J. Chem. SOC., Faraday Trans. I, 1989, 85(6), 1357-1363 Measurement of Activity Coefficients, Mass-transfer Coefficients and Diffusion Coefficients in Multicomponent Liquid Mixtures by Reversed-flow Gas Chromatography? Pericles Agathonos and George Karaiskakis" Physical Chemistry Laboratory, University of Patras, Patras, Greece Reversed-flow gas-chromatography sampling is a new method for studying heterogeneous catalysis, diffusion, adsorption and evaporation. We report its application to the simultaneous determination of mass-transfer coefficients for the evaporation of multicomponent liquid mixtures and the diffusion coefficients of vapours from these liquid mixtures into the carrier gas. Using suitable mathematical equations the gaseous equilibrium concentration of each component under study, in the pure state and in the mixture, has been determined. Using these equilibrium concentrations, together with the mass-transfer coefficients and diffusion coefficients already determined, activities and activity coefficients were calculated.The liquid mixtures used were : n-hexane-n-heptane, n-hexane-methanol, n-hexane-ethanol, n- hexane-butan-2-01, n-hexane-acetone, n-hexane-methanol-butan-2-01, n- hexane-ethanol-butan-2-01 and n-hexane-methanol-butan-2-01-water. All the activity coefficients found were compared with those calculated by the UNIFAC (universal quasichemical functional-group activity coefficients) method. Most problems in chemical-engineering design are concerned with separation operations (e.g. distillation and extraction). For a rational design of such separation processes, we require quantitative information on phase equilibria and on interface transport in multicomponent liquid mixtures.Satisfactory experimental equilibrium and interface transport data are seldom available for the particular conditions of composition, temperature and pressure required in a particular design problem. A new method, termed reversed-flow gas chromatography (r.f.g.c.), which gives information not only on phase equilibria,. but also on interphase transport in multicomponent liquid mixtures, is presented in the present work. This method is applied to various physico-chemical measurements. It consists of reversing the direction of flow of the carrier gas from time to time. If other gases are contained in the carrier gas and their concentration depends on a rate process within the chromatographic column, each flow reversal creates a perturbation in the chromatographic elution curve in the form of extra peaks (termed 'sample peaks').Then, by repeatedly reversing the flow of the carrier gas, a repeated sampling of this rate process is performed. and then to the dehydration of alcohols3 and the deamination of primary arnine~.~ The method has been extended to the determination of adsorption equilibrium con~tants,~ rates of catalyst dehydration,6 Lennard-Jones parameters,' molecular diameters and critical volumes in gases8 and diffusion coeffi~ients.~-'l The method has recently been reviewed.12 Recently r.f.g.c. has been used to determine activity coefficients for the alcohol component in binary liquid mixtures of alcohol and water at a constant temperature and (Greece).R.f.g.c. was first applied to kinetic studies in heterogeneous Presented in part at the 1988 World Chromatography and Spectroscopy Conference, May 16-17, Corfu 46 1357 FAR 11358 Thermodynamics of Solutions oven 2 I I ' I Fig. 1. Experimental set up for measuring mass- transfer coefficients, diffusion coefficients and activity coefficients in multicomponent liquid mixtures by the reversed-flow gas-chromatography method. various alcohol mole fractions, and also to estimate excess partial molar thermodynamic functions of mixing for alcohols in water.13 In addition, r.f.g.c. has been used to determine mass-transfer coefficients for the evaporation of the alcohol component at various mole fractions from alcohol-water mixtures and diffusion coefficients of the alcohol vapour into the carrier gas.l* The objective of this work was to determine activity coefficients, together with mass- transfer coefficients, for the evaporation of multicomponent (with two, three or four substances) liquid mixtures.The difference from previous studies is that, while the flame ionization detector used