J. Phys. Chem. 1902, 86, 2967-2974
stant of a cathodic electron-transfer process changes with overvoltage q according to the Tafel equation
ket log - = -(1- ) a
4,"
nF 2.303RTq
where n is the number of electrons involved in the transfer step, F is the Faraday constant, k,," is the rate constant for electron transfer at the standard potential of the redox couple, and a is the transfer coefficient. At room temperature (T = 298 K) eq 11 assumes the form
The overvoltage can be expressed in terms of the standard redox potential of the MV2+/+couple and the potential of the Ti02 conduction band 7 = ECB(Ti02) - Eo(MV2'/+)
(13)
Inserting the values for E C B and E0(MV2+/') into eq 13 gives = 0.315 - 0.059(pH)
log
(14)
ket a = (1 - a)(pH) - (1- ~~)(5.34)
(15) ket where 5.34 is the pH at which q = 0. Equation 15 predicts a linear relationship between log k , and pH as is observed experimentally over a large pH range. The value for the transfer coefficient derived from the slope of the straight line is a = 0.52 which indicates a symmetrical transition state as observed for the majority of electrochemical reactions. According to eq 15 the intercept with the ordinate at pH 0 is given
2967
by -[(l- a)(5.34) - log ke,O]. The parameter k,,O corresponds to the rate constant of electron transfer a t the standard potential of the MV2'/+ couple, which the colloidal TiOz particle reaches at pH 5.34. From Figure 3 one cm/s, which corresponds to a obtains he: = 4 X moderate electrochemical rate constant. From this value, the intercept is predicted to be at -4.9, which is in fair agreement with the experimental value of -4.2 determined by extrapolating the linear portion of the curve in Figure 3 back to the ordinate axis. At lower pH values, the experimental points in Figure 3 exhibit a negative deviation from the Tafel line. Such a behavior is commonly observed in electrochemical processes when the overvoltage approaches zero and is attributed to anodic reoxidation of MV+. At lower pH, the latter reaction contributes significantly to the overall process of relaxation of the system into equilibrium after pulsed-laser excitation. In conclusion, the analysis presented here provides new physical insight into the phenomenon associated with light-activated charge transfer across the semiconductor solution interface. Kinetic equations are derived which allow the determination of important electrokinetic parameters, such as the heterogeneous rate constant for electron transfer and the transfer coefficient. In the case of MV+ reduction by conduction-band electrons from TiOz particles, the interfacial electron-transfer step controls the rate at lower pH when the overvoltage available to drive the reaction is small. At higher pH, mass-transfer effects become increasingly important and determine the overall reaction rate.
Acknowledgment. We gratefully acknowledge the informative discussions with A. J. Nozik. A.J.F. was supported by the Office of Basic Energy Sciences, Division of Chemical Energy, US Department of Energy (Contract EG-77-(2-01-4042);M.G. was supported by the Swiss National Foundation and Ciba-Geigy, Switzerland.
Nonisothermal Matter Transport In Sodium Chloride and Potassium Chloride Aqueous Solutions. 1. Homogeneous System (Thermal Diffusion) F. S. Gaeia,'t 0. Pema,t 0. Scala,t and F. Bellucclt International Institute of Genetics and Biophysics of C.N.R., 80125 Naples, Ita&, and Istituto dl Principi di Ingegneria Chimica, Faculty of Engineering, University of Naples. Piazzale Tecchlo, BO 125 Naples, Italy (Received: February 3, 198 1; I n Final Form: December 11, 1981)
Thermal diffusion of sodium chloride and potassium chloride aqueous solutions exhibits anomalous behavior in solute-specificconcentration and temperature ranges. An accurate analysis is presented showing that these results are not instrumental artifacts. A physical interpretation of the experimental results is advanced, within the frame of reference of the radiation-pressure theory of thermal diffusion. Current ideas on the structure of electrolytic solutions can be fruitfully used, within the proposed approach, to describe some important thermodynamic parameters in terms of molecular organization in the liquid phase.
(1) Introduction
The redistribution of the components of an initially homogeneous solution, under the action of a temperature gradient-effect of thermal diffusion-has been studied by many authors, over more than a century.1-6 However, 'International Institute of Genetics and Biophysics of C.N.R. Istituto di Principi di Ingegneria Chimica.
*
0022-3654/82/2086-2967$01.25/0
temperature and concentration dependencies of the effect in liquids are still largely unexplored. A suitable molecular theory capable of interpreting already known data at (1)C. Ludwig, Sonder-Berichte Akad. Wiss. Wien, 20, 539 (1856). (2)Ch. Soret, Arch. Sci. Phys. Nat., 2,48 (1879). (3)Ch. Soret, C. R. Hebd. Seances Acad. Sci., 91,289 (1880). (4)Ch. Soret, Arch. Sci. Phys. Nat., 4, 209 (1880). (5) Ch. Soret, Ann. Chim. Phys., 22,239 (1881).
