Soret Coefficients in Some WaterâMethanol, WaterâEthanol, and

11524

J. Phys. Chem. B 2007, 111, 11524-11530

Soret Coefficients in Some Water-Methanol, Water-Ethanol, and Water-Isopropanol Systems J. K. Platten,*,†,‡ M. M. Bou-Ali,‡ P. Blanco,‡ J. A. Madariaga,§ and C. Santamaria§ UniVersity of Mons-Hainaut, B-7000 Mons, Belgium, Manufacturing Department, Mondragon Unibertsitatea, Loramendi, 4. Apartado 23, 20500 Mondragon, Spain, and Departamento de Fı´sica Aplicada II, UniVersidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain ReceiVed: May 31, 2007; In Final Form: July 3, 2007

In this article, we re-examine the published experimental values of the Soret coefficients of a few wateralcohol systems at a mean temperature of 37.5 °C, because we feel that these published values (including microgravity values) are doubtful. The main technique applied is the use of thermogravitational columns to determine the thermodiffusion coefficient. The obtained values did not agree with the published values; worse, sometimes the sign was different.

Introduction When a binary mixture is subjected to a temperature gradient, molecular separation occurs, and this is known as the thermodiffusion or the Ludwig-Soret effect. In that case, the mass diffusion flux is given by1

J ) -FD∇c - FDTc0(1 - c0)∇T

(1)

where F is the specific mass (density), c is the mass fraction of the reference component (we always take the denser component as the reference, but the opposite choice could equally well be made) with initial (or mean) value equal to c0, D is the isothermal molecular diffusivity, DT is the thermodiffusion coefficient, and ∇c is the gradient of mass fraction induced by the temperature gradient ∇T. Under steady-state conditions (J ) 0), we have

∇c ) -

DT c (1 - c0)∇T D 0

(2)

and the ratio DT/D is called the Soret coefficient, ST. The Soret coefficient of the reference component can be positive or negative depending on the sense of migration of this component, which can go to the hot or to the cold. Its absolute value remains rather small and usually does not exceed 10-2 K-1. Therefore, the mass fraction gradient induced by the Soret effect remains small, but it has far-reaching implications in many situations ranging from the operation of solar ponds2 to the microstructure of the ocean3 and perhaps convection in stars4 or even mass transport across membranes induced by small thermal gradients in living matter where thermal diffusion could assume a sizable magnitude for an ensemble of cells with the dimensions of an organ or a tumor.5 As a supplementary example, several works have shown6,7 the importance of thermal diffusion in thermodynamic studies on the knowledge of the initial state of natural hydrocarbon reservoirs under gravitational and geothermal effects. The oil industry is particularly interested in multicom* To whom correspondence should be addressed. E-mail: Jean.platten@ skynet.be. † University of Mons-Hainaut. ‡ Mondragon Unibertsitatea. § Universidad del Paı´s Vasco.

ponent systems, but before starting experiments, even in a ternary system, one has to be sure of the correctness of the measured Soret coefficient in binary systems. The direct determination of ST based on eq 2 could be delicate, given that it implies the absence of convection, which is not always easy under terrestrial conditions (normal gravity), but convection can be avoided in well-conducted experiments.8 Alternatively, experiments in microgravity have been proposed and performed9 to alleviate the difficulty that could result from unexpected convective currents. Nevertheless, the accuracy of these microgravity experiments finally rests on the accuracy of the analysis on earth of the recovered samples sometime after several months of flight; thus, the postprocessing of the samples on earth is also crucial. On the other hand, instead of trying to avoid convection, convective coupling in thermogravitational cells (with a horizontal temperature gradient perpendicular to the gravity field, a situation that always induces convection) can be used to determine DT.10-13 A separate isothermal experiment could give the isothermal molecular diffusion coefficient D,10 and finally the ratio DT/D can be computed. In the summer of 1999, several research groups from different universities and research institutes met in Fontainebleau, France, in order to establish a reliable database of Soret, thermodiffusion, and isothermal diffusion coefficients, using different techniques such as thermogravitational cells and laser-Doppler velocimetry to analyze the influence of the Soret-induced solutal contribution to the buoyancy force and its consequence on velocity amplitudes, or very precise optical techniques such as thermal diffusion forced Rayleigh scattering (TDFRS). They used binary mixtures of dodecane, isobutylbenzene, and 1,2,3,4-tetrahydronaphtalene (components of interest in the oil industry), and because the values produced by each group did not differ too much from each other, the participating teams proposed “benchmark” values for the Soret coefficients and declined the need to go to microgravity at least for organic mixtures around room temperature.14-19 The present work is in the same spirit, but it concerns less exotic mixtures chosen from among systems that were already investigated in the microgravity environment.9 The article is organized as follows: In section 2, we discuss some systems and their properties. In section 3, we present the results for the thermodiffusion coefficients obtained from the separation in three different thermogravitational columns: one

