Langmuir 2001, 17, 4261-4266
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Extended Langmuir Isotherm for Binary Liquid Mixtures A Ä ngel Pin˜eiro, Pilar Brocos, and Alfredo Amigo* Departamento de Fı´sica Aplicada, Facultad de Fı´sica, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain
J. Gracia-Fadrique* and M. Guadalupe Lemus Laboratorio de Superficies, Departamento de Fı´sica y Quı´mica Teo´ rica, Facultad de Quı´mica, Universidad Nacional Auto´ noma de Me´ xico, Me´ xico D. F. 04510, Me´ xico Received August 21, 2000. In Final Form: April 10, 2001 A new model for describing the surface tension of binary liquid mixtures as a function of the bulk composition over the whole concentration range is presented. We first derive an equation relating surface and bulk volume fractions that generalizes the Langmuir isotherm so as to cover the entire range of concentrations. By combining this isotherm with a new mixing rule for nonideal solutions, we obtain an equation with two adjustable parameters, one measuring the lyophobicity of one component and the other accounting for the effect of molecular interactions. The model provides an excellent description of surface tension data for a wide variety of solutions with π0 ) σA - σB values ranging from 2.2 to 51.0 mN/m.
Introduction Molecular interactions in liquid mixtures are generally studied in terms of bulk thermodynamic properties such as excess volumes, excess enthalpies, excess heat capacities, and excess Gibbs energies, for which many theories have been developed in the past century (notably the Flory, TK, UNIQUAC, UNIFAC, ERAS, and SAFT models). Less attention has traditionally been paid to the theory of surface tension. In fact, most studies of surface tension have focused on the effects of strong surfactants, generally in aqueous solution.1-4 Such systems may often be dealt with using the Langmuir isotherm
βxB ) Γmax 1 + βxB Γ
derived a thermodynamic model for ideal mixtures. Guggenheim7 gave a statistical mechanics treatment of regular solutions that was later applied by others.8,9 Eberhart10 presented a one-parameter equation for binary liquid mixtures assuming surface tension to be a linear function of surface layer mole fraction. Sprow and Prausnitz11 proposed a model relating the surface tension of regular solutions to activity coefficients and requiring vapor-liquid equilibrium data, and in a subsequent paper12 extended the theory to complex mixtures, but both versions rely on the assumption
Ai ) vi2/3NAv1/3
(2)
(1)
where xB is the mole fraction of surfactant B in the bulk solution, Γ is the number of moles of surfactant per unit area at the surface, Γmax is the value of Γ when the surface is saturated, and β is a measure of the lyophobicity of the surfactant. However, this equation becomes inconsistent at high xB unless β . 1, and is therefore unable to model the behavior of mixtures of mutually miscible nonelectrolytes with very similar surface tensions, for which the surface is not saturated and no component undergoes aggregation. Models of the surface tension of nonaqueous solutions have been limited in scope to particular sets of mixtures such as regular or ideal solutions, and many require measurements of other thermophysical properties, such as activity coefficients.5 Belton and Evans,6 for example, * To whom correspondence should be addressed. E-mail:
[email protected];
[email protected]. (1) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley-Interscience: New York, 1997. (2) Chattoraj, D. K.; Birdi, K. S. Adsorption and the Gibbs Surface Excess; Plenum Press: New York, 1984. (3) Gracia-Fadrique, J. Langmuir 1999, 15, 3279. (4) Aveyard, R.; Haydon, D. A. An Introduction to the Principles of Surface Chemistry; Cambridge University Press: Cambridge, UK, 1973. (5) Strey, R.; Viisanen, Y.; Aratono, M.; Kratohvil, J. P.; Yin, Q.; Friberg, S. E. J. Phys. Chem. B 1999, 103, 9112. (6) Belton, J. W.; Evans, M. G. Trans. Faraday Soc. 1945, 41, 1.
