Scientific paper
Diffusion Coefficients of Sodium Fluoride in Aqueous Solutions at 298.15 K and 310.15 K
Ana C. F. Ribeiro,1'* Victor M. M. Lobo,1 Abilio J. F. N. Sobral,1 Helder T. F. C Soares,1 Ana R. J. Leal1 and Miguel A. Esteso2
1 Department of Chemistry, University of Coimbra, 3004 - 535 Coimbra, Portugal
2 Departamento de Química Física, Facultad de Farmacia, Universidad de Alcalá 28871.
Alcalá de Henares (Madrid)
* Corresponding author: E-mail: anacfrib@ci.uc.pt Tel: +351-239-854460; fax: +351-239-827703
Received: 03-11-2009
Abstract
Mutual diffusion coefficients (interdiffusion coefficients) have been measured for sodium fluoride in water at 298.15 K and 310.15 K at concentrations between 0.003 mol dm-3 and 0.05 mol dm-3. The diffusion coefficients were measured using a conductimetric cell. The experimental mutual diffusion coefficients are discussed on the basis of the Onsager-Fuoss model. The limiting molar conductivity of the fluoride ion in these solutions at 310.15 K has been estimated using these results.
Keywords: Diffusion, electrolytes, solutions, sodium fluoride, transport properties.
1. Introduction
The knowledge of electrolytes diffusion data is important for essential reasons, helping to understand the nature of aqueous electrolyte structure, and for its practical application in many everyday technical fields, for instance, dental corrosion. However, the magnitude and behaviour of these transport properties for electrolytic systems in the oral cavity are poorly known, even though this is a prerequisite to obtain adequate understanding and solution of these corrosion problems.
For some time, the research interest of our group has been focused on dental restoration and therefore in obtaining data (not available in the literature) of the transport properties for ionic systems involved in the dental damage processes in the oral cavity.1-3 In fact, since oral restoration involves various dental metallic alloys, and the oral cavity is a wet environment, those systems provide favourable conditions for corrosion. This phenomenon has been minimized by the use of fluoride compounds, under different systemic and topical forms, which prevent an anti-caries action.4,5 This has provided the impetus for the present study of the diffusion of these fluoride ions in aqueous solutions.
As far as the authors know, no data on mutual diffusion coefficients of sodium fluoride have been published at physiological temperature.6 Consequently, in the present study mutual diffusion coefficients, D, (interdiffusion coefficients) are reported at this temperature, together with those at 298.15 K, for aqueous solutions of sodium fluoride in the concentration range from 0.003 to 0.05 mol dm-3, by using an open-ended conductimetric capillary cell.7-29 These results are discussed on the basis of the On-sager-Fuoss mode.30-33 At this stage, no attempt is made to split mutual diffusion data into their individual contributions since for practical purposes, such as the chemistry in the oral cavity, is the global value what is required.
2. Experimental Section
2. 1. Materials
Sodium Fluoride (J. T. Baker, pro analysis > 99%) was used without further purification.
The solutions for the diffusion measurements were prepared in calibrated volumetric flasks using bi-distilled water. The solutions were freshly prepared and de-aerated for about 30 min before each set of runs.
2. 2. Diffusion Measurements
An open-ended capillary cell, which has been used to obtain mutual diffusion coefficients for a wide variety of electrolytes,7-9 is described in great detail in previous papers.7-29 and so only some relevant points concerning this method on the experimental determination of binary diffusion coefficients, D are described.