10.1021/jp074206z CCC: $37.00 © 2007 American Chemical Society Published on Web 09/07/2007

Soret Coefficients in Some Water-Alcohol Systems

J. Phys. Chem. B, Vol. 111, No. 39, 2007 11525

parallelepiped column in Mons, Belgium, with a five-point sampling technique;10,17 a more conventional cylindrical column with mass fraction determination at the top and bottom of the column as performed by the group of Bilbao, Spain;11-13,18 and finally a cylindrical column built in Mondragon, Spain, (MGEP) to operate at high pressure20 (up to 500 bar) but used in this work at atmospheric pressure. Because the different experiments do not differ too much from each other, we propose benchmark values for the thermodiffusion coefficient. In section 4, we describe the determination of the isothermal molecular diffusivity in an independent experiment, and in section 5, we present new values of the Soret coefficients that should be considered, at least in our opinion, as benchmark values. 2. Investigated Systems and Their Properties In a previously published work on Soret coefficients of organic solutions,9 11 systems were investigated (Table 1 of ref 9) by three experimental techniques: the Thomaes flow cell method,21 the onset of convection in a Benard cell (see, e.g., ref 22), and the SCM microgravity approach. A rapid inspection of Table 1 of ref 9 shows an extraordinary discrepancy between the different results (that sometimes differ by a factor 10), where the microgravity values are claimed to be the “true” values, because the experiments are convection-free. However, the samples were recovered after a flight of 9 months and analyzed on the ground by densitometry. Densitometry is also indirectly used for the determination of the Soret coefficient based on the onset of free convection in a Rayleigh-Benard configuration. Indeed, the important parameter in such an experiment is the solutal contribution to the buoyancy, β∆c, compared to the thermal contribution, -R∆T (where β and R are the mass and the thermal expansion coefficient, respectively, of the solution). Moreover, because the ratio ∆c/∆T is given by eq 2, the socalled separation ratio β∆c/(-R∆T) is given by

Ψ)

β DT c (1 - c0) R D 0

(3)

The critical Rayleigh number for the onset of convection in a horizontal layer heated from below is a function of this separation ratio. For positive values of Ψ, the solutal and thermal contributions to the buoyancy (or to the density gradient) are of the same sign and add together. Because the temperature field is destabilizing, so is the solutal contribution, and convection starts earlier than without the Soret effect. Therefore, the critical temperature difference for the onset of convection is very small, and convection can be detected only by very accurate local measurements of the velocity field,23 e.g., by laser-Doppler velocimetry, and certainly not by old techniques that rely on global measurements such as the total heat flow (or Nusselt number), which is not substantially increased until the Rayleigh number reaches the classical value of 1708. When the separation ratio Ψ is negative, the two contributions are of opposite sign, and therefore, the mass fraction field is stabilizing (the denser component migrates to the hot wall, to the bottom of a Benard cell). In that case, the critical Rayleigh number becomes greater than its “one-component” value of 1708, as does the critical temperature difference, so that experiments based on the critical Rayleigh number become possible. However, in that case, we have an inverted Hopf bifurcation, characterized by its “Hopf frequency”, also related to the separation ratio. It has been known for a long time24-26,22 that the determination of the separation ratio based on the Hopf frequency is more accurate than that based on the critical Rayleigh number. However, one

