Differential mutual diffusion coefficients of binary and ternary aqueous systems measured by the open ended conductometric capillary cell and by the Taylor technique

Journal of Molecular Liquids 156 (2010) 58–64

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Journal of Molecular Liquids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m o l l i q

Differential mutual diffusion coefficients of binary and ternary aqueous systems measured by the open ended conductometric capillary cell and by the Taylor technique☆ Ana C.F. Ribeiro a,⁎, Joaquim J.S. Natividade a, Miguel A. Esteso b a b

Department of Chemistry, University of Coimbra, 3004 - 535 Coimbra, Portugal Departamento de Química Física, Facultad de Farmacia, Universidad de Alcalá, 28871 Alcalá de Henares, Madrid, Spain

a r t i c l e

i n f o

Available online 4 May 2010 Keywords: Diffusion Solutions Electrolytes Drugs

a b s t r a c t Binary and ternary mutual diffusion coefficients of aqueous solutions of electrolytes and non-electrolytes have been measured at different conditions (concentration, temperature, techniques, etc.), having in mind a contribution to a better understanding of the structure of these solutions, behaviour of electrolytes and nonelectrolytes in solution, as well as supplying the scientific and technological communities with new data on these important parameters in solution transport processes. By using different techniques of the open ended capillary conductometric method and the Taylor dispersion technique we achieved, for different systems, experimental data of diffusion coefficients, presented and discussed in this review. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The supervision and guidance of Prof Victor Lobo were no doubt of utmost importance for the evolution of the professional career of one of the authors (Ana Ribeiro), and for the definition of the research options of his co-workers throughout the last decades. In the last years, Lobo and his group have been particularly dedicated to the study of mutual diffusion behaviour of binary electrolyte solutions [1–20], helping to go deeply in the understanding of their structure, including practical applications in fields as diverse as corrosion or therapeutic uses. In particular, Lobo et al. have been interested in data on diffusion coefficients of chemical systems occurring in the oral cavity, to understand and resolve corrosion problems related to dental restorations in systems where data were not available. Bearing in mind that oral restorations involve various dental metallic alloys, we have been particularly interested in those involving metal ions such as copper (II) [18], cobalt (II) [11], chromium [17] and aluminium [19]. These studies are justified once the properties and behaviour of chemical systems in the oral cavity are poorly known, even though this is a prerequisite to obtain adequate understanding, and consequent solution, of these wear and corrosion problems.

☆ This paper is dedicated to Prof. Victor Lobo on the occasion of his 70th birthday. ⁎ Corresponding author. Tel.: +351 239 854460; fax: +351 239 827703. E-mail addresses: [email protected] (A.C.F. Ribeiro), [email protected] (M.A. Esteso). 0167-7322/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2010.04.020

Using the open ended capillary cell, our group has also been focused on the study of the diffusion behaviour of pseudo-binary systems such as SDS/sucrose/water [9], SDS/cyclodextrin/water [12], PbNO3/HNO3/water [13], CuCl2/sucrose/water [21], CuCl2/fructose/ water [21], CuCl2/glucose/water [21], CuCl2/β-cyclodextrin/water [22], NH4VO3/β-cyclodextrin/water [15], NH4VO3/carbohydrates/ water [14]. These systems are actually ternary systems, and we really have been measuring only main diffusion coefficients (D11)1. However, from experimental conditions, we may consider those systems as pseudo-binary ones, and, consequently, take the measured parameters as binary diffusion coefficients, D. A few years ago (2002), we have installed the Taylor technique, being now able to study in our laboratory the diffusion behaviour of multicomponent chemical systems under different conditions. As a consequence, we have been measuring diffusion coefficients of some binary systems (e.g., lactic acid [23], carbohydrates [24–26], and drugs [27], in aqueous solutions), and some ternary systems involving electrolytes in different media (e.g., CuCl2/caffeine/water [28]) or drugs, alone or in combination with cyclodextrins (β-CD/caffeine/ water [29] and HPβCD/caffeine/water [30]). On the other hand, from the data obtained with the Taylor technique, it has also been possible to estimate some other parameters, such as diffusion coefficient at infinitesimal concentration, dissociation degree, activation energy, activity coefficient and effective hydrodynamic radius. In addition,

1

See Section 2.2 Taylor technique.