Basically, this consists of two vertical capillaries, each closed at one end by a platinum electrode, and positioned one above the other with the open ends separated by a distance of about 14 mm. The upper and lower tubes, initially filled with solutions of concentrations 0.75 c and 1.25 c, respectively, are surrounded with a solution of molar concentration, c (defined in mol dm 3). This ambient solution is contained in a glass tank (200 x 140 x 60) mm immersed in a thermostat bath at 298.15 K. Perspex sheets divide the tank internally and a glass stirrer creates a slow lateral flow of ambient solution across the open ends of the capillaries. Experimental conditions are such that the concentration at each of the open ends is equal to the ambient solution value c, that is, the physical length of the capillary tube coincides with the diffusion path. This means that the required boundary conditions described in the literature8,9 to solve Fick's second law of diffusion are applicable. Therefore, the so-called Al effect8,9 is reduced to negligible proportions. In our manually operated apparatus, diffusion is followed by measuring the ratio w = R/Rb of resistances Rt and Rb of the upper and lower tubes by an alternating current transformer bridge. In our automatic apparatus, w is measured by a Solartron digital voltmeter (DVM) 7061 with 6 1/2 digits. A power source (Bradley Electronic Model 232) supplies a 30 V sinusoidal signal at 4 kHz (stable to within 0.1 mV) to a potential divider that applies a 250 mV signal to the platinum electrodes in the top and bottom capillaries. By measuring the voltages V' and V'' from top and bottom electrodes to a central electrode at ground potential in a fraction of a second, the DVM calculates w.
In order to measure the differential diffusion coefficient D at a given concentration c, the bulk solution of concentration c is prepared by mixing 1 L of "top" solution with 1 L of "bottom" solution, measured accurately. The glass tank and the two capillaries are filled with c solution, immersed in the thermostat, and allowed to come to thermal equilibrium. The resistance ratio w = w^ measured under these conditions (with solutions in both capillaries at concentration c) accurately gives the quantity T = 104 / (1 + wj.
The capillaries are filled with the "top" and "bottom" solutions, which are then allowed to diffuse into the "bulk" solution. Resistance ratio readings are taken at various recorded times, beginning 1000 min after the start of the experiment, to determine the quantity t= 104/(1 + w) as t approaches t.
Considering the quantities Yt and Y^defined by the equations
Yt = (W - 1) / (W + 1) Y = (w - 1) / w + 1)
OO	^ OO	'	OO	'
(1) (2)
and t > 1000 min, it is possible to evaluate the diffusion coefficient by using the corresponding slope of the equation (3)
ln (Yt - YJ = constant - At A = (n2 D) / (4 a2) (2)
(3)
(4)
Therefore, plotting the experimental results ln(Yt -Y^ ) as a function of time, and using an adequate programme of successively interactions coupled with the minimum square deviations method, a straight line equation is obtained, and the corresponding slope is directly proportional to the diffusion coefficient eq (4).
3. Results and Discussion
Mutual diffusion coefficients, D, of sodium fluoride in aqueous solutions at 298.15 K and 310.15 K are shown in Table 1, where D is the mean value of, at least, three independent measurements. The standard deviations of the means are shown in Table 1. Previous papers reporting data obtained with this conductimetric cell support our view that the inaccuracy of our results should not be much larger than the imprecision. That is, we believe that our uncertainty is not much larger than (1 to 3) %.
For the purposes of this study, it was not necessary to extend the limits in concentration beyond those indicated in Table 1.
A linear dependence on the concentration c has been found
D = a0 + a1 c
(5)
where the coefficients Sq and a1, are adjustable parameters obtained by fitting the experimental data. Table 2 shows the coefficients a0 and a1 of eq 5. These may be used to calculate values of diffusion coefficients at specified concentrations within the range of the experimental data shown in Table 1. The goodness of the fit (obtained with a confidence interval of 98%) can be assessed by the excellent correlation coefficients, R2, and the low standard deviation (< 1%).
To understand the transport process of this electrolyte in aqueous solutions, as a first approach the experimental mutual diffusion coefficients at 298.15 K were compared with those estimated by using Onsager-Fuoss equation suitable for dilute solutions [eq 6 (Table 3)]
D = M
Js,\+\z.
12,\
RT
c
1 + c-
Sln y.
