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Scientific paper
Determination of the Limiting Conductances
and the Ion-Association Constants of Calcium
and Manganese Sulfates in Water
from Electrical Conductivity
Measurements
Marija Be{ter-Roga~*
Faculty of Chemistry and Chemical Technology, University of Ljubljana, SI-1000 Ljubljana, Slovenia.
* Corresponding author: E-mail: marija.bester@fkkt.uni-lj.si Tel: +386 1 2419 410; Fax: +386 1 2419 425
Received: 03-12-2007
Abstract
Electric conductivities of diluted calcium and manganese sulfate solutions aqueous solutions were measured from 5 to 35 °C (steps of 5 °C) in the concentration range 2 × 10–4 < c/mol dm–3 < 5 × 10–3. Evaluation of the limiting molar conductivity ?? and the association constant KA is based on the chemical model of electrolyte solutions, including short-range forces. From the temperature dependence of the limiting molar conductivities Eyring’s enthalpy of activation of charge transport was estimated. The standard Gibbs energy, enthalpy and entropy of the ion-pairing process were calculated from the temperature dependence of the ion-association constants. The comparison with some divalent metal sulfates aqueous solutions was made.
Keywords: Divalent metal sulfates, calcium sulfate, manganese sulfate, aqueous solutions, ion association, electric conductivity
1. Introduction
Divalent metal ions are among the major inorganic species in natural waters and are also of considerable practical importance as they have significant biological functions.
Recent investigations on divalent metal sulfates in aqueous solutions1–4 have demonstrated strong interactions occurring between all particles, ions and solvent molecules. Examination of the results obtained from different techniques1,2,5–8 shows the interplay of ion association and solvation. The understanding of the nature of the association process is complicated by strong hydration of the free ions. Ultrasonic absorption measurements9–11 lead to the formulation of a three-stage association process: initially hydrated ions form a double solvent-separated ion pair (2SIP), in which both ion hydration shells remain essentially intact. This step is followed by the formation of a solvent-shared (SIP) and then of a contact (CIP) ion pair. Recently the simultaneous existence of all three ion pair
types in MgSO4, CoSO4 and NiSO4 aqueous solutions was demonstrated using dielectric relaxation spectroscopy.2,4
In the present work we extended our former investigation on 2,2 electrolytes in dilute aqueous solutions1,3 to calcium sulfate and manganese sulfate solutions. Many of the physicochemical properties of CaSO4 and MnSO4 are tolerably well known.12 Nevertheless relatively little reliable information is available on the extent and nature of their association. Some electrical conductance studies of CaSO413–15 and MnSO4 (aq)16–17 solutions have been reported previously. However, the measurements were carried out over limited ranges of temperatures mainly. In this work precise measurements on both systems in water were carried out covering the temperature range from 278.15 K 5 to 308.15 K, and electrolyte concentrations from ?0.2 × 10–3 to ?5 × 10–3 mol dm–3.
The experimental data were treated in the framework of the low concentration chemical model (lcCM)18 and the obtained results were compared with the reported values for other divalent metal sulfates aqueous solutions.
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2. Experimental 2. 1. Materials
Calcium sulfate (CaSO4 × 2H2O, p.a. Merck) and manganese sulfate (MnSO4 × H2O) were stored under dry nitrogen and used without preceding purification. Demineralised water was distilled in a quartz bi-distillation apparatus (DESTAMAT Bi18E, Heraeus). The final product with specific conductivity of less than 5 × 10–7 S cm–1 was distilled into a flask allowing storage under a nitrogen atmosphere.