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59

using the ternary diffusion coefficients, we can conclude if the presence of a certain solute affects the diffusion of the other one. In order to avoid some common confusion, we would like to highlight the true meaning of the quantity we have been measuring, that is, “mutual diffusion coefficient”, usually denoted as “D” or “Dij” for binary and ternary systems, respectively [31,32]. It is necessary to distinguish between two distinct processes: self-diffusion D* (intradiffusion, tracer diffusion, single ion diffusion, and ionic diffusion) and mutual diffusion D (interdiffusion, concentration diffusion, and salt diffusion) [31,32]. Many techniques are used to study diffusion in aqueous solutions. Methods such as NMR, polarographic, and capillary-tube techniques with radioactive isotopes, measure selfdiffusion coefficients (“intradiffusion coefficients”). However, for bulk ion transport, the appropriate parameter is the mutual diffusion coefficient, D. Relationships derived between intradiffusion and mutual diffusion coefficients, D* and D, have had limited success, and consequently experimentally determined mutual diffusion coefficients are absolutely necessary. 2. Experimental 2.1. The open ended conductometric capillary cell An open ended capillary cell2 developed by Lobo, and which has been used to obtain mutual diffusion coefficients of a wide variety of electrolytes [1–19], is described in great detail in previous papers (theory of method, description of the different cell components, streamlined flow and Δl) [2,4]. Basically, it 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 0.75c and 1.25c, respectively, are surrounded by a solution of concentration c. This ambient solution is contained in a glass tank 200 × 140 × 60 mm immersed in a thermostat bath at 25 °C. 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 (Fig. 1). 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 literature [2] to solve Fick's second law of diffusion are applicable. Therefore, the so-called Δl effect [2] is reduced to negligible proportions. In our manually operated apparatus, diffusion is followed by measuring the ratio w = Rt / 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, accurately measured. 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 τ∞ = 104 / (1 + w∞). The capillaries are filled with the “top” and “bottom” solutions, which are then allowed to diffuse into the “bulk” solution. Resistance 2 Gold Medal at the 15th International Exhibition of Inventions and New Techniques, Geneva, Switzerland, 3–12 April, 1987.

Fig. 1. Schematic representation of the open ended capillary cell. TS, BS: support capillaries; TC, BC: top and bottom diffusion capillaries; CE: central electrode; PT: platinum electrodes; D1, D2: perspex sheets; S: glass stirrer; P: Perspex block; G1, G2: perforations in Perspex sheets; A, B: sections of the tank; L1, L2: small diameter coaxial leads.

ratio readings are taken at various recorded times, beginning 1000 min after the start of the experiment, to determine the quantity τ = 104 / (1 + w) as τ approaches ∞ (τ∞). The diffusion coefficient is evaluated using a linear least-squares procedure to fit the data and, finally, an iterative process is applied using 20 terms of the expansion series of Fick's second law for the present boundary conditions.

2.2. Taylor technique The theory of the Taylor dispersion technique is well described in the literature [31,33,34] and so, in the present paper, we only point out some relevant points concerning the experimental determination of binary and ternary diffusion coefficients. The above dispersion method is based on the dispersion of small amounts of solution injected into laminar carrier streams of solvent or solution of different composition, flowing through a long capillary tube [31,33,34]. The length of the Teflon dispersion tube was measured directly by stretching the tube in a large hall and using two high quality theodolytes and appropriate mirrors to accurately focus on the tube ends. This technique gave a tube length of 3.2799 (±0.0001) × 104 mm, in agreement with less-precise control measurements using a good-quality measuring tape. The radius of the tube, 0.5570 (±0.00003) mm, was calculated from the tube volume obtained by accurately weighing (resolution 0.1 mg) the tube when empty and when filled with distilled water of known density.

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At the start of each run, a 6-port Teflon injection valve (Rheodyne, model 5020) was used to introduce 0.063 mL of solution into the laminar carrier stream of slightly different composition. A flow rate of 0.17 mL min− 1 was maintained by a metering pump (Gilson model Minipuls 3) to give retention times of about 1.1 × 104 s. The dispersion tube and the injection valve were kept usually at 298.15 K (±0.01 K) in an air thermostat. Dispersion of the injected samples was monitored using a differential refractometer (Waters model 2410) at the outlet of the dispersion tube. Detector voltages, V(t), were measured at accurately 5 s intervals with a digital voltmeter (Agilent 34401 A) with an IEEE interface. Binary diffusion coefficients were evaluated by fitting the dispersion equation 1=2

