Article pubs.acs.org/Langmuir

Partial Surface Tension of Components of a Solution George Kaptay* University of Miskolc, Egyetemvaros, Miskolc, 3525 Hungary Bay Zoltan Nonprofit Ltd on Applied Research, 2 Igloi, Miskolc, 3519 Hungary ABSTRACT: First, extending the boundaries of the thermodynamic framework of Gibbs, a definition of the partial surface tension of a component of a solution is provided. Second, a formal thermodynamic relationship is established between the partial surface tensions of different components of a solution and the surface tension of the same solution. Third, the partial surface tension of a component is derived as a function of bulk and surface concentrations of the given component, using general equations for the thermodynamics of solutions. The above equations are derived without an initial knowledge of the Gibbs adsorption equation and without imposing any restrictions on the thickness or structure of the surface region of the solution. Only surface tension and the composition of the surface region are used as independent thermodynamic parameters, similar to Gibbs, who used only the surface tension of the solution and the relative surface excesses of the components. The final result formally coincides with the historical Butler equation (1932), but without its theoretical restrictions. (Butler used too many unnecessary model restrictions during his work: he started from the Gibbs adsorption equation, and he assumed the existence of a surface monolayer.) Thus, the renovated Butler equation has gained general validity in this article. It was applied to derive both the Langmuir equation and the Gibbs adsorption equation, but the latter two equations do not follow from each other. Thus, it is shown that logically (not historically) the renovated Butler equation is a root equation for surface tension and the adsorption of solutions. It can be used to perform calculations for specific systems if the corresponding specific experimental data/models are loaded into it. In this case, both surface tension and surface composition can be calculated from the renovated Butler equation, which cannot be done using the Gibbs adsorption equation alone.

1. INTRODUCTION Modeling surface tension of solutions and surface adsorption of different components of solutions is an ongoing problem in surface science.1−5 The major steps in the development of this field of science in historical perspective are shown in Figure 1. It is all based on the bulk thermodynamics of heterogeneous systems developed by Gibbs and also on his definition of surface tension.6 The first and still the most basic achievement in the surface tension of solutions is the Gibbs adsorption equation,6 describing the relationship among the surface tension of the solution, the surface excesses of its components, and bulk chemical potentials of the components. Even in the simplest case of binary solutions and by assuming that the bulk chemical potentials of the components are known, this equation contains two unknowns: the surface tension of the solution and the relative surface excess of the second component (if the dividing surface of Gibbs is selected such that the surface excess of the first component is zero). Thus, for actual calculations by the Gibbs adsorption equation, a further independent equation is needed, describing how the surface concentrations (or relative surface excess values) depend on bulk concentrations. Such a relationship was first derived by Langmuir in a meaningful way,7,8 although only for dilute solutions and for surfaces forming ideal solutions. (Important note: The Langmuir equation and the Gibbs adsorption equation do not follow from © 2015 American Chemical Society

each other.) Combining the Gibbs adsorption equation with the Langmuir equation, the surface tension can be described as a function of bulk composition at any fixed temperature. The resulting equation essentially coincides with the semiempirical Szyszkowski equation, derived earlier.8 This connection was first revealed by Langmuir.7 In the original thermodynamics of Gibbs (1875−1878), the partial surface tension of components in a solution is not defined.6 However, starting from the paper of Butler (1932), this value appears, although without proper definition.10 According to www.google.scolar, Butler’s paper has been cited since 1932 in about 500 papers. (For comparison, the Gibbs paper is cited more than 8000 times and used without citation in a much larger number of papers.) From the 500 citations of the Butler equations, let us mention a few;11−42 to the best knowledge of the author, none of them provides a general thermodynamic definition of the “partial surface tension” physical quantity extending the general thermodynamic framework created by Gibbs.6 The goal of this article is to fill this definition/knowledge gap and, as a consequence, to “renovate” the historical Butler equation. Received: January 19, 2015 Revised: March 27, 2015 Published: May 5, 2015 5796

DOI: 10.1021/acs.langmuir.5b00217 Langmuir 2015, 31, 5796−5804

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the surrounding equilibrium vapor phase, the equilibrium shape of a liquid droplet is a sphere because this shape provides the minimum surface area per unit volume among all 3D shapes.

