Scientific paper
Global Thermodynamic Analysis of Conductivity Data
Matjaž Boncina* and Marija Bester-Rogac
Faculty of Chemistry and Chemical Technology, University of Ljubljana, SI-100 Ljubljana Slovenia * Corresponding author: E-mail: matjaz.boncina@fkkt.uni-lj.si Received: 01-02-2012 Dedicated to Prof. Dr. Gorazd Vesnaver on the occasion of his 70h birthday
Abstract
Conductivity can serve as a tool to study ion properties in different solvents. In the current study, principles of the global modelling of the conductivity data measured at different concentrations and temperatures are demonstrated. Global model was developed on the basis of two modern conductivity theories; the low concentration chemical model and the Quint-Viallard model. It was tested on three different systems; NaCl in water, NaCl in water-1,4 dioxane mixture and MgSO4 in water. It turned out, that such approach successfully describes all the measured experimental data simultaneously. The applied global analysis of the experimental data allows determination of the thermodynamic profile of ion pairing process and the transport properties of ions at the same time.
Keywords: Conductivity, lcCM model, Quint-Viallard, global model
1. Introduction
The temperature and concentration dependence of the electrolyte conductivity is one of the most appropriate and precise methods for studying ion-ion, ion-solvent and solvent-solvent interactions in solutions. It has been intensively applied since the beginning of 20th century, but a significant progress has been made during the last decades. Evaluating the conductivity within the framework of present electrolyte theories requires namely high accuracy in measurements which is difficult to achieve over a sufficiently large temperature range. This problem has been successfully solved recently with the help of contemporary equipment.1'2 Thus, determination of precise temperature-dependent electric conductivity data is nowadays more user-friendly what makes this method interesting and appealing again.
In recent years, important theoretical progress has been made by the implementation of chemical models of electrolyte sotutions. Through the years, they have been progressively developed for dilute solutions, starting with the Kohlrausch and the Onsager models to more sophisticated Barthel's low concentration chemical model (lcCM)3 and the Quint-Viallard (QV)4 model. First two models can provide information only about limiting molar conductivities while the last two take into account also ion pairing.
Apelblat recently re-examined the conductivity data of diverse electrolyte systems by help of the QV conductivity equations.5-9 It was observed for symmetrical electrolytes that the limiting molar conductivities of electrolytes at infinite dilution, are very similar irrespective of the applied conductivity equation, whereas the derived ion association equilibrium constants, KT, can vary considerably.
Despite modern computers and programs, data evaluation is still quite a time consuming procedure. First, one has to analyse data at each temperature separately to obtain AT and KT. Then, from temperature dependence of Kt, the standard Gibbs free energy, AGT enthalpy, AHT and entropy, AS°T, of the ion pairing process can be calculated. In order to reduce the steps in the estimation of the thermodynamic functions of ion association we propose a global analysis of the conductivity data. Global models are usually applied to the set of experimental data obtained from different techniques at different temperatures, concentrations etc. They improve the ratio between the number of experimental points and the number of adjustable parameters and to some extend reduce correlations between adjustable parameters. They have been frequently and successfully used in biophysical field as shown by Lah and Vesnaver.10-12
In this paper, we demonstrate a global analysis of the literature conductivity data for diluted solutions of
NaCl in water and in water-1,4 dioxane mixture13 and MgSO4 in water14 reported in the temperature range from 278.15 to 308.15 K. The results, (A°, KT and thermodynamic functions of ion pairing process) obtained by the global analysis, are compared to those obtained from the individual fits and discussed in the light of possible advantages and disadvantages of both approaches.