V ðt Þ = V0 + V1 t + Vmax ðtR =t Þ

h i 2 2 exp –12Dðt−tR Þ = r t

ð1Þ

to the detector voltages, where r is the internal radius of our Teflon dispersion tube. The additional fitting parameters are the mean sample retention time tR, peak height Vmax, baseline voltage V0, and baseline slope V1. Diffusion in a ternary solution is described by the diffusion equations (Eqs. (2) and (3)), ∂c ∂c −ð J1 Þ = ðD11 Þv 1 + ðD12 Þv 2 ∂x ∂x

least, 3 independent measurements. In general, the error of such average results is lower than 1 %. These data were fitted using the polynomial equation D = a0 + a1 c

1=2

+ a2 c + a3 c

3=2

2

where the coefficients a0, a1, a2, a3 and a4 are adjustable parameters. This fit was obtained using the mean square deviation method and the goodness of the fit can be assessed by the correlation coefficient (R2), using a confidence interval of 95%, calculated with a minimum of 4 experimental data points. Table 4 shows the coefficients a0 to a4 of Eq. (5). They may be used to calculate values of diffusion coefficients at specified concentrations within the range of the experimental data. The estimation of diffusion coefficients from some models describing the movement of matter in electrolyte solutions is, in the end, a process contributing to the knowledge of their structure, provided we have accurate experimental data to test these equations. Assuming that each ion of diffusing electrolyte can be regarded as moving under the influence of two forces ( i — a gradient of the chemical potential for that ionic species; and ii — an electrical field produced by the motion of oppositely charged ions), we come up to the Nernst–Hartley equation [35]      0 0 0 0 2 D = ðν1 + ν2 Þλ1 λ2 = ν1 jZ1 j λ1 + λ2 RT = F 1 + ðd lnγ = d ln cÞ

½



ð3Þ

∂c1 ∂c and 2 are the molar fluxes and the gradients in the ∂x ∂x concentrations of solutes 1 and 2, respectively. Main diffusion coefficients, D11 and D22, give the flux of each solute produced by its own concentration gradient. Cross diffusion coefficients D12 and D21 give the coupled flux of each solute driven by a concentration gradient in the other solute. A positive Dik cross-coefficient (i ≠ k) indicates cocurrent coupled transport of solute i from regions of higher concentration of solute k to regions of lower concentration of the same solute k. However, a negative Dik coefficient indicates countercurrent coupled transport of solute i from regions of lower to higher concentration of solute k. Extensions of the Taylor technique have been used to measure ternary mutual diffusion coefficients (Dik) for multicomponent solutions. These Dik coefficients, defined by Eqs. (2) and (3), were evaluated by fitting the ternary dispersion equation (Eq. (4)) to two or more replicate pairs of peaks for each carrier stream

where J1, J2,

½



ð6Þ

ð2Þ

∂c ∂c −ð J2 Þ = ðD21 Þv 1 + ðD22 Þv 2 ∂x ∂x

ð5Þ

+ a4 c

where λ0 are the limiting ionic conductivities of the ions (subscripts 1 and 2 for cation and anion, respectively), Z1 is the algebraic valence of the ion, v1 is the number of ions formed upon complete ionisation of one solute “molecule”, T is the absolute temperature, R and F are the gas and Faraday constants, respectively, and γ± is the mean molar activity coefficient. Eq. (6) is often written as 0

D = D ½1 + ðd lnγ = d ln cÞ

ð7Þ

where D0 is the Nernst limiting value of the diffusion coefficient. Onsager and Fuoss [35–38], improved Eq. (7) by taking into account the electrophoretic effects (Eq. (8)):   0 D = D + ∑Δn ½1 + ðd lnγ = d ln cÞ

ð8Þ

The difference between Eqs. (7) and (8) can be found in the electrophoretic term, Δn, given by    n 0 n 0 2  n Δn = kB T An Z1 t2 + Z2 t1 = a jZ1 Z2 j

ð9Þ

1=2

V ðt Þ = V0 + V1 t + Vmax ðtR =t Þ "

2

12D1 ðt−tR Þ W1 exp − r2 t

!

