Fig. 5 shows the results one obtains forK,Fe(CN),, which may confirm the previous Vo evaluation (108.5 cm3 mol-1)4within z +_ 0.5 cm3 mol-' [the usual Redlich or Owen extrapolationare ineffective for K,Fe(CN),].As for specific? non-electrostatic interactions in dilute and very dilute solutions ofhighly charged electrolytes,' one cannot at present reach any definitive conclu-sion. Remarkable negative deviations in 4L8.27 and very small positive deviations indV5 were noticed for LaFe(CN),, and they can hardly be justified in terms of merelyelectrostatic interactions (DHLL + B2 and IBBE predict very large positive devi-ations for 3:3 salts).On the other hand, the positive deviations observed at lowconcentrations in 4) and & of 2:4 salts (as well as in 4, of 2:317 and 2:216 salts)are satisfactorily justified in terms of the approximate treatments of the primitivemodel by DHLL + B2 and IPBE, and this suggests that interactions are fundamen-tally electrostatic for such salts, Further studies, for other highly charged electro-lytes, are necessary in order to provide a better understanding of such mattersF .MALATESTA A N D R . ZAMBONI 251711210600 o o 8 .-----?@---V+7~ v v v0+-V VI I 0.4,(I10.2FIG. 5.-Plots of $\. - (IPBE) against ,j;I for K,Fe(CN), [from the data of ref. (4)]: d In Gldp = 0,ii = 4 8, (0); d In ildp = ,8 [ A = 4.2 (e), i = 3.5 8, (O)]. Dash-and-dot line: 108.5 cm3 mol-'.The authors thank Prof. A. Indelli for many valuable discussions concerning thisresearch. This work was supported by the Consiglio Nazionale delle Ricerche(C.N.R.).F. Millero, Chem. Rev., 1971, 71, 147 and references therein.2F. H. Spedding, M. J. Pika1 and B. 0. Ayers, J . Phys. Chem., 1966, 70, 2440.3L.G. Hepler, J. H. Stokes and R. H. Stokes, Trans. Faraday Soc., 1965, 61, 20.4A. Billi, F. Malatesta, R. Zamboni and A. Indelli, J . Chem. Phvs., 1974, 61, 4787.5A. Indelli and R. Zamboni, J . C. S. Faraday I, 1972,68, 1831.,A. Indelli and R. De Santis, J . Chem. Phys., 1969, 51, 2782.J. E. Desnoyer, M. Arel, G. Perron and C. Jolicoeur, J . Phys. Chem., 1969, 73, 3346; G. Perron, N. R.Desroisiers and J. E. Desnoyer, Canad. J . Chem., 1976, 54, 2163; A. ROUX, G. M. Mustbally, G. Perronand J. E. Desnoyer, Canad. J . Chem., 1978, 56, 24.E. A. Guggenheim, and R. H. Stokes, Equilibrium Properties of Single Strong Electrolytes (Pergamon,London, 1969).'E. 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Phys., 1972, 56, 3071.F. Malatesta, Gazzetta, 1979, 109, 3252518 APPARENT MOLAL VOLUMES OF STRONG ELECTROLYTES260. Redlich and P. Rosenfeld, 2. phys. Chem. (Leipzig), 1931,155,65; Z . Elektrochem., 1931,37, 705; 0.Redlich and D. M. Meyer, Chem. Rev., 1964,64,221; B. B. Owen and S. R. Brinkley, Proc. N . X Acad.Sci., 1949, 51, 753.27E. Lange and W. Miederer, Z. Elektrochem., 1956, 60, 362.(PAPER 9,4934