J. Chem. Soc., Faraday Trans. I , 1989, 85(6), 1217-1231 The Orthobaric Surface Tensions of the Three Binary Mixtures formed by Krypton, Ethane and Ethene Benilde S. Almeida,* Virgilio A. M. Soares, Ian A. McLuret and J. C. G. Calado Centro de Quirnica Estrutural, Complex0 I, Instituto Superior Tecnico, Avenida Rovisco Pais, 1096 Lisbon Codex, Portugal The surface tensions of krypton-thane and krypton-ethene mixtures have been determined as a function of composition from 116.0 to 124.0 K and for ethene-ethane mixtures from 117.0 to 135.0 K. In each case the deviations from mole-fraction linearity are negative, although least so for the mixture of hydrocarbons. The results have been analysed in terms of a number of theories that take no explicit account of the quadrupole nature of the hydrocarbons.Agreement with simple theories could be achieved by some empirical corrections of the Berthelot and Lorentz combining rules. However, the values of these differ significantly from those obtained with similar treatments applied to the bulk properties of these mixtures. Despite the importance of a knowledge of the surface tensions of simple liquid mixtures for testing both molecular theories and computer simulations of liquid-vapour interfaces, reports of experiments furnishing such results are rare. Essentially the entire stock rests upon the measurements of Fuks and Bellemans' on krypton-methane mixtures, of Sprow and Prausnitz2 on nitrogen-argon, nitrogen-carbon monoxide, nitrogen-methane, argon-methane and methane-carbon monoxide mixtures and our own measurements on methane-tetrafluoromethane mixture^.^ In addition, measure- ments on the mixture argon-krypton have recently been completed at Cornell Uni~ersity.~ With this background and paying heed to the limited number of available mixtures of truly simple substances, we have determined the surface tensions of the three binary mixtures containing krypton, ethane and ethene over a range of temperature using the method of differential capillary rise. The hydrocarbons have quadrupole moments of opposite sign, 3.54 x e.s.u.cm-3 for ethene and -0.80 x e.s.u. cmP3 for ethane; thus the results ought to afford a good test of theories of surface tension that take these factors properly into account, although this is a task that so far we have not undertaken.Experimental Apparatus The three-capillary appartus and the results of our measurements of the surface tensions of pure methane, krypton, tetrafluoromethane, ethane, ethene and dimethylether have already been r e p ~ r t e d . ~ Detailed accounts of the equipment and the experimental procedure are also a ~ a i l a b l e . ~ Materials The pure substances were obtained from the following sources with the indicated purities given by the manufacturers : krypton (Matheson, 99.997 YO), ethane (Air Liquide, i Permanent address: Department of Chemistry, The University, Sheffield S3 7HF. 12171218 Mixture Surface Tensions at Low Temperatures Table 1. Surface tensions, y, and excess surface tensions, yE, as a function of mole fraction, x,, at temperatures, T ~~ x, y/mNm-’ y”/mNrn-’ x, y/mNm-’ y”/mNm-’ 0 0.151 0.249 0.346 0.448 0 0.140 0.233 0.332 0.433 0 0.129 0.2 18 0.3 I3 0.414 0 0.21 1 0.306 0.454 0 0.202 0.295 0.441 0 0.191 0.282 0.425 0 0.149 0.298 0.400 0.499 0 0.149 0.298 0.399 0.498 25.91 21.53 20.70 19.50 18.79 24.96 20.38 19.50 18.38 17.72 24.20 19.22 18.29 17.25 16.66 27.36 23.70 22.00 20.33 26.69 22.52 20.93 19.27 26.07 21.34 19.87 18.22 27.35 26.85 26.50 26.24 26.08 25.69 25.29 24.90 24.65 24.46 krypton( 1 bthene(2) T = 116.0 K 0 0.553 -2.77 0.653 - 2.70 0.885 - 2.47 1 -2.72 T = 120.0 K 0 0.539 - 3.25 0.641 - 3.25 0.880 - 3.43 1 -3.13 T = 124.0 K 0 0.521 - 3.75 0.675 - 3.28 0.873 - 3.95 1 - 3.58 krypton( 1 )-ethane(2) T = 116.0 K 0 0.559 - 1.33 0.71 1 - 1.98 0.886 - 2.07 1 T = 120.0 K 0 0.546 - 1.90 0.700 - 2.45 0.88 1 - 2.47 1 T = 124.0 K 0 0.530 -2.50 0.685 - 2.93 0.874 - 2.96 1 ethene( 1 ethane(2) 0 0.600 - 0.093 0.700 -0.201 0.849 - 0.282 1 -0.287 T = 116.0 K T = 126.0 K 0 0.600 -0.118 