J. Chem. SOC., Faraday Trans. 1, 1988, 84(2), 647-656 Differential Thermo-osmotic Permeability in Water-Cellophane Systems Crist6bal Fernandez-Pineda and M. Isabel Vazquez-Gonzalez Department of Physics, Faculty of Sciences, University of Malaga, Malaga, Spain The thermo-osmosis of pure water through two different cellophane membranes at temperatures ranging between 33 and 47 "C has been studied. Two types of experiments have been carried out: one to determine the dependence of the phenomena on stirring rate, with the average temperature and the temperature difference between the two bulk phases being kept constant while varying the stirring rate; the other varying the temperature difference between the two bulk phases, with the temperature of the cold bulk phase being kept as close to a fixed value as possible and employing a constant stirring rate.In all experiments the thermo-osmotic flow was from the hot to the cold side, and thermo-osmotic permeability was found to decrease with the mean temperature. The experimental values of the global thermo-osmotic permeability were corrected by taking into account the temperature polarization, and from these corrected values the differential thermo-osmotic permeabilities were calculated for the two membranes employed. The relationship between the differential thermo-osmotic permeability and the temperature was found to be different for the two membranes, being linear for the 600P membrane and quadratic for the 500P membrane. However, within the range of temperatures used in the present study, in each case, the differential thermo-osmotic permeability decreased with temperature.For comparison, the differential thermo-osmotic per- meabilities of two analogous membranes mentioned in the literature were calculated, and their behaviour was shown to be very similar to those of the membranes used in the present study within the same temperature range. The phenomena of mass transport through a membrane due exclusively to a temperature difference between the two membrane surfaces is known as thermo-osmosis and has been widely studied.'Y2 A comparison of the published results of a large number of authors, the majority of whom employed cellophane or cellulose acetate membranes with different acetyl contents, reveals qualitative and quantitative differences between the transport coefficients.The recorded differences in mass transport behaviour arise from the different nature of the membranes, and some authors have demonstrated that thenno-osmotic phenomena can only occur in suitably dense membrane^.^-' However, recent reports show that thenno-osmotic flows across porous membranes can Thermo-osmosis is also reported in systems employing 'charged ' or ' activated ' membranes and electrolyte solutions. 13-16 Inspection of the literature also reveals contradictory results for analogous membranes. Carr and S01lner~~ state that thermo-osmotic flow across cellophane membranes is undetectable when water and electrolyte solutions are employed, while Alexander and Wirtz,l* Rastogi et al.,374 Hasse et al.,19-22 and more recently Vink and C h i ~ t h i ~ ~ have obtained marked thermo-osmotic effects in experiments with similar cellophane membranes.Rastogi et al.,374 working with Du Pont 600 membranes, that had first been washed in progressive dilutions of NaOH, then in dilute HCl, and finally repeatedly in conductivity water, found that the thermo-osmotic permeabilities for water 647648 Thermo-osmosis in Water-Cellophane Systems are independent of AT, up to AT = 19 "C, when measurements are made at a constant mean temperature. In ref. (3) the reported constant mean temperature was 40 "C. When the measurements are made at a constant temperature difference, AT, thermo-osmotic permeability decreases as a function of mean temperature up to a minimum value and thereafter increases with temperature.In ref. (3) again, AT = 14 "C, the mean temperature ranged from 40 to 55 "C and the minimum value of the