Sc
(6)
Table 1: Diffusion coefficients, D of NaF in aqueous solutions at various concentrations, c, and the standard deviations of the means, SD. at 298.15 K and 310.15 K.
c/mol dm 3	T =	298.15 K		T =	310.15 K
	D/10-9 m2 s-1 a	Sfl/10-9 m2 s-1 b	AD/DLit <	D/10-9 m2 s-1 a	SD /10-9 m2 s-1 b
0.000	1.370 d	-	-2.2 e	2.038 d	-
0.003	1.368	0.015	1.0	2.017	0.010
0.005	1.365	0.020	1.1	2.009	0.029
0.008	1.364	0.019	1.1	2.005	0.017
0.010	1.353	0.015	1.0	2.003	0.010
0.030	1.328	0.020	0.6	1.899	0.009
0.050	1.300	0.005	-0.8	1.804	0.005
aD is the mean diffusion coefficient for 3 experiments
cAD/DLit represent the relative deviations bet-
ween our diffusion coefficients, D (Table 1), and experimental D values obtained from Taylor technique, respectively. D extrapolated using our data (Table 1). eRelative deviations between D extrapolated using our data and the Nernst value (eq 10) using data from ref 39.
b
SD is the standard deviation of that mean.
Table 2: Fitting coefficients (a0 and a^ of the linear dependence [D/(m2 s-1) = a0 + a1 (c/mol dm-3) to the mutual differential diffusion coefficients for sodium fluoride in aqueous solutions at 298.15 K and 310.15 Ka
T/K	a0/109 m2 s-1	a1/109 m2 s-1	R2 a
298.15	1.372	-1.449	0.991
310.15	2.038	-4.647	0.993
a See page 6.
where D is the mutual diffusion coefficient of the electrolyte in m2 s-1, R is the gas constant in J mol-1 K1, T is the absolute temperature, z1 and z2 are the algebraic valences of a cation and of an anion, respectively, and the last term in parenthesis is the activity factor, with y± being the mean molar activity coefficient, c the concentration in mol m-3, and M, in mol2 s m-3 kg-1 , given by
M = '

0 i o
Xi X
/ ~2
v j, zJA ,IzJA 21 21 ! 11 II
c+AM'+dM'
(7)
In eq 3, the first-and second-order electrophoretic terms, are given by
(m *
Ahr = --
l'2

2 v. +v„ für îi0(i + >L-a}
(8)
o
(9)
where n0 is the viscosity of the water in N s m 2, NA is the Avogadro's constant, e0 is the proton charge in coulombs, n1 and n2 are the stoichiometric coefficients, and are the limiting molar conductivities of the cation and anion, respectively, in Q-1 m2 mol-1, Kis the "reciprocal average radius of ionic atmosphere" in m-1 (see e.g.,34), a is the mean distance of closest approach of ions in m, (¡>(ka) = |e2kaEj(2ka)/(1 + Ka)| has been tabulated by Harned and Owen,34 and the other letters represent well-known quantities.34 In this equation, phenomena such as hydrolysis,35,36 complexation and/or ion association,37 are not taken into consideration. There is no direct method for measuring the ion size parameter a, "mean distance of closest approach" from the Debye-Huckel theory, but it may be estimated from different methods by using experimental and theoretical approaches for sodium salts (e.g., from mean ionic activity coefficients and diffusion coefficients, and ab initio calculations and molecular mechanic studies (MM2)).38
Concerning the values of parameter a obtained by adjustment of Onsager-Fuoss equation to the experimental data of diffusion coefficients, we see that, in general, we should note that calculations based on eq 2 are not greatly affected by the choice of the ion size parameter a, within the limits indicated (Table 3). For example, comparing the calculated diffusion coefficients of sodium fluoride, Dof (Table 3), using the three different values of this parameter a, with the related experimental values at 298.15 K (Table 1), a reasonable agreement is observed between the experimental data and this model (deviations < 3%). The deviation between the limiting D0 value calculated by extrapolating experimental data to c ^ 0 (Table 2) and the Nernst value (Table 3) is also acceptable (2.2%). The decrease of the diffusion coefficient, when the concentration increases, may be interpreted, among other factors, on the basis of species resulting from the eventual formation of ion pairs, increasing these phenomena with concentration. In relation to the effect of temperature on diffusion, an increase in the experimental D