has to be able to record the velocity field, say, every 2 s with a resolution of 5 µm/s to catch the oscillatory onset of free convection. Thus, the method based on heat flux diagrams (socalled Schmidt-Milverton plots27) was used in the past28 but cannot today compete with modern techniques. This is not the only reason why the results from the Benard cell presented in Table 1 of ref 9 are doubtful: In addition to the use of equations valid for stress-free boundary conditions and applied without any change to rigid boundaries even though it is well-known that empirical correction factors have to be introduced for both the variation of the critical Rayleigh number and the Hopf frequency,22,26 the experimental strategy relies on density measurements, because from eq 3, one must know the two expansion coefficients R and β to determine ST from Ψ. A few examples of expansion coefficients are given in Table 2 of ref 9. These results do not stand up to a critical survey. It is wellknown that the thermal expansion coefficient of pure water is 20.68 × 10-5 K-1 at 20 °C and increases with temperature, reaching 36.58 × 10-5 K-1 at 37.5 °C, the working temperature of the experiments reported in ref 9. These values were deduced from density measurements of pure water at different temperatures that are in all tables of numerical constants and data (e.g., the famous CRC tables). On the other hand, alcohols (at least those with a small number of carbon atoms) expand more than water. Therefore, by adding a small amount (say, 10 wt %) of isopropyl alcohol to water, how can one believe that the expansion coefficient will drop from the value of 36.58 × 10-5 K-1 for pure water to the value of 13 × 10-5 K-1 given in Table 2 of ref 9. In fact, the densities of mixtures of water and isopropyl alcohol were published at different mass fractions (in steps of 1 wt %) and at different temperatures between 0 and 30 °C and are now available on the Internet.29 An interpolation equation was derived for this system,30 and the deduced thermal expansion coefficient at 21 °C is 31.78 × 10-5 K-1. Because the mean temperature in the experiments reported in ref 9 was around 37 °C, the expansion coefficient should be greater than the latter value. In fact, the correct value is 45.37 × 10-5 K-1 (see Table 1). Thus, we cannot believe the reported value of 13 × 10-5 K-1. Therefore, we decided to re-examine the expansion coefficients around 37.5 °C for all of the systems listed in Table 2 of ref 9. Table 1 presents our results, also reported in Figure 1 for water-methanol as an example. The densities were measured with a well-calibrated quartz vibrating U-tube densimeter (model DMA5000) manufactured by Antoon PAAR, having an accuracy of 2 × 10-6 g/cm3. For the system water (10 wt %)-methanol (90 wt %), the densities given in the second column of Table 1 produce an expansion coefficient of 11.32 × 10-4 K-1. Thus, we cannot accept the published value in ref 9 of an expansion coefficient for the water (10 wt %)methanol (90 wt %) system thought to be equal to 8 × 10-5 K-1 instead of 11.32 × 10-4 K-1, i.e., a value that differs from ours by a factor of close to 15. This is also true for the other systems listed in Table 2 of ref 9, namely, the system water (97 wt %)-n-butyl alcohol (3 wt %) for which the reported value of the expansion coefficient is equal to 7 × 10-5 K-1. Indeed because the expansion coefficient of pure water is 36.58 × 10-5K-1 at 37.5 °C, how can one believe that it will drop to 7 × 10-5K-1 by adding 3 wt % n-butyl alcohol? Table 1 reports our measurements. In fact, all of the values reported in ref 9 implying density measurements are false, and because the analysis in the Soret experiment of the samples that were recovered after the flight were done by densitometry, we feel that it is necessary to reconsider the published values of the Soret coefficients. In

11526 J. Phys. Chem. B, Vol. 111, No. 39, 2007

Platten et al.

TABLE 1: Densities (in g/cm3) and Expansion Coefficients of Some Water-Alcohol Systems temp (°C)

water (10 wt %)methanol (90 wt %)

water (10 wt %)ethanol (90 wt %)

36.5 37 37.5 38 38.5 slope ∂F/∂T (g/cm3 K) expansion coefficient R ) -(1/F0)(∂F/∂T) (K-1)

0.805002 0.804550 0.804093 0.803638 0.803183 -0.0009100 11.32 × 10-4 a 8 × 10-5 b

0.803430 0.802980 0.802529 0.802078 0.801627 -0.00090160 11.23 × 10-4 a c

temp (°C)

water (97 wt %)n-butanol (3 wt %)

water (95 wt %)n-butanol (5 wt %)

36.5 37 37.5 38 38.5 slope ∂F/∂T (g/cm3 K) expansion coefficient R ) -(1/F0)(∂F/∂T) (K-1)

0.988596 0.988405 0.988216 0.988022 0.987828 -0.0003838 38.8 × 10-5 a 7 × 10-5 b

0.985432 0.985233 0.985030 0.984827 0.984621 -0.0004056 41.18 × 10-5 a 7 × 10-5 b

temp (°C) 36.5 37 37.5 38 38.5 slope ∂F/∂T (g/cm3 K) expansion coefficient R ) -(1/F0)(∂F/∂T) (K-1) a

Figure 2. Density of water-methanol as a function of the mass fraction in water.

TABLE 2: Properties of Water-Alcohol Systems (10 wt % in Water) at 37.5 °C mixture

water (90 wt %)water (80 wt %)isopropanol (10 wt %) isopropanol (20 wt %) 0.976561 0.976340 0.976121 0.975899 0.975674 -0.0004429 45.37 × 10-5 a 13 × 10-5 b

0.960115 0.959811 0.959503 0.959194 0.958887 -0.0006146 64.05 × 10-5 a 17 × 10-5 b

From the present work. b From ref 9. c Not reported in ref 9.