2. Global Model Analysis of Conductivity Data
The ion pair formation of a symmetrical electrolyte XY can be written as
X;+ + Y;

(1)
The equilibrium concentrations are equal to cXz+ = cYz- = ac and cXz+Yz- = (1 - a)c where c is the total electrolyte concentration and a the degree of ion pair dissociation. Thus, the equilibrium association constant, KT, can be expressed as
K j — "
V-	Y--
^v-"' y y." ■ T V-
jl "K V
(2)
The activity coefficients of the individual species in dilute solutions, yi, can be approximated by the Debye-Huckel equation
r
7¡ = exp
Kq
1 + ATÄ,
K = I (iKNAqI\ q =
%7iseJtT
(3)
where ^ is the Debye parameter, I the ionic strength, Ri the ion size parameter,15 e0 the proton charge, e the relative permittivity of the solvent, e0 the permittivity of vacuum and T the absolute temperature. The other symbols have their usual meaning. It should be noted, that ion pairs are neutral species and therefore yXz+Yz_ is assumed to be unity.
According to the lcCM model, the dependence of molar conductivity ATon concentration at a given temperature T is defined as3
— = A" - S(ac)"2 + Eaeln(ac)+Jtac—J2(ae)3'2. (4) a
Similar expression is used in the Quint-Viallard (QV) model where AT is expressed as the sum of ionic molar conductivities \ T 4
A7 = ykM,
i c
4j = À.°T~SiIin + E.J In f + JuI -JTf
(5)
where X[T is the limiting molar conductivity of the ion at a given temperature. In the case of 1:1 and 2:2 electrolytes,
the ionic strength is equal to I = ac and I = 4ac, respectively. The coefficients S, E, J1, J2 and Si, E;, J1i, J2i are rather complex functions of temperature, viscosity,16 the dielectric constant of pure water,17 distance parameter R (or ai in the QV model) and valences z. The explicit expressions for these coefficients are given in detail in the literature.3,4,18 The distance parameter R (a;) defines the distance to which ions are regarded as ion-pairs (the upper limit of ion association), R = at = a + n-s. The lower limit of the association is the distance of closest approach of cation and anion (contact distance), a = a+ + a , where a+ and a represent the radii of cation and anion,19 s is the length of an oriented solvent molecule and n is an integer. If we assume contact ion pairs n = 0, for solvent-shared n = 1 and for solvent-separated ion pairs n = 2.In water, s is the length of an OH-group, s = 0.28 nm.
In the global analysis an expression for temperature dependence of limiting molar conductivity AT, is needed. In principle, Walden's rule3 or Eyring approach20 could be used for this purpose but unfortunately both approaches fail to successfully describe temperature dependence of limiting molar conductivity (see supporting information for details). Therefore, A0T or X\T are approximated by a simple expression21
A^Al+Aa-W + BiT-T,)2 Kt=Kt^A(T-T0) + B.í(J-T<>)Z
(6)
where A°o is the limiting molar conductivity at the reference temperature T0 = 298.15 K and A and B are the temperature independent coefficients. In the QV model the temperature dependence of limiting molar conductivity of cation and anion, X°iT and coefficients A. and B. (i = Xz+,
_	Mo	11
Yz-) must be defined.
The ion association process can be described in terms of changes of standard thermodynamic parameters: the Gibbs free energy, AG°, and enthalpy, AH°, at the reference temperature T0 = 298.15 K and the corresponding heat capacity ACp (assumed to be temperature independent). These parameters define AG° and AH° at any T through the Gibbs-Helmholtz ret ation and Kirchhoff's law:
AH°T=AH'TI+AC°P(T-TIJ)
T-Tt-Tm —
(7)
Thus, a single set of (adjustable) parameters AG°, AH'°o and AC° defines KT = exp(-AG° / RT) and a (eq. 2) at any T. Global fitting of the model function (eq. 4 for the lcCM model and eq. 5 for the QV model) to the experimental conductivity data, measured at various T, was based on the non-linear regression procedure by minimizing
^(AexpftT) - AcalcftT))2 where Aexp(i,T) is the i-th eXperimental molar conductivity at temperature T and A,
calc(i,T)
is
the i-th calculated value from the model at temperature T. Since it is known that AC° is close to zero in such processes,22-23 five adjustable parameters were used in the lcCM minimization process: AG°, AH°, A1 in the QV: AG°, AH°, A and Bi (i
54.