Table 3: Diffusion coefficients of sodium fluoride calculated from Onsager-Fuoss theory, DOF, at 298.15 K.3'
c/mol dm 3	d'op	Ad/d'op	D'' " op	AD/D''op	D''' " op	AD/D'''op
	/10-9 m2 s-1	/ %	/10-9 m2 s-1	/ %	/10-9 m2 s-1	/ %
	a)	b)	c)	b)	d)	b)
0.000	1.401	-2.5 e)	1.401	-2.5 e)	1.401	-2.5 e)
0.003	1.367	0.07	1.365	0.4	1.364	0.3
0.005	1.360	0.4	1.361	-0.1	1.363	-0.1
0.008	1.352	0.9	1.351	1.0	1.352	0.9
0.010	1.346	0.5	1.350	0.2	1.354	-0.1
0.030	1.329	-0.1	1.335	-0.5	1.339	1.0
0.050	1.310	-0.8	1.326	-2.0	1.340	-2.9
2.2 X 10 10 m obtained from the sum of the ionic radii (obtained from diffraction methods).40
ra,ot;,m	n	^Of, D"of and D'"0F values, respectively.
Notes: a) a
represent the relative deviations between D (Table 1) a MM2.41 d) a = 7.1 X 10-10 m estimated using MM2.41
c) a = 4.6 X 10 10 m estimated using e) Relative deviations between D extrapolated (Table 1) and the Nernst value (eq 10).
values was found at all sodium fluoride concentrations. Also, the decrease of the diffusion coefficient was obtained when the concentration increases. However, given the absence of the values of parameters for estimations of Dof, only the diffusion coefficient of sodium fluoride at infinitesimal concentration and the equivalent conductance of the fluoride ion were estimated.
From the following equation for analysis of the data, shown in Table 2, we estimated the diffusion coefficient of sodium fluoride at infinitesimal concentration as D0 = 1.372 x 10-9 m2 s-1 and D0 = 2.038 x 10-9 m2 s-1 at 298.15 K and 310.15 K, respectively. To estimate AF_ at 298.5 K and 310.15 K, we may assume that the above D0 value coincides with the Nernst value26 from
I 111
d„ rtK+I+M V V . (10)
F2 ]zNa+ZfH X° +|zF-[+ >*°JzNa+|
Na	F
where ZNa+ and ZF- represent the algebraic valences of a cation and of an anion, respectively. ^Na+ is the molar conductance of Na+ at infinitesimal concentration, estimated at 310.15 K by using a polynomial equation fitted to experimental data from reference 39 (that is, ^Na+ = 59.0 x 10-4 Q-1 m2 mol-1). At 298.15 K, the authors used the value found in this reference (that is, ^Na+ = 50.10 x 10-4 Q-1 m2 mol-1). Therefore, from eq 6, we have = 53.30 x 10-4 Q-1 m2 mol-1 and = 98.10 x 10-4 Q-1 m2 mol-1 at 298.15 K and 310.15 K, respectively. The deviation between the value calculated by this method at 298.15 K and the value found in the literature is also acceptable (3%).
4. Conclusions
In conclusion, we have measured mutual diffusion coefficients (interdiffusion coefficients) for sodium fluoride in water at 298.15 K and 310.15 K at concentrations
between 0.003 mol dm-3 and 0.05 mol dm-3' These diffusion coefficients have been measured having in mind a better understanding of the structure of these systems.
5. Acknowledgements
Financial support from FCT (FEDER)-PTDC/AAC-CLI/098308/2008 is gratefully acknowledged. One of the authors (MAE) is grateful to the University of Alcalá (Spain) for the financial assistance (Mobilty Grants for Researchers Program).