2

12D2 ðt−tR Þ + ð1−W1 Þ exp − r2 t

!# ð4Þ

Two pairs of refractive-index profiles, D1 and D2, are the eigenvalues of the matrix of the ternary Dik coefficients. W1 and 1 − W1 are the normalized pre-exponential factors. In these experiments, small volumes of ΔV of solution, of ― ― composition  c1 + Δc1 and  c2 + Δc2 are injected into carrier solutions of composition, c̄1, c̄2 at time t = 0. 3. Experimental results and discussion In Tables 1–3 some differential isothermal diffusion coefficients of various electrolytes (1:1, 2:2, 2:1 and 3:1), measured using the open ended conductometric capillary cell, are given. The mutual differential diffusion coefficients shown in these tables are average results of, at

where kb is the Boltzmann's constant, An is a function of the static dielectric permittivity, and the viscosity of the solvent, of temperature, and of a dimensionless concentration-dependent quantity (κa), where κ is the reciprocal of average radius of the ionic atmosphere and a is a distance parameter (defined by Robinson and Stokes [36], such as the distance within which no other ions can penetrate and within the electrophoretic velocity remains constant); t01 and t02 are the limiting transport numbers of the cation and anion, respectively. Since the expression for the electrophoretic effect has been derived on the basis of the expansion of the exponential Boltzmann function because that function had been consistent with the Poisson equation, we only, in major cases, would have to take into account the electrophoretic term of the first order (n=1). For symmetrical electrolytes we can consider the second term. Thus, the experimental data can be compared with the calculated D on the basis of Eqs. (10) and (11)   0 D = D + Δ1 + Δ2 ½1 + cðd lnγ = dcÞ

ð10Þ

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61

Table 1 Mutual differential diffusion coefficients, D, [3] of some electrolytes, 1:1 and 2:2, at 25 °C. Concentration, c, is given in molarity, mol dm− 3, and the diffusion coefficient should be read as D / (10− 9 m2 s− 1). (Binary systems). BeSO4

c D c D c D c D c D c D c D c D c D c D c D c D c D c D c D c D

CdSO4

HCl KCl

KClO4 KSCN

LiCl LiClO4

MnSO4 NaC2H3O8 NH4VO3

0.005 1.309 0.5 0.668 0.001 0.648 0.08 0.543 0.005 3.218 0.001 1.964 0.5 1.867 0.01 1.827 0.001 1.800 0.2 1.748 0.1014 1.294 0.001 1.208 0.1 1.333 0.03 0.565 0.003 1.147 0.001 1.264

0.008 0.936

0.01 1.043

0.03 0.805

0.05 0.816

0.08 0.736

0.1 0.778

0.2 0.742

0.3 0.718

0.002 0.642 0.1 0.523 0.008 3.177 0.002 1.983 1.0 1.891 0.02 1.798 0.005 1.756 0.5 1.734 0.2000 1.288 0.002 1.277

0.003 0.635

0.005 0.628

0.008 0.623

0.01 0.602

0.02 0.591

0.03 0.588

0.05 0.550

0.001 3.165 0.003 1.985

0.02 3.149 0.005 1.963

0.03 3.116 0.008 1.945

0.05 3.060 0.01 1.891

0.08 3.031 0.05 1.893

0.1 3.017 0.1 1.894

0.2 1.830

0.03 1.763 0.008 1.746 1.0 1.740 0.2976 1.285 0.003 1.292

0.08 1.676 0.010 1.736

0.1 1.633 0.02 1.725

0.03 1.714

0.05 1.739

0.08 1.764

0.1 1.743

0.6971 1.312 0.005 1.312

0.9961 1.322 0.008 1.298

0.02 1.313

0.03 1.362

0.05 1.343

0.08 1.360

0.05 0.525 0.008 1.106 0.005 1.174

0.08 0.525 0.01 1.102 0.010 1.135

0.1 0.501 0.02 1.091 0.020 1.116

0.03 1.087 0.030 1.110

0.05 1.073 0.050 1.090

0.08 1.076

0.1 1.096

  0 D = D + Δ1 ½1 + cðd lnγ = dcÞ

ð11Þ

for symmetrical and non-symmetrical electrolytes, respectively.