previously did not detect water, now all the vapour components are detected and so a separation chromatographic column, filled with an adsorbent, is necessary. The extension of the reversed-flow gas-chromatography technique to multicomponent mixtures is extremely attractive because, while the number of pure fluids of interest in chemical technology is large, the number of different mixtures is much larger.Experimental The solutes used were n-hexane, n-heptane, methanol, ethanol and acetone (all Uvasol grade from Merck A.G.) and butan-2-01 (laboratory-reagent grade from B.D.H.). The carrier gas was helium of 99.99% purity from Linde (Athens). The apparatus used and the experimental procedure followed have been described elsewhere ;I5 only slight modifications were made. A conventional gas chromatograph (Pye Unicam, series 104) contained in its oven (fig. 1, oven 1) two sections of lengths I' and 1 of a stainless-steel chromatographic column [(loo+ 100) cm x 4 mm i.d.1 containing no chromatographic material. A stainless-steel diffusion column of length L (100 cm x 4 mm i.d.) was connected perpendicularly at its upper end to the middle of the column I' + 1.At the lower end of column L a 4 cm glass tube containing 0.5 cm3 of the pure liquid or liquid mixture was connected. The end D, of the column of length I'+Z was connected to the carrier-gas supply while the other end D, was connected to the separation column of length L', which was placed in oven 2 in another gas chromatograph. The end of this column was connected to the flame ionization detector via a six-port valve. The length of the separation column L' for the experiments with theP. Agathonos and G. Karaiskakis 1359 66 60 54 ro Imin Fig. 2. A reversed-flow chromatogram, showing two sample peaks for the diffusion of butan-2-01 (a), ethanol (b) and n-hexane (c) vapours into helium at 333.2 K and 1.2 atm.n-hexane-n-heptane mixture was 51 cm and it was packed with y-Al,O, (100-120 mesh from MCB), while for the experiments with alcohols and other substances it was of length 43 cm and was packed with 20% Carbowax 20 M on Chromosorb P, 60-85 mesh. At a given time after the liquid or the liquid mixture was placed in position, an asymmetric concentration us. time curve for the vapours of the pure liquid or the liquid mixture was recorded, and was seen to rise continuously and approach a limiting plateau. This concentration us. time curve consists of one, two or more numerous substances. During the rise period, and also when the plateau was reached, flow reversals for 12 s [from the x direction (fig. 1) to the opposite one and vice uersa] were effected by means of the six-port valve.This time period was shorter than the gas hold-up time in column sections Z’, I and L’. When the gas flow was restored to its original direction, sample peaks (one, two or more) were recorded (fig. 2). This reversal of the flow of the carrier gas was repeated several times with the same duration of backward flow. This gave rise to a series of peaks corresponding to various times to from the beginning of the experiment. The pressure drop along column I’ + 1 + L’ was ca. 0.2 atm.? The working temperature range was 298.2-338.6 K for the evaporation of the mixtures (oven 1) and 343.2- 473.2 K for the chromatographic material (oven 2). The volumetric carrier-gas flow rate, V, at ambient temperature was 0.55 cm3 s-l. t 1 atm = 101 325 Pa.46-21360 Thermodynamics of Solutions Results and Discussion In a previous paper15 it was shown that each sample peak produced by a short flow reversal is symmetrical, and its maximum height h from the final baseline is given by 2k, Dc, v(k, L + 0) h = 2c(l', to) = { 1 - exp [ - 2(k, L + D ) t,/L2]) where c ( t , to) is the vapour concentration at x = I' (cf. fig. l ) , the time to is measured from the moment of placing the liquid at the bottom of column L to the last backward reversal of gas flow, kc is the mass-transfer coefficient for solute evaporation, D is the diffusion coefficient of the solute vapour into the carrier gas, c, is the concentration of the vapour in equilibrium with the bulk liquid phase at the working temperature and u is the linear velocity of the carrier gas.Using eqn (1) the values of D and k, can be determined for the substances under study, as described in