where Ai is the partial molar area occupied at the surface by component i, vi is its molar volume, and NAv is Avogadro’s number; this assumption implies that vi is the same for both bulk and surface phases and for both the pure liquid and the mixture, and that all molecules are spherical. Shereshefsky13 developed a three-parameter equation for surface tension which was shown by Hansen and Sogor14 to be inconsistent with the Gibbs adsorption equation, although its three parameters allowed it to correlate data well. Other authors15,16 have modified and/ or tested some of the models cited above. Finally, Reid et al.17 noted that many thermodynamical approaches to the surface tension of mixtures, whether based on classical or statistical thermodynamics, arrive at expressions like (7) Guggenheim, E. A. J. Phys. Chem. 1945, 41, 150. (8) Evans, H. B.; Clever, H. L. J. Phys. Chem. 1964, 68, 3433. (9) Schmidt, R. L.; Randall, H. C.; Clever, H. L. J. Phys. Chem. 1966, 70, 3912. (10) Eberhart, J. G. J. Phys. Chem. 1966, 70, 1183. (11) Sprow, F. G.; Prausnitz, J. M. Trans. Faraday Soc. 1966, 62, 1105. (12) Sprow, F. B.; Prausnitz, J. M. Can. J. Chem. Eng. 1967, 45, 25. (13) Shereshefsky, J. L. J. Colloid Interface Sci. 1967, 24, 317. (14) Hansen, R. S.; Sogor, L. J. Colloid Interface Sci. 1972, 40, 424. (15) Suri, S. K.; Ramakrishna, V. J. Phys. Chem. 1968, 72, 3073. (16) Schmidt, R. L. J. Phys. Chem. 1967, 71, 1152. (17) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987.
10.1021/la001210s CCC: $20.00 © 2001 American Chemical Society Published on Web 06/16/2001
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∑ i)1
() xiγi γiS
Ai(σ - σi)
exp
RT
Pin˜ eiro et al.
so that at equilibrium
)1
(3) φBS )
where xi is the mole fraction of component i in the bulk liquid, γi is the activity coefficient of i in the bulk liquid (normalized so that γi f 1 as xi f 1), γiS is the activity coefficient of i in the surface phase (normalized so that γiS f 1 as the surface phase becomes identical with that of pure i), σ and σi are the surface tensions of the mixture and of pure i, respectively, and Ai is approximated by eq 2. In this paper we present a straightforward model based on a generalized Langmuir equation. Instead of mole fractions xB and surface coverages Γ/Γmax, it uses bulk and surface volume fractions φB and φBS; since volume fractions take into account the relative sizes of molecules as well as their number, we thus avoid the introduction of additional parameters to account for size effects (as, for instance, in the Volmer equation18). The new model has two adjustable parameters, one representing the lyophobicity of one of the components and the other representing the effect of interaction between components A and B. It is able to describe the surface tension of a wide variety of binary mixtures as a function of their bulk composition over the whole concentration range. The Model Surface and Bulk Compositions. The Langmuir isotherm19 for a solution of B in A can be derived by considering equilibrium between the processes of adsorption and desorption of B at the surface. The adsorption and desorption velocities are
1 + (β - 1)φB β′φA
(8)
1 + (β′ - 1)φA
where β′ ) 1/β. Note that eqs 8 hold over the whole concentration range in the sense that φB ) 1 w φBS ) 1, whereas in eq 5 xB ) 1 w θ ) β/(1 + β) < 1; the assumption that xB does not limit desorption of B means that the Langmuir isotherm is only consistent for small xB or β . 1. Henry’s law is also verified at infinite dilution by the new isotherm:
φBS ) βφB
(9)
It is worth noting that eq 8 can be rewritten in the form
φBS )
β(φB/φA) 1 + β(φB/φA)
(10)