The theory of mutual diffusion in binary electrolytes, developed by Pikal [39], includes the Onsager–Fuoss equation, but has new terms resulting from the application of the Boltzmann exponential3. The electrophoretic correction appears as the sum of two terms L

Table 2 Mutual differential diffusion coefficients, D, [3] of some electrolytes 2:1 at 25 °C. Concentration is given in molarity, mol dm− 3, and the diffusion coefficient should be read as D / (10− 9 m2 s− 1). BaBr2

Ba(ClO4)2 Ca(NO3)2

CdBr2 CdCl2

CdI2 Cd(NO3)2 CoCl2 Mg(NO3)2 MnCl2 MnSO4 NiCl2

c D c D c D c D c D c D c D c D c D c D c D c D c D c D D D

0.001 0.842 0.080 1.131 0.005 1.265 0.001 1.164 0.1 0.973 0.001 1.337 0.001 1.139 0.080 0.921 0.001 1.203 0.001 1.035 0.001 1.330 0.001 1.118 0.001 1.032 0.03 0.565 1.032 1.114

0.002 0.893 0.100 1.121 0.008 1.240 0.002 1.191

0.003 0.005 0.008 0.010 0.020 0.030 0.050 0.846 0.823 0.886 0.840 0.948 1.189 1.142

0.01 1.221 0.003 1.156

0.02 1.148 0.005 1.144

0.03 1.148 0.008 1.090

0.05 1.129 0.01 1.086

0.002 1.220 0.002 1.101 0.100 0.902 0.002 1.194 0.002 1.020 0.003 1.320 0.003 1.155 0.002 0.994 0.05 0.525 0.994 1.112

0.003 0.995 0.003 1.091

0.005 0.890 0.005 1.048

0.010 0.905 0.008 1.027

0.020 0.954 0.010 0.020 0.030 0.050 1.025 1.003 0.982 0.946

0.003 1.186 0.003 0.993 0.005 1.318 0.005 1.166 0.003 0.991 0.08 0.525 0.991 1.083

0.005 1.075 0.005 0.971 0.008 1.290 0.008 1.145 0.005 0.988 0.01 0.501 0.988 1.074

0.008 0.933 0.008 0.951 0.01 1.269 0.01 1.084 0.008 0.974

0.010 0.869 0.020 0.970 0.05 1.217 0.02 1.059 0.01 0.976

S

Δνj = Δνj + Δνj

where ΔvLj represents the effect of electrostatic interactions of longrange, and ΔvjS represents them as short-range. Designating the solute thermodynamic mobility M (Pikal [39]) by 12

0.08 1.062 0.02 1.032

0.10 1.048 0.03 0.08 1.020 0.983

M = 10 L = c

0.2 1.091 0.08 1.052

0.3 1.086 0.1 1.020

0.974 0.976 0.050 0.080 0.100 1.059 1.044 1.012 0.974 0.953

ð13Þ

where L is the thermodynamic diffusion coefficient, ΔM can be represented by the equation     0 0 1− ΔM = M ð1 = MÞ = 1 = M

ð14Þ

where M0 is the value of M for infinitesimal concentration, and ΔM = ΔM

0.1 1.108 0.05 1.080

ð12Þ

0F

+ ΔM1 + ΔM2 + ΔMA + ΔMH1 + ΔMH2 + ΔMH3 ð15Þ

The first term on the right hand in Eq. (15), ΔM0F, represents the Onsager–Fuoss term for the effect of the concentration in the solute thermodynamic mobility, M; the second term, ΔM1, is a consequence of the approximation applied on the ionic thermodynamic force; the other terms result from the Boltzmann exponential function (see page 9).

3 These equations are adequately described in the literature [39], and consequently we only report a summary concerning the relevant points of this model.

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Table 3 Mutual differential diffusion coefficients of some electrolytes 3:1, at 25 °C. Concentration is given in molarity, mol dm− 3, and the diffusion coefficient should be read as D / (10− 9 m2 s− 1) [3]. Al(NO3)3 c D

0.002 0.885

0.003 0.897

0.008 0.908

0.020 0.901

0.030 0.885

0.050 0.862

0.08 0.845

0.1 0.827

The relation between the solute thermodynamic mobility and the mutual diffusion coefficient is given by 3

D = ðL = cÞ 10 RTν½1 + cðd lnγ = dcÞ

ð16Þ

From Eqs. (16) and (13), a new version of Pikal's equation (Eq. (17a) and (17b)) is obtained D=