detail e1~ewhere.l~. l5 As has been shown in a previous paper13 the equilibrium concentration, c,, of a solute is given by c, = vh, [(LID) + (1 / k c ) ] (2) 2 where h, is the maximum value (at infinite time) for the peak height. This concentration can be calculated using the above equation, as all quantities on the right-hand side are known. If experiments are performed with liquid mixtures giving c, and with pure solutes leading to c:, the ratio C,/C: is equal to p / p * which is the activity a of each component in the liquid mixture, assuming that the deviation of the solute vapour from ideal behaviour is small. Thus where co, h,, D and k, refer to the component in the mixture and c t , h z , D* and k,* to the pure component.We can also calculate the activity coefficients, y , for the components of the mixture using the relationship Several experiments were performed with various binary liquid mixtures, in order to determine mass-transfer coefficients for the evaporation of the compounds of the mixtures, diffusion coefficients of the vapours of these compounds into helium, as well as activities and activity coefficients of the components of the mixtures. These results are summarized in table 1. All values of the diffusion coefficients were calculated using the Hirschfelder- Bird-Spotz (HBS) equation,16 but the diffusion coefficient for butan-2-01 was calculated using the Fuller-Schettler-Giddings (FSG) method.17 A comparison of the diffusion coefficients found with those calculated theoretically permits the calculation of the method's accuracy, which is defined as If we disregard the high value of the mass-transfer coefficient for n-hexane in solution with methanol and the high value of the diffusion coefficient of methanol vapour into the carrier gas helium in the same solution, both of which can be attributed to accidental errors, the following conclusions hold for all other systems.The mass-transfer coefficients for the liquids in solution are smaller than those in pureP. Agathonos and G. Karaiskakis 1361 Table 1. Mass-transfer coefficients of pure liquids and liquid mixtures, diffusion coefficients into a carrier gas (helium) and activity coefficients (experimental, calculated and literature values) at various temperatures and a pressure of 1.2 atm pure substance or mixture D/ 1 OP3 cm2 s-I accuracy kc this T/K .'c /lop4 cm s-I work calcd (YO) Y,,,~, ycalcd ylit" n-hexane methanol butan-2-01 n- hexane acetone n-hexane ethanol n-hexane- methanol n-hexane- butan-2-01 n-hexane- acetone n-hexane- ethanol ethanol- n-hexane n- hexane- n-heptane 333.2 333.2 333.2 318.2 3 18.2 298.2 298.2 333.2 333.2 3 18.2 298.2 298.2 318.2 1 1 1 1 1 1 1 0.054 0.946 0.094 0.906 0.1 0.9 0.1 0.9 0.1 0.9 0.097 0.903 162 355 49 67 123 50 44 497 279 79 16 66 29 38 21 31 38 34 25 320 317 0.9 794 604 23.9 337 317 5.9 305 292 4.3 402 397 1.2 247 261 5.7 392 398 1.5 308 317 2.9 1246 604 51.5 460 317 31.1 424 317' 25.2 295 292 1.0 393 397 1.0 246 261 6.1 346 398 15.0 394 398 1.0 249 261 4.8 297 294 1.0 262 251 4.2 12.46 13.06 12.21 1.88 3.27 3.34 6.20 3.76 3.40 4.83' 6.01 5.35 5.92 9.20 8.01 7.13 0.64 0.97 - 0.88 0.96 - ~ " Literature values given by Fredenslund et a l l 8 other values using the HBS method.values using the UNIFAC method. Theoretical values using the FSG method; all Values calculated from the empirical eqn (6); all other liquids, as observed previously.14 The values of the diffusion coefficients of the vapours are very close to the theoretical ones. All the diffusion coefficients show that the effective diffusion coefficient in a multicomponent mixture is approximately equal to the diffusion coefficient of each component in the pure carrier gas. The presence of chromatographic materials (y-Al,O, and 20 % Carbowax 20 M on Chromosorb P) in column L' does not seem to influence the values of the diffusion coefficients found.The values of the activity coefficients for the liquids in solution (especially the values for the solutions of n-hexane and simple alcohols) are close to those determined theoretically by the UNIFAC (universal quasichemical functional-group activity coefficients) method based on the quasichemical theory of liquid solutions, and described by Fredenslund et a1.l' and Oishi and Prausnitz.lg These values, ycalcd, and those that Fredenslund et al.la present as typical experimental values, ylit, are given in the last two columns of table 1 . Our experimental values (especially those for n-hexane in solutions with methanol and ethanol) are closer to the literature experimental values than the theoretical ones.Apart from the UNIFAC method, there are several other methods and many empirical equations for predicting activity coefficients in binary mixtures. For instance, the activity coefficients at infinite dilution, 7 7 , of alkanes in ketones can be estimated by the empirical equation given by Reid et al.