thermo-osmotic permeability was near 51 "C. This behaviour is contrary to that reported by Haase et al.,19-22 who used cellophane membranes with water as the permeant. They employed two types of membranes in ref. (19), 0.3 and 0.6 kg m-2, manufactured by Kalle and Co. (Wiesbaden). For some experiments, these were pretreated by washing in water and subsequently impregnated with a deposit of copper ferrocyanide. In ref. (20)-(22) the membranes were manufactured by Kalle AG Wiesbaden-Biebrich. In all these published studies the thermo-osmotic permeabilities measured at constant AT values were found to decrease with average temperature, with the flux going from the hot to the cold side of the membrane.Moreover, these authors found20-22 that above t,, x 55 "C the direction of the flux changes, i.e. water goes from the cold to the warm compartment. The measurements were made within two temperature ranges, between 11 and 77 "C20 and between 10 and 90 0C.21v22 In ref. (20)-(22), AT x 1.3 "C. Most of the reported differences have been explained by Belluc~i~~ using the concept of temperature polarization, first suggested by Vink and C h i ~ t h i ~ ~ and used implicitly by Dariel and Kedem.5 However, because a number of contradictory observations of thermo-osmotic phenomena taking place in cellophane membranes have not been completely explained or eliminated, we have designed new experiments to measure thermo-osmotic phenomena in the water-cellophane systems, to provide more information on these conflicting results, and these are reported in the present paper.The first set of experiments attempted to study the influence of stirring rate on thermo- osmosis in order to obtain the correction factor introduced by Belluc~i~~ and to calculate the true global thermo-osmotic permeability coefficient, which could then be used to interpret experimental results for all values of the temperature difference between the bulk components and all average temperatures. In the second set of experiments, using a constant stirring rate, the bulk temperature in one of the half-cells was varied while maintaining the bulk temperature in the other almost constant. Values of the thermo-osmotic permeabilities obtained for each experimental run were then corrected by the method explained above, and from these the differential thermo-osmotic permeability, b( t), was calculated by a procedure analogous to that used to obtain the differential diffusion c o e f f i ~ i e n t ~ ~ ! ~ ~ and previously used for other types of measurements in thermo-osmotic experiments.' Experimental Membranes Two different commercial membranes were used, 600P and 500P, supplied by Cellophane Espaiiola S.A. Both membranes were pretreated before the experiments by being washed in twice-distilled and deionized water for 72 h. The air remaining in the membrane matrix was eliminated by immersing the samples in water in an Erlenmeyer flask and submitting them to a partial vacuum provided by a Buchner funnel attached to a water tap. The thickness, 6, of the wet membranes was determined by a Millitron-Compact measuring instrument (1202-IC) to an accuracy of f 2 pm.The fractional void volume, E , was determined by the procedure described by Fernindez-Pineda and Serrano;27 the results are shown in table 1.C. Fernandez- Pineda and M . I. Vazquez-Gonzakz Table 1. Thickness (S), density @) and fractional void volume ( E ) of the membranes employed membrane 6/pm p/103 kg m-3 & 600P 6 2 f 4 1.49 k 0.23 0.75 +O. 14 500P 51 _+4 1.26 _+ 0.15 0.74 2 0.10 649 Permeant Pure water (twice-distilled and deionized) was used for all experiments. To eliminate dissolved air and avoid the possible formation of air bubbles in the thermo-osmotic cell, the water was boiled and then filtered through a Millipore filter of 0.45 pm nominal pore size before being introduced into the measuring apparatus.Apparatus All the measurements were made with an experimental device similar to those employed by Mengual et a1.' and Fernandez-Pineda and Ser~ano.