1 M0

  1   103 RTν ½1 + cðd lnγ = dcÞ ΔM 1− M0

ð17aÞ

The applicability of these models to our experimental curves for different electrolytes in dilute aqueous solutions is reported in the literature [1,3]. We see that for symmetrical uni–univalent electrolytes, both theories give similar results, consistent with the experi-

mental ones. In fact, if Pikal's theory is valid, ΔM0F must be the major term, all other terms being much smaller, partially cancelling each other. Concerning symmetrical, but polyvalent electrolytes, we can well see that Pikal's theory is a better approximation than the Onsager–Fuoss theory. In polyvalent non-symmetrical electrolytes, agreement between experimental data and Pikal calculations is not so good, eventually because of the full use of Boltzmann's exponential in Pikal's development. In conclusion, no theory on diffusion in electrolyte solutions is capable of giving generally reliable data concerning the magnitude of the diffusion coefficient, D. However, for estimating purposes when no experimental data are available, we can use the Onsager–Fuoss equation (Eq. (10)) for symmetrical uni–univalent electrolytes (1:1) in aqueous dilute solutions (or Pikal's Eq. (16), once the results are similar). For symmetrical polyvalent electrolytes, we use the Pikal equation Eq. (16). For non-symmetrical electrolytes, in general, we can see a reasonable agreement between our results and those obtained by Eq. (11). For higher concentrations (c N 0.01 M), our experimental data and those from other researchers are in good agreement, but the results from the above models differ considerably from experimental observation (e.g. see [11]). This is not surprising if we take into account the formation of complexes, the change with concentration of parameters such as viscosity, mean distance of closest approach of ions [40], and static dielectric permittivity and hydration, factors not taken into account in Onsager–Fuoss' and Pikal's models.

Table 4 Fitting coefficients (a0 to a4) of a polynomial equation to the mutual differential diffusion coefficients of different electrolytes at 25 °C. The diffusion coefficients should be read as: D/ (m2 s− 1) = [a0/(m2 s− 1)] + [a1/(m2 s− 1 M− 1/2)] (c / M)1/2+[a2/(m2 s− 1 M− 1)]](c / M) + [a3/(m2 s− 1 M− 3/2](c / M)3/2+[a4/(m2 s− 1 M− 2] (c / M)2, being M the molarity (mol dm− 3) [3]. Electrolyte

R2

a0

a1

a2

a3

a4

Al(NO3)3 BaBr2a BaBr2b Ba(ClO4)2 BeSO4 Ca(NO3)2 CdBr2 CdCl2 CdI2 Cd(NO3)2 CdSO4 CoCl2 CsI HCl KClc KCld KClO4 KSCNe KSCNf LiCl LiClO4g LiClO4h Mg(NO3)2i Mg(NO3)2j MnCl2 MnSO4 NaC2H3O8 NiCl2 NH4VO3

1.00 1.00 1.00 0.98 0.98 0.98 0.98 0.99 1.00 0.99 1.00 0.98 0.99 0.99 0.99 1.00 1.00 0.98 1.00 1.00 0.98 1.00 1.00 1.00 1.00 1.00 0.99 0.99 1.00

7.965e−10 − 1.685e−9 − 1.436e−8 1.629e−9 2.024e−9 1.193e−9 8.432e−10 1.259e−9 1.285e−9 9.657e−10 8.755e−10 1.350e−9 2.052e−9 3.450e−9 1.323e−9 1.895e−9 1.480e−9 1.824e−9 1.644e−9 1.168e−9 1.997e−9 2.780e−9 7.909e−10 1.723e−9 1.642e−9 2.132e−9 1.309e−9 1.203e−9 1.399e−9

3.004e−9 1.829e−7 2.689e−7 − 7.999e−9 − 1.403e−8 2.029e−10 5.381e−8 − 4.840e−9 − 1.140e−8 6.724e−9 3.317e−9 − 0.497e−9 − 2.762e−9 − 5.681e−9 4.471e−8 − 8.165e−11 9.227e−9 − 5.707e−10 5.866e−10 1.123e−9 − 9.577e−9 1.280e−8 2.207e−8 − 1.360e−8 − 4.018e−8 − 1.950e−8 − 4.651e−9 − 2.768e−9 − 4.175e−9

− 2.627e−8 − 4.619e−6 − 1.707e−6 5.227e−8 5.518e−8 − 2.207e−8 − 1.683e−6 3.549e−8 4.816e−7 − 1.934e−7 − 8.062e−8 1.808e−9 0 4.276e−8 − 1.091e−6 5.900e−10 − 8.219e−8 − 9.324e−9 − 1.031e−9 − 3.456e−9 1.257e−7 5.213e−8 − 5.395e−7 9.832e−8 9.118e−7 7.852e−8 3.772e−8 2.194e−8 2.766e−8