2. DERIVATION OF THE RENOVATED BUTLER EQUATION BY EXTENDING THE THERMODYNAMIC FRAMEWORK OF GIBBS 2.1. Simple Case of a One-Component Pure Phase. To start from basics, let us consider a macroscopic liquid phase consisting of component i only. The amount of matter within this pure liquid phase is denoted as ni (mol). This pure liquid phase is surrounded by a vapor phase also consisting of only component i with a vapor pressure that provides equilibrium with this liquid phase. This macroscopic pure liquid phase has its own equilibrium molar volume. Coupled with its final amount of matter, it will have a final and specific equilibrium volume. If its shape is defined, then it will lead to its well-defined surface area, denoted as Aoi (m2). In the absence of gravity and other phases, the liquid droplet will be spherical at equilibrium. However, the liquid phase is allowed to take other shapes as well, especially under the influence of gravity and other force fields. The standard partial absolute (not molar) Gibbs energy of the pure liquid phase is denoted as Goi (J). Following eq 1, the surface tension of the pure liquid phase i can be defined as Figure 1. History of development of different equations on surface tension and the adsorption of solutions.

⎛ dG o ⎞ σio ≡ ⎜ io ⎟ ⎝ dA i ⎠ p , T , n

i

For the subject of this article, the most important achievement of Gibbs is that he redefined the surface tension of solutions in a thermodynamic way, as6 σ≡

⎛ dG ⎞ ⎜ ⎟ ⎝ dA ⎠ p , T , n

i

where σoi (J/m2) is the surface tension of a pure liquid phase i. Physically, eq 3 corresponds to the very slow gradual change in the shape of the pure liquid droplet, leading to the gradual change in its surface area and thus the gradual change in its Gibbs energy. The derivation according to eq 3 should be performed at constant pressure p (Pa), constant absolute temperature T (K), and constant amount of matter of component i within the liquid phase. Equation 3 can be integrated using the following two boundary conditions: (i) at Aoi = 0, Goi = Gob,i, where Gob,i (J) is the standard partial absolute bulk Gibbs energy of macroscopic pure liquid phase i and (ii) at Aoi = Aoi , Goi = Goi . The result of the integration of eq 3 is

(1)

where σ (J/m ) is the surface tension of a liquid solution phase, G (J) is its absolute (not molar) Gibbs energy, and A (m2) is its absolute surface area. Physically, eq 1 corresponds to the very slow gradual change in the shape of the liquid droplet, leading to a gradual change in its surface area and thus leading to the gradual change in its Gibbs energy. The derivation according to eq 1 should be performed at constant pressure p (Pa), constant absolute temperature T (K), and constant amount of matter for all components i within the liquid phase ni (mole). Equation 1 can be integrated using the following two boundary conditions: (i) at A = 0: G = Gb, where Gb (J) is the bulk Gibbs energy of the macroscopic liquid phase and (ii) at A = A: G = G. The result of the integration of eq 1 is 2

G = G b + Aσ

(3)

Gio = G b,o i + Aioσio

(4)

Equations 3 and 4 are boundary cases of the most general equations (eqs 1 and 2). Let us mention that Gibbs did not specifically define the surface tension of a pure phase i and did not specifically write eqs 3 and 4. Nevertheless, eqs 3 and 4 obviously fit into the framework of the thermodynamics of Gibbs and are widely used in the literature. This point was important to mention here because the fact that Gibbs did not specifically mention a quantity 140 years ago does not mean that this quantity does not exist in nature and cannot be defined today (as frequently claimed in the literature and in the anonymous reviews criticizing partial surface tension and related quantities). Before going on, let us mention that if C different pure liquid phases are positioned side by side without a common interface, then the standard integral absolute Gibbs energy of this system is denoted by Go (J). It is connected to the standard partial absolute Gibbs energies of the components in an obvious way:

(2)

As follows from eq 2, the total Gibbs energy of a liquid phase consists of two terms: the bulk term and the surface term. Because nature tends to minimize the Gibbs energy and both the surface tension and surface area can have positive values only, nature will tend to minimize all terms in eq 2: the bulk Gibbs energy, the surface tension, and the surface area. The bulk Gibbs energy is minimized by adopting the equilibrium volume and equilibrium bulk structure for the liquid phase as a function of p, T and ni. The surface tension is minimized by adopting an equilibrium structure and equilibrium composition of the surface region of the liquid phase (also as a function of p, T, and ni), the latter called surface excess by Gibbs. Surface area is minimized by adopting an equilibrium shape for the liquid phase; in the absence of gravity and other phases except

C

Go =

∑ Gio i

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where ΔG (J) is the integral absolute mixing Gibbs energy of the solution, defined as

Similarly, the standard integral bulk absolute Gibbs energy of the system can be defined and denoted as Gob (J). It is connected to standard partial bulk absolute Gibbs energies of the components as

G = Go + ΔG

Each atomic/molecular position along the surface is occupied by a different component. The atoms/molecules of these components are distributed statistically along the surface. The absolute value of the partial surface area occupied by component i is denoted as Ai (m2). The absolute value of the total surface area of the liquid solution phase will be the sum of all of the partial surface areas:

C

G bo

∑

=

G b,o i

i

(6)

2.2. Case of a Multicomponent Liquid Solution. Now, let us consider a C-component liquid solution phase, with ni being the amount of matter of component i within this liquid phase. The total amount of matter within this liquid solution phase is written as

C

A=

∑ ni i

(7)

⎛ dG ⎞ σi ≡ ⎜ i ⎟ ⎝ dA i ⎠ p , T , n

Gi = G b, i + Ai σi

(8)

C

G = Gb +

∑ Aiσi i

C

Gb =

∑ G b,i

i

(16)

Now, let us note that the left-hand sides of eqs 2 and 15 are identical; both describe the total absolute Gibbs energy of the solution. Then, the right-hand sides of eqs 2, and 15 must also be equal. After canceling out Gb from both sides of this new equality, the following equation is obtained:

(9)

C

∑ ΔGi

(15)

where Gb of eq 2 is written as the sum of partial absolute bulk Gibbs energies of the components:

Similarly, the integral absolute mixing Gibbs energy and the partial absolute mixing Gibbs energies are connected as ΔG =

(14)

Equations 13 and 14 are boundary cases of the general equations 1 and 2. Let us mention again that although Gibbs did not specifically define the partial surface tension of component i in the solution, this fact does not mean that such a physical quantity cannot be defined, extending the thermodynamic framework developed by Gibbs. Now, let us substitute eq 14 into eq 9

i

i

(13)

where σi (J/m ) is the partial surface tension of component i. The creation of this definition was the primary goal of this article. Let us explain its physical meaning: when the shape of the liquid solution is changed slowly and gradually at constant pressure, constant temperature, and constant amounts of all components in the solution, the partial surface area of component i (Ai) will gradually change, leading to a gradual change in the partial Gibbs energy of the same component i (Gi); the ratio of these two quantities is the partial surface tension of component i. Equation 13 can be integrated using the following two boundary conditions: (i) at Ai = 0, Gi = Gb,i, where Gb,i(J) is the partial absolute bulk Gibbs energy of component i in the liquid solution and (ii) at Ai = Ai, Gi = Gi. The result of the integration of eq 13 is

C

∑ Gi

i

2

where ΔGi (J) is the partial absolute mixing Gibbs energy of component i, i.e., the Gibbs energy change connected to the transfer of ni amount of component i from its standard pure liquid state into the liquid solution consisting of the abovementioned C components. Although the discussion of ΔGi is beyond the scope of this article, let us mention that it includes the configurational entropy term and excess Gibbs energy term, corresponding to attraction/repulsion between the components within the solution phase, relative to the attraction within the pure liquid phase (eq 2443−46). The integral absolute Gibbs energy of the solution and the partial absolute Gibbs energies of the components in the same solution are connected as G=