0.699 - 0.236 0.849 - 0.294 1 - 0.298 18.42 -2.11 17.43 - 1.66 16.34 -0.87 16.33 0 17.22 -2.63 16.63 - 2.25 15.43 -1.18 15.48 0 16.0 1 - 3.20 15.34 - 2.87 14.32 - 1.52 14.46 0 19.44 18.17 16.95 16.33 18.36 17.04 15.90 15.48 17.27 15.91 14.85 14.62 25.93 25.93 25.68 25.53 24.30 24.16 23.98 23.82 - 1.75 - 1.35 - 0.64 0 -2.21 - 1.80 -0.91 0 - 2.7 I - 2.30 - 1.21 0 - 0.265 -0.196 -0.102 0 - 0.27 1 - 0.222 -0.127 0B.S. Alrneida, V. A . M. Soares, I. A . McLure and J . C. G. Calado 1219 Table 1. (cont.) x, y/mN m-' y"/mN m-' x, y/mNm-' y"/mN m-I T = 135.0 K 0 24.18 0 0.598 22.67 - 0.280 0.148 23.73 -0.146 0.698 22.49 - 0.249 0.296 23.30 - 0.274 0.848 22.28 -0.155 0.397 23.05 - 0.309 1 22.12 0 0.496 22.85 -0.310 Table 2.Parameters of Redlich-Kister equations A , B and C and standard deviations o for krypton-thane and ethene-ethane at various temperatures T T/K A/mN m-I B/mN m-' C/mN m-' a/mN m-' ~ ~~ ~~ krypton( lethane(2) 116.0 -7.742 1.821 0.4091 0.19 120.0 -9.503 3.077 - 2.439 0.12 124.0 - 11.34 4.498 - 5.496 0.06 ethene( 1 )-ethane(2) 117.0 - 1.135 -0.01037 0.7898 0.016 126.0 -1.186 0.006 87 0.4709 0.0 12 135.0 - 1.245 0.035 64 0.1200 0.016 99.9 YO) and ethene (Matheson, 99.98 YO). The purities quoted are considered satisfactory for the work carried out and so, save for deaeration according to techniques described by Calado,' no further purification was undertaken.Results The fluid sample is partitioned between a volume entirely within the cryostat that contains the two-phase system, and therefore the interfaces of interest, and a volume, larger by a factor of at least ten at essentially room temperature, that contains vapour only. The actual composition of the liquid in the capillaries was calculated from the total amounts of substance in the system, obtained by gas volumetry, and the known vapour-liquid equilibria results for these mixtures.' The composition of the liquid phase thus becomes a function of temperature. Since in practice it was difficult to reproduce precisely equal sets of temperatures for each different mixture, the final results listed for a given temperature were obtained by linear interpolation of the experimental quantities.The densities of the liquid mixtures were calculated from the known densities of the components : krypton,s ethane9 and ethene'O and from the excess volumes,' presumed temperature-independent. The method for calculating the correct liquid composition and the surface tension at any given temperature is described el~ewhere.~,'' A detailed statistical analysis of the likely sources of error leads to an expected error of 0.01 mN m-l in the determination of the surface tension and 0.001 in the mole fraction. The results are listed in table 1. The excess surface tensions yE were obtained from the expression Y E = y-xx,y,*-xx,y,*1220 Mixture Surface Tensions at Low Temperatures 0 XKr 0.5 1 0 - 1 w -3 -4 0 XKr 0.5 1 0 - 1 -3 XCIHJ 0 0.5 1 0 -0.1 - IE % -0.2 \ w x -0.3 -0.4 Fig.1. Excess surface tension,y", as a function of mole fraction, x, for (A) krypton-thene mixtures at (a) 116, (b) 120 and (c) 124 K; (B) krypton-ethane mixtures at (a) 116, (6) 120 and (c) 124 K ; and (C) ethene-ethane mixtures at (a) 117, (b) 126 and (c) 135 K. where y, 7;' and y t are the surface tensions of the mixture, component 1 and component 2, respectively, and x , and x, are the mole fractions in the liquid state of components 1 and 2, respectively. The definition implies the acceptance henceforward on grounds of convenience of an ideal surface tension linearly dependent on mole fraction rather than a more fundamentally based concept of surface ideality.B. S. Alrneida, V. A.M . Soures, I. A . McLure and J. C. G. Caludo 1221 Table 3. Excess Gibbs functions, GE, excess enthalpies, H E , and excess surface tensions, yE, for equimolar mixtures of krypton-ethene, krypton-ethane and ethenexthane mixture GF/J mol-I H E J/mol-' yE/mN m-' ~ _ _ kryp ton-e thene 240.4 3 15.0 - 2.42 krypton- t hane 79.8 49.1 - 1.94 et hene-e thane 98.9 192.7 -0.31 (115.8K) (117.7K) (116.0K) (1 15.8 