Figure 1. Density of the water (10 wt %)-methanol (90 wt %) system versus temperature around 37.5 °C.

particular, for the two systems of Table 1, at the same temperature of 37.5 °C and at the same mass fraction of water (10%), the published values of the Soret coefficients exhibit a change in sign, simply upon addition of one more carbon atom to the alcohol; this is hard to believe. The proposed microgravity (SCM) values in ref 9 are

water-methanol (10 wt % water at 37.5 °C): ST ) -25 × 10-3 K-1 water-ethanol (10 wt % water at 37.5 °C): ST ) +3.7 × 10-3 K-1 We understand that, for a given system, the Soret coefficient can change sign with concentration, and at a given concentration, it can even change sign with temperature, but here, we are

property



water (90 wt %)isopropanol (10 wt %)

R (10-4 K-1) β ν (10-6 m2/s) F0 (g/cm3)

11.32 0.351 0.79 0.804093

11.23 0.329 1.37 0.802529

4.54 0.172 1.05 0.976121

operating at the same mass fraction and at the same temperature. Moreover, a value of ST ) -25 × 10-3 K-1 seems (in absolute value) to be very large for an organic mixture. We also decided to investigate the system water (90 wt %)-isopropanol (10 wt %) at 37.5 °C, a third system published in ref 9. This system already has been investigated under standard gravitational conditions in the past24,31 around room temperature (with some 30-40% discrepancy in the reported results). However, one factor that seems firmly established is the negative sign of the Soret coefficient. Thus, we believe that ST is also negative at 37.5 °C. In any case, we give in Table 1 the densities and thermal expansion coefficients for the water-isopropyl alcohol systems. In addition to a precise knowledge of the thermal expansion coefficient R, we also must know the mass (or solutal) expansion coefficient β. To this end, we measured the densities of samples having a composition close to the mean density c0, i.e., 1 and 2 wt % below and above c0. Samples were prepared by weighing given amounts of the pure substances, which were then mixed together. An example of the variation of the density F of the mixture with the mass fraction c of water is given in Figure 2, and from the slope, we get the mass expansion coefficient β ) (1/F0)(∂F/∂c), given in Table 2. Finally, as we shall see in the next section, in the frame of the use of thermogravitational columns (see eq 4 below), we also need to know the kinematic viscosity ν ) µ/F. The dynamic viscosity µ was measured by a standard falling-ball (Ho¨ppler) viscosimeter with an accuracy of 1%. Because the densities were measured, we report kinematic viscosities in Table 2. These values were re-examined using of an Ubbeholde viscosimeter in order to certify the values of Table 2. 3. Thermodiffusion Coefficients As already stated in the Introduction, thermodiffusion coefficients were obtained from separations in different types of


J. Phys. Chem. B, Vol. 111, No. 39, 2007 11527

thermogravitational columns: a parallelepiped column in Mons, Belgium, with the five-point sampling technique;10 a more conventional cylindrical column with mass fraction determination at the top and at the bottom of the column performed by the Bilbao (Spain) group;11-13 and finally, the cylindrical Mondragon (Spain) (MGEP) column also with five-point sampling. In all cases, the thermodiffusion coefficient DT was deduced from Furry-Jones-Onsager-Majumdar theory (see the working equation in refs 10 and 11-13, where the references to the original works are given). For convenience, let us also recall here the fundamental working equation

∆c ν 1 ) 504 DTc0(1 - c0) 4 H Rg e

(4)

where ∆c is the mass fraction difference over the column length H, c0 is the mean value of c, e is the gap width, R is the thermal expansion coefficient, ν is the kinematic viscosity, and DT is the thermodiffusion coefficient that we want to measure. In Mons, Belgium, we used a parallelepiped column (H ) 53 cm, e ) 1.50 mm) with five sampling positions along the column height and analyzed the five removed samples by measuring the density. Thus, we obtained the vertical density gradient ∆F/H. We need, of course, a calibration curve for transforming densities into mass fractions, so we measured F(c) around c0 as explained in section 2 and were able to calculate the mass expansion coefficient β ) (1/F0)(∂F/∂c). By dividing this density gradient ∆F/H by F0β, we obtained the mass fraction gradient ∆c/H needed in eq 4. In Bilbao, Spain, two columns with concentric cylinders were used (H ) 50 cm, e ) 1.02 mm or H ) 52 cm, e ) 1.93 mm). Samples were taken at the top and bottom of the columns, and the removed samples were analyzed by taking their index of refraction n with a Pulfrich refractometer having a resolution of 10-5. Thus, ∆n/H was known, and by dividing this quantity by ∂n/∂c as obtained from the calibration curve n(c), the same mass fraction gradient ∆c/H was found. In Mondragon, Spain, the dimensions of the cylindrical column were H ) 50 cm and e ) 1.00 mm, and the five removed samples were analyzed by densitometry. Thus, different columns and techniques were employed in different laboratories, so that, if the obtained results agree with each other, then one can have confidence in the proposed values. Before presenting the results, we would like to add some comments on the difficulties encountered during this research. Both the water (10 wt %)-ethanol (90 wt %) and water (10 wt %)-methanol (90 wt %) systems have indexes of refraction that exhibit a maximum in the vicinity of the particular composition that we wanted to investigate. In other words, the index of refraction of the samples was not very sensitive to the small variations of c between the top and bottom of the column. Therefore, we had to increase the difference in mass fraction c in order to increase the difference in th refraction index ∆n, and according to eq 4, this is made possible by the use of a small-gap column (e ≈ 1 mm), resulting in a mass fraction difference 16 times greater than the one we would have had using the usual gap of ∼2 mm because the separation behaves as 1/e4. Note here that, for the same reason (small value of ∂n/∂c) the TDFRS technique, successfully applied to organic systems,14-16 would have a poor accuracy in the present case. This also is true for any other technique based on the variation of the index of refraction, such as the beam-bending technique.26 This difficulty does not exist when samples are analyzed by densitometry. A second difficulty was encountered for the water (90 wt %)-isopropanol (10 wt %) system, where we expected a