0 1982 American Chemical Society
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Gaeta et al.
present appears to be a distant goal. Experimental studies are conveniently carried out with either a Soret cell or a thermogravitational column. In the first one the solution is contained between two horizontal, 2. metallic plates, kept at different temperatures, the liquid being gravitationally stable.= The concentration gradient Y produced is measured by appropriate means. In the 0thermogravitational column, fmt developed by Clusius and 9 Dicke16-'and applied to liquids by Korsching and W i r t ~ , ~ ~ c~n . the hot and cold plates are inclined, and thermal convection is advantageously exploited to speed up thermodiffusive component separation. i When one works with Soret cells, much care is necessary -4 to avoid convective mixing and to obtain precise meaI 16' ' 10-2 '10-3 1 10 suremenh of small concentration differences. The thermogravitational column, in turn, is a notoriously tricky &,mol mi-' device calling for considerable ability from the experiFlgure 1. Soret coefficients for NaCl and KCI in aqueous solutions as menter to avoid interference of thermohydrodynamical a function of initial concentratlon at T , = 30 'C: (A)NaCl and (e) instabilities which may seriously affect experimental reKCI, present measurements; (A)NaCl and (0)KCI, from ref 19, sults.'@-'2 interpolated; (A)NaCl and (0)KCI, from ref 22; (0)NaCl from ref 15; Whatever the apparatus employed, the magnitude of the (W) KCI from ref 17, interpolated; (+) KCI from ref 15; (X) KCI from ref 16. effect is conveniently expressed by the Soret coefficient s (K-l), defined as TABLE I: Values of s, D ,and D' for NaCl and KCl 1 dn 1 dn Aqueous Solutions, at Tav= 30 "C, Calculated from s = -- -- Literature Dataa nn' d T n dT L I
where n and n'are molar fractions of solute and solvent, respectively. In dilute solutions (n' N l),the second expression of eq 1 applies. Typically the steady-state fractional concentration difference Anln for a AT of 1 K is of the order of 0.1%. Solute fluxes can alternatively be represented by the thermal diffusion coefficient D' (cm2 s-l K-l). This quantity represents the drift velocity of a component as produced by a temperature gradient of 1K cm-' applied to the solution. Aqueous electrolytic solutions have attracted a good deal of attention from students of thermal diffusion. Still, a systematic study of concentration and temperature dependencies of the effect has not yet been done. Surprisingly enough, a survey of the pertinent literature evidences systematic gaps in the existing data concerning the concentration dependence of the Soret coefficient in aqueous solutions. We shall be concerned elsewhere with the general aspects of this circumstance. In the case of NaCl and KC1 solutions, which are the object of the present study-and which have been most extensively investigated-the concentration gap is between about and 0.7 M. This range would be all the more interesting since these solutions are too concentrated to allow treatment in terms of the Debye-Hiickel theory. Accordingly, every piece of experimental information concerning them would be particularly useful. Incidentally, observable effects stemming from the coalescence of the circumionic Coulombic hydration spheresl3J4should appear at these concentrations. Their observation might yield new valuable insight into the structure of aqueous solutions. (6)K. Clusius and G. Dickel, Naturwissenschaften, 26, 546 (1938). (7)K.Clusius and G. Dickel, Naturwissenschaften, 27,487 (1939). (8)H.Koreching and K. Wirtz, Naturwissenschaften,27,267(1939). (9)H.Koraching and K. Wirtz, Ber. Dtsch. Chem. Ges., 73,249(1940). (10)S. R. De Groot and C. J. Gorter, Physica, 9,923 (1942). (11)I. Prigogine, L.De BrouckBre, and R. Amand, Physica, 16, 579 (1950). (12)I. Prigogine, L.De BrouckBre, and R. Amand, Physica, 16, 851 (1950). (13)H. S. Frank and W.-Y. Wen, Discuss. Faraday Soc., 24, 133 (1957). (14)R. A. Horne, Adu. Hydrosci., 6, 107 (1970).
solute and
-'
109~' 105D,b cm' s ref for values of cm' s-l K Soret coeff
concn, M
lo's, K-'
NaCl 0.0025
3.20
1.76
5.63
interpolated
0.005
3.01
1.75
5.27
interpolated
0.01
2.70
1.74
4.70
from ref 1 9 interpolated from ref 1 9
0.02
2.30
1.73
3.98
0.05 2.0
2.01 1.50
1.72 1.70
3.46 2.55
from ref 1 9
interpolated
from ref 1 9 22
interpolated
from ref 1 5
KCl 0.0025
2.50
2.18
5.45
0.005 0.01 0.02 0.05 2.0 2.0
2.35 2.15 2.00 1.30 1.20 1.05
2.16 2.14 2.12 2.14 2.20 2.20
5.08 4.60 4.24 2.78 2.64 2.31
interpolated
2.0
0.75
2.20
1.64
from ref 1 5 17
interpolated from ref 19
a
See references in the last column.
22 19
From ref 23.
Inspection of published data on thermal diffusion of these solutes evidences peculiar thermodiffusive behavior on both sides of the neglected concentration range. The concentration dependence exhibits opposite slopes at low and high concentrations, indicating the existence of an interposed minimum. This circumstance by itself should have stimulated interest to investigate the anomalous region. In Figure 1 a summary of the best published values of Soret coefficients for NaCl and KCl solutions'k22 is re(15)C.C.Tanner, Trans. Faraday SOC.,23, 75 (1927). (16)C.C. Tanner, Trans. Faraday SOC.,49,611 (1953). (17)L. G. Longsworth, J. Phys. Chem., 61,1557 (1957). (18)L. G. Longsworth in "The Structure of Electrolytic Solutions",W. J. Marner, Ed., New York, 1959,p 183. (19)J. N.Agar and J. C. R. Turner, Proc. R . SOC.London, Ser. A, 255, 307 (1960).
