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Langmuir 2002, 18, 3604-3608
Activity Coefficients at Infinite Dilution from Surface Tension Data J. Gracia-Fadrique* 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
Pilar Brocos, A Ä ngel Pin˜eiro, and Alfredo Amigo Departamento de Fı´sica Aplicada, Facultad de Fı´sica,Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Received December 6, 2001 A method for calculating activity coefficients at infinite dilution from surface tension data is derived from the conditions for equilibrium between surface and bulk phases rather than from the conditions for liquid-vapor or liquid-liquid equilibrium, as is more usual. Specifically, the method combines the Volmer surface equation of state and the Gibbs adsorption equation to derive an expression for the surface chemical potential. The key operation of this treatment is the choice of the same reference state for both bulk and surface phases. The γ∞ values calculated for systems for which suitable data are available in the literature agree well with values obtained by other methods.
Introduction The concepts of fugacity and activity represent the transition from pure thermodynamics to the theory of intermolecular forces, and hence the approach to the behavior of real fluids. These properties are of primary importance in a great variety of research fields, notably the study of dilute solutions. For instance, the denaturization of proteins in certain solvents has been modeled1,2 using solubilities calculated from activity coefficients at infinite dilution, γ∞. In industry, solvents for liquid extraction or extractive or azeotropic distillation can be chosen on the basis of knowledge of the ratio of the activity coefficients of the key components at infinite dilution in the solvent.3 Knowledge of γ∞ also throws light on solutesolvent interactions, since it characterizes the behavior of the solute molecules when surrounded by the solvent. Equilibrium compositions in liquid-vapor and liquidliquid systems have hitherto been the main tools for determining activities and activity coefficients. The general procedure uses the bulk compositions of the two phases. However, for substances of low volatility such as polymers or proteins, the widely employed vapor-liquid equilibrium (VLE) technique faces the problem of measuring vapor-phase concentrations. Strey et al.4 recently noted that the activity coefficients of most genuine surfactants have not been determined because of the imprecision of the VLE method for these sorts of molecules and the insensitivity of other techniques at low concentrations. Since the Gibbs equation requires knowledge of activities, these authors pointed out the urgent need for * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Hovorka, S.; Dohnal, V.; Carrillo-Nava, E.; Costas, M. J Chem. Thermodyn. 2000, 32, 1683-1705. (2) Dohnal, V.; Costas, M.; Carrillo-Nava, E.; Hovorka, S. Biophys. Chem. 2001, 90, 183-202. (3) Gmehling, J.; Menke, J.; Schiller, M. Activity coefficients at infinite dilution; Chemistry Data Series; Dechema: Frankfurt, 1994; Vol. IX, part 3. (4) Strey, R.; Viisanen, Y.; Aratono, M.; Kratohvil, J. P.; Yin, Q.; Friberg, S. E. J. Phys. Chem. B 1999, 103, 9112-9116.
precise determinations of the activity coefficients of surfactants, the so-called shorter-chain surfactants in particular. In the absence of experimental data, semiempirical tools such as the NRTL, Wilson, or UNIQUAC equations have been employed to predict γ values.5 In this paper it is shown how thermodynamics furnishes an alternative way of deriving activity coefficients, starting from the chemical potentials of the surface and bulk phases. Although thousands of γ data are listed in collections such as ref 3, as far as we know none of them has been determined from surface tension values. Our aim here is to provide a method for calculation of the infinite dilution activity coefficients of the components of binary mixtures from experimental surface tension data. Several authors have previously related bulk and surface properties, activity coefficients included, through either a surface equation of state or empirical mixing rules,6-9 but we are not aware of any paper showing a direct link between surface tensions and activity coefficients that allows predictions in both directions (although attempts were made by Hiranuma and Kuog10 using statistical surface thermodynamics and by Hammers et al.11 in the framework of interfacial tension). The key advantage of surface tension for our purposes is its ability to reflect changes at the surface caused by very small solute concentrations; for instance, 1 ppm of any commercial surfactant is able to decrease the surface tension of water by 20 mN‚m-1. In the case of mixtures of simple liquids, (5) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 1999. (6) Sprow, F. G.; Prausnitz, J. M. Trans. Faraday Soc. 1966, 62, 1105-1111. (7) Sprow, F. B.; Prausnitz, J. M. Can. J. Chem. Eng. 1967, 45, 2528. (8) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987. (9) Nath, S. J Colloid Sci. 1999, 209, 116-122. (10) Hiranuma, M.; Kuog, M. Kagaku Kogaku 1966, 30, 797-800. (11) Hammers, W. E.; Meurs, G. J.; De Ligny, C. L. J. Chromatogr. 1982, 246, 169-189.