Figure 3. Variation of density with elevation in the parallelepiped thermogravitational column for water-methanol (10 wt % in water) at 37.5 °C.

negative Soret coefficient for water, our reference component. According to eq 2, when DT < 0, ∇c and ∇T have the same sign: c increases when T increases. Therefore, the reference component (the denser component, water) migrates to the hot wall and, in thermogravitational columns, is then advected to the top of the column, creating a potentially unstable vertical density stratification in the gravity field. In cylindrical columns, it has been experimentally shown32 that an adverse concentration gradient can be sustained if the imposed temperature gradient (or its dimensionless equivalent, the Grashof number) exceeds some critical value. In contrast to the behavior of cylindrical columns, in parallelepiped columns, we were never able to sustain an adverse concentration gradient, regardless of the imposed temperature difference.33 These experimental findings, raising an interesting stability problem that is far from being resolved, are not absurd owing to the fact that the conservation equations are quite different in cylindrical and Cartesian coordinates (particularly the Laplace operator), together (probably) with the most dangerous mode of instability, which could not be the same in both systems of coordinates. Therefore, the results for the water (90 wt %)-isopropanol (10 wt %) system are only given using the cylindrical column. Note that, for this last system, we were considering the low-alcohol-content case, far from any possible maximum in the index of refraction, and therefore, the sensitivity of the technique allowed for smaller separations and the use of a larger-gap column (∼2 mm). Moreover, for stability reasons, this inverse density gradient had to remain small in order to operate in the stable region, and this imposed the use of large-gap columns. Also, the TDFRS technique continued to work.34 In Figures 3 and 4, we give an example of the variation of density with elevation along the parallelepiped column for water-methanol (10 wt % in water) at 37.5 °C and for waterethanol (10 wt % in water) at the same temperature. From the linear dependence F(z), we obtained ∆F/H. Together with the mass expansion coefficient β given in Table 2, we deduced ∆c/H and, finally, from eq 4, DT. The values of the kinematic viscosities are given in Table 2. In fact, each experiment was repeated several times with the same accuracy as in Figures 3 and 4, and we give only the mean value of DT in Table 4 below. The experiments performed in Bilbao, Spain, with the smallgap (1.02-mm) cylindrical column are summarized in Table 3a,b for the water (10 wt %)-ethanol (90 wt %) and water (10 wt %)-methanol (90 wt %) systems (the water-isopropanol

11528 J. Phys. Chem. B, Vol. 111, No. 39, 2007

Platten et al. TABLE 5: Water (90 wt %)-Isopropanol (10 wt %): Steady Separation for Various Values of ∆Ta

a

∆T (K)

∆c × 103

10 15 20 25 30

1.15 1.88 2.72 2.51 3.04

Mean value of ∆c for ∆T ) 20, 25, and 30 K: 2.76 ( 0.28.

Figure 4. Variation of density with elevation in the parallelepiped thermogravitational column for water-ethanol (10 wt % in water) at 37.5 °C.

TABLE 3: Results Obtained with the Small-Gap (1.02-mm) Cylindrical Column for (a) Water (10 wt %)-Ethanol (90 wt %) and (b) Water (10 wt %)-Methanol (90 wt %)a (a) Water (10 wt %)-Ethanol (90 wt %) run

∆n × 105

1 2 3 4

12.3 11.1 10.9 12.9

b

∆t

∆T (K)

2 h 30 2 h 30 5 h 00 2 h 30

10 10 10 10

(b) Water (10 wt %)-Methanol (90 wt %)c run

∆n × 105

1 2 3

34.0 33.4 32.4

∆t 9 h 00 3 h 30 7 h 00

∆T (K) 5 10 10

Mean temperature, 37.5 °C; ∆T, imposed temperature difference; ∆t, time before sampling; ∆n, difference in index of refraction; and ∆c, mass fraction between top and bottom sampling ports. b Mean value of ∆n (×105) ) 11.8. Calibration value ∂n/∂c ) -1.55 × 10-2. Mean value of ∆c ) 0.761 × 10-2. c Mean value of ∆n (×105) ) 33.3. Calibration value ∂n/∂c ) -4.42 × 10-2. Mean value of ∆c ) 0.753 × 10-2. a