Thermal Diffusion of NaCl and KCI Aqueous Solutions
The Journal of Physical Chemistry, Vol. 86, No. 15, 1982 2969
ported in graphical form. Also very good data on the e ~that ~~~~~~ ordinary diffusion coefficient D are a ~ a i l a b l so the thermal diffusion coefficient D’ = sD also can be calculated (Table I). In this paper we present a systematic experimental study of the thermodiffusive behavior of NaCl and KC1 aqueous solutions, throughout the concentration range 0.005-1 M, at various average temperatures. This study extends our previous investigationu on these solutes. From the values of Soret coefficients, the heats of transport have been calculated. Also presented is an attempt to discuss these data in terms of thermodynamic phenomenology and of a molecular approach based on the notion of thermal radiation forcea acting in nonisothermal liquids. A tentative physical interpretation of the so-called heat of transport is also advanced. (2) Experimental Section (2.1) Apparatus, Materials, and Methods. As already mentioned, the apparatus’adopted in this investigation is the thermogravitational column. The extensive study by De Groat% cleared the way of most uncertainties previously existing in the column’s phenomenology. Details concerning apparatus design and experimental procedures have already been given elsewherewm so that only a very concise description is needed here. We used two columns of identical design-one made of brass, the other of stainless steel. Runs were often repeated, under identical conditions, interchanging apparatuses; reproducibility of results was fair (f5%) and independent of choice of column. The system of steady convective currents developed in the core of the apparatus-represented in Figure 2constantly carries fresh solution into the working area between the plates, where the temperature gradient causes thermodiffusive separation. Components diffusing to the cold wall are carried into the lower reservoir, and the others into the upper one. Component concentrations in the cold and warm reservoirs, C, and C,, change with time, according to the e x p r e s s i ~ n ~ ~ ? ~ ~
where H is a complex quantity, which describes the thermohydraulic behavior of the column. If the influence on the local density of the liquid from component separation is neglected, the third, approximate expression of eq 2 is obtained, where q, p , and /3 are solution viscosity, density, and coefficient of thermal expansion, V is the volume of each reservoir, g is the intensity of the gravitational field, b is the width of the plates, while a and AT (20)J. N. Agar and J. C. R. Turner, J. Phys. Chem., 64,lo00 (1960). (21)P. N. Snowdon and J. C. R. Tumer, Trans. Faraday SOC.,56,1409 (1960). (22)K.F. Alexander, 2.Phys. Chem., 203, 213 (1954). (23)H. S. Harned and B. B. Owen, ‘The Physical Chemistry of Electrolytic Solutions”, 3rd ed., Reinhold, New York, 1958,p 700. (24)F.S.Gaeta, D. G. Mita, G. Perna, and G. Scala, Nuovo Cimento B, 30,163 (1975). (25)S. R. De Groot, ‘L’ Effect Soret, Diffusion Thermique dans les Phasea Condens&”, Academiech Proefschrift, North-Holland Publishing Co., Amsterdam, 1945. (26)H. J. V. Tyrrell, ‘Diffusion and Heat Flow in Liquids”, Butterworth, London, 1961. (27)F. S. Gaeta and N. G. Cursio, J. Polym. Sci. Part A-1, 7 , I697 (1969). (28)F.S.Gaeta, A. Di Chiara, and G. Perna, Nuouo Cimento B, 66, 260 (1970). (29)F. S.Gaeta, G. Scala, G. Brescia, and A. Di Chiara, J. Polym. Sci., Polym. Phys. Ed., 13, 177 (1975).
2
-Convective *lharmo
currants dlffuslva fluxes
Figure 2. Core of thermogravitationai column, with hydrodynamic fluid circulation and thermodiffusive matter flux schematically represented. Under working conditions the apparatus is inclined 3-5’ with respect to the vertical position, warm side up. Dimensions are not to scale; real values are given In the text.
are the distance and the temperature difference between hot and cold plates; finally, t is the duration of the run. In both of our columns one has b = 8 cm and V = 15 cm3, plate separation being found to be optimal at a = 0.045 cm. Run duration has been set at t = 5 h. This choice of running time was adopted on the basis of the results of preliminary runs, lasting from 2 to 12 h, designed to establish the time dependence of the effect. This was found to be linear up to 7 h, thus operatively defining the field of validity of eq 2 which holds only for “short” running times.25 A further reason which induced us to carefully investigate the time dependence of the concentration ratio C,/C, weU beyond the linear range will be discussed in the following. The analysis of concentrations in this study has been carried out by different methods for the sake of comparison. The concentration of Na+ and K+ ions has been determined in the depleted and concentrated fractions by a microproceasor ion analyzer Model 901 of Orion Research Inc. equipped with sodium- and potassium-selective electrodes. These measurements were cross-checked with a flame photometer (Flammeledron 11). The concentrations of chlorine were separately assessed by chemical methods. All chemicals used were pure pro-analysis grade; double-distilled water was used as solvent and in rinsing operations. All details pertaining to handling of apparatus, thermostation, temperature measurements, and calibration procedures can be found e l a e ~ h e r e . ~ ~ - ~ ~ (2.2) Results. Our experimental results on thermal diffusion of NaCl and KCl aqueous solutions are displayed
The Journal of Physical Chemistry, Vol. 86, No. 15, 7982
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&eta et ai.