T, A and B and eight ° X+ Yz-).
3. Results and Discussion
Global model analysis based on the lcCM or the QV set of equations was tested on three sets of conductivity data: NaCl in water, NaCl in water-1,4 dioxane mixture with 20 mass percent of 1,4 dioxane (xD = 0.20)13 and MgSO4 in water.14 The conductivity data for all the investigated systems are reported in the temperature range from 278.15 to 308.15 K in steps of 5 K. The results obtained from individual and global analysis for NaCl in water and water-1,4 dioxane mixture based on the lcCM are presented in Table 1 and Figure 1. The distance parameter R was set to R = a + 2s = 0.839 nm (a = 0.279 nm). Other parameters needed to calculate the model function are presented in Ref. 13. Table 1 reveals that individual and global fit give almost identical limiting molar conductivity of NaCl, A°To, quite similar constant of association, Kt , but in some examples (NaCl in water) very different values for AH° and TAS° accompanying the ion pairing process.
52
50-
o cn
46-
44-
1 1 1 s	1 1 1 ■ ---individual fit
s. "v N. N. N \\	-global fit
	% -
	
0.00
0.02
0.04
c"21 moP'2 dm-3*
0,06
0.08
Figure 1. Molar conductivities of NaCl in water-1,4dioxane mixture at 298.15 K. Experimental data are presented by symbols while the lines refer to the individual and global analysis based on the lcCM.
agreement with the experimental molar conductivities which is evident from Figure 2. Comparison of the results obtained from individual and global fit give the same conclusions as for NaCl systems, i.e. almost identical limiting molar conductivity, h?To, and slight differen-
Table 1. The thermodynamic and transport properties of NaCl in water and water-1,4dioxane mixture (Xp individual13 and global analysis of conductivity data. The results correspond to lcCM model.
: 0.20) at T0 = 298.15 K obtained from
water
individual fit1
global fit
water-1,4 dioxane mixture individual fit13	global fit
Kt
AG ° / kJ mol-1 l° 1 AH°/ kJ mol-1 T i TAS°/ kJ mol-1
Ai / SS cm2 mol-1
T o
A / S cm2 mol-1 K-1 B / S cm2 mol-1 K-2
2.38 ± 0.44 -2.92 8.88 11.80 126.54 ± 0.02 / /
2.63 ± 0.08 -2.40 ± 0.07 1.5 ± 2.2 3.9 ± 2.3 126.50 ± 0.01 2.635 ± 0.001 (9.81 ± 0.07) 10-
23.7 ± 1.0 -8.55 7.72 16.27
52.61 ± 0.03 / /
18.9 ± 0.2 -7.29 ± 0.03 8.1 ± 1.1 15.4 ± 1.1 52.06 ± 0.03 1.270 ± 0.003 0.0077 ± 0.0001
In Table 2 we present the results obtained from individual and global analysis (the lcCM and the QV model) for MgSO4 in water. The results correspond to the distance parameter equal to R = a + 2s = 0.896 nm (a = 0.336 nm). In the QV model the limiting molar conductivity of cation and anion may be treat ed as adj ustable parameters. However, these two parameters are highly correlated therefore the exact physical meaning may be questionable when both of them are used as adjustable parameters. For this reason, the limiting molar conductivity of cation at 298.15 K (1/2 ■ AMg2+,To = 52.7 cm2 Q-1 mol-1)5 was set as fixed parameter and only AMg2+ and BMg2+ coefficients describing its temperature dependence were fitted. The proposed global model gives excellent
ces in AG°, AH°o, TAS° accompanying the ion pair formation. However, there is one major difference between individual and global analysis: the former gives AC° = 0.37 kJ mol1 K1 while the latter AC° = 0.00 kJ mol1 K1. It should be noted that according to the literature data ACp effects accompanying the ion pair formation are close to zero.22 Moreover, global analysis allows us to reduce high correlation between adjustable parameters, close to 1 or -1. For instance, correlation between Kt and A0To in case of individual fit of MgSO4 in water is -0.93 which can be reduced to -0.88 with global analysis. Correlation matrix corresponding to the lcCMg lo-bal model analysis of the MgSO4 conductivity data is pre sen ted in Tab le 3.