TABLE 4: Values of DT in K-1 m2 s-1

Bilbao Mons MGEP mean value from TDFRS34



6.06 × 10-12 7.14 × 10-12 6.59 × 10-12 (6.60 ( 0.5) × 10-12 (5.4 ( 0.3) × 10-12

3.63 × 10-12 4.00 × 10-12 3.68 × 10-12 (3.77 ( 0.2) × 10-12 (4.0 ( 1.8) × 10-12

water (90 wt %)isopropanol (10 wt %) -8.4 × 10-12 instability problem instability problem -8.4 × 10-12 (-7.8 ( 0.4) × 10-12

system requires a separate comment). In Table 4, we give the mean values of DT from each laboratory (Bilbao, Spain; Mons, Belgium; and Mondragon, Spain), together with the mean values from the three laboratories, and we compare with those from the TDFRS technique34 (even if the accuracy is poor). We consider the mean values given in Table 4 as “benchmark values”. Concerning the water (90 wt %)-isopropanol (10 wt %) system, as already mentioned before, we used a 1.93-mm column. The experiment was repeated for several temperature differences ∆T until the separation became independent of ∆T

Figure 5. Sketch of the OEC technique.

within experimental errors. Recall that a stable adverse vertical concentration gradient is obtained for a Grashof number larger than some critical value.32 Table 5 shows the separations for five temperature differences: For ∆T > 20 °C, the stationary separation ∆c becomes independent of ∆T. Taking a mean value of ∆c ) (2.76 ( 0.28) × 10-3, we can evaluate DT reported in Table 4. The value that we found is compared to that deduced from TDFRS experiments (see Table 5.5 of ref 34). Because the difference is 7%, we believe that the present result is quite acceptable. 4. Isothermal Diffusion Coefficients For the two alcohol-rich systems (10 wt % water-90 wt % alcohol), the isothermal diffusion coefficients were measured by the so-called “open-ended capillary (or tube)” (OEC) technique, which is very easy to set up, inexpensive, rather accurate, and explained in detail and used for producing benchmark values for organic systems.10,14 Recall here that a tube of given length L (see Figure 5), initially containing a mixture with initial mass fraction of water c0 (in the present case, c0 ≈ 0.12), is immersed at time t ) 0 in an infinite bath having a smaller water mass fraction c∞ ≈ 0.08 (the reason for this is to avoid problems with free convection by putting a heavier mixture in the tube). By infinite, we mean a vessel having a volume much greater than that of the tube, such that, during the diffusion process (when water diffuses outside the tube and alcohol inside the tube), the composition of the bath remains almost constant. The mass fraction of water, averaged over the whole tube length 〈c(t)〉 is followed in time, and this gives the diffusion coefficient D corresponding to the mean mass fraction of water of 0.10. The diffusion process stops when the mass fraction of water in the tube approaches 0.08. Under such circumstances, the solution of Fick’s law, given in detail in ref 10, is

ln

{

}

π2[〈c(t)〉 - c∞] 8(c0 - c∞)

)-

π2 Dt 4L2

(5)

In fact, the mass fraction evolution was not followed in a single tube; at time t ) 0, 14 tubes (of 2.26 cm3 mean volume) were immersed in two double-jacket thermostated vessels (7 per vessel), each having a volume of 600 cm3. These vessels were, of course, covered to avoid evaporation. At given time intervals,


J. Phys. Chem. B, Vol. 111, No. 39, 2007 11529 the values of D in Table 6. Also, we added the available results from the literature for the studied systems at 37.5 °C. We did not feel it necessary to add results around room temperature, because the Soret coefficient is temperature-dependent and one could always argue that “discrepancies” are due to a different mean temperature. Reference 9 includes the SCM (Soret coefficient in microgravity) results, together with ground-based measurements. The additional work (ref 35) concerns experiments performed in a horizontal Soret cell, packed with a porous medium. Transport coefficients are sensitive to the packing, but because the Soret coefficient is a ratio of two transport coefficients, it is likely that the corrections in both DT and D (e.g., the square of the tortuosity) cancel, as experimentally shown for a particular example of an organic mixture.36 Therefore, the value reported in ref 35 could serve as a possible comparative value in the discussion.

Figure 6. Determination of the isothermal diffusion coefficient. Time variation of -ln[〈F(t)〉 - F0] for the water (10 wt %)-ethanol (90 wt %) system at T ) 37.5 °C.