Stock solutions of MnSO4 was prepared by adding weighed amount of water to weighed amount of salts and the concentration was checked by titration with EDTA (Merck). Due to the low solubility of CaSO4 in water in this case the stepwise concentration of the solution in the measuring cell was achieved by successive additions of weighed samples of the electrolyte by help of the special designed inlet cap.19,20
After measurement of the solvent conductivity at all temperatures of the programme a weighed amount of a stock solution or a solid electrolyte was added and the temperature programme was repeated. A home-developed software package was used for temperature control and acquisition of conductance data. The measuring procedure, including corrections and the extrapolation of the sample conductivity to infinite frequency is described in the literature.22
The densities of the solutions were determined by the method of Kratky et al.23 using of a Paar densimeter (DMA 60, DMA 601 HT) at 25 °C combined with a precision thermostat. From the weights and the corresponding solution densities d the molar concentrations c were determined. As usual, a linear change of density with increasing salt content for dilute solutions was found, d = ds + Dm, where dsis the density of solvent, given in Table 1. m is the molonity of the electrolyte (moles of electrolyte per
2. 2. Conductivity Measurement
Conductivity was recorded with a PC-interfaced LCR Meter Agilent 4284 A connected to a three-electrode measuring cell described elsewhere.19,20 The cell was calibrated with dilute potassium chloride solutions21 and immersed in the high precision thermostat described previ-ously.22 The oil bath was set to each temperature of a temperature programme with reproducibility within 0.005 K. The temperature was additionally checked with a calibrated Pt100 resistance thermometer (MPMI 1004/300 Merz) connected to an HP 3458 A multimeter.
Table 1: Densities, viscosities and dielectric constants of pure water and limiting ionic conductances of SO4– in water.a
T	d0	7J • 103c	é	A°°(l/2 S04)e
278.15	0.99997	1.5192	85.897	46.00
283.15	0.99970	1.3069	83.945	53.68
288.15	0.99910	1.1382	82.039	61.81
293.15	0.99821	1.0020	80.176	70.47
298.15	0.99705	0.8903	78.358	80.00
303.15	0.99565	0.7975	76.581	89.47
308.15	0.99404	0.7195	74.846	99.90
aUnits: T, K; d0 kg dm 3; 7), Pa s; X°°, S cm2 mol 1 bRef.24; cRef.25; dRef.26; eRef.1
Table 2: Equivalent conductivities of CaSO4 and MnSO4 in aqueous solutions.a
in • 10 3      278.15      283.15      288.15     293.15      298.15      303.15      308.15
CaSO4, D = 0.137
0.67949     71.744      82.427      93.641     105.324    117.364     129.493     142.212
0.93075     69.574      79.962      90.816     102.104    113.634     125.671     137.624
1.22236     67.532      77.576       88.126      99.018     110.145     121.294     127.758
1.71082     64.870                                      94.967     105.682     116.600     122.254
2.24404     62.505       71.765       84.561      91.400     101.574     111.982     118.409
2.75238     60.690      69.649       81.441      88.615      98.460      108.549     114.386
3.31797     58.840      67.528      78.989      85.892      95.393      104.980     110.921
3.99319     57.225       65.655       74.416      83.324      92.485      101.675     142.212
MnSO4, D = 0.147
0.29609     71.915       82.889      94.455     106.584    119.152     132.171     145.684
0.61672     68.302      78.637       89.522     100.866     112.611     124.769     137.310
0.90185     66.034                      86.426      97.300     108.544     120.142     132.062
1.22932                     73.441       83.544     93.984     104.786     115.826     127.197
1.59206     61.929      71.186       80.898      90.956     101.320     111.929     122.796
2.02372     60.024      68.935       78.283      87.949      97.919      108.099     118.508
2.4056       58.573       67.266      76.358      85.774      95.443      105.294     115.356
2.83019     57.175       65.641       74.518      83.647      93.024      102.627     112.351
3.53324     55.271       63.415       71.952      80.700      89.689      98.849      108.163
4.57376     52.935                       68.831      77.171      85.698      94.367      103.156
aUnits: T, K; in, mol kg 1; A, S cm2 mol 1; D, kg2 dm 3 mol 1
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kilogram of solution). The density gradient D, given in Table 2, was considered to be independent of temperature. The measured conductivity data of all investigated systems are given in Table 2 as a function of the temperature independent molonities, m, which can be converted to the temperature dependent molarities by use of the relationship
c = md
(1)
Taking into account the sources of error (calibration, titration, measurements, impurities) the specific conductivities are accurate to within 0.05% at MnSO4 and 0.2% at CaSO4 solutions.