8.615e−8 4.886e−5 4.712e−6 − 1.575e−7 − 9.033e−8 1.147e−7 1.697e−5 − 1.238e−7 − 7.555e−6 1.764e−6 5.130e−7 4.267e−9 0 − 1.588e−7 1.145e−5 − 1.321e−9 2.808e−7 7.695e−8 5.411e−10 4.237e−9 − 7.244e−7 − 6.998e−8 6.200e−6 − 2.931e−7 − 8.986e−6 − 1.049e−7 − 1.376e−7 − 8.387e−8 − 5.596e−8

− 1.016e−7 − 1.845e−4 − 4.788e−6 1.700e−7 5.167e−8 − 1.707e−7 − 5.465e−5 1.541e−7 3.464e−5 − 5.174e−6 − 1.054e−6 − 1.647e−9 0 2.110e−7 − 4.441e−5 8.093e−10 − 3.426e−7 − 1.400e−7 0 − 1.750e−9 1.576e−6 0 − 2.718e−5 3.035e−7 3.219e−5 0 1.837e−7 1.084e−7 0

a b c d e f g h i j

0.001 ≤ c ≤ 0.01. 0.02 ≤ c ≤ 0.1. 0.001 ≤ c ≤ 0.01. 0.01 ≤ c ≤ 1.0. 0.001 ≤ c ≤ 0.1. 0.1 ≤ c ≤ 1.0. 0.001 ≤ c ≤ 0.03. 0.03 ≤ c ≤ 0.1. 0.001 ≤ c ≤ 0.01. 0.01 ≤ c ≤ 0.1.

A.C.F. Ribeiro et al. / Journal of Molecular Liquids 156 (2010) 58–64

63

Table 5 Ternary diffusion coefficients, D11, D12, D21 and D22, for aqueous HPβCD (1) + caffeine (2) solutions and the respective standard deviations, SD, at 25 °C, obtained by the Taylor technique and estimated by using our model.a c1b

c2b

(D11 ± SD)/ (10− 9 m2 s− 1)

(D12 ± SD)/ 10− 9 m2 s− 1)

(D21 ± SD)/ 10− 9 m2 s− 1)

(D22 ± SD)/ (10− 9 m2 s− 1)

0.0000

0.0000

− 0.0615 ± 0.002

− 0.036 ± 0.003

0.0000

0.0100

+ 0.010 ± 0.003

0.031 ± 0.003

0.0100

0.0000

0.323 ± 0.003 (0.322) 0.322 ± 0.004 (0.320) 0.321 ± 0.003 (0.322)

+ 0.020 ± 0.003

0.042 ± 0.003

0.0000 0.0000 0.0000 0.0000 0.0000 0.0005 0.0010 0.0025 0.0050 0.0100 0.0005

0.0005 0.0010 0.0025 0.0050 0.0100 0.0000 0.0000 0.0000 0.0000 0.0000 0.0005

0.760 ± 0.003 (0.764) 0.738 ± 0.003 (0.771) 0.600 ± 0.003 (0.438) 0.770c 0.760d 0.749c 0.738d 0.703d

− 0.003 ± 0.015

0.040 ± 0.025

0.0010

0.0010

− 0.0029 ± 0.014

0.050 ± 0.026

0.0025

0.0050

− 0.004 ± 0.004

0.067 ± 0.013

0.0050

0.0050

− 0.011 ± 0.006

0.054 ± 0.024

0.0050

0.0025

− 0.003 ± 0.013

0.051 ± 0.015

0.0100

0.0100

− 0.021 ± 0.016

0.103 ± 0.022

0.321e 0.321e 0.318e 0.314e 0.307f 0.305 ± 0.010 (0.322) 0.300 ± 0.015 (0.320) 0.316 ± 0.004 (0.320) 0.293 ± 0.006 (0.321) 0.292 ± 0.003 (0.321) 0.283 ± 0.009 (0.320)

0.744 ± 0.043 (0.715) 0.730 ± 0.033 (0.686) 0.676 ± 0.016 (0.660) 0.665 ± 0.013 (0.573) 0.690 ± 0.039 (0.573) 0.640 ± 0.036 (0.522)