1. 2-
01
0
6
L
10
8
& . l o 3 .mol
I2
IL
.I-'
Flgwe 3. &ret coefficients for NaCl aqueous soiutlons as a function T,, = of initial concentration at different average temperatures: (0) 27 "C; (e)Tav = 30 " C (El) T,, = 35 "C; (A)T,, = 43 "c.
KC I
IP I 0
01
(13
02
c,
mol
Or
05
li'
Figure 4. Soret coefficients for KCI aqueous solutions as a function of initial concentration at different average temperatures: (0)T,, = 30 "C; (0) T,, = 35 "C; (A)T, = 38.5 "C.
in Figures 1, 3, and 4, as a function of initial solute concentration at different average temperatures. In Figure 1 the values of Soret coefficients of sodium chloride and potassium chloride calculated from our experimental data, through eq 2, are plotted against solute initial concentration. These data refer to runs performed at T,, = +30 "C and AT = 16 "C. Because of this choice of temperatures, a direct comparison is possible with some results of Tanner15 for 1 M NaCl a t T,, = 31 "C and AT = 14 "C and with those of the same author for 1 M KC1 at T,, = 32 "C and AT = 14 "C. Comparison is also feasible-upon interpolation-with Longsworth's relative to 1M KC1 at T,, = +25 and +35 "C, respectively, at AT = 10 "C. In Figure 1, where all of these data are compiled, the concentration dependence of s found by these authors can be appreciated. At the low-concentration side the values of s calculated from the data in the literature for NaCl and KCl at +30 "C have been obtained by interpolation from the data of Agar and T ~ r n e r ' in ~ .the ~ ~range 10-3-10-2 M, at average temperatures +25 and +34.7 "C. Also a result reported by Snowdon and Turnerz1 can be used in this same range. Again, the concentration dependence of published values unmistakably emerges. An additional point for each one, KC1 and NaC1, at 0.05 M and T,, = 30 "C, AT N 17 "C by AlexandeP is added to the other data. Our own data are also compiled in this same figure and fill the gap left among preexisting measurements. In the regions where our data overlap with the others, a nice fit generally occurs. The agreement is particularly good in the case of results by Agar and Turner-in the dilute
range-and with those of Tanner at the opposite end. More interestingly, the slopes of our plots match those found by the quoted authors. The existence of a minimum in the plot of Soret coefficients against concentration in both NaCl and KCI aqueous solutions could in fact be predicted from already published data. What could not be inferred was instead the precise position of the minima within the unexplored concentration range, nor their characteristics, as for instance evidenced from the proposed aspect of these minima, according to From the present data it can be seen that the minimum for NaCl falls at much smaller concentrations than the one for KC1 and also appears to be much sharper and deeper than the other. Rather striking is the circumstance that in both cases there is a concentration range, around the minimum, where the sign of the Soret coefficient inverts, that is, a region in which the solute is enriched in the warmer solution rather than in the cooler one, as generally found to occur. Soret coefficients of NaCl aqueous solutions in the range 5 X 10-3-13 X M at various T,, values are reported in Figure 3. The temperature difference between the plates was AT = 20 "C throughout this series of runs. The principal observation relative to these results is that the position of the minimum shifts to lower concentrations when the average temperature is raised. Furthermore, the depth of the minimum increases with temperature, while the overall shape of the curves does not seem to change, all the plots looking alike. Figure 4 shows the results of a similar investigation on KCI aqueous solutions. The concentration range between 0.05 and 0.5 M has been studied at various average solution temperatures. Also in this case the temperature difference has been set to AT = 20 "C, as with NaCl. Also the other adjustable parameters-angle of tilt of apparatus and run duration-as well as all details of apparatus handling, sampling procedures, concentration measurements, etc., were the same. These data accordingly can be directly compared with the ones of Figure 3. Apart from the systematically higher values of the concentrations at which the minima are observed to occur, relative to NaCI, these plots closely resemble the others, in general features. The positions of the minima are displaced to lower concentrations with increasing average temperature; the depth of the minima exhibits a temperature dependence analogous to the one of sodium chloride. The absolute depth of these minima, however, seems to be smaller than with NaC1. In Table I1 the values of Soret coefficients calculated from our experimental results of Figure 1 are reported together with the corresponding values of the coefficient of thermal diffusion D'. These values can be obtained either by multiplying s by the value of the ordinary diffusion coefficient D or by calculating the magnitude of thermodiffusive solute fluxes in the working area between the plates and dividing these fluxes by the applied temperature gradient. This calculation is easily done by assuming the working area of the column t o be practically equal to bh, where h is the distance between the inner brims of the two reservoirs (Figure 2). The values of D' calculated according to the above procedure are reported in the sixth column of Table 11. These values are in good agreement with the corresponding values obtained from the product sD. (2.3) Concentration Dependence and the Forgotten Effect. Before proceeding any further, we must discuss the possible occurrence of an instrumental effect which (30)J. J. Chanu, J. Chin. Phys., Phys.-Chim. Bid., 55, 743 (1958).