~' It is a cell which consists of two equal 0.1 m long cylindrical chambers with a diameter of 0.06 m. They are surrounded by concentric, cylindrical walls of the same length but with a diameter of 0.09 m. The arrangement was temperature-stabilized by circulating water between the two concentric cylinders, and each water jacket was connected to a different thermostat. The water was stirred by a chain-drive magnetic-cell stirrer assembly2' to ensure the uniformity of water temperatures in each chamber.Each chamber was connected to a glass tube placed vertically and inserted in the superior part of the half-cell. Each chamber, with its corresponding glass tube, could be filled separately by means of a two-way stopcock on the rubber tube connecting the chamber with a water reservoir located at a higher level. Each arrangement, consisting of the half-cell, glass tube, rubber tube and water reservoir was enclosed in its own large air-bath which provided a temperature-controlled environment. The membrane was placed between the two chambers in a methacrylate holder with two stainless-steel grids between which the membrane was fixed. The membrane surface area exposed to the flow was q = (20.4k0.6) x m2, and the cross-sections of the glass tubes were (20k 1) x lo-' and (1 9 3) x m2 (determined by the mercury-drop technique).The glass tubes were pretreated with Rhodorsil-240 obtained from Rh6ne-Poulenc to prevent meniscus formation and to obtain a flat interface which would facilitate accurate measurement of the differences in pressure. Measurement of Thermo-osmotic Permeability The evolution with time of the difference in hydrostatic pressures, AP, in the thermo-osmotic experiments towards steady state is determined by an equation of the where and AP AP APm B ATb '- ATb ATb A y =--, y ---A, y =--=- where t is the time, ATb is the temperature difference measured between the bulk phases, APo and APm are the differences in hydrostatic pressure at t = 0 and t = GO, respectively,650 Thermo-osmosis in Water-Cellophane Systems A and B are integral, global or average phenomenological coefficients termed permeability and thermo-osmotic permeability, respectively, qo is the cross-section of the glass tube used for the pressure measurements, 6 is the membrane thickness, g is the acceleration of gravity, M is the molar mass and q is the membrane surface area exposed to the flow.In each experimental period of 7-10 days, the meniscus heights of the water in the glass tubes were measured, using an Ealing cathetometer with 1 x lo-' m accuracy, in runs comprising at least 15 different periods. The flows, in all cases, were from the hot to the cold side of the membrane. The sequences of height differences obtained were transformed into units of pressure which were subsequently divided by the corresponding temperature difference and fitted to a curve with the shape of eqn (1) following the method proposed by Lybanon?' From these fits values of z and yW were obtained.The hydraulic permeability, A, was obtained from z through eqn (2b), then the value of B was obtained from A and ym using eqn (2a). To eliminate errors originating from a bulging of the membrane that occurred at the beginning of each experiment, the initial pressure difference, APo(t = 0), used in the calculations was always greater than the initial experimental value. The values of AP, ranged between 1.5 x lo-, and 2 x lo-, rnH,O.T Results and Discussion In a previous phase of this experimental work, the relationship between the temperatures at which the thermostats were set, the chamber water temperatures and the stirring rates were investigated.'