K) ( 1 17.0 K) (1 16.0 K) (161.4 K) (161.4 K) (135.0 K) For krypton-ethane and ethane-ethene mixtures the excess surface tensions are well represented at different temperatures by Redlich-Kister expressions yE = x1 x2[A + B(x1 - x,) + C(X, - x,),] (2) with the least-squares-fitted coefficients and standard deviations 0 that are listed in table 2. The high asymmetry of the surface tension for krypton-ethene mixtures made it impossible to fit the results in the same way, and the fitting equation of Myers and Scott" devised specifically for the representation of highly skewed excess quantities was no more successful.Fig. 1 illustrates the excess surface tensions as a function of mole fraction at a series of convenient temperatures. Discussion Common features of the surface tensions of the three mixtures are the uniformly negative sign of the deviations from ideality and the far from linear dependence on mole fraction. The former is predicted by the quasi-crystalline regular solution theory of Guggenheim, which associates a negative deviation of the surface tension with positive values of the exchange energy. Values of this energy can be obtained from the positive sign of the excess Gibbs function GE and the excess enthalpy H E determined in our laboratories.' The non-linearity is very different from the near-linearity of the surface tensions of argon-krypton mixtures at temperatures far from the critical point of argon that is found both from experiment4 and from computer ~ i m u l a t i o n .' ~ ' ~ ~ It suggests that these mixtures cannot be treated as comprising simple spherical molecules, a point to which we later return. In table 3 we compare G", H E and yE for the equimolar mixtures at temperatures that, except for ethene-ethane, are close enough for purposes of comparison. For that mixture the results that are available for the bulk properties refer to temperatures well above the range of the surface-tension measurements.However, faute de mieux, they serve as a qualitative guide to the behaviour of the mixture at similar temperatures. The table shows that the signs of the excess surface tensions [eqn (l)] mirror those of the bulk excess properties in the normal way, i.e. they are opposite. The most non-ideal mixture, krypton-ethene, does exhibit the largest excess surface tension. The surface behaviour of the two other mixtures is less regular. Krypton-ethane has the higher excess surface tension but the lower excess Gibbs function, even after estimating from the excess enthalpy the effect of the difference in temperatures for ethene-ethane. The irregularity in behaviour with mixtures of such simple molecules is a reminder of the importance of the structure of the interface on the surface-tension behaviour of mixtures. Several statistical theories have been used in the study of interfaces, and the1222 Mixture Surface Tensions at Low Temperatures 0 XKr 0.5 1 0 - -1 IE " ! -2 $ -3 XKr 0.5 0 - I E -1 3 x -2 0 -0.1 - 'E -Oa2 x -0.3 -0.4 -0.5 1 0 1 Fig.2. Excess surface tension, y e , as a function of mole fraction, x, according to the lattice the'ory for (A) krypton-ethene at 116 K (B) krypton-ethane at 116 K and (C) ethene-ethane at 135 K. experimental values of the surface tension may be interpreted in terms of the models used in the theories. From the various treatments, the lattice theory of Guggenheim, the regular solution approach of Sprow and Prausnitz, the corresponding-states treatment and Flory's theory have been the most frequently applied.Quasi-crystalline Theories The lattice theory of homogeneous liquid solutions was applied by Schuchowitsky, l5 Belton and Evans1' and Guggenheiml' to the study of interfaces. The followingB. S. Almeida, V. A. M. Soares, I . A . McLure and J . C. G. Calado 1223 Table 4. W values for krytpon-ethene, krypton-ethane and ethene-thane Wopti rn i z ed / W from H E / W from G"/ mixture J mol-' J mol-I J mol-' krypton-thene 1500 1260 962 kryp ton- thane 400 196 3 19 et heneeet hane 450 77 1 396 equations were devised for the surface tension