The Journal of Physical Chemistry, Vol. 86, No. 15, 1982 2971
Thermal Dlffuslon of NaCl and KCI Aqueous Solutions TABLE 11: Some Values of s, D', and Q* Calculated from Our Own Experimental Results in NaCl and KCI Aqueous Solutions at T, = 30 "Ca 10'D' from solute lO'D',' fluxes, Q*, cm' s-' solute and lo%, 109~,,,b cm' s-' concn,M K-' K' ' R' cal mol-' K" NaCl 5.2 5.13 1028.42 0.005 +2.95 +2.955 773.72 3.9 3.89 0.022 +2.25 +2.253 271.42 1.4 1.41 0.040 +0.80 +OB02 -4.47 -856.60 -4.4 0.046 -2.62 -2.618 -1.50 -2.53 -503.24 -2.6 0.06 -1.499 -3.34 -0.02 0.15 -0.01 -0,009, -0.017 0.80 +0.92 +0.921 1.54 329.78 1.6 KC1 5.24 734.06 5.3 0.007 + 2 . 2 0 +2.207 495.40 3.2 0.10 + 1 . 5 0 +1.505 3.11 +1.00 +1.001 2.04 0.20 327.10 2.1 0.30 -0.73 -0.729 -1.48 -238.57 -1.6 0.32 -1.35 -1.348 -2.75 -791.01 -2.8 0.40 -0.015 -0.014, -0.03 -4.91 -0.03 0.80 + 0 . 4 0 +0.403 0.83 129.26 0.8 1.25 +0.65 +0.655 1.38 206.22 1.4 a Soret coefficients in second column have been calculated by means of eq 2, where the concentration effect on liquid density has been neglected. When this factor is accounted for, through eq 3 and 4, the values reported in the third column are obtained. The values of D employed in the calculation of D' from the relation D'= SD (fourth column) have been taken from the literature (ref 23). Values of D' in the sixth column of the table have been , , calculated from the experimental values of C, and C through the expression D' = (l/AT)[2aV/(bht)](Cw- C,)/ (C, + C,) where h = 4.8 cm while all other quantities Corrected for "forhave been defined in section 2.1. gotten effect". e D'= sD.
may affect results obtained with thermogravitational columns. During a run, fluid density in the column varies in the y direction (Figure 2), owing to the temperature gradient, and this causes convective circulation in the z direction. Apparatus design is such to ensure a sufficiently slow laminar flow not to appreciably alter either the temperature gradient or thermodiffusive matter flux. A concentration gradient is gradually built up along y, which contributes its share to the density variation, thus affecting the calculation of the Soret coefficient. These effects have been treated in detail by Tyrrell.26 Here in the following we concisely deal with those aspects of the problem which are essential to our analysis. According to Tyrrell the quantity H figuring in eq 2 can be written in a more general form which accounts also for the influence of component concentration on the local density of the liquid. This expression is
H = -SAT-(gba3
-dap )
6!n dy
(3)
where (4)
In eq 2 the term snn'(dp/dn) has been neglected, as is common practice in these measurements. This approximation holds good at small solute concentrations and K-l. values of s of the order of Neglecting the concentration effect, however, is not permissible in the present study, since the minima could
be suspected to be an i n s t m e n t a l artifact. Furthermore, the position and depth of the minima might be affected by the approximation introduced into the calculation of Soret coefficients by eq 2. For these reasons the coefficients were recalculated by using in place of eq 2 the expression obtained by employing eq 3 and 4. These values are reported in Table 11, third column. The exclusion of an instrumental cause of the observed thermodiffusive anomalies, however, is of such importance in this context that it calls for direct experimental investigation. The rationale of these control experiments is the following. Since fluid densities may be affected either in the same or in opposite senses by temperature and concentration variations, an acceleration or a slowing down of the hydrodynamical currents with time may follow from combination of the above effects. This in turn leads to an increase or to a decrease in the observed separation rates of the components in the two reservoirs of the apparatus. In the particular case where the density changes induced by solute migration are of opposite sign and exceed the effects of thermal expansion, convective circulation is first brought to a halt and then inverts. Under these conditions inversion of the concentration ratio CJC, may follow. This behavior-attention to which was first drawn by De Groot and Gorterlo-has been called the "forgotten effect" owing to the fact that previously the influence of concentration variation on thermogravitational column circulation has been generally disregarded. Prigogine and co-workersl1J2systematically investigated this effect experimentally, proving the occurrence of separation inversion induced by the forgotten effect. Solution density increases with concentration in both NaCl and KC1, throughout the range experimentally explored, with the exception of the most dilute sodium chloride and potassium chloride aqueous solutions, where anyhow the concentration is so small that its influence on hydrodynamical circulation is negligible. Therefore, dp/dn is to be considered always positive in the range of interest, and, in the cases where solute concentrates at the cold side of the column, the separation rate of components will increase with time in consequence of the forgotten effect. This situation will be evidenced by a concentration vs. time plot of the two reservoirs like that in Figure 5a. When solute concentrates on the hot side, the density of the warmer, ascending fluid gradually increases, so that a slowing down and eventually an inversion of convective motion follows. This would result in a concentration vs. time plot like that in Figure 5b. The presence of such effects can therefore be experimentally ascertained, by performing a series of runs of variable duration and plotting the final concentrations in the two column reservoirs against time. Since the forgotten effect is expected to be greater where Cois higher and/or where separations are bigger, these runs should be performed both at the high limit of the concentration range investigated and at the concentrations yielding the greatest separation ratios. Accordingly, a series of runs of variable duration, comprising those between 2 and 10 h, was performed for each solute, at initial concentrations corresponding respectively to the minimum in the plots and to the highest concentrations employed. In Figure 6 the results of such a series of measurements for potassium chloride aqueous solutions are given. Even if this plot is of the kind of Figure 5b, the influence of the forgotten effect can be appreciated only after over 6 h of running. Results of similar runs performed at the upper limits of the explored concentration ranges show that the separations (and the Soret coefficients) found there may
2872
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Gaeta et ai.