* 239 30 The chamber water temperatures were measured by platinum resistance thermometers (100 Q at 0 "C).The stirring rate was measured with a digital tachometer (On0 Sokki, model HT-430). Two types of experiments were carried out. In one we kept constant the temperature values of the thermostats (44.9 and 34.3 "C for the 600P membrane; 47.2 and 3 1.7 "C for the 500P membrane) and varied the stirring rate (between 0 and 346 r.p.m. for the 600P membrane and between 0 and 330 r.p.m. for the 500P membrane). In the other we kept constant the stirring rate (277 r.p.m. for the 600P membrane and 220 r.p.m. for the 500P membrane) and varied the temperature values of the thermostats.The results obtained for both membranes are analogous. Inspection of the data reveals the following. (a) The values of the differences in temperature between the water in the two bulk phases correspond closely to the temperature differences between the two thermostats. (The literature reports a greater divergence between the thermostat and chamber temperatures.'* 23) (b) For stirring rates > 80 r.p.m., ATb and Tb, the average temperature in the bulk, are constant for a given AT,(AT, = 10.6f0.2 "c, AT, = 8.5k0.2 "c and i+b = 39.4f0.2 "c for the 600P mem- brane; AT, = 15.5k0.2 "C, AT, = 12.2k0.2 "C and T b = 39.4k0.2 "C for the 500P membrane) within experimental error for the two membranes employed. At 0 r.p.m., for the 600P membrane ATb = 10.4k0.2 "c and Tb = 39.9k0.2 "c, while for the 500P membrane AT, = 12.8f0.2 "C and Tb = 41.0k0.2 "C.(c) For a given stirring rate, ATb depends on AT,. Two other groups of experiments were carried out. In the first, the measurements were made at different stirring rates for each membrane while maintaining constant the water temperatures in both chambers. For the 600P membrane these temperatures were t, = 43.7 "C and t, = 35.2 "C; for the 500P membrane they were C, = 45.5 "C and t, = 33.2 "C. In fig. 1 plots of I API us. time at different stirring rates are shown for a few illustrative cases. Tables 2 and 3 show the calculated values of A and B at different stirring rates for the 600P and 500P membranes, respectively. The values of the hydraulic permeability, A, oscillate about an average value in both cases.In each case a linear fit by the least- squares method of the data for A vs. stirring rate was obtained which gave correlation t 1 mH,O = 980.665 Pa.C. Fernandez- Pineda and M. I. Vazquez-Gonzalez 65 1 L I I I I 50 100 150 tlh Fig. 1. Evolution with time of the difference in hydrostatic pressure, I AP I, for different stirring rates: A, 198; 0, 314; x , 340 r.p.m. The solid lines represent the curves fitted to eqn (1) for membrane 600P. The water temperatures in each chamber were tl = 43.7 "C and f , = 35.2 "C. Table 2. Values of A and B at different stirring rates for membrane 600P (tl = 43.7 "C, f , = 35.2 "C and AT, = 8.5 "C) stirring B/ 10-lo mol m-' rate/r.p.m. A/10-12 mol s kg-' K-1 s-1 346 1.8f0.3 314 2.7 f 0.4 277 3.6 f 0.5 230 1.7 f 0.3 198 2.9 & 0.4 157 4.1 k0.8 109 2.8 f 0.4 3.7 k0.4 3.6 f0.4 3.4 f0.3 3.4 k0.3 3.2 k0.3 2.68 _+ 0.25 2.27 f 0.21 Table 3.Values of A and B at different stirring rates for membrane 500P (tl = 45.5 "C, t, = 33.2 "C and AT, = 12.3 "C) stirring B/ rnol m-' rate/r.p.m. A/10-12 mol s kg-' K-1 s-1 330 280 270 220 185 140 120 85 1.3k0.3 4.1 f 0.9 3.0 & 0.6 4.1 f 0.9 5.4k 1.1 2.3 f0.5 2.8 f 0.6 4.8 f 1.2 4.7 k 0.7 4.2 k 0.6 3.9 f 0.6 3.5 f0.5 3.3 & 0.4 3.1 f 0.4 2.7 k 0.4 2.2 f 0.3652 Thermo-osmosis in Water-Cellophane Systems coefficient values of -0.42 for the 600P membrane and -0.37 for the 500P membrane; their respective slopes were -4.12 x mol s kg-l. Student’s t- test was applied to determine whether these slopes were significantly different from zero.In both cases a significance level of 0.05 was set, so that the t-test gave strong evidence of zero slopes. This confirms that, within the range of stirring rates studied, the hydraulic permeability was independent of stirring rate7 and so may be considered to have a constant value. In each case the mean value was chosen : for the 600P membrane it was (2.8 0.4) x mol s kg-l. The values of the thermo-osmotic permeability, B, shown in tables 2 and 3 were calculated using the average values of A given above and those of I