of binary mixtures in the case of regular solutions : y = y A + (RT/s)/ln (x",xk) + ( W / s ) [(x:)~ - (~3~1 - Wm(x;)'/s (3) y = yB + (RT/s)/ln (x;/xb) + ( W1/s) [(xi)2 - - W ~ ( X ; > ~ / S (4) where yi is the surface tension of pure liquid i, xs and xi are the mole fractions of component i at the surface and in the bulk, respectively, s is the molar surface area obtained from the molar volume V and Avogadro's number NA through the expression s = V2/3Ni'3 ; 1 and rn are geometrical parameters which take the values 1 /2 and 1 /4 for a close-packed lattice, W is the energy of mixing, related to the exchange energy w and the coordination number z by W = NAwz.The description of the interface with only the surface monolayer different in composition from the bulk of the liquid is inherently inconsistent with the Gibbs adsorption equation ; however, it gives sufficiently precise results for the surface tension. The calculated curves for the excess surface tension us.composition for the mixtures krypton-ethene and krypton-ethane at 116.0 K and ethene+thane at 135.0 K obtained through eqn (3) and (4) are presented in fig. 2 together with the experimental points. The dotted lines refer to the value of Wcalculated from H E and the full line to those with empirically optimised values of W. These two sets of values are presented in table 4 along with those obtained from GE. The agreement between the optimised values of W and the values from G" for krypton+thane and ethene-ethane is good. On the contrary, the optimised W for krypton-ethene is closer to the value obtained from H E . Overall, we may conclude that Guggenheim's lattice treatment yields, despite its simplicity, a reasonable empirical description of the surface behaviour for the mixtures studied.Regular-solution Theory The regular-solution approach of Sprow and Prausnitzl' considers the surface region as a regular solution in the Scatchard-Hildebrand sense. The regular-solution theory equations provide reasonable estimates of both surface and bulk properties for solutions of non-polar components. In fact Sprow and Prausnitz compared the calculated and experimental values for the surface tension of a series of binary mixtures involving argon, nitrogen, carbon monoxide and methane and found good agreement. According to this treatment the surface tension is given by where sy is the molar surface area and si the partial molar surface area of component i, fP and are the surface and bulk activity coefficients of i, respectively, and the other symbols enjoy their usual meanings.The data needed for the determination off,' were1224 28 26 24 22 - 'E z 20 E 'h 18 16 14 0 Mixture Surface Tensions at Low Temperatures 26 I I 24 22 20 E k 18 3 16 14 0 0.5 XKr 1 0.5 -YKr 1 26 - 'E 5 'h 24 22 Fig. 3. Surface tension, y, as a function of mole fraction, x, according to the regular-solution theory approach for (A) krypton-thene at (a) 116, (b) 120 and (c) 124 K; (B) krypton-ethane at (a) 116, (6) 120 and (c) 124 K ; and (C) ethene-thane at (a) 117, (b) 126 and (c) 135 K.B. S. Almeida, V. A. M. Soares, I. A . McLure and J . C. G. Calado 1225 taken from ref. (6). The following equation has been used to calculate the surface activity coefficients. where RTlnfl = SP 0; [(6; - d;), + 216; 41 s, = x;S;/x x;q (6) i and 6; is the surface solubility parameter defined as the square root of the surface cohesive energy density cii.The cross-energy density cii is the geometric mean of cii and cii for the pure components, connected through the empirical factor 1 ci. = (1 - r) (c.. c. )1'2. 