Table 2. The thermodynamic and transport properties of MgSO4 in water at T0 = 298.15 K obtained from individual14 and global analysis of conductivity data.
individual fit
14
global fit lcCM
global fit QV
Kt
T00 1 AGtJ kJ mol-1
AH'/ kJ mol-1 T0
T AS°/ kJ mol-1
o T0	11
ACp/ kJ mol-1 K-1 1/2 AT/ S cm2 mol-1 A / S cm2 mol-1 K-1 B / S cm2 mol-1 K-2
157.2 ± 2.1 -12.55 6.65 19.2 0.37 132.70 ± 0.08 / /
176.3 ± 0.8 -12.83 ± 0.01 4.1 ± 0.3 16.9 ± 0.3 0.00 133.31 ± 0.06 2.906 ± 0.005 0.0116 ± 0.0002
138.8 ± 1.6 -12.23 ± 0.03 3.9 ± 0.7 16.2 ± 0.7 0.00 130.7 ± 0.2 / /
Comparison of the results obtained from individual and global analysis reveals differences in thermodynamic quantities, mostly in AH°, TAS°o and ACp accompanying the ion pairing process. It should be noted that these contributions were obtained from the temperature dependence of AG°13'14'24'25 in the case of individual analysis and such procedure may lead to large errors.
0.00 0.01 0.02 0.03 0.04 0.05 0.06
c"21 mol1/2 dmK
Figure 2. Molar conductivities of aqueous MgSO4 solutions from 278.15 to 308.15 K (steps of 5 K) in the concentration range 2 • 10-4 < c/mol dm-3 < 3 • 10-3. Experimental data are presented by symbols while the lines refer to the corresponding best global fit of the model function (the lcCM model eq. 4 and the QV model eq. 5). a) Comparison of individual and global analysis at 298.15 K. b) Results of the lcCM global fit.
Table 3. Correlation matrix corresponding to the lcCM global model analysis of the MgSO4 conductivity data.
	K	AHt	AT	A	B
KT	1.00				
AHT	-0.05	1.00			
AT	-0.88	0.04	1.00		
A	-0.19	-0.83	0.12	1.00	
B	-0.01	-0.37	-0.25	0.68	1.00
4. Conclusions
Global model analysis based on the low concentration chemical model (lcCM) and also the Quint-Viallard set of equations was successfully applied to the conductivity data for the first time. Model was tested on three different systems and results were compared to those obtained from the individual fits. It turned out, that this approach successfully describes all the measured experimental data simultaneously. Therefore, the applied global analysis of experimental data allows determination of the thermodynamic profile of ion pairing process (Gibbs free energy, enthalpy, entropy, heat capacity) and transport properties of ions at the same time. Moreover, the global model improves the ratio between the number of fitting parameters and the number of experimental points, reduces correlation between adjustable parameters and therefore gives more reliable results. The weakest point of the global model is that the temperature dependence of limiting molar conductivity is not based on thermodynamic relationship like association constant which is based on the Gibbs-Helmholtz relation and Kirchhoff's law but it is simple described with a second order polynomial.
As it has been already observed by Apelblat,5 the limiting conductivities of electrolytes at infinite dilution are very similar irrespective to the applied conductivity equation, whereas the derived ion association constants can vary considerably. The ion association constants and thus also all the derived thermodynamic functions are therefore of doubtful physical value and should be rather treated just as fitting parameters.