where (φB/φA) ) (xBvB/xAvA) can be understood as the ratio of occupations. The Langmuir isotherm for B is obtained upon setting vA ) vB and xB f 0. The bulk composition needed in order to obtain a given ratio of the components at the surface is derived from eq 8 by solving for φB:
φB )
φBS βφAS + φBS
(11)
From eq 11 the value of φB at which φBS ) φAS, denoted φBC, is
vads ) kads(1 - θ)xB vdes ) kdesθ
(4)
where kads and kdes are rate constants, θ is the surface coverage Γ/Γmax, and xB is the mole fraction of B in the bulk. Note that the concentration of B in the bulk does not limit its desorption. At equilibrium, vads ) vdes, so that
θ)
φAS )
βφB
βxB 1 + βxB
(5)
where β ) kads/kdes. For very small xB, eq 5 reduces to the Henry isotherm:
θ ) βxB
(6)
Under these conditions, β is Henry’s constant for two dimensions and measures the lyophobicity of the solute, i.e., its tendency to be adsorbed by the surface. To take the relative sizes of A and B into account, instead of using the surface coverage θ and the mole fraction xB we use volume fractions: we consider the surface to consist of a thin layer of finite depth, and we denote by φBS the fraction of the volume of this layer occupied by B and by φB the fraction of the bulk volume occupied by B. We also give A and B the same status by allowing for the concentration of B to limit its desorption. With these changes, eqs 4 become
vads ) kads(1 - φBS)φB vdes ) kdes(1 - φB)φB
S
(7)
φBC ) 1/(β + 1)
(12)
The equilibrium point of the adsorption/desorption process is given by the value of β. If β ) 1 (kads ) kdes), then φB ) φBS and φA ) φAS; if β < 1 (kads < kdes), then φB > φBS; if β > 1 (kads > kdes), then φB < φBS. Deviation of Surface Tension from Ideal Mixing Behavior. We assume that the surface tension of an ideal solution (one obeying Raoult’s law) is given by
σ ) φAσA + φBσB
(13)
We first modify eq 13 to account for differences between φA (respectively φB) and φAS (respectively φBS) in nonideal mixtures:
σ ) φASσA + φBSσB
(14)
A similar assumption was made by Eberhart10 and by Connors and Wright,20 with the difference that they employed surface mole fractions instead of surface volume fractions. We next account for structural changes and modifications in the cohesion forces by introducing a Margules type interaction term:
σ ) φASσA + φBSσB - λφASφBSπ0
(15)
where π0 ) σA - σB and we assume, without loss of (18) Volmer, M. Z. Phys. Chem. 1925, 115, 253. (19) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (20) Connors, K. A.; Wright, J. L. Anal. Chem. 1989, 61, 194.
Langmuir Isotherm for Binary Liquid Mixtures
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Table 1. Values of the Parameters r and β Obtained by Fitting Eq 16 to Surface Tension Data Taken from the Literature, Together with the Corresponding Values of π0 and of the objective function χ2 A THF THF THF THF THF ethanol ethanol 1-propanol 1-propanol benzene benzene benzene dodecane acetone water water water a
B hexane heptane octane nonane decane hexane heptane hexane heptane hexane isooctane hexafluorobenzene isooctane isooctane methanol 2-propanol 1-propanol
π0/(mN‚m-1) 8.80 7.00 5.54 4.36 3.40 3.95 2.16 5.34 3.55 10.45 9.64 7.1 6.58 4.67 49.80 51.01 49.06
R
χ2
β
1.00 1.00 1.00 1.48 ((0.05) 1.60 ((0.03) 2.06 ((0.03) 2.09 ((0.03) 2.01 ((0.02) 1.89 ((0.03) 1.00 1.00 1.00 1.00 2.23 ((0.04) 1.00 1.59 ((0.06) 1.63 ((0.05)
10-5
7× 6 × 10-5 2 × 10-5 4 × 10-5 2 × 10-5 8 × 10-5 1 × 10-4 4 × 10-5 5 × 10-5 2 × 10-4 1 × 10-4 2 × 10-4 5 × 10-5 2 × 10-4 4 × 10-5 5 × 10-5 3 × 10-5
1.82 ((0.03) 1.82 ((0.02) 1.85 ((0.01) 1.24 ((0.07) 1.22 ((0.04) 0.98 ((0.03) 1.34 ((0.04) 0.86 ((0.02) 1.17 ((0.04) 2.19 ((0.06) 1.82 ((0.04) 2.55 ((0.07) 1.56 ((0.02) 1.56 ((0.06) 3.16 ((0.03) 6.59 ((0.41) 12.57 ((0.73)
refa 21 21 21 21 21 22 22 22 22 23b 8c 24c 8c 23 25b 25b 25b
T ) 298.15 K unless indicated. b T ) 293.15 K. c T ) 303.15 K.