of the identification of a plausible physical basis for this striking behavior. The close similarity of results for both NaCl and KC1 aqueous solutions suggests some general mechanism of solute-solvent interaction rather than a situation peculiar to a particular ionic species. We want to proceed now to consider some quantitative information which can be obtained from our results and will hopefully help us in their interpretation. It is well-known that from the Soret coefficients the heats of transport Q* can be calculated. If Q*+ and Q*are the heats of transport of cation and anion, for uniunivalent salts Q* = Q*+ + Q*-. The calculation of Q* from the Soret coefficient s is easily done through the following expression:
\.
-._,_._._._.-.-.-.-I
1.
Figure 5. Typical plot of concentrations in warm and cold reservoirs of a thmmgavitational column as a function of time in the presence of the forgotten effect. In case a no inversion occurs; in case b the separation decrease may lead to inversion (11) or not (I) depending on the system's physical conditions (from ref 11 and 12).
15
i
0
KCI
2
i
6
a
Tav: 30'C
1 0 1 2
time(hr)
Flgwe 6. One of our experimental plots where the incidence of the forgotten effect is greatest: 1.25 M KCI; T,, = 30 O C . Behavior of type b is evident in this case.
have been increased by about 1.2% only, due to behavior of the kind exemplified in Figure 5b. These experimental findings agree very well with results of the calculations done by substituting expressions 3 and 4 into eq 2.
In conclusion, the thermodiffusive anomalies evidenced by Figures 1, 3, and 4 cannot be instrumental artifacts produced by the forgotten effect. The occurrence of these anomalies, accordingly, must be attributed to an intrinsic cause. The sense of thermodiffusive drift of solute relative to the temperature gradient is really inverted, presumably owing to changes occurring in the physical properties of the medium. (3) Discussion of results From the above analysis it follows that the thermodiffusive anomalies found by us are not an instrumental artifact. Accordingly, we are confronted with the problem
where R is the gas constant, m is the molality of the solution, and y+ is the mean activity coefficient. The quantities [a In yJ2 In mIT can be calculated from published data for our solutions and are summarized in Table I1 together with the values of Q* obtained through eq 5. Noteworthy are the high negative values assumed by Q* corresponding to the minima. For ions moving through water, the heat of transport represents energy liberated along the path of the particles which interact with the surrounding water. Three mechanisms appear to contribute to Q*: (a) local energy fluctuations associated with the interactions of the moving particle with its nearest neighbors; (b) electrostatic interactions between ionic charges and the dipolar water molecules (this interaction extends well beyond nearestneighbor distance; ( c ) structure-making or structurebreaking effects of the ions on water.31 High positive heats of transport are considered to be indicative of low entropy in the free water surrounding the ions. Conversely, negative values of Q*, such as those found in the anomalous regions, indicate an exceptionally high entropy in the surrounding fluid. Accordingly two concentration regions can be operatively identified. The first one ranges up to C*, this being the critical concentration where the minimum of the Soret coefficient occurs; the second one extends above C*. In the former region free water, i.e., water not bound to ions, is increasingly disordered with increasing salt concentration, particularly owing to long-range interactions. Above C* the heats of transport increase with increasing concentration. An ordered configuration, consisting of polarized solvent only, is now prevailing in the liquid. This entails the observed rise of Q* back to positive values in concentrated solutions. Congruous with this interpretation is the observation that a temperature increase displaces the minimum to lower concentrations. Thermal disorder indeed is synergic with the disordering effect due to the introduction of more ions. To proceed further toward the identification of molecular mechanisms responsible for the observed thermodiffusive behavior, it is necessary to discuss these results with reference to a definite model of the structure of electrolytic solutions and to treat thermal diffusion effects in terms of the thermal radiation forces which are their Particularly useful for our purposes is the (31) W.-Y. Wen, "Water and Aqueous Solutions", R. A. Horne, Ed., Wiley, New York, 1972, p 644. (32) F. S. Gaeta, Phys. Reu., 182, 289 (1969). (33) F.S.Gaeta and A. Di Chiara, J. Polym. Sci., Polym. Phys. Ed., 13, 163 (1975). (34) F. S.Gaeta, 'Radiation Forces Produced by CompressionalWaves in Non-Isothermal Liquids", in preparation.