AP,/ATb I obtained from the fits of IAP/ATbI us. time data at each stirring rate, for both membranes. In the second group of experiments the water temperature in one of the bulk phases was varied while keeping almost fixed, in most cases, the water temperature in the other, and employing a constant stirring rate; for membrane 600P this was 277 r.p.m.and for membrane 500P it was 220 r.p.m. The evolutions of I AP I with time are analogous to those given in fig. 1 . From the values of z obtained from the fits, the values of the hydraulic permeability were obtained in the same way as in the previous set of experiments. Again it was found that the values of the hydraulic permeability, A , oscillated about an average value. A linear fit by the least-squares method was made to determine A as a function of the average temperature for each membrane. A correlation coefficient of - 0.38 was obtained for membrane 600P, and of -0.36 for membrane 500P; the values of their slopes were -4.65 x mol s kg-l, respectively.Student’s t-test was again applied: in both-cases a significance level of 0.05 was set, so that the t-test gave strong evidence of zero slopes. These results confirm that the hydraulic permeability was independent of the mean temperature and may be considered as constant within the temperature range studied. The average values for the 600P and 500P membranes were (2.04 _+ 0.1 1) x 10-l2 and (3.6 f 0.4) x The values of the thermo-osmotic permeability, B, were calculated from the average values of hydraulic permeability given above and values of I AP,ATb I obtained from the fits of I AP/ATb 1 us. time data at each value of ATb for both membranes. The values of A and B are shown in tables 4 and 5. and -6.09 x mol s kg-l, and for the 500P membrane it was (3.5 f 0.5) x and - 1.79 x mol s kg-’, respectively.The Correction Factor Following the procedure proposed by Belluc~i,’~ the ratio between the flow which would occur at an infinite stirring rate and the flow at a given stirring rate, R, for the same average temperature in the bulk fluids, i“,, and with the same temperature difference between them, ATb, is given by where (ATm)R is the actual temperature difference between the membrane sides at a stirring rate R, K is the thermal conductivity of the membrane-permeant system, 6 is the membrane thickness, h is the convection coefficient in the half-cells andfis a correction factor for the temperature polarization. Note thatfis independent of Tb and ATb, and thus the same factor may be used to interpret experimental results for all the values of ATb and Tb, with the sole restriction that the stirring rate, R, remains constant.Considering the empirical description given by Haase,28 eqn (3) is transformed (in the present case, the hydraulic permeability, A , is independent of the stirring rate) in theC. Fernandez- Pineda and M . I. Vazquez-Gonzalez 653 Table 4. Values of hydraulic permeability (in 10-l2 mol s kg-') and thermo-osmotic permeability (in 1 O-'O mol m-' s-l K-l) (experimental, corrected and calculated) for different temperature dif- ferences, AT,,, at a stirring rate of 277 r.p.m. for membrane 600P. AT,/"C AT," = ATb f '/"C A Bexptl B = f'exptl Bcalc, 36.8 - 33.4 38.3 - 33.6 39.8 - 33.9 41.8 - 34.3 43.6 - 34.7 45.3 - 34.9 40.8 - 37.2 42.3 - 37.6 43.9 - 37.9 45.8 - 38.2 42.7 - 39.3 44.4 - 39.7 46.0 -40.0 44.5 - 41.1 46.4 - 41.5 36.19 - 34.01 37.46 - 34.44 38.74- 34.96 40.45 - 35.65 42.00 - 36.30 43.43 - 36.78 40.15 - 37.85 41.46 - 38.44 42.82 - 38.98 44.44- 39.56 42.09 - 39.91 43.56 - 40.54 44.92 - 41.08 43.89 - 41.7 1 45.52 - 42.38 2.1 f0.5 2.6 f 0.4 2.2 f 0.3 1.6 f 0.2 2.3 f0.3 2.1 f 0.3 1.8f0.3 2.0 f 0.3 3.1 f0.3 2.3 f 0.3 1.5f0.2 2.4 f 0.4 1.9 f0.3 1.7f0.3 1.8f0.3 3.17 f 0.25 3.14f0.25 3.13 & 0.24 2.90 f 0.23 2.83 f0.22 2.62 f 0.21 2.61 f 0.24 2.64 f 0.2 1 2.44 f 0.18 2.27 f 0.18 2.42 f 0.19 2.30 & 0.19 2.17 f0.18 2.28 f 0.28 2.01 f0.16 4.9 f 0.8 4.9 f 0.8 4.9 f 0.8 4.5 fO.8 4.4 f 0.8 4.1 f0.7 4.1 f0.7 4.1 f0.7 3.8 