No hydrolysis correction was made in data analysis.
association represented by the upper limit of the association constant, KA and the distance parameter of the mean activity coefficient, y I (Eqs. 3 and 4).
Analysis of the conductivity data was carried out by setting the coefficients S, E and J1 of Eq. (2) to their calculated values.18 Then three-parameter fits were used to obtain the limiting values of molar conductivity A°°, the association KA and the coefficient J2 by non-linear least squares iterations. The input data for the calculation of the coefficients S, E, J1 J2, y"+ are the solvent properties d(T), e(T), r\(T) given in Table 1 and the distance parameter R.
The lower limit a of the association integral is the distance of closest approach of cation and anion (contact distance)
3. Data Analysis
The analysis of conductivity data in the framework of the low concentration chemical model (lcCM) given in Ref.18 and the literature quoted there, uses the set of equations
— = /f -Si[äc + Eacln{ac)+ J}ac + J2{ac)~2 (2)
(3a-d)
(4)
where A and A°° are the molar conductivities at molarity c and infinite dilution, (1-a) is the fraction of oppositely charged ions acting as ion pairs, and KA is the equilibrium constant of the lcCM with upper association limit R; y\ is the corresponding activity coefficient of the free ions, (y±)2 = y\ y\, k is the Debye parameter, e is the proton charge, e is the relative permittivity of the solvent, eo is the permittivity of a vacuum and T the absolute temperature. The other symbols have their usual meaning. W* is a step function for the potential of mean force between cation and anion due to non-coulombic interactions in the association process, see section 4.2.
The coefficients of Eq. (2) are explained in Ref.18. The parameters S and E, which do not depend on the distance parameter R, are estimated with the help of the solvent parameters. The coefficients Jl and J2 additionally depend on the distance parameter R representing the distance up to which oppositely charged ions are counted as non-conducting ion pairs, and hence is the upper limit of ion-pair
(5)
calculated from the ionic radii of metal ions (a+ = 0.106 and 0.091 nm for Ca2’ and Mn2+, respectively) and SO42 – (a– = 0.258 nm),18 giving the contact distance a = 0.364 and 0.349 nm for CaSO4 and MnSO4, respectively.
From extended investigations on electrolyte solutions in amphiprotic hydroxylic solvents (water, alcohols) it is known that the upper limit of association is given by an expression of the type
R = a + n x s
(6)
where s is the length of an oriented solvent molecule, n is an integer, n = 0, 1, 2, ... Here, s is the length of an OH-group, d0H, and s = d0H = 0.28 nm.
4. Results and Discussion
4. 1. lcCM Calculations
Figures 1 and 2 show the experimental data and the results of the lcCM calculations executed by the use of Eqs. 2-6 under the assumption n = 2 for Eq. 6 encompassing
Figure 1:. Equivalent conductivities of aqueous CaSO4 solutions from 5 to 35 °C (steps of 5 °C). (O) this work; (¦) Ref.13; (A) Ref.14; (D) Ref.15; full line: lcCM calculations.
a = a  + a
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three types of ion pairs: contact ion pairs, solvents-shared and solvent-separated ion pairs.
The limiting conductivities, A°°, and association constants, KA, resulting from the data analysis based on the lc-CM model are given in Table 3. It shows the comparison of the calculated lcCM data for the cases I (n = 1 for CIP and SIP ion pairs) and II (n = 2 for all investigated ion pairs, CIP, SIP and 2SIP). The limiting conductivities do not differ significantly, as can be expected. R(J2) calculated from J2 which is obtained by the data analysis is not necessarily found identical to R which is the input-parameter, R = a + ns, chosen for the calculation of Jl and y±, but it may be used as a compatibility control.18 Here R(J2) and R are in reasonable agreement for the case I and for the case II as well.
Figure 2: Equivalent conductivities of aqueous MnS04 solutions from 5 to 35 °C (steps of 5 °C). (O) this work; (O) Ref.16; (A) Ref.17; full line: lcCM calculations.