6. References
1.	N. M. Taher, A. S. Al Jabab, Dent. Mater. 2003, 19, 54-59.
2.	L. Reclaru, R. Lerf, P. Y Eschler, A. Blatter, Biomaterials 2002, 23, 3479-3485.
3.	E. Brambilla, Caries Res. 2001, 35, 6-9.
4.	M. Petzold, Caries Res. 2001, 35, 45-51.
5.	L. L Demos, H. Kazda, F. M. Cicuttini, M. I. Sinclair, C. K. Fairley, Aust. Dent. J. 2001, 46, 80-87.
6.	L. Ruanhui, D. G. Leaist, J. Chem. Soc., Faraday Trans. 1998, 94, 111-114.
7.	V. M. M. Lobo, Handbook of Electrolyte Solutions, Elsevier, Amsterdam, 1990.
8.	J. N. Agar, V. M. M. Lobo, J. Chem. Soc., Faraday Trans. 1975, 171, 1659-1666.
9.	V. M. M. Lobo, Pure Appl. Chem. 1993, 65, 2613-2640.
10.	V. M. M Lobo, A. C. F. Ribeiro, L. M. P. Verissimo, Ber. Bunsenges. Phys. Chem. 1994, 98, 205-208.
11.	V. M. M. Lobo, A. C. F. Ribeiro, L. M. P. Verissimo,. J. Chem. Eng. Data 1994, 39, 726-728.
12.	V. M. M Lobo, A. C. F. Ribeiro, A. J. M. Valente, Corros. Prot. Mat. 1995, 14, 14-21.
13.	V. M. M Lobo, A. C. F. Ribeiro, S. G. C. S. Andrade, Ber. Bunsenges. Phys. Chem. 1995, 99, 713-720.
14.	V. M. M. Lobo, A. C. F Ribeiro, L. M. P. Verissimo, J. Mol.
Liquids 1998, 78, 139-149.
15.	A. C. F Ribeiro, V. M. M Lobo, J. J. S. Natividade, J. Mol. Liquids 2001, 94, 61-66.
16.	A. C. F. Ribeiro, V. M. M Lobo, E. F. G Azevedo, M. G Miguel, H. D. Burrows, J. Mol. Liq. 2001, 94, 193-201.
17.	A. C. F Ribeiro, V. M. M. Lobo, E. F. G. Azevedo, J. Sol. Chem. 2001, 30, 1111-1115.
18.	A. C. F. Ribeiro, V. M. M. Lobo, J. J. S. Natividade, J. Chem. Eng. Data 2002, 47, 539-541.
19.	A. C. F Ribeiro, V. M. M. Lobo, E. F. G. Azevedo, M. G. Miguel, H. D.Burrows, J. Mol. Liq. 2003, 102, 285-292.
20.	A. J. M Valente, A. C. F. Ribeiro, V. M. M. Lobo, A. Jiménez, J. Mol. Liq. 2004, 111, 33-38.
21.	A. C. F. Ribeiro, A. J. M. Valente, V. M. M. Lobo, E. F. G. Azevedo, A. M. Amado, A. M. A. Costa, M. L. Ramos, H. D. Burrows, J. Mol. Struct. 2004, 703, 93-101.
22.	A. C. F. Ribeiro, V. M. M. Lobo, A. J. M. Valente, E. F. G. Azevedo, M. G. Miguel, H. D. Burrows, Colloid and Poly. Sci. 2004, 283, 277-283.
23.	A. C. F Ribeiro, V. M. M. Lobo, R. C. Oliveira, H. D.Burrows, E. F. G. Azevedo, S. I. G. Fangaia, P. M. G. Nicolau, F. A. D. R. Guerra, J. Chem. Eng. Data 2005, 50, 1014-1017.