f 0.6 3.5 f 0.6 3.8 f0.6 3.6f0.6 3.4k0.6 3.6 f 0.6 3.1 f0.5 5.1 4.9 4.7 4.5 4.2 4.0 4.3 4.1 3.9 3.6 3.8 3.6 3.4 3.4 3.2 ~~ a AT, is the temperature difference between the sides of the membrane.f is the correction fact or. Table 5. Values of hydraulic permeability (in 10-l2 mol s kg-') and thermo-osmotic permeability (in 1O-lo mol m-' s-' K-' ) (experimental, corrected and calculated) for different temperature dif- ferences ATb, at a stirring rate of 220 r.p.m. for membrane 500P. 35.9 - 32.3 37.4 - 32.6 39.1 - 32.8 40.8-33.1 42.6 - 33.4 44.3 - 33.9 45.5 - 34.1 39.7 - 36.3 41.6-38.1 43.5 -40.2 45.5-42.1 47.4 -44.2 35.05 - 33.14 36.28 - 33.72 37.63 - 34.27 39.00 - 34.90 40.45 - 35.55 41.87 - 36.33 42.83 - 36.77 38.90 - 37.10 40.78 - 38.92 42.73 -40.97 44.70 - 42.90 46.65 - 44.95 2.3 & 0.5 3.0 f0.7 6.2f 1.3 5.1 & 1.1 4.0 f 1 .O 5.3f 1.1 4.6f 1.1 3.5 f0.7 1.7 f 0.4 1.8 f 0.4 2.2 f 0.5 2.7 f 0.6 5.5 & 0.7 5.5 k0.7 4.8 f 0.6 4.7 k 0.6 4.3 f 0.5 4.1 f 0.5 4.0 f 0.5 4.6 & 0.6 4.0 f 0.5 4.1 f0.5 3.8 k0.5 3.8 f0.5 10.3 & 2.5 10.3 f 2.5 9.1 f2.2 8.8 f 2.1 8.1 f 1.9 7.8 f 1.9 7.6 f 1.9 8.6 f 2.0 7.6 f 1.8 7.8 f 1.9 7.1 f 1.7 7.2 f 1.7 10.4 9.8 9.3 8.9 8.5 8.1 8.0 8.3 7.6 7.2 7.1 7.2 To obtain (lAPm/ATbl)R=m a linear fit was made of l/(lAPm/ATbl)R against reciprocal stirring rate.The value of 0 r.p.m. was not used, as it would have modified the conditions of the same T,, or AT,,. The value of the intercept of the linear fit give the value of (AP,/ATb I)R=oo, which corresponds to an infinite stirring rate. From this value the correction factors for temperature polarization are obtained. For membrane 600P (stirring rate 277 r.p.m.) f= 1.56f0.09, while for membrane 500P (stirring rate 220 r.p.m.) f= 1.88 0.12.The values of the 'corrected' thermo-osmotic permeability, P O r r (= B), are calculated by multiplying the values of the experimental thermo-osmotic permeability, (BexptJR, by the correction factor. The results for each membrane are shown in tables 4 and 5. The ratio [AT,/(AT,),] (= the correction factor) was calculated in a similar way by 22 FAR 1654 Thermo-osmosis in Water-Cellophane Systems Mengual et al.,' while Tasaka and Futamura3' used a different method of estimation. In the case of cellulose acetate membrane^,^ the factors for the two membranes used were 1.41 and 1.45. For various types of membranes ranging in thickness from 3 x to 7 x m the values of the correction factor ranged from 1.45 to 1 .35.30 These values are of the same order of magnitude as those presented in this paper.Differential Thermo-osmotic Permeability To establish the relationship between the values of P O r r , obtained for different temperature differences between the two sides of the membrane, with the corresponding differential coefficient, b(t), the latter is expressed in a similar way to the differential diffusion coefficient : 7 9 25 1 I't, J - b( t ) d t p o r r - t2--1 t , where t, and t, are the Celsius temperatures on each side of the membrane. As the form of b(t) is unknown, it was assumed that a development in the form of power series may be admitted, following the method proposed by Hammond and Stokes:26 (6) where a,,, a,, a,, ...are constants. Then, substituting b(t) in eqn ( 5 ) by eqn (6), the following result is obtained : P O r r = a, + (a,/2) ( t , + t,) + (a,/3) ( t i + t, t, + tt) + (a3/4) (ti + ti t, + t, tt + t;) + (a4/5)(t; + ti t, + tf ti + t , t! + tt) + .-- (7) To determine the type of dependence that b(t) has on temperature, the data contained in tables 4 and 5 were fitted to an equation with the form of eqn (7) by multiple regression analy~is,~, in which the number of terms was varied from 2 to 5. Once the fits were made, to select the optimum, the following inspection procedure was used. (i) Observe the corrected determination coefficient for each fit, R2. (ii) Test, by applying the standard Student's t-test at the significance level a = 0.05, whether the individual coefficients may be assumed equal to zero for every fit.