The conductance data at 298.15 K from the literature and from this work are compared in Table 4. The table compiles data resulting from our redetermination of the authors’ original A(c) data with the help of lcCM calculations. After réévaluation according to the lcCM the earlier measurements are in good agreement with our results.
Calorimetrie measurements on the heat of dilution27 of aqueous solutions at 298.15 K and their data evaluation with the lcCM (n = 2) yield KA values of 192 (183) and 194 (191) for CaS04 and MnS04, respectively, and are in good agreement with the values obtained by our conductivity measurements, given in parenthesis.
Combining the limiting conductivities A°° of Table 3 and known limiting values of A°°so2-(T)
(7)
Table 4: Comparison of the results at 298.15 K with literature
data.a
	1/2A"	KA
CaS04	138.09±0.28b	183.0±5.5b
	138.4±0.2C	270.5±0.3C
	136.4±0.3d	273±9d
	136.4±0.1e	239±3e
MnS04	132.05±0.07b	190.6±1.6b
	132.3±0.1f	291±4f
	132.5±0.2f	276±5f
Values are redeterminated from the reported data by lcCM (n = 2).
aUnits: ??, S cm2 mol–1; KA, dm3 mol–1.
bThis work, cRef.15, dRef.13, eRef.14, fRef.17, gRef.16
Table 3: Limiting equivalent conductivities, 1/2??, association constants, KA, and R(J2) para-
1/2??              KA          R(J2)         1/2??
I : n= 1
KA             R(J2)
II : n = 2
CaS04
278.15 283.15 288.15 293.15 298.15 303.15 308.15
R = 0.644 83.62±0.12 96.25±0.21 109.42±0.18 123.26±0.15 137.65±0.28 152.20±0.71 168.09±2.21
140.2±5.7 142.2±6.1 141.1±4.5
142.82±3.2 147.5±5.5
144.60±12.5 168.9±36.4
0.672 0.672 0.708 0.739 0.743 0.797 0.701
R = 0.924 83.87±0.18 96.55±0.22 109.75±0.0.5 123.65±0.15 138.09±0.28 152.45±0.72 168.68±2.27
173.5±6.0
176.4±6.5
175.4±1.4
177.6±3.3
183.0±5.5
180.4±12.9
206.8±38.4
0.968 0.967 0.991 1.025 1.029 1.075 0.997
MnSO
4
278.15 283.15 288.15 293.15 298.15 303.15 308.15
R = 0.629 78.95±0.05 91.20±0.05 104.04±0.07 117.63±0.08 131.76±0.09 146.53±0.09 161.96±0.10
129.3 ± 1.7      0.674
137.9±1.9 141.0±1.8 149.2±1.9 156.7±2.1 166.8±1.9 178.7±1.9
0.631 0.664 0.658 0.606 0.654 0.641
R = 0.909 79.11±0.03 91.34±0.04 104.25±0.05 117.88±0.06 132.05±0.07 146.85±0.06 162.32±0.07
160.8 ± 1.3 168.1±1.5
173.5±1.4 182.4±1.5 190.6±1.6 201.5±1.3 214.2±1,3
0.979 0.967 0.975 0.972 0.975 0.972 0.969
a Units: T, K; ??, S cm2 mol–1; K , dm3 mol–1; R, nm.
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yields the limiting cation conductivities A^2+ and their temperature dependence (Table 5)
lable 5: Single ion conductivities.
T	A°°(l/2 Ca2+)	A°°(l/2 Mn2+)
278.15	37.87	33.11
283.15	42.87	37.66
288.15	47.94	42.44
293.15	53.18	47.41
298.15	58.09	52.05
303.15	62.98	57.38
308.15	68.78	62.42
aUnits: T, K; X°°, S cm2 mol 1.
The temperature dependence of limiting conductivity yields Eyring’s enthalpy of activation of charge trans-port28
(8)
where ds is the density of the solvent and B is a constant. The obtained values are Alf = 13.94 and 14.94 kJ/mol for Ca2+ and Mn2+ ion, respectively. It could be assumed that for the jump of the Ca2+ in a prepared vacancy in the wa-ter-or to produce such a vacancy-less energy is required than for Mn2+.