(12)

At this point the reader might feel that the assumption of the monolayer is hidden behind eq 12. In fact, the same result follows for an arbitrary thickness of the surface layer, with a homogeneous distribution of all components within the surface layer in accordance with their surface mole fractions. (For the definition of the latter, see the text after eq 25.) Now, let us apply eq 1 to component i of the solution

The liquid phase is surrounded by a vapor phase consisting of the same C components. The partial pressures of the components in the vapor phase are such that they remain in equilibrium with the components of the macroscopic liquid solution phase. The total integral absolute Gibbs energy of the liquid solution is denoted as G (J). At given p, T, and ni values, it has its equilibrium (minimum) value corresponding to its equilibrium structure and equilibrium molar volume. The latter, combined with ni, provides an equilibrium volume to the phase. For a specific shape it leads to the specific value of the surface area of the solution, denoted as A (m2). The surface tension of this liquid solution phase is defined by Gibbs (eq 1). The total integral Gibbs energy of this phase is described by eq 2 as having two terms: the bulk term and the surface term. Each component i is distributed statistically within this liquid solution phase. Any component i within the solution phase can be characterized with its own partial absolute Gibbs energy, denoted as Gi (J). It is connected to the standard partial absolute Gibbs energy of the same component through the following equation Gi = Gio + ΔGi

∑ Ai i

C

n=

(11)

C

Aσ = (10)

∑ Aiσi i

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Langmuir ⎡ d(Gio + ΔGi) ⎤ σi ≡ ⎢ ⎥ dA i ⎣ ⎦ p,T ,n

Combining eq 17 with eq 12, the following equation results: C

∑ Ai(σ − σi) = 0 i

(21)

i

(18)

The first term on the right-hand side of eq 21 is developed as (see also eq 3)

Equation 18 has an infinite number of mathematical solutions. Among them, the simplest solution is σ = σi (19)

⎛ dGio ⎞ ⎜ ⎟ ⎝ dA i ⎠

Equation 19 means that all partial surface tensions equal each other and also equal the surface tension of the solution. For a C-component system, there are C equations hidden behind eq 19: σ = σ1 (19a) σ = σ2 .........

(19b)

σ = σC

(19c)

p , T , ni

⎛ dG o dA o dns, i ⎞ i ⎟⎟ = ⎜⎜ i o d A d A ⎝ i i dns, i ⎠ p,T ,n

i

⎛ dG o dA o dns, i ⎞ ⎟⎟ = ⎜⎜ io i ⎝ dAi dns, i dAi ⎠ =

p , T , ni

ωo σio i ωi

(22)

where ns,i (mol) is the amount of component i in the surface region, ωoi (m2/mol) is the molar surface area of component i in pure liquid i (which is defined as the derivative of the standard partial surface area of pure liquid i by the amount of component i in the surface region), and ωi (m2/mol) is the molar surface area of component i in the liquid solution (which is defined as the derivative of the partial surface area of component i in the liquid solution by the amount of component i in the surface region). Now, let us discuss the second term of eq 21. When the shape of the liquid solution changes slowly (through equilibrium steps) such that its total surface area A is increased, the partial surface area Ai will also proportionally increase. This means that some part of component i that was originally in the bulk of the solution will appear in the surface region of the solution. Thus, a corresponding small amount of component i is transferred from the bulk of the solution to the surface of the solution (in the case of positive surface excess) or vice versa (in the case of negative surface excess). Thus, the second term of eq 21 can be written as