24.	A. C. F. Ribeiro, M. A. Esteso, V. M. M. Lobo, A. J. M. Valente, S. M. N. Simoes, A. J. F. N. Sobral; H. D. Burrows, J. Chem. Eng. Data 2005, 50, 1986-1990.
25.	A. C. F. Ribeiro, M. A. Esteso, V. M. M. Lobo, V. M. M.; A. J. M Valente, S. M. N. Simoes, A. J. F. N. Sobral, L. Ramos, H. D. Burrows, J. Carbohydr Chem. 2006 25, 173-185.
26.	A. C. F. Ribeiro, V. M. M. Lobo, A. J. M. Valente, S. M. N. Simoes, A. J. F. N. Sobral, L. Ramos, H. D. Burrows, Polyhedron 2006, 25, 3581-3587.
27.	A. C. F. Ribeiro, M. A. Esteso, V. M. M. Lobo, A. J. M. Valente, S. M. N. Simoes, A. J. F. N. Sobral, H. D. Burrows, J. Mol. Struct. 2007, 826, 113-119.
28.	A. C. F. Ribeiro, M. A. Esteso, V. M. M. Lobo, A. J. M. Valente, A. J. F. N. Sobral, H. D. Burrows, Electrochim. Acta. 2007, 52, 6450-6455.
29.	A. C. F. Ribeiro, M. C. F. Barros, A. S. N. Teles, A. J. M. Va-lente, V. M. M. Lobo, A. J. F. N. Sobral, , M. A. Esteso, Elec-trochim. Acta. 2008,54, 192-196.
30.	L. Onsager, R. M. Fuoss, J. Phys. Chem. 1932, 36, 26892778.
31.	R. A. Robinson, R. H. Stokes, Electrolyte Solutions, 2nd Ed., Butterworths, London, 1959.
32.	V. M. M. Lobo, A. C. F. Ribeiro, Port. Electrochim. Acta
1993,	11, 297-304.
33.	V. M. M. Lobo, A. C. F. Ribeiro, S. G. C. S. Andrade, Port. Electrochim. Acta 1996, 14, 45-124.
34.	H. S. Harned, B. B. Owen, The Physical Chemistry of Electrolytic Solutions, 3rd Ed., Reinhold Pub. Corp, New York, 1964.
35.	C. F. Jr. Baes, R. E. Mesmer, The Hydrolysis of Cations, John Wiley and Sons, New York, 1976.
36.	J. Burguess, Metal Ions in Solution, John Wiley & Sons, Chichester, Sussex, England, 1978.
37.	V. M. M. Lobo, A. C. F. Ribeiro, Port. Electrochim. Acta
1994,	12, 29-41.
38.	A. C. F. Ribeiro, M. A. Esteso, V. M. M. Lobo, H. D. Burrows, A. M. Amado, A. M. da Costa Amorim, A. J. F. N. Sobral, E. F. G. Azevedo, M. A. F. Ribeiro, J. Mol. Liq. 2006, 128, 134-139.
39.	D. Dobos, Electrochemical Data. A Handbook for Electroc-hemists in Industry and Universities, Elsevier, New York, 1975.
40.	Y. Marcus, Chem. Rev. 1988, 88, 1475-1498.
41.	HyperChem v6.03 software from Hypercube Inc., 2000, USA. MM+ molecular mechanics force field calculation using a Po-lak-Ribiere conjugated gradient algorithm for energy minimization in water with a final gradient of 0.05 kcal/A mol.
Povzetek
Z merjenjem električne prevodnosti v kapilarni celici smo določili difuzijski koeficient natrijevega fluorida v vodi pri 298.15 in 310.15 K v koncentracijskem območju med 0.003 in 0.05 mol dm-3. Eksperimentalne vrednosti smo obravnavali z Onsager-Fuossovim modelom, s pomočjo katerega smo določili tudi limitno vrednost molske prevodnosti fluorid-nega iona v vodi pri 310.15 K.