5. Acknowledgments
M. B. and M. B. R. would like to acknowledge Prof. Dr. Gorazd Vesnaver for his appreciated collaboration and helpful scientific advices. The authors also acknowledge the financial support of the Slovenian Research Agency (ARRS) through Program P1-0201.
6. References
1.	J. Barthel, R. Wachter, H.-J. Gores, in: B. E. Conway, J. O'M. Bockris (Ed.): Modern Aspects of Electrochemistry No. 13, Plenum Press, New York, 1979, pp. 1-79.
2.	M. Bester-Rogac, D. Habe, Acta Chim. Slov. 2006, 53, 391-395.
3.	J. Barthel, H. Krienke, and W. Kunz, Physical Chemistry of Electrolyte Solutions-Modern Aspects, Steinkoppf/Darm-stadt, and Springer, New York, 1998.
4.	J. Quint, A. Viallard, J. Solution Chem. 1987, 7, 533-548.
5.	A. Apelblat, J. Solution Chem. 2011, 40, 1209-1233.
6.	A. Apelblat, J. Solution Chem. 2011, 40, 1724-1734.
7.	A. Apelblat, J. Solution Chem. 2011, 40, 1291-1316.
8.	A. Apelblat, J. Solution Chem. 2011, 40, 1234-1257.
9.	A. Apelblat, J. Solution Chem. 2011, 40, 1544-1562.
10.	J. Lah, I. Drobnak, M. Dolinar, G. Vesnaver, Nucleic Acids Res. 2008, 30, 897-904.
11.	I. Drobnak, G. Vesnaver, J. Lah, J. Phys. Chem. B2010, 114, 8713-8722.
12.	J. Lah, M. Bešter-Rogač, T. M. Perger, G. Vesnaver, J. Phys. Chem. B 2006, 110, 23279-23291.
13.	M. Bešter-Rogač, R. Neueder, J. Barthel, J. Solution Chem. 1999, 28, 1071-1085.
14.	M. Tomšič, M. Bešter-Rogač, A. Jamnik, R. Neueder, J. Barthel, J. Solution Chem. 2002, 51,19-31.
15.	J. Kielland, J. Am. Chem. Soc. 1937, 59, 1675-1678.
16.	L. Korson, W. Drost-Hansen, F. J. Millero, J. Phys. Chem. 1969, 73, 34-39.
17.	B. B. Owen, R. C. Miller, C. E. Miller, H. L. Cogan, J. Phys. Chem. 1961, 65, 2065-2070.
18.	A. Apelblat, R. Neueder, J. Barthel, Electrolyte Data Collection, Electrolyte Conductivities, Ionic Conductivities and Dissociation Constants of Aqueous Solutions of Organic Monobasic Acids. Chemistry Data Series, vol. XII, Part 4a. Dechema, Frankfurt am Main, 2004.