10.1021/la011761y CCC: $22.00 © 2002 American Chemical Society Published on Web 03/29/2002
Calculating Activity Coefficients
Langmuir, Vol. 18, No. 9, 2002 3605
Table 1. Experimental γ∞ Values for the Solute of Binary Mixtures Reported To Exhibit Ideal Behavior at the Vapor-Liquid Interface surface tension data ref
T/(K)
γ∞ (exptl)a
chlorobenzene nitrobenzene water ethanol
toluene 1,4-dioxane toluene 1,4-dioxane water-d methanol
b c b c d b
298.15 293.15, 303.15 283.15, 313.15 293.15, 303.15 298.15 273.15, 303.15
carbon tetrachloride chloroform
chloroform carbon tetrachloride
h
298.15
0.92 0.97 0.96 0.82 1.04 1.00e 1.01f 0.99g 1.13i 1.16i
solvent
solute
benzene
a Data from ref 3, the temperature being 298.15 K unless otherwise indicated. b Shereshefsky, J. L. J. Colloid Interface Sci. 1967, 24, 317-321. c Suri, S. K.; Ramakrishna, V. J. Phys. Chem. 1968, 72, 3073-3079. d Belton, J. W.; Evans, M. G. Trans. Faraday Soc. 1945, 41, 1-12. e T ) 351.35 K. f T ) 382.15 K. g T ) 423.95 K. h Reference 7. i T ) 328.15 K.
such as water + ethanol, surface tension changes of 1 mN‚m-1 can be caused4 by a solute mole fraction of 1 × 10-3.
Table 2. Comparisona of the Experimental Activity Coefficients at Infinite Dilution γ∞ for Aqueous Solutions of 1-Alkanols, with the (π*/x)xf0 Values Calculated by fitting Equation 1 to Surface Pressure Data (in mN‚m-1)
Theory
solute
(π/x)xf0b
(π*/x)xf0c
γ∞ d
We wish to find a function relating γ∞, the activity coefficient at infinite dilution, to surface tension. Denoting by x the mole fraction of solute and by π the surface pressure of the solution (i.e., the difference σ10 - σ, σ10 being the surface tension of the pure solvent and σ the surface tension of the solution), we begin by noting that in the dilute region the surface pressure is a linear function of composition
methanol ethanol 1-propanol 1-butanol 1-pentanol
444 1342 4270 12330 35800
8.9 27.0 88.4 259.1 767.4
1.66 3.92 13.76 50.50 197.50
(π/x)xf0 ) R
(1)
where R characterizes a system for a given temperature and pressure. Let us consider the Gibbs adsorption equation for a binary system
dπ ) Γ1 dµ1 + Γ2 dµ2
(2)
which reduces under the Gibbs convention (Γ1 ) 0) to
dπ ) Γ dµ2
(3)
In the subsequent text, the symbol µ without subscript will be reserved to refer to the solute or component 2. By using the ideal bulk chemical potential, the Gibbs adsorption equation takes a more familiar form
Γ)
x dπ 1 ) A RT dx
( )
T
(4)
where A is surface area per mole of solute at the surface, T is temperature, and R is the gas constant. Just at the dilute region eq 4 can be rewritten as
Γ)
π Rx ) RT RT
(5)
This last expression reveals the ideal two-dimension surface equation of state πA ) RT (the surface analogue to the ideal gas equation), which generally keeps just at infinite dilution. The corresponding surface chemical potential is obtained by coupling eq 3 and eq 5
dπ )
π dµS RT
(6)
a All data at T ) 298.15 K. b From ref 12. c π0 values required for transformation taken from: Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents, 4th ed.; Wiley-Interscience: New York, 1986. d From ref 3.