Thermal Diffusion of NaCl and KCI Aqueous Solutions
n‘
Figure 7. Water structure near a Ne+ ion. Reprinted with permission from ref 31. Copyright 1972 John Wlley and Sons.
theory of water structure proposed by Frank and Wen13 involving the existence in the liquid of metastable structures numbering 35-45 molecules.36 When an ionic solute is added, it causes the formation of a complex circumionic atmosphere, whose member water molecules cannot participate in the original water structures. Two main regions compose the ionic atmosphere, as already pointed out by Bockris,= a close hydration zone constituted of some nearest-neighbor water molecules and a region of distant hydration dominated by polarization effects. As shown by Mikhailov et aL3’ and confirmed by Vdovenko et al.,= this second region, on the whole, contains a higher proportion of disordered structures than pure water. The complexity of the Coulombic hydration sphere is illustrated by Horne,14whose model is reproduced in Figure 7 , using Na+ as an example. A small group of primary hydrated water molecules (region A) is surrounded by a Frank-Wen cluster (region B); furthermore, some water, whose structure has been partly broken, surrounds the cluster (region C). Region A is strongly dependent on the specific nature of the central ion and is only mildly affected by concentration and temperature variations. On the other hand, regions B and C are affected by these changes. The free water comprising neighboring ionic hydration spheres will also be drastically affected by temperature and concentration variations. A critical stage is reached when the volumes unaffected by polarization become too small to allow formation of the structural clusters characteristic of water in bulk. Here a small increase of either temperature or concentration will induce an order-disorder transition in the free water. This event constitutes a second-order phase transition, and once it has taken place further concentration increments will lead to progressive coalescence of the Coulombic atmospheres, entailing a gradual increase of the degree of the configurational order. This ordered phase prevailing at relatively high concentration will be determined by polarization forces and will be compatible with a smaller number of hydrogen bonds per water molecule than in free water. While a temperature increase at low concentration favors the disordering of water structure, at concentrations above (35)G. Nemethy and H. A. Scheraga, J. Chem. Phys., 36,3382(1962). (36)J. O’M. Bockris, Quart. Rev. Chem. SOC.,3, 173 (1949). (37)I. G. Mikhailov and Yu. P. Symikov, Zh. Strukt. Khim., 1, 12 (1960). (38)V. M. Vdovenko, Yu.V. Guryikov, and Yc.K. Legin, Zh. Strukt. Khim., 10,576 (1969).
The Journal of Physlcal Chemlstry, Vol. 86, No. 15, 1982 2973
the order-disorder transition it will favor the transition from hydrogen-bonding-controlled configurations into more compact ones determined by polarization effects. It is useful at this point to introduce the radiationpressure theory of thermal d i f f ~ s i o n to ~ ~quantitatively -~ interpret our experimental results. According to the radiation-pressure theory the flux of thermal energy through a nonisothermal liquid is coupled with a flux of momentum, and as a consequence on every particle contained in the liquid a thermal radiation force will act, given by
where % is a numerical constant (% < l),K is the thermal conductivity, u is the velocity of propagation of high-frequency elastic (Debye) waves, and up is the particle cross section. Subscripts 1 and p are for liquid bulk properties and “particle” properties, respectively. The density of momentum flux [(K/u)(dT/dy)Il depends upon the bulk (average) momentum conductivity of the liquid (K/u)l. On the other hand, the local value ( K / u ) ~ determines the change of momentum flux at the particle surface, thus generating the thermal radiation forces. “Particle” here refers to any local structure with properties different from the solution’s average, be it a small foreign body suspended in the liquid, a hydrated ion or molecule, or a cluster of water molecules characterized by a significant fluctuation of the K/u ratio relative to its surroundings. Accordingly, this particle will thermodiffuse along the direction of the applied temperature gradient in a sense dependent on the sign of the quantity (K/u)l - (K/u),. The value of the Soret coefficient of the particle can be s h o ~ n to~ be ~ given - ~ ~ by s
-2[(K/u)l- (K/u)plup/(kTav)
(7)
where k is Boltzmann’s constant. Experimental verifications of this approach to thermal diffusion in liquidsthere including the inversion of sense of the thermodiffusive drift in coincidence with change of sign of the quantity ( K / U ) ~ (K/u),-are by now amply in our hands.24~27-29~39.40 We have measured thermal radiation forces acting on macroscopic plungers suspended in nonisothermal liquids and ascertained the validity of the predictions of eq 6-for both magnitude and sense of force-in dependence of the K/u of solid and surrounding l i q ~ i d . We ~ ~ have * ~ ~checked eq 6 and 7 by using macromolecules in liquids.n-29v40Finally, we have suggested that the origin of inversions in thermodiffusive drift found by us in NaCl and KC1 aqueous solutions24could be explained on the same grounds. We presently want to reconsider this last point in more detail. We assume that the thermodiffusive anomalies found by us are due to an order-disorder transition occurring in the solution, as described above. In dilute solutions the circumionic polarized regions, widely apart one from the other, drift to the cold side of the cell, indicating a positive value of (K/U)~ - (K/u),. The positive value of the heat of transport in these dilute solutions is indicative of the highly ordered state of the surrounding water. This water also has a high percentage of hydrogen bonding. Here the drifting particle is the ion plus its close hydration sphere (region A). Its movement involves extensive breaking of hydrogen bonds in the (39)G.Brescia, E.Grossetti, and F. S. Gaeta, Nuouo Cimento, SOC. Ztal. Fis. B , 8B,329 (1972). (40) F. S.Gaeta, G. Perna, and G. Scala, J.Polym. Sci., Polym. Phys. Ed., 13,203 (1975).