where i = 1, 2, ...., C. Mathematically, eq 18 has an infinite number of further solutions in addition to eq 19. However, according to these further possible mathematical solutions at least two components of the solution will not obey eq 19a−c. Thus, different atomic/molecular sites along the surface will have their own different partial surface tension values: one of them with a larger value and another one with a smaller value. However, nature tends to minimize the total Gibbs energy of the liquid solution, including the minimization of its surface tension, with all its partial surface tension values corresponding to all atomic/ molecular sites along the surface. Thus, different atomic/ molecular sites along the surface cannot have different partial surface tension values in equilibrium. If they do have different partial surface tension values in a nonequilibrium situation, then Marangoni flow will start along the surface to correct the situation. During Marangoni flow, subsurfaces with lower surface tension will flow toward subsurfaces with higher surface tension to replace them and to achieve global equilibrium of the surface when all partial surface tension values are equal and have values that are as low as possible. Thus, surface equilibrium is proven to correspond to eq 19. Equation 19 is developed here by extending the framework of thermodynamics of Gibbs, without any assumption of the structure of the bulk or surface of the liquid solution. Thus, eq 19 has general validity. With its total of three characters (plus the equals sign), eq 19 is most probably the shortest and simplest equation of physical chemistry of surfaces. Its simplicity is similar to the famous condition of heterogeneous equilibrium of Gibbs, written here for two-phase system α and β containing the same component i6 μi(α) = μi(β) (20)

⎛ dΔGi ⎞ ΔGs, i − ΔG b, i = ⎜ ⎟ ωi ⎝ dA i ⎠ p , T , n

(23)

i

where ΔGs,i (J/mol) is the partial molar mixing Gibbs energy of component i in the surface region of the solution, corresponding to the transfer of 1 mole of i atoms/molecules from the surface region of the pure i phase to the surface region of the solution, ΔGb,i (J/mol) is the partial molar mixing Gibbs energy of component i in the bulk of the solution, corresponding to the transfer of 1 mole of i atoms/molecules from the bulk of the pure i phase to the bulk of the solution. According to the thermodynamics of solutions,43−46 ΔG b, i = RT ln xi + ΔG b,E i = RT ln ai

where μi(α) (J/mol) and μi(β) (J/mol) are chemical potentials of component i in the two corresponding phases. In the same way that eq 20 is often called the basic thermodynamic equation of bulk equilibrium of heterogeneous systems, eq 19 can be called one of the basic thermodynamic equations of surface equilibrium. 2.3. Derivation of a More Useful Form of Equation 19. It should be admitted that eq 19, as it is now, is not yet ready to perform practical calculations. Therefore, let us derive its more useful alternative. For this purpose, let us substitute eq 8 into eq 13:

(24)

where R = 8.3145 J/mol K is the universal gas constant, xi (dimensionless) is the bulk mole fraction of component i in the solution defined as ratio ni/n, ΔGEb,i (J/mol) is the partial molar bulk excess Gibbs energy of component i, corresponding to the bulk mole fraction of xi, and ai (dimensionless) is the bulk activity of component i, corresponding to its bulk mole fraction xi. Formally a similar equation can be applied to the partial molar mixing Gibbs energy of component i in the surface region ΔGs, i = RT ln xs, i + ΔGs,Ei = RT ln as, i 5799

(25)

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should be selected first. Then, models should be created for the partial molar bulk excess Gibbs energies of all of the components of the solution as a function of their bulk mole fractions and temperature. Similar models should be created for the surface partial molar excess Gibbs energies of all of the components of the solution as a function of their surface mole fractions and temperature. Further models should be created for the molar surface areas of all components as a function of their surface mole fractions and temperature. Finally, the surface tensions and molar surface areas of all pure components should be known as a function of temperature. Then, for one round of calculations, temperature and the bulk mole fractions of all components should be fixed. For a C-component solution (C-1), equations of type 32 should be written. Combining these equations with the materials balance equation (eq 26), the C unknowns, i.e., the surface mole fractions of the C components of the solution, can be found. Substituting these values into eq 28, the partial surface tensions of all components are found. They will be equal or at least approximately equal if the numerical solution is used to solve the C equations above. Substituting these values into eq 19, the surface tension of the solution follows, being identical to the partial surface tension of each component or their average if they are not exactly the same due to approximations in the numerical solution. Thus, a knowledge of eqs 19 and 30−32 is indeed sufficient to find the surface tension of the solution and its surface composition if model parameters of the specific solution are known. 2.4. On the Butler Equation. It should be recognized that eqs 19 and 30−32 coincide with the historical Butler equation. However, Butler derived his equation with an assumption of a surface monolayer, and he started his derivation from the Gibbs adsorption equation. Therefore, the Butler equation is usually considered in the literature to be one of the useful but simplified model equations, certainly not the basic equation of surface science. In the above general thermodynamic derivation, no mention of the Gibbs adsorption equation and no model restrictions are made. Thus, the above general thermodynamic derivation lifts the Butler equation from the rank of being one of the useful model equations to the rank of being one of the basic equations of surface science. As such, the renovated Butler equation is an alternative to the Gibbs adsorption equation6