Integration affords
µS ) µ0,S + RT ln
π π0
(7)
where π0 is a reference surface pressure related to the value of the standard chemical potential and usually selected as 1 mN‚m-1. At equilibrium conditions, the equality of eq 7 with the ideal bulk chemical potential µb ) µ0,b + RT ln x, leads to the classical definition for the standard free energy of adsorption
-∆G0 ) µ0,S - µ0,b ) RT ln(π/x)xf0
(8)
This important quantity in surface chemistry, built from different standard states for bulk and surface, translates the bulk/surface transport processes taking place in the dilute region into energy values. If instead of the ideal bulk chemical potential we select the real chemical potential with the corresponding activity coefficient µb ) µ0,b + RT ln(γx), and then we take limits at infinite dilution, we are able to calculate activity coefficients at infinite dilution from surface tension data, or inversely, we can predict surface activities from the more common data of infinite dilution activity coefficients reported in the literature. A key operation is to choose the same standard state in both phases: µ0,b ) µ0, S, in such a way that
RT ln(γx) ) RT ln π*
(9)
where π* ) π/π0 is called the reduced surface pressure and π0 ) σ10 - σ20 denotes the selected reference surface pressure π0, reached as x f 1. Thus, we have the agreement of standard states, as defined for conventional activity coefficients under symmetric normalization: γ f γ∞ as x
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Gracia-Fadrique et al.
Table 3. Comparison of the Experimental and Predicted (Eq 16) Activity Coefficients at Infinite Dilution γ∞ of Various Solutes in Watera γ∞ experimentalc refb
π range
methanol
d, e
ethanol
solute
predicted eq 16
at 298.15 K
ln(π*/x)xf0
ΓmRT
π0
2.5-32
2.4218
26.38
49.99
1.69
1.53 1.64 1.65 1.66 1.74 5.92
f
1.5-36.5
3.5167
22.89
49.72
3.84
3.27 3.55 3.73 3.76 3.91 3.92 4.03 4.79f
1-propanol
f
2-41
4.6346
24.17
48.31
13.95
10.90 11.20 13.76 13.80f 14.10 15.00 133.50
1-butanol
d g
7-46.5 9.5-26.5
5.9495 5.7627
24.30 26.03
48.4 47.59h
52.32 51.15
45.10 50.50 51.60 52.80 53.70 205.60
1-pentanol
d g
17-45.5 7.5-31.5
7.4148 6.9778
20.51 34.26
48.73 46.65h
2-butanol
d
6.5-33
5.7483
20.32
49.1
28.00
22.40 24.80 26.20 26.32
2-methyl-2-Propanol
d i
6-45 6-46.5
5.8113 5.7667
16.11 15.46
52.0 51.87
13.25 11.14
12.20 12.27 12.90
2-propanone
d, j
17-41
4.1071
16.26
50.70
2.69
7.31 7.56 21.06 61.86
2-butoxyethanol
n
4.5-12.5
7.5210
10.35
45.2
23.42
25.63o
154.3 274.8
other temp 1.77e 2.53e
192.00 197.00 197.50 212.50 338.40
2.32k 5.83l 7.96m
a The range of surface pressures for which ln(π*/x) is a linear function of π is specified, as well as π0 data and the parameter values resulting from the adjustment to eq 19. Units of π, π0, and ΓmRT are mN‚m-1. b Source of surface tension data, the temperature being 298.15 K unless otherwise indicated. c Data from ref 3 unless otherwise indicated. d Gammon, B. E.; Marsh, K. N.; Dewan, A. K. R. Transport properties and related thermodynamic data of binary mixtures; Design Institute for Physical Property Data: New York, 1993; Part 1.e T ) 303.15 K. f Reference 4. g Posner, A. M.; Anderson, J. R.; Alexander, A. E. J. Colloid Sci. 1952, 7, 623-644. h Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents, 4th ed.; Wiley-Interscience: New York, 1986. i Glinski, J.; Chavepeyer, G.; Platten, J.-K. J. Chem. Phys. 1995, 102, 2113-2117. j T ) 273.15 K. k T ) 273.15 K, extrapolated. l T ) 288.15 K. m T ) 297.25 K. n Trejo, L. M. Doctoral Thesis, Universidad Nacional Auto´noma de Me´xico, Me´xico D. F., 1995. o Gmehling, J.; Onken, U.; Rarey-Nies, J. R. Vapor-liquid equilibrium data collection; Chemistry Data Series; Dechema: Frankfurt, 1981; Suppl. 1, Vol. I, part 1A).