TABLE 6: Values of D in m2 s-1 system

D (m2 s-1)

water (10 wt %)-methanol (90 wt %) water (10 wt %)-ethanol (90 wt %)

35.1 × 10-10 13.9 × 10-10

a tube was removed from a particular vessel, and its concentration was analyzed by densitometry. In contrast to what has been written before,10 there is even no need to have a very precise calibrating curve, because from eq 5, we need the ratio of two mass fraction differences. When the density F is linearly related to the mass fraction c (F ) A + Bc) as shown in Figure 2, eq 5 can be rewritten as

ln

{

}

π2[〈F(t)〉 - F∞] 8(F0 - F∞)

)-

π2 Dt 4L2

(6)

where the indices 0 and ∞ have the same meaning as for c. Even better, dropping all of the unnecessary constants, eq 6 can be written as

ln[〈F(t)〉 - F∞] ) -

π2 Dt + constant 4L2

(7)

showing that one has only to plot the difference between the mean density in the tube and that of the surrounding liquid (which represents the driving force) or its logarithm versus time (see Figure 6 as an example). From the slope of such a graph and the mean value of the tube length (58.7 mm ( 0.25), we can obtain the isothermal diffusion coefficient, as reported in Table 6. 5. Soret Coefficients The Soret coefficients, ST ) DT/D, given in Table 7 were obtained by taking the values of DT in Table 4 and dividing by

6. Conclusions There is certainly not a universal technique for measuring a Soret coefficient, and preferably, different techniques and analysis methods should be applied to obtain reliable data. In this respect, techniques that employ variations of the index of refraction should be avoided near a maximum of the index of refraction with the composition of the system, unless the experiment is conducted in such a way to have a sufficient separation. This is achieved by the use of small-gap columns when analyzing the samples by refractometry, whereas larger gaps (and smaller separations) can be used when analyzing the samples by densitometry. Also, when suspected, instability problems should be avoided. Considering to the stability diagram in the solutal Rayleigh number-Grashof number plane, published in ref 32, one must conduct experiments in the stable region, or in other words, one must use small solutal Rayleigh numbers (small separations, achieved by the use of large-gap columns, i.e., 1.93 mm in this work) and large Grashof numbers (or large temperature differences, larger than 20 °C; see Table 5). Therefore, all of the results presented in Table 4 represent a compromise between simplicity, efficiency, and accuracy. Even if there is a difference between results originating from different laboratories using different techniques and/or analysis methods, we believe that the mean values given in that Table 4 can be regarded as safe. The main result of the present work is perhaps to have demonstrated that the previously published results in ref 9 are false and that there is no change of sign in the Soret coefficient when going from the water-methanol to the water-ethanol system at the same mass fraction and temperature. For the water-isopropanol system, the proposed value of ST ) -8.32 × 10-3 K-1 relies only on cylindrical thermogravitational columns and is not too far from that obtained in the completely different TDFRS technique or in the elmentary Soret cell filled with a porous medium. These differences are quite acceptable. On the other hand, the discrepancy for this system between the values published in ref 9 ranging from -2.2 × 10-3 to -21 × 10-3 K-1 is completely unacceptable.

TABLE 7: Values of the Soret Coefficients in K-1 at 37.5 °C water (10 wt %)methanol (90 wt %) proposed values (this work) Table 1 of ref 9 Table 5.5 of ref 34 Table 6 of ref 35 a


water (90 wt %)isopropanol (10 wt %)

1.88 × 10-3

2.71 × 10-3

-8.32 × 10-3 a

from -1.4 × 10-3 to -26 × 10-3 4.27 × 10-3

from 3.7 × 10-3 to 4.0 × 10-3 3.0 × 10-3

from -2.2 × 10-3 to -21 × 10-3 -7.67 × 10-3 -7.5 × 10-3

We used the value of D from Table 5.5 of ref 34.