structural changes. Equation 14 is valid if structural changes during the mixing are insignificant and the energies required to bring molecules of A and B from the bulk to the surface are the φiS-based averages of those of pure compounds. Coupling the New Isotherm with the Mixture Rules. For comparisons between different systems it is convenient to consider the reduced surface pressure π* ) π/π0, where the surface pressure π is defined as (σA - σ) when σA > σB. Using eq 8 in eq 15 affords
π* )
β[β + R(φA/φB)] [β + (φA/φB)]2
) φBS(RφAS + φBS)
(16)
where R ) λ + 1. Note that this expression for π* has been made possible by factoring the coefficient of φASφBS in eq 15 into λ and π0. Equation 16 has two adjustable parameters: β, which relates surface and bulk compositions and measures lyophobicity; and R, which accounts for the effect of interaction between A and B. When this interaction causes insignificant changes of structure and cohesion forces during the mixing (R ) 1), eq 16 reduces to
π* ) φBS
(17)
which describes the surface tension behavior when the deviation from ideality is due only to β, i.e., to the difference between the affinities of A and B for the surface. This equation can also be written as
(σ - σA)/(σB - σ) ) βφB/φA
Figure 1. (a) Linear dependence of [(σ - σA)/(σB - σ)] on (φB/ φA) for {c-C4H8O + CH3(CH2)6CH3} mixtures at 298.15 K. (b) As for (a), but using a ln-ln plot.
generality, that σA g σB. In most cases λ > 0 and eq 14 is an upper limit for σ. Since surface tension is the energy per unit area required to bring molecules from the bulk to the surface, π0 measures dissimilarity between the cohesion forces of the pure compounds. In addition, π0 depends on the difference in the size between A and B. The factor λ represents interaction effects related to
(18)
Figure 1a shows this proportionality between [(σ - σA)/ (σB - σ)] and (φB/φA) for {tetrahydrofuran (THF) + octane} mixtures. The corresponding ln-ln plot, Figure 1b, is included for clarity. Results and Discussion To test the model, high-quality σ data for nonelectrolyte binary mixtures were extracted from the literature.8,21-25 Table 1 lists the corresponding values of π0, the values of the adjustable parameters R and β obtained by fitting eq (21) Pin˜eiro, A Ä .; Brocos, P.; Amigo, A.; Pintos, M.; Bravo, R. J. Chem. Thermodyn. 1999, 31, 931. (22) Papaioannou, D.; Panayiotou, C. J. Chem. Eng. Data 1994, 39, 457.
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Figure 2. Plots of the reduced surface pressure π* against the volume fraction of component B for the following binary mixtures: (a)-(e), {c-C4H8O + CH3(CH2)n-2CH3} at 298.15 K, n ) 6-10; (f)-(i), {CH3(CH2)m-1OH + CH3(CH2)n-2CH3} at 298.15 K, m ) 2 or 3, n ) 6 or 7; (j)-(l), {H2O + CH3OH}, {H2O + CH3(CHOH)CH3} and {H2O + CH3(CH2)2OH} at 293.15 K. b, experimental data; s, calculated curve (eq 16).
Langmuir Isotherm for Binary Liquid Mixtures
Figure 3. Plots of the surface volume fractions of both components against the bulk volume fraction of component B for {c-C4H8O + CH3(CH2)n-2CH3} at 298.15 K: (a) n ) 6 to n ) 8; (b) n ) 9 and n ) 10.
16, and the values of the objective function χ2 for the fits. In all cases B was taken to be the component of the mixture with the lower surface tension. The calculated curves fit the experimental data very well in all cases. By way of example, Figure 2 shows plots of π* against φB for some binary mixtures of types {THF or 1-alkanol + n-alkane} and {water + alkanol}. For the systems {THF + hexane, heptane, or octane} we found that R ) 1, which implies that the deviation of surface tension from ideal behavior is attributable exclusively to lyophobicity differences. Furthermore, β is nearly the same for all three systems; i.e., the dependence of surface composition on bulk composition is practically independent of the hydrocarbon chain length. Thus a single pair of φiS-φi curves (i ) A, B) with a unique φBC corresponding to their crossing point describes all three cases (Figure 3a). The π*-φB curves, Figure 2a-c, are of course also identical, and we can say that a corresponding states equation has been found for these mixtures. The remaining systems of this series, {THF + nonane or decane}, show a different behavior. A value of β closer to (23) Papaioannou, D.; Panayiotou, C. J. Colloid Interface Sci. 1988, 130, 432. (24) McLure, I. A.; Edmonds, B.; Lal, M. J. Colloid Interface Sci. 1983, 91, 361. (25) Va´zquez, G.; A Ä lvarez, E.; Navaza, J. M. J. Chem. Eng. Data 1995, 40, 611.