f 0 and γ f 1 as x f 1. Let us point out that on taking π0 ) π0, we have assumed implicitly a linear behavior of surface tension vs mole fraction on the whole concentration range, i.e., π ) π0x. In this way, for consistency with the conditions from which eq 7 was derived, eq 9 concerns to binary mixtures obeying Raoult’s law in the surface phase. In particular, at the very dilute region it can be rewritten for convenience as
γ∞ ) (π*/x)xf0
(10)
Taking into account this expression, we should find γ∞ )
1 when π is a linear function of x in the entire domain. In Table 1 we have collected from the literature some binary systems reported to behave approximately according to eq 1 over the whole concentration range, and having available γ∞ data at least for one of the components. The experimental values of the activity coefficients at infinite dilution approach the unity and therefore agree well with eq 10. To check the predictive ability of that equation when dealing with highly nonideal surfaces, we have taken from the literature surface activities of 1-alkanols in water measured by the pendant drop method,12 together with
Calculating Activity Coefficients
Langmuir, Vol. 18, No. 9, 2002 3607
selected values of activity coefficients at infinite dilution,3 as shown in Table 2. Although we have found a linear correlation between (π*/x) values and activity coefficients at infinite dilution, we have not obtained the expected equality (eq 10), the π*/x values being indeed about four times greater than the activity coefficients. This suggests an insufficient ability of the ideal surface chemical potential to describe a real behavior. That’s why we have decided to make use of the first natural nonideal surface equation of state of the van der Waals type, the Volmer equation, in which the factor A in the ideal EOS is modified in the same way as the volumetric factor of the ideal gas equation is modified in the van der Waals equation
π (A - A0) ) RT
(11)
where A0 is the surface area per mole of solute at surface saturation conditions. In terms of relative adsorption Γ, the last equation can be expressed as
1 1 RT ) + Γ π Γm
(12)
The replacement of Γ in the Gibbs adsorption equation allows evaluation of the surface chemical potential derived from the Volmer equation
∫µµ dµS ) RT ∫ππ (d ln π) + ∫ππ Γdπm 0
0
0
µS ) µ0,S + RT ln π* -
π0 - π Γm
(13)
(14)
Under equilibrium conditions, µb ) µS. Again we have chosen the same reference state at the surface and into the bulk, that is, the state in which x ) 1, so that µ0,b ) µ0, S and therefore
RT ln(γx) ) RT ln π* -
π0 - π Γm
(15)
Our region of interest is just the infinite dilution zone where the equality of chemical potentials is kept. At the limit x f 0, π f 0, and γ f γ∞
ln γ∞ ) ln
(π*x )
xf0
π0 ΓmRT
(16)
This is the relationship we require, since π0 is experimentally determinable and the parameters ln(π*/x)x f0 and ΓmRT can be obtained from the same surface equation of state employed in the construction of the surface chemical potential. In fact, although many surface equations of state provide the characteristic parameters needed in eq 16, for thermodynamical consistency we employed those derived from the Volmer equation. In this way, by combining Gibbs adsorption equation with Volmer equation, and applying equilibrium conditions
RT
dπ dπ + ) RT d(ln γx) π Γm
(17)
(12) Clint, J. H.; Corkill, J. M.; Goodman, J. F.; Tate, J. R. Hydrophobic Surf., Kendall Award Symp. 1969, 5, 180-188.
Figure 1. Examples of aplication of the proposed method for aqueous solutions of ethanol (a), 2-methyl-2-propanol (b), or 1-pentanol (c). Equation 19 has been fitted to data in the zone of linear behavior with the aim of get the parameters needed in eq 16.