AHA° = AGA° +TASA° = A0 + + 298.15 A j + A2 (298.152-T2)
(11)
(12)
Table 6 compiles the coefficient of the polynomial, Eq. (10). The values of AGA0 and ASA at 298.15 K are AGA0 = A0 and ASA = A1. The comparison of AHA0 at 298.15 K with the values obtained by calorimetric measurements is also presented.
Figure 3 shows the Gibbs energies AG°A, enthalpies A-HA and entropies ASA of ion-pair formation in CaSO4 and MnSO4 aqueous solutions.
The enthalpies of ion-pair formation show reasonable agreement with the values from the calorimetric measurements. The entropies are in the range between 60 and 70 J mol–1 K–1 as reported for other divalent sulfate aqueous solutions and increase at increasing temperature (Figure 3) suggesting that ion pair formation in the investigated systems is an entropy driven process.
According to the Eq. (4) the Gibbs’ energy of ion-pair formation, AGA, can be split into two terms
AG0=AG/
+ AGA,
(13)
where AGA = NAW* and AGAoul is obtained from the “cou-lombic” part of the association constant KA in Eq. (4)
4. 2. Thermodynamics of the Ion-Pair Process
The association constant KJT) is linked to the Gibbs’ energy of ion pair formation, AGA°(T), by
AGAcoul = -RT ln KAC
(14)
(15)
AGA(T) = -RT In KA(T).
(9)
AGA°(T) is represented for the measured temperature-dependent association constants of Table 3 with the help of a polynomial
AGA°(T) = A0+ Aj (298.15-T) + + A (298.15-T)2.
(10)
Enthalpy and entropy of association here are obtained as follows
A polynomial of type of Eq. (9) AGA*(T) = B0+ Bj (298.15-T) +
+ B (298.15-T)2
(16)
is used for evaluation of the AGA, AHA and ASA of the ion-pair formation presented in Table 7, together with the coefficients B0, B1 and B2 of AGA polynomials. It is evident that AG* is small in comparison to that of total value of AGA, indicating a preference of strong electrostatically bound ion pairs.
Table 6: Coefficients of polynomials AG°A(T) = A0 + A1 (298.15-T) + A2 (298.15-T)2 and AH% (298.15 K) for
CaSO4 and MnSO4 in water.a			
A0 = AGA (298.15 K)	Ax = ASA (298.15K)	A2	AHA (298.15 K)
CaSO4                   -12 897 MnSO4                  -13 022	61.48 69.38	-0.736 -0.304	5420         6670b 7670         7770b
a Units: A0, J mol 1; A1, J mol 1 K 1; A2, J mol 1 K 2; AH%, J mol 1    b Ref.27 from calorimetry
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Figure 3: Temperature dependence of thermodynamic functions of association of CaSO4 and MnSO4. (D) AGA
(•) AH0; (O) TAS0.
Table 7: Coefficients of polynomials ?GA*(T) = B0 + B1 (298.15-T) + B2 (298.15-T)2 and ?HA* (298.15 K) for
B0 = ?GA* (298.15 K)          B1 = ?SA* (298.15 K)          B2
?H* (298.15 K)
CaSO4 MnSO4
1042 1465
-17.5 1.0
0.155 -0.140
–4232 1784
aUnits: B , J mol–1; B , J mol–1 K–1; B , J mol–1 K–2; ?H * J mol–1
5. Comparison of Some Divalent Metal Sulfates
In some ways this work means a conclusion of our investigations of divalent metal sulfates dilute aqueous solutions. Thus it is reasonable to make a summary and to present a comparison between studied systems as it is made in Table 8.
The most remarkable feature of the present results is the surprising uniformity of the properties of the aqueous solutions of divalent metal sulfates. It can in large part be ascribed to the strong hydration of the ions (and ion pairs) found in these solutions, which tends to attenuate the effects of differences in ion size and electronic configuration.
The hydrated radius is much larger than similar ions and the large solvation shell around the ion also causes its low mobility and low limiting conductivities ?? in solution.