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unity implies greater resemblance between bulk and surface compositions, i.e., better solubility. Moreover, A-B interactions appear to have a significant influence in these solutions, which is ascribed to destruction of the orientational order of the pure n-alkanes26 and to steric effects caused by the unlike sizes and shapes of molecules of A and B. These interactions appear to be stronger for B ) decane (λ ) 0.60) than for B ) nonane (λ ) 0.48), but the lyophobicity of the alkane is nearly the same for these two systems, which accordingly share the same plots of φiS as a function of φi (Figure 3b). The lyophobicity of isooctane in benzene is of the same magnitude as that of the shorter alkanes in THF. In the other systems containing benzene β is larger, but all three have λ ) 0, showing that the energies required to bring A and B molecules from the bulk to the surface are the φiS-based averages of those of pure compounds. Isooctane has the same lyophobicity in dodecane as in acetone, but whereas λ ) 0 in dodecane (as expected for a mixture of two inert solvents), λ ) 1.23 in acetone, which may be attributed to the breaking of the strong dipolar A-A interactions upon formation of the mixture. The disruption of alkanol polymers upon mixing with an alkane explains why λ * 0 for all these systems. As expected, λ is slightly greater for {ethanol + alkane} than for {1-propanol + alkane}, since the destruction of structure is more marked when the alkanol chain is shorter and the density of -OH groups higher. The lyophobicity of the alkane is also greater in ethanol than in 1-propanol, and appears to increase with the alkane chain length in both cases. Note that the surface of {ethanol + hexane} solutions has almost the same composition as the bulk: β ) 0.98. For {1-propanol + hexane} β ) 0.86, i.e., 1-propanol is expelled from the bulk to the surface upon addition of hexane, despite having a greater surface tension than hexane. In the highly nonideal mixtures {water + methanol, 1-propanol, or 2-propanol} the mixing process involves the breaking of A-A and B-B hydrogen bonds and the formation of new hydrogen bonds between A and B molecules. These interactions should have a significant effect on π* only when the sizes of A and B are very dissimilar. This explains why λ ) 0 for {water + methanol} but λ ) 0.63 and 0.59 for 1-propanol and 2-propanol, respectively. The strong hydrophobicity of alkanols seems to depend on their size and position of their inactive groups: β ) 3.16 for methanol, 6.59 for 2-propanol, and 12.57 for 1-propanol. The position of the OH group in 2-propanol, at the center of the hydrocarbon chain, results in its hydrophobicity being similar to that expected for ethanol. Conclusion A new model describing the surface tension of binary liquid mixtures as a function of the bulk composition over the whole concentration range is obtained by coupling a generalization of the Langmuir isotherm with a mixing rule for the surface tension of nonideal solutions that includes an interaction parameter. The final equation fitted to experimental data has two adjustable parameters reflecting physically meaningful aspects of the behavior of the molecules in the bulk and at the surface. The parameter β, which relates bulk and surface compositions, has the same meaning as in the Langmuir equation since both isotherms match at infinite dilution of the component B, and it is a measure of the lyophobicity of the component (26) Patterson, D. Pure Appl. Chem. 1976, 47, 305-314.
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with the lower surface tension. The parameter R takes into account the effect of the A-B interactions on surface tension. The new model faithfully correlates surface tension data over the entire concentration range for a wide variety of solutions with π0 ranging from a few units for typical nonelectrolyte mixtures to several tens for aqueous solutions of alkanols. Even though the interactional Margules type term of the mixing rule is symmetrical with respect to the surface volume fractions, the model is able to correlate surface tension data for mixtures with very dissimilar components, such as {water + alkanol}. Although the final equation cannot be used for systems with positive aneotropes27 (typically mixtures involving complexation and π0 values lower than 1 mN/m) because (27) McLure, I. A.; Edmonds, B. Nature Phys. Sci. 1973, 241, 71.
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surface pressure is not well-defined in the aneotropic region, work is in progress to extend the present model so as to include such cases, which we have recently tackled using the real associating solution model.28 Acknowledgment. The authors are grateful to Dr. Miguel Costas for helpful advice. This work was supported by the Xunta de Galicia (Consellerı´a de Educacio´n e Ordenacio´n Universitaria) under Project No. XUGA20606B98. A Ä . P. thanks Intercambio Acade´mico UNAM and the University of Santiago de Compostela for financial support. LA001210S (28) Pin˜eiro, A.; Brocos, P.; Bravo, R.; Amigo, A. Fluid Phase Equilib. 2001, 182, 337.