11530 J. Phys. Chem. B, Vol. 111, No. 39, 2007 Acknowledgment. This work presents results partly obtained within the framework of the following projects: GOVSORET (Departamento de educacioń Universidades e investigacioń del GV), SORETAQUI (Programa de cooperacioń Euskadi-Aquitania del Departamento de presidencia, del GV), TERMOFLU (Programa de Ciencia, Tecnologıá e Innovacioń de la Diputacioń Foral de Guipu´zcoa), and TESBLUR (Plan nacional e I+D+I del Ministerio de Educacioń y Ciencia del Gobierno Espanõl). References and Notes (1) de Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; North-Holland Publishing Co.: Amsterdam, 1962. (2) Weinberger, W. Solar Energy 1964, 8, 45. (3) Gregg, M. Sci. Am. 1973, 228, 64. (4) Spiegel, E. A. Annu. ReV. Astron. Astrophys. 1972, 10, 269. (5) Bonner, F. J.; Sundelo¨f, L. O. Z. Naturforsch. 1984, 39 (6), 656. (6) Montel, F. Entropie 1994, 184/185, 86. (7) Montel, F. Entropie 1998, 214, 7. (8) Wiegand, S.; Ko¨hler, W. Thermal Nonequilibrium Phenomena in Fluid Mixtures; Ko¨hler, W., Wiegand, S., Eds.; Lecture Notes in Physics; Springer: Berlin, 2002; Vol. 584, p 189. (9) Van Vaerenbergh, S.; Legros, J. C. J. Phys. Chem. B 1998, 102, 4426. (10) Dutrieux, J. F.; Platten, J. K.; Chavepeyer, G.; Bou-Ali, M. M. J. Phys. Chem. B 2002, 106, 6104. (11) Ecenarro, O.; Madariaga, J. A.; Navarro, J.; Santamaria, C.; Carrion, J. A.; Saviron, J. M. Macromelecules 1994, 27, 4968. (12) Bou-Ali, M. M.; Ecenarro, O.; Madariaga, J. A.; Santamaria, C.; Valencia, J. J. Phys.: Condens. Matter 1998, 10, 3321. (13) Bou-Ali, M. M.; Ecenarro, O.; Madariaga, J. A.; Santamaria, C.; Valencia, J. J. Non-Equilib. Thermodyn. 1999, 24, 228. (14) Platten, J. K.; Bou-Ali, M. M.; Costese`que, P.; Dutrieux, J. F.; Ko¨hler, W.; Leppla, C.; Wiegand, S.; Wittko, G. Phil. Mag. 2003, 83 (1718), 1965. (15) Wittko, G.; Ko¨hler, W. Phil. Mag. 2003, 83 (17-18), 1973. (16) Leppla, C.; Wiegand, S. Phil. Mag. 2003, 83 (17-18), 1989. (17) Platten, J. K.; Bou-Ali, M. M.; Dutrieux, J. F. Phil. Mag. 2003, 83 (17-18), 2001.

Platten et al. (18) Bou-Ali, M. M.; Valencia, J. J.; Madariaga, J. A.; Santamaria, C.; Ecenarro, O.; Dutrieux, J. F. Phil. Mag. 2003, 83 (17-18), 2011. (19) Costese`que, P.; Loubet, J. C. Phil. Mag. 2003, 83 (17-18), 2017. (20) Urteaga, P.; Bou-Ali, M. M.; Madariaga, J. A.; Santamaria, C.; Blanco, P.; Platten, J. K. Thermodiffusion: Basics and Applications; BouAli, M. M., Platten, J. K., Eds.; Mondragon Unibertsitateko Zerbitzu Editoriala: Mondragon, Spain, 2006; p 449. (21) Thomaes, G. J. Chim. Phys. 1956, 53, 407. (22) Platten, J. K.; Dutrieux, J. F.; Chavepeyer, G. Thermal Nonequilibrium Phenomena in Fluid Mixtures; Ko¨hler, W., Wiegand, S., Eds.; Lecture Notes in Physics; Springer: Berlin, 2002; Vol. 584, p 313. (23) Lhost, O.; Platten, J. K. Phys. ReV. A 1989, 40 (11), 6415. (24) Lhost, O.; Platten, J. K. Phys. ReV. A 1988, 38 (6), 3147. (25) Touiri, H.; Platten, J. K.; Chavepeyer, G. Eur. J. Mech. B: Fluids 1996, 15 (2), 241. (26) Kolodner, P.; Williams, H.; Moe, C. J. Chem. Phys. 1988, 88 (10), 6512. (27) Schmidt, R. J.; Milverton, S. W. Proc. R. Soc. London Ser. A 1935, 152, 589. (28) Platten, J. K.; Chavepeyer, G. J. Fluid Mech. 1973, 60, 305. (29) Washburn, E. W., Ed. International Critical Tables of Numerical Data, Physics, Chemistry and Technology; Knovel: New York, 1926-30, 2003; Vol. 3, p 120. Also available online at http://www.knovel.com/ knovel2/Toc.jsp?BookID)735. (30) Lhost, O. Contribution expe´rimentale a` l’e´tude de la convection libre induite par effet Soret. Master’s Thesis, University of Mons, Mons, Belgium, 1986. (31) Poty, P.; Legros, J. Cl.; Thomaes, G. Z. Naturforsch. 1974, 29A, 1915. (32) Bou-Ali, M. M.; Ecenarro, O.; Madariaga, J. A.; Santamarıá, C.; Valencia, J. Phys. ReV. E 1999, 59, 1250. (33) Dutrieux, J. F. Contributions a` la me´trologie du coefficient Soret. Ph.D. Thesis, University of Mons, Mons, Belgium, 2002. (34) Leppla, C. Thermodiffusion in Mischungen aus Lo¨sungsmitteln, Alkanen und Oligostyrol/Essigester mittels TDFRS. Ph.D. Thesis, University of Mainz, Germany, Mainz, 2002. (35) Costese`que, P.; Gaillard, S.; Gachet, Y.; Jamet, Ph. Phil. Mag. 2003, 83 (17-18), 2039. (36) Platten, J. K.; Costese`que, P. J. Porous Media 2004, 7 (4), 317.

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