22 3i (7) Unlike Sprow and Prausnitz, who found that the geometric-mean rule gave a good description of their systems, we had to use the corrected combination rule (7) with 1 taking random, temperature-dependent values. Comparison between theory and experiment for the three systems under discussion is shown in fig. 3. The value 1 = 0.02 12 obtained for the ethene-ethane mixture agrees well with the parameter k,, = 0.016 (correcting the Berthelot combining rule) calculated' through the fitting of experimental G" values to the Frisch-Longuet-Higgins-Widom equation of state. However, for the two other systems there is no correlation between the two sets of values.This is unsurprising since the values of the surface solubility parameter used to get they; are inconsistent with the observed behaviour. The calculation of 6: = c:i was made through the following expression for the cohesive energy density : where AH:ap represents the enthalpy of vaporization of component i, (Htd - Hyat) is the correction for the real gas enthalpy relative to the saturation conditions at which pp and up are the vapour pressure and molar volume, respectively, of component i. The term ( y i - Tdy,/dT) represents the change in energy on forming the surface from the pure liquid.The outcome of the calculations suggests a greater difference between the solubility parameters of krypton and ethane than between krypton and ethene, which implies a greater deviation from ideality (measured by parameter 1) for the former system, in contradiction of the experimental evidence. The main reason for the failure of this treatment when applied to mixtures involving spherical and bicentric molecules has almost certainly to do with the relatively empirical definition of the surface cohesive energy density and the neglect of the shape effects associated with the presence of the quadrupole moment of ethene and ethane. Corresponding-states Treatments More modern approaches to the prediction of the surface tensions of pure liquids and their mixtures are based on the principle of corresponding states (CSP).We discuss el~ewhere~.'~ the applicability of the CSP to the surface tensions of krypton, ethane and ethene and some other liquids confirming that a three-parameter CSP, in particular that suggested by Brock and Bird,20 leads to a fairly good description of the surface tensions of such relatively simple fluids. The generalisation of the CSP to mixtures requires the introduction of mixing rules for the CSP parameters, in this case the critical temperature T, and pressure p , as well as Pitzer's acentric factor cu. Since for simple molecules p r cc &lo3, where E and o are the characteristic energy and size, respectively, associated with the molecular pairwise energy of interaction, and T, cc E, the average values ( p c ) and (T,) for the mixture may be obtained from the average molecular parameters ( E ) and (a).For estimating surface1226 Mixture Surface Tensions at Low Temperatures tensions it seemed appropriate to consider the two-dimensional van der Waals mixing rules (&a2) = x; E l , a;, + 2x1 x , E l , a;, + x f E,, Of2 ( 0 2 ) = Xf O;, + 2x1 x , a;, + X f Of, (9) where E~~ and oij are the energy and the size parameters corresponding to interactions between the pair of unlike molecules i and j. The cross- term parameters are given by the following relations : 0 1 2 = (1 +A,) (011 + O2,>/2 (12) where k,, and j12 are empirical factors connecting the Berthelot and Lorentz combining rules. The main problem is how to relate the value of o for the mixture with the ~o values for the components. In the absence of a well founded relation we took the easiest assumption : Lo = x , Lo, + x , Lo,.Other authors2, have used an alternative approach defining a pseudo-critical volume for one of the components in order to assure conformality with the other component. The reduced surface tension for the mixtures obtained through the expression of Brock and Bird is a dynamic value corresponding to a freshly formed surface. The calculation of the equilibrium value which accounts for the selective adsorption at the surface, the static surface tension, depends on the statistical model of the interface. We chose the monolayer model developed by Prigogine and Defay,,, which correlates ydynamic with ystatic as follows : Ystat = Ydyn - x : (dYdyn/dxl)2/2RT (14) where xi is the bulk mole fraction and s is the molar surface area already defined as V2/3 N,1/3 where V represents now the average molar volume of both components and N , is