The predicted values of D11 and D22 (using KD = 158 L mol− 1 [30]) obtained from the proposed model are indicated in parentheses. c1 and c2 are in units of mol dm− 3. c Taylor binary D value estimated for aqueous caffeine at 25 °C using the linear equation D / 10− 9 m2 s1 = D0 [1 + A(c/mol dm3)] (standard deviation b 1%) [30]. D0 is the diffusion coefficient at infinite dilution. d Taylor binary D value for aqueous caffeine at 25 °C [16]. e Taylor binary D value estimated for aqueous HP-β-CD [30 ] using the linear equation D / 10− 9 m2 s1 = D0 [1 + A(c/mol dm3)] (standard deviation b1%) and our data. f Taylor binary D value for aqueous HP-β-CD [30]. a

b

As an example of some data for ternary systems, Table 5 shows the mutual diffusion coefficients measured by Taylor dispersion method (D11, D22, D12 and D21) reported for aqueous solutions of 2hydroxypropyl-β-cyclodextrin(HPβCD) + caffeine at 25 °C at carrier concentrations from (0.000 to 0.010) mol dm3, for each solute, respectively [30]. From these data, and using a particular model [30], it was possible to estimate some parameters, such as the number of moles of each component transported per mole of the other component driven by its own concentration gradient, and the fraction of associated species (x1) and caffeine (x2) in this complex, the D monomer and dimmer fractions, xM 2 andx2 , respectively, and the 0 limiting diffusion coefficients of the HPβCD, DHPβCD , of the dimmers 0 caffeine entities, DD, and of the complexes (1:1) between HPβCD and caffeine, D0complex, with the stability constant K = 287 ± 60 mol− 1 dm+ 3 at 25 °C [30]. That is, 0

0

0

D11 = x1 Dcomplex + ð1−x1 ÞDHPβCD h i 0 0 0 0 D22 = x2 Dcomplex + ð1−x2 Þ DM Y + 2 ZDD

ð17bÞ ð18Þ

whereD011 andD022 are the experimental tracer ternary diffusion coefficients of HPβCD dissolved in supporting caffeine solutions, and the tracer ternary diffusion coefficients of caffeine in supporting HPβCD solutions, respectively, and Y and Z are given by the following Eqs. (19) and (20), M

Y = x2

ð19Þ

D

Z=

x2 2

ð20Þ

assuming M

D

x2 + x2 = 1

ð21Þ

Table 5 compares experimental results with calculations on the basis of the model here proposed, similar to the model proposed by Leaist [41]. From the present experimental conditions, i.e., for [HPβCD]/[caffeine] ratio values b 1, and for cases as c1 = c2 ≤ 5 × 10− 4 mol dm− 3, the experimental data and the results predicted from the above models are in reasonable agreement (that is, the deviations are in the range 2% to 3%). However, in general, at higher concentrations (for [HPβCD]/[caffeine] ratio values ≥ 1), our experimental data and the results from the above model [30] differ considerably (deviations between 4% and 20%). This is not surprising if we take into account all approximations involved in this model. For example, among them, we have the motion of the solvent and the change of parameters such as viscosity, static dielectric permittivity and change of hydration with concentration, neglected in dilute solutions, but that can be relevant for concentrated solutions. Similar conclusions have also been carried out by Albright, Miller, Vitagliano, Sartorio and Paduano [42–44]. In this context, taking now only in consideration the effect of the viscosity on main ternary diffusion coefficients for a similar system (that is, β-CD + caffeine + H2O [29]), we obtain more significant differences between theoretical results and our data, leading us to conclude that the viscosity factor seems to have a negligible role on the diffusion of this system. Thus, it is not possible to accurately know

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the reasons for deviations between our data and the predicted values, as we would desire. However, we could say that what eventually can be more important for some areas of interest (e.g. pharmaceutical applications) is the experimental thermodynamic behaviour of the involved species, not so much the complex question of interpretation of these data (such as the nature of their internal binding forces). 4. Conclusions Diffusion coefficients measured for aqueous solutions of systems involving electrolytes, drugs and cyclodextrins, provide transport data necessary to model the diffusion for various chemical and pharmaceutical applications. However, once more we conclude that theoretical approaches do not lead us to obtain useful values, mainly for very concentrated solutions, taking into account their deviations from the experimental results. Thus, the experimental work on the determination of mutual diffusion coefficients remains essential. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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