Only CaSO4 and CuSO4 exhibit slightly higher values of ??. Calcium, however, belongs to a group of ions with much weaker hydrogen bonding.30,31 Recent study of the solvation structures of Mg2+ and Ca2+ ions in water by applying a Car-Parinello-based constrained molecular dynamics method suggests that the hydration structure of Ca2+ is shighly variable.32 Therefore the higher value of ?? (1/2 Ca2+) can be explained by the assumption that the binding of water molecules in the hydrate shell of Ca2+ ion weaker than at other divalent metal ions and thus the motion of Ca2+ ions in aqueous solutions is slightly less hindered.
Table 8. Comparison of ionic radii, a+, ionic limiting conductivities, X°° (1/2 M2+), constants, KA, entropies, ASA, and enthalpies, AHA, of ion pair formation, and Eyring’s enthalpy of activation of charge transport, AH#, for divalent sulfates in aqueous solutions.a
	a+b	A°°(1/2 M2+)	KA	A S0 A	AH0 A	AH#
CaSO	0.106	58.09	183	61.5	5.42	13.94
CdSOc	0.103	54.06	245	72.0	7.87	14.77
CoSO d	0.082	53.64	208	66.5	6.59	14.82
CuSO4d	0.072	56.39	249	67.7	6.52	14.99
MgSO4e	0.078	53.03	157	64.3	6.62	14.90
MnSO4	0.091	52.05	191	69.4	7.67	14.94
NiSO d	0.067	54.02	211	59.9	4.61	14.74
ZnSO4d	0.083	53.04	199	60.9	5.05	14.81
aUnits: a+, nm; X°°, S cm2 mol-1; KA, dm3 mol-1;			?SA0, ?HA0,	AH#, kJ mol–1	bRef.18; cRef.29; dRef.3; eRef.1	
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In the same sense can be interpreted the lower value for Eyring’s enthalpy of activation of charge transport found for Ca2+ (AH# = 13.94 kJ mol–1).
The higher value of X°° (1/2 Cu2+) could be interpreted as a consequence of the more expressed hydrolysis. The equilibrium constant for the hydrolysis reaction of Cu2+ ion is the biggest among the divalent metal cations (3 × 10–8) namely.33
Neglecting the value obtained for CaSO4 aqueous solutions the average value of Eyring’s enthalpy of activation of charge transport is, AH# = 14.8 ± 0.1 kJ mol–1 for all other discussed divalent metals sulfates. This finding is in agreement with the fact that AH# usually depends on the solvent properties mainly.
Entropies ASA0 of ion pair formation, resulting form the temperature dependence of KA are very close together for all discussed salts. Otherwise constant, KA, and enthalpies AHA0 of ion pair formation show more deviations but are still in reasonable agreement.
6. Concluding Remarks
It is evident that all investigated divalent metal sulfates exhibit very similar properties in diluted aqueous solutions. Recent investigations of the concentrate aqueous solutions of MgSO4, CoSO4, NiSO4 and CuSO4 by dielectric relaxation spectroscopy2,4,34 suggest that the chief difference among these solutions could be in the formation of triple ions. They form only in concentrated solutions therefore it is necessary to extend the conductivity measurements on the concentrated solutions to determine whether some more definitive rules can be derived.
7. Acknowledgment
This work was supported by Slovenian Research Agency (P1-0201).
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Povzetek
Izmerili smo elektri~no prevodnost razred~enih vodnih raztopin kalcijevega in manganovega sulfata v temperaturnem obmo~ju med 5 in 35 °C v obmo~ju koncentracij 2 × 10–4 < c/mol dm–3 < 5 × 10–3. Na osnovi kemijskega modela smo dolo~ili vrednosti molskih prevodnosti pri neskon~nem razred~enju, ??, ter konstante asociacije ionov, KA, v posameznem sistemu. S pomo~jo znanih vrednosti limitnih prevodnosti sulfatnega iona smo ocenili limitne prevodnosti kalcijevega in manganovega kationa. Dobljene vrednosti smo primerjali z vrednostmi ostalih dvolentnih kovinskih kationov.
Be{ter-Roga~:  Determination of the Limiting Conductances and the Ion-Association Constants ...