Since γ is constant at the extremes of the concentration range,4 it can be stated for the solute dilute region that
RT
dπ dπ dx + ) RT d(ln γ∞) + RT d(ln x) ) RT π Γm x (18)
and finally we get after integration
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Langmuir, Vol. 18, No. 9, 2002
(π*x ) ) ln(π*x )
ln
-
xf0
π ΓmRT
Gracia-Fadrique et al.
(19)
The required parameters are obtained from the plot of experimental ln(π*/x) versus π. Equation 19 predicts a linear behavior at least in the dilute region, its slope being the inverse of the parameter ΓmRT and its intercept the value of ln(π*/x) at infinite dilution. We can then return to eq 16 and evaluate the desired activity coefficient at infinite dilution. Let us point out that the Gibbs convention has been employed in the derivation of eqs 16 and 19 just for the sake of simplicity. In fact, it can be shown that the use of this convention is not necessary, since our interest is restricted to very dilute solutions: By combining the Gibbs-Duhem equation for the bulk solution (1 - x) dµ1 + x dµ2 ) 0 with the original Gibbs adsorption eq 2 we get
(
dπ ) Γ2 -
x Γ dµ2 1-x 1
)
(20)
We are interested in the dilute region, so when x f 0 an expression identical in form to the eq 3 is obtained
dπ ) Γ2 dµ2
(21)
Results There are relatively few published experimental data of good quality with which to test eq 16. Surface tension measurements are generally not available for mixtures of compounds of low molecular weight, at least in the very dilute regions, while experimental activity coefficients, for the reasons noted in the Introduction, are unavailable for genuine surfactants. The γ∞ values of the systems for which we have found appropriate data in the literature (all of them aqueous solutions of small molecules) are listed in Table 3 together with the predicted values of γ∞, the π range of the data to which eq 19 was fitted in each case, the experimental values of π0, and optimized values of ln(π*/x)xf0 and ΓmRT used in making these predictions. Although the experimental γ∞ data for any given system vary considerably because of differences in experimental technique, in general the predicted values agree quite satisfactorily with what appear to be the most reliable experimental values. Besides, results for aqueous mixtures
of 1-butanol, 1-pentanol, and 2-methyl-2-propanol suggest that the quality of σ measurements plays a critical role. Figure 1 illustrate the method for ethanol, 2-methyl2-propanol, or 1-pentanol as solutes, showing how the ln(π*/x)-π plot can be approached by a straight line in the solute dilute region. According to our experience, the range of surface pressures where data could be used for this purpose would be roughly 0-10 mN‚m-1 for genuine surfactants in water. Let us note that the steep change in the slope of ln(π*/x)-π plots in parts a and b of Figure 1 is due to the approach to surface saturation conditions. Conclusions Since surface tension measurements reflect very small quantities of solute, they are a convenient means of studying the thermodynamic properties of solutions in the very dilute region, where other techniques fail. In this work we have tackled the building of a comprehensive method for evaluation of activity coefficients at infinite dilution from high-quality surface tension data: A surface equation of state (EOS) is combined with the Gibbs adsorption equation and then equilibrium conditions between bulk and surface phases are applied, after a certain choice of the reference states. In relation to the dependence of ln(π*/x) on π, two cases have become definite: (i) When the plot of ln(π*/x) versus π is a straight line of negative slope at least in the range 0-10 mN‚m-1, the use of the Volmer equation (eq 11) is recommended and γ∞ is given by eq 16. (ii) When the plot of ln(π*/x) versus π is a straight line of slope zero over the whole surface pressure range, the corresponding surface EOS is the ideal one (eq 5) and γ∞ ) 1 according to eq 10. If the slope of ln(π*/x)-π at low surface pressures is positive or the plot is nonlinear, then the present approach fails, but we are currently working on its extension starting from a more realistic surface equation of state. Acknowledgment. This work was supported by Xunta de Galicia, Secretarı´a Xeral de Investigacio´n e Desenvolvemento, under the project PGIDT01PXI20601PR. A Ä .P. thanks Intercambio Acade´mico UNAM, DGAPA project IN105300, and the University of Santiago de Compostela for financial support. LA011761Y