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The Journal of Physical Chemistry, Vol. 86, No. 15, 1982
surrounding water. A high activation energy is accordingly to be expected for the process of thermal diffusion in dilute NaCl and KC1 aqueous solutions. In 0.005 M NaCl solutions, for instance, the value of E* is about 5.4 kcal mol-' as calculated from our results. Literature data l9 for 0.01 M NaCl and KC1 aqueous solutions, respectively, are 6.6 and 11 kcal mol-'. An increase of salt concentration brings about a decrease of the (K/u),ratio. This decrease becomes dramatic when the order-disorder transition is induced throughout the free water still remaining. The state of disorder of this medium-indicated by the negative values of Q*-involves a drastic decrease of the ratio (K/u),and possibly also an increase in the ( K / u ) , brought about by changes in the compactness of the polarized regions in response to the modifications occurring in the surrounding fluid. The extent of hydrogen bonding in the disordered water will be small, and a drastic decrease of the height of the energy barriers opposing thermodiffusive drift is to be expected. The change of the momentum conductivities is so great that sign inversion of the quantity (K/u),- (K/u),follows, and the sense of drift is inverted. The change in the thermal radiation force driving thermal diffusion complicates the calculation of the activation energies in the anomalous regions. From the experimental temperature dependence reported in Figures 3 and 4, the calculation of E* thus is not directly feasible. This particular aspect of the problem will be extensively dealt with in future work. Increasing solution concentration beyond C* leads to the total disapperance of free water, due to the coalescence of the circumionic spheres. The drifting particle-the ion with its primary hydration water molecules-is now surrounded by polarized water only, which is a comparatively ordered medium, albeit possessing configurations different from the ones of bulk water. This medium has once again a ( K / U> ) ~(K/u),, and the heats of transport turn out to be positive. The energy barriers are smaller than those in dilute solutions. For instance, in 4 M solutions E* amounts to 5.3 kcal mol-' for KC1 as from Longsworth's re~u1ts.l~ It is interesting to observe that the critical concentration C* for potassium chloride is higher than the one for sodium chloride. An explanation for this circumstance can be found within our proposed frame of reference as the consequence of the greater intensity of the Coulombic field of the Na+ ion relative to the one of the bulkier K+ ion. It would be interesting at this point to calculate the values of the term (K/u),- (K/u),throughout the explored concentration range. From eq 7 this is not feasible, owing to the unknown value of the cross section CJ, of the hydrated drifting particle. However, the very plausible assumption can be made that the solvated ion is a spherical particle. By introducing the Stokes-Einstein relationship 61rrq = kT,,/D, from eq 7 one gets
where q is solution viscosity, which can be independently measured. The values of the term in square brackets calculated from eq 8 for NaCl and KC1 solutions turn out to have the same order of magnitude as those which can be calculated from literature values of thermal conductivity
and velocity of propagation of elastic waves for common liquids and solid^.^,^^ For very dilute solutions the (Klu), is practically the same as in pure water. Hence, the (K/u), relative to the hydrated ions can be easily evaluated. Also these quantities turn out to have the expected values, thus providing an encouraging check for the theory. (4) Conclusions
The phenomenology of thermal diffusion in aqueous solutions of sodium chloride and potassium chloride found by us seems to open a field of experimental and theoretical investigation still unexplored. Some preliminary conclusions can be drawn anyhow. The thermal radiation pressure approach allows a coherent representation of experimental findings, within which the two endsconstituted of the thermodynamic description of thermal diffusion and the molecular theory of electrolytic solutions-can be brought to meet. The circumstance that adequate quantitative values of K/u can be derived from the theoretical approach, without introducing any fudge constant, is encouraging. This indicates that the theory of thermal radiation forces may be adopted as a useful tool in the interpretation of experimental results and as a guideline for planning future research. One objective of the next investigations should be the extension to other aqueous and nonaqueous solutions, to assess the possible generality of the observed phenomena. Another should be to separately measure ( K / Uand ) ~ (K/u),throughout the anomalous concentration ranges, to learn how much the anomaly has to be attributed to alteration of the properties of the bulk liquid, and how much to a change in the structure of the hydrated particles. We assume that this result may ultimately help to clarify the very important and still mysterious molecular mechanism of biological membrane transport of sodium and potassium. In the third place it should be understood whether the reported anomalies can be observed in an ordinary Soret cell, or whether gravitational instability of the horizontal concentrated layer occurring at C = C* would not make the effect impossible to study with this kind of apparatus. A last point, perhaps susceptible of broad generalization, is the following. In dilute NaCl and KC1 solutions the hydrated ions are transported to the cold wall, in the same sense that thermal energy is transported. Thus, matter transport by thermal diffusion increases the rate of entropy production beyond the one due to thermal energy transport. When, however, the increase in salt concentration leads to the cooperative transition which makes free water the most disordered phase, this one drifts to the cold side. This maximizes again the rate of entropy production. At higher concentrations the hydrated ions, constituting once again the phase with the higher entropy density, are transported to the cold wall. In other words, the microscopic state of order in the solutions seems to rearrange always in such a way that the forces acting on the system may attain the highest rate of entropy production compatible with the system's characteristics. It is interesting to point out the role played by thermal radiation forces in this context. In sodium chloride and potassium chloride aqueous solutions these forces are proportional to the entropy density of the local domains. We expect that this circumstance will be proved to be absolutely general by future studies.