*O1362 Thermodynamics of Solutions Table 2. Mass-transfer coefficients of liquids in multicomponent mixtures, diffusion coefficients into a carrier gas (helium) and activity coefficients (experimental and calculated values) at constant temperature (333.2 K) and pressure (1.2 atm) D/ I 0-3 cm2 s-' kc this accuracy Yexptl Ycalcd mixture X cm s-l work calcd ( O h ) butan-2-01- ethanol- n-hexane butan-2-01- methanol- n-hexane butan-2-01- methanol- n-hexane- waterb 0.807 0.172 0.02 1 0.807 0.172 0.021 0.360 0.077 0.009 0.554 103 229 20 1 112 222 140 135 383 230 320 493 315 322 717 318 30 1 60 1 320 317" 484 317 317" 604 317 317" 604 317 0.9 1.8 0.6 1.6 15.8 0.3 5.3 0.5 0.9 0.678 1.047 2.143 0.789 1.053 2.337 1.427 1.121 0.999 1.084 4.167 1.002 1.185 4.018 1.557 0.8 19 - " Theoretical values using the FSG method; all other values using the HBS method.detectable by a flame ionization detector. Not In this equation N , and N2 are the total numbers of carbon atoms in the alkane and ketone, respectively, Ni and N i are the numbers of carbon atoms in respective branches of the ketone, and E , n and 0 are constants depending on the system and the temperature, which are given by Reid et aL2' The application of this equation to the system n-hexane-acetone (x, = 0.1 and T = 318.2 K) resulted in a value of y? of 4.83, instead of 3.76 calculated by the UNIFAC method, which is closer to our experimental value (cf. table 1).Following the experiments with binary mixtures, experiments were performed with two ternary mixtures, and with one quaternary mixture. The results of these experiments are listed in table 2. With respect to the results of table 2 the following tentative conclusions can be drawn. (i) While in binary mixtures the mass-transfer coefficients of the two components are lower than those of the same compounds in the pure state, in ternary and quaternary mixtures the mass-transfer coefficients are in some cases higher than those of the pure compounds.This is probably due to the irreversible adsorption of these components on the solid material contained in column L' in the presence of the other components of the mixtures. (ii) The diffusion coefficients of the vapours from these liquid mixtures into the carrier gas helium are very close to those determined theoretically for the diffusion of the pure compounds into helium. (iii) The activity coefficients for the alcohol compounds in the multicomponent mixtures found by the present method are relatively close to those determined by the UNIFAC method. The abnormally large divergences for the activity coefficients of n- hexane might be attributed again to the separation material in column L'. For instance, while the adsorption of pure n-hexane on the separation material might be a reversible process, the adsorption of n-hexane in the presence of the other components of the mixtures might not be reversible, because of the appearance of competitive effects.Clearly, further work is necessary regarding the simultaneous study of the properties of the two interfaces, e.g. liquid-vapour and vapour-solid, related to this problem. It is hoped that the present paper, together with previous ones,13-15 will help to introduce workers in various fields of separation processes (e.g. distillation andP. Agathonos and G. Karaiskakis 1363 extraction) to the technique of reversed-flow gas chromatography, which is a new tool for studying the thermodynamic properties of solutions.We acknowledge the help of Mrs M. Barkoula. References 1 N. A. Katsanos and I. Georgiadou, J. Chem. SOC., Chem. Commun., 1980, 242, 640. 2 N. A. Katsanos, J. Chem. SOC., Faraday Trans. I , 1982, 78, 1051. 3 G. 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