Avogadro’s number.The results of the application of this treatment to our systems are presented graphically in fig: 4. The full lines denote the composition dependence of the excess surface tension obtained by optimising the fitting to the experimental points through the introduction of arbitrary values for k,, andj,,. The dotted lines were obtained at the lowest temperature using the bulk values for k,, andj,, taken from ref. (23) and (24). Both sets of values are presented in table 5 for each system at a given temperature.The effect of parameter k,, on surface tension is much more important than that of j,,.. The same dependence was found by Soares and McLure in relation to other mixtures. 25 The agreement between the experimental points and the calculated curves shown in fig. 4 is poor even with optimized values fork,, andj,,. In fact these adjusting parameters take up all the shortcomings of the model, in particular its inability to deal with shape factors. The van der Waals mixing rules may constitute a source of error. In addition the surface composition is not considered explicitly in this model, although in the definition of the surface tension it is implicitly taken into account. Conformal Solution Theory embodying Shape Factors An alternative approach to the corresponding-states treatment of surface tension was presented by Murad.26 The theoretical basis of this treatment is well established for conformal fluids.The generalisation to non-conformal fluids is possible through the introduction of empirical shape factors, which depend on the critical constants andB. S. Almeida, V. A . M . Soares, I. A. McLure and J . C. G. Calado 1227 0 -1 1 x - 3 -4 0 -1 -3 0 -0.1 - 'E 5 -0.2 \ W x -0.3 Fig. 4. Excess surface tension, .'.Kr 0 0.5 1 0 XKr 0.5 1 0 xC? HJ 0.5 1 I I I I I I I I I yE, as a function of mole fraction, x, using the corresponding-states theory for (A) krypton-thene mixtures at (a) 116, (b) 120 and (c) 124 K ; (B) krypton+thane mixtures at (a) 116, (b) 120 and (c) 124 K; and (C) ethene-thane mixtures at 117 K.1228 Mixture Surface Tensions at Low Temperatures Table 5.Bulk and surface values for the parameters k,, andj,, bulk values surface values system k,, J,, k12 j , , krypton-ethene at 116.0 K” 0.063 0.00434 0.11 0.012 krypton-ethane at 1 16.0 K1* 0.038 0.0066 0.08 0.005 ethenexthane at 135.0 K18 0.016 0 0.016 0 acentric factors of each substance. The surface tension y,(T) of a fluid a at temperature T is related to the corresponding value yo of a reference fluid by the expression: (15) where T;, V; and T,“, V,“ refer to fluid a and the reference fluid, respectively, and flz,o and $ a , o are state-dependent shape factors which can be obtained using the empirical method described by Hanley et al.27 Eqn (1 5 ) may also be applied to the calculation of the surface tension of mixtures considering fluid a as a substance equivalent to the mixture with critical constants obtained from the values for the pure components by the method described for the CSP treatment.The outcome of the application of Murad’s approach to pure components19 is encouraging and justifies its generalisation to mixtures. The results obtained for the mixtures krypton-ethene, krypton-ethane and ethene-ethane are presented in fig. 5. The agreement between the theoretical and experimental points is good for ethene-ethane mixtures but apparently poor for the mixtures involving krypton, perhaps because the change in the surface composition is not taken into account. Y,(T) = ( f l a , o TYT,“) (V,“/4,,o V,“)”I”Y~(TT~/B,.