(iii) Test, by using the F-test at the same significance level, a = 0.05, whether all coefficients in the model are zero for each fit. When these foregoing tests had been applied, the best fit for the 500P membrane was found to be quadratic and the best fit for the 600 P membrane was linear. Table 6 shows the corresponding functions b(t)/6 for the two membranes. With the object of comparing the results of the present work with those reported in the literature for similar cellophane membranes, the differential thermo-osmotic permeabilities were calculated using the data of Rastogi et aL3 and of Haase et al.,l Note that the experimental values of thermo-osmotic permeability used in these two papers are not corrected for temperature polarization. All the values in the above published papers, and in the present work, are expressed as b(t)/6 in mol rn-, s-' K-l t o permit a comparative analysis between the results obtained from previous data and those of the present work. After carrying out the same type of fits and tests as for the membranes in the present work, the optimum fit for the membrane used by Haase et aL21 was found to be cubic [four terms in eqn (7)]; for the membrane used by Rastogi et aL3 it was quadratic [(three terms in eqn (7)].The results of the above optimal fits are shown in table 6. Inspection of table 6 (and the corresponding graphical representations) suggests the following. (1) Within the temperature range studied in the present work, all the differential coefficients decreased with increasing Celsius temperature.(2) The behaviour of membrane 500P, within the measured temperature range and when extrapolated, is similar to that reported by Rastogi et aL3 Both membranes show a decrease in the differential coefficient with increasing temperature for a given range of temperatures, b(t) = a,+a, t + a , t 2 + ...C. Fernandez- Pineda and M. I. Vazquez-Gonzalez 655 Table 6. Optimal differential thermo-osmotic permeability divided by membrane thickness, b(t)/6, for the different membranes as a function of Celsius temperature membrane b(t)/S/mol m-2 s-' K-' -~ ~ 600P 2.04 x -0.35 x 10-9 500P 6OO2l 6003 39.5 x 15.22 x 10-6t+ 149.2 x 10-'t2 14.3 x 10-5-5.90x 10-6t+67.5 x 1.75 x 10-5+0.11 x 10-6t-8.32 x 10-9t2- 14.57 x 10-12t3 after which they exhibit an increase with temperature.The values of the function b(t)/d for the 600P membrane resemble, within the measured temperature range and when extrapolated, the results of Haase et aL2' since they exhibit a decrease with increasing temperature (up to ca. 50 "C); they reach zero at between 50 and 60 "C, after which they change sign and continue to decrease with higher temperatures. (3) Above a temperature close to 50 "C, the behaviour of the b(t)/d functions for the membranes of Rastogi et aL3 and Haase et aL21 differs significantly. Those of Rastogi et aL3 exhibit a minimum and thereafter increase with temperature, while those of Haase et decrease to zero and continue decreasing with increasing temperature.In our opinion, these differences may result from the different membrane pre-treatment procedures employed by the various investigators. Note that it was not possible to obtain experimental results for temperatures outside the range 33-47 "C because the experimental periods required are so great that the membranes deteriorate. However, investigations using higher temperatures are planned for the future. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I5 16 17 18 19 20 21 22 23 24 25 G. Lippmann, C. R. Acad. Sci. (Paris), 1907, 145, 104. M. Aubert, Ann. Chim. Phys., 1912, 26, 145. R. P. Rastogi, R. L. B. Blokhra and R. K. Agarwal, J. Electrochem. Soc., 1962, 109, 616. R. P. Rastogi and K. Sing, Trans. Faraduy SOC., 1966, 62, 1754, M. S. Dariel and 0. Kedem, J. Phys.Chem., 1971, 79, 1773. J. I. Mengual, J. 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Kmenta, Elements of Econometrics (MacMillan, New York, 1971). Paper 71867; Received 18th May, 1987