19.	V. M. Goldschmidt, Chem. Ber. 1927, 60, 1263-1296.
20.	S. B. Brummer, G. J. Hills, J. Chem. Soc. Faraday Trans. 1961, 5, 1816-1837.
21.	H. S. Harned, B. B. Owen, The Physical Chemistry of Electrolytic Solutions, 3rd edn. Reinhold, New York, 1958.
22.	M. Bončina, J. Lah, J. Reščič, and V. Vlachy, J. Phys. Chem. B 2010, 114, 4313-4319.
23.	K. A. Sharp, B. Madan, J. Phys. Chem. B 1997, 101, 43434348.
24.	M. Bončina, A. Apelblat, M. Bešter-Rogač, J. Chem. Eng. Data 2010, 55, 1951-1957.
25.	M. Bončina, A. Apelblat, J. Barthel, M. Bešter-Rogač, J. Solution Chem. 2008, 37, 1561-1574.
Povzetek
Z merjenjem električne prevodnosti raztopin elektrolitov spomočjo različnih modelov proučujemo lastnosti ionov v različnih topilih. V tem delu smo prikazali možnost uporabe globalne analize podatkov za električno prevodnost razredčenih raztopin elektrolitov v širokem temperaturnem območju. Globalni model smo razvili na podlagi dveh modernih teorij s področja prevodnosti, t.i. kemijskega modela (lcCM) in Quint-Viallardovega modela. Ustreznost globalnega modela smo preverili na treh različnih sistemih: NaCl v vodi, NaCl v mešanici vode in 1,4 dioksana ter MgSO4 v vodi. Izkazalo se je, da globalni model uspešno popiše eksperimentalne podatke izmerjene pri različnih koncentracijah in temperaturah in določi termodinamski profil procesa tvorbe ionskih parov skupaj s transportnimi lastnostmi ionov.
Supporting information
As already mentioned in the main text, temperature dependence of limiting molar conductivity may be described with Walden's rule
A °Trj = const	(1-SI)
or Eyring approach
where n is the viscosity of pure water,1 d the density of pure water,2 AH* temperature independent Eyring's enthalpy of activation of the charge transport which depends only on the solvent properties and const temperature independent constant which has no physical meaning. Equations apply to limiting molar conductivity of the salt used in lcCM model, however, same expressions can be used also for modeling temperature dependence of limiting molar conductivity of cation and anion needed in QV model. If Walden's rule was used to model temperature dependence of AT, three adjustable parameters were used in the global lcCM minimization process: AGT, AHT and const while in the case of Eyring approach four adjustable parameters were used: AGT, AHT ,AH* and const.
Models describing temperature dependence of limiting molar conductivity do not affect the thermodynamics of ion pairing process. However, they have big impact on limiting molar conductivities (Table 1-SI) and quality of the fit, especially at low (278.15 K) and high (308.15 K) temperatures (Table 2-SI). Table 1-SI reveals that Walden's rule is the least suitable for this purpose. Same conclusion may be drawn from all three investigated systems.
Table 2-SI. Experimental and calculated molar conductivities obtained from global analysis of NaCl in water at 278.15 K.Temperature dependence of Aj was described with polynomial expression, Walden's rule and Eyring approach.The results correspond to lcCM model.
		polynomial	Walden's	Eyring
		expression	rule	approach
		eq. 6	eq.1-SI	eq. 2-SI
103 m /mol kg1	Aexp	Acalc/ S cm2 mol-		1
0.1778	77.00	77.03	74.32	78.52
0.3359	76.81	76.77	74.08	78.26
0.5695	76.53	76.48	73.83	77.96
0.8645	76.27	76.19	73.58	77.67
1.2066	75.94	75.91	73.35	77.39
1.5044	75.74	75.70	73.18	77.18
1.9519	75.47	75.42	72.96	76.90
2.5393	75.15	75.10	72.72	76.57
References
1.	L. Korson, W. Drost-Hansen, F.J. Millero, J. Phys. Chem. 1969, 75, 34-39.
2.	E. F. G. Herington, Pure Appl. Chem. 1976, 45, 1-9.
Table 1-SI. Limiting molar conductivities of NaCl in water obtained from individual and global analysis of conductivity data.In the case of global analysis, temperature dependence of Aj was described with polynomial expression, Walden's rule and Eyring approach.The results correspond to lcCM model.
individual	global
polynomial Walden's Eyring expression	rule approach
eq. 6	eq. 1-SI eq. 2-SI
T / K		AT / S	cm2mol 1	
278.15	77.71	77.74	74.98	79.23
283.15	89.12	89.19	87.16	89.43
288.15	101.09	101.14	100.08	100.53
293.15	113.57	113.58	113.68	112.59
298.15	126.54	126.50	127.95	125.65
303.15	139.95	139.92	142.84	139.74
308.15	153.78	153.83	158.32	154.90
Best fit parameters obtained from global analysis: Walden's rule: const = 1.139 • 10-2 S kg m mol-1 s-1 Eyring approach: AH* = 15.8 kJ mol-1; const = 11.218