~ T,“) The Flory-Patterson Treatment A formal correlation of our results was obtained by combining Flory’s theory of mixtures with the corresponding-states principle formulated by Patterson and Rastogi.28 Lam and BensonZ9 among 0 t h e r s ~ ~ 3 ~ ~ obtained values for the surface tension of pure fluids and then mixtures using Patterson’s equation for the reduced surface tension yR: yR( VR) = MV;15/3 - [( V;l3 - I)/ V;] In [( Vk’I” - 0.5)/( V;/I” - l)] where M is the fractional reduction in the number of nearest neighbours for a cell on the surface relative to one in the bulk of the liquid, and VR is the reduced volume. The reduced surface tension is defined as where k , is Boltzmann’s constant and p* and T* are the characteristic reduction factors for pressure and temperature, respectively.They depend on the thermal expansion coefficient a and the isothermal compressibility, K .The reduced volume in Flory’s theory is given by the following expression: VR = [(I +4aT/3)/(1 +aT)I3. The treatment maybe generalised to mixtures by replacing the pure-component characteristic values with appropriate averages dependent upon the adjustable parameters s12, the ratio of molecular surface area of contact per segment for each species, and X12, a cross-interaction parameter. We applied this theory to the calculation of the surface tensions of pure krypton andB. S. Almeida, V. A. M . Soares, I. A. McLure and J. C. G . Calado 1229 0 0.5 1 XKr 0 0.5 -4- K r 1 0 0.5 -vC. HJ 1 Fig. 5. Surface tension, y, as a function of mole fraction, x, using Murad’s approach for (A) krypton-thene at 116 K, (B) krypton-ethane at 116 K and (C) ethene-thane at 117 K.42 FAR I1230 Mixture Surface Tensions at Low Temperatures Table 6. Experimental and calculated values (Flory's theory) for the excess surface tension yE of krypton( lethene(2) at 1 16.0 K 0.148 - 2.77 -0.388 0.241 - 2.70 -0.383 0.337 - 2.97 - 0.380 0.437 -2.72 - 0.377 0.541 -2.11 -0.369 0.642 - 1.66 - 0.369 0.879 - 0.87 -0.342 ethene and their mixtures. The predictions for the pure substances are poor, as the comparison between the experimental values at T = 116.0 K for the surface tension of krypton y = 16.33 mN m-l and ethane y = 25.72 mN m-' with the calculated values 14.17 and 2 1.60 mN m-', respectively, confirms. The calculated excess surface tensions for the krypton-ethene mixtures at 116.0 K are compared in table 6 with the experimental results.Again the success of the treatment is limited since although the negative sign of yE is correctly predicted its order of magnitude is not. These discrepancies are not unexpected, since even the bulk properties of this system are at best only qualitatively described by Flory's Furthermore, the treatment ignores adsorption which may explain the low values obtained for yE. The mediocre description of the experimental results, despite heavy parameterisation, leaves this theory relatively unappealing. Conclusion Although this review of the success of the different theories in describing our results is not exhaustive, it is clear that no treatment adequately describes the surface tensions of binary mixtures of nearly simple substances whose molecules are but slightly non- spherical and whose polarity is restricted to relatively small quadrupole moments, albeit in one mixture of opposite signs.Only the non-realistic lattice approach gave a reasonably quantitative account of our results without recourse to correcting parameters of unknown physical significance. The accurate prediction of the bulk properties of mixtures of polar liquids is possible only by taking specific account of multipole moments and p~larisabilities.~~~ 33 These effects are equally important at the interfaces of mixtures of polar liquids, and we intend to pursue the study of such mixtures both from the experimental point of view arid through the interpretation of the results using modern theories of interfaces, for example those based on perturbation that incorporate the influence of polarity on interfacial structure.We gratefully acknowledge receipt of the NATO research grant no. 194.80 in support of this work. References 1 S. Fuks and A. Bellemans, Physica, 1966, 32, 594. 2 F. B. Sprow and J. M. Prausnitz, Trans. Faraday SOC., 1966, 62, 1097. 3 V. A. M. Soares, B J. V. S. Almeida, I. A. McLure and R. A. Higgins, Fluid Phase Equilibria, 1986, 32, 9.B. S. Almeida, V. A. M. Soares, I. A . McLure and J . C. G. Calado 1231 4 K. C. Nadler, J. A. Zollweg, W. B. Streett and I. 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