where xs,i (dimensionless) is the mole fraction of component i in the surface region of the solution defined as ratio ns,i/ns, where ns,i (moles) is the amount of component i in the surface region of the solution and ns (moles) is the total amount of all components in the surface region of the solution, ΔGEs,i (J/mol) is the partial molar excess Gibbs energy of component i in the surface region corresponding to the surface mole fraction of xs,i, and as,i, (dimensionless) is the activity of component i in the surface region corresponding to its mole fraction xs,i. Note that in the above definitions of amount of matter, mole fraction, excess partial Gibbs energy, and activity in the surface region no restriction is made on the thickness or the structure of the surface region; it is declared here only that in general these quantities are different from those corresponding to the bulk of the solution (which is obvious and is in agreement with Gibbs). Thus, the treatment herewith is still of general validity; specific details on the structure and thickness of the surface region are left for a specific model corresponding to a specific liquid solution. The only generally valid equation connecting the above surface properties follows from the materials balance in the surface region: C

∑ xs,i = 1

(26)

i

This equation is a logical analogue of the material balance equation of the bulk: C

∑ xi = 1

(27)

i

Let us mention that the mole fraction of component i in the surface region is related to the surface excess of the same component i, as defined by Gibbs. The relationship between them is revealed below. Now, let us substitute eqs 22−25 into eq 21 in two different versions, according to the two alternate ways that eqs 24 and 25 were written: σi = σio

E E ωio RT ⎛ xs, i ⎞ ΔGs, i − ΔGi + ln⎜ ⎟ + ωi ωi ⎝ xi ⎠ ωi

(28)

σi = σio

ωio RT ⎛ as, i ⎞ ln⎜ ⎟ + ωi ωi ⎝ ai ⎠

(29)

C

dσ = −∑ Γi dμi i

By substituting eqs 28 and 29 into eq 19, the following equations are obtained: σ=

ωo σio i ωi

σ = σio

RT ⎛ as, i ⎞ + ln⎜ ⎟ ωi ⎝ ai ⎠

E E ωio RT ⎛ xs, i ⎞ ΔGs, i − ΔGi + ln⎜ ⎟ + ωi ωi ⎝ xi ⎠ ωi

where Γi is the surface excess of component i. (For more details, see the original work of Gibbs.6) At first sight, eqs 19 and 33 have little in common. However, a deeper analysis shows that the two are actually identical to each other (ref 13 and below). Although theoretically identical in terms of practical usage, the renovated Butler equation is superior to the Gibbs adsorption equation for at least two reasons. First, eq 19 is far easier to explain, to teach, and to remember for a newcomer to the field (student) compared to eq 33. Second, eqs 19, 31, and 32 are sufficient to find the surface tension of the solution and all of the surface concentration (surface excess) values of the components (see above). On the other hand, eq 33 alone is not sufficient to do so. Otherwise eqs 19 and 30−33 are similar in the sense that they provide only the theoretical framework; to use them for practical calculations for real solutions, they should be loaded by experimental data/model equations regarding different properties of the specific liquid solution: the excess molar Gibbs energy of

(30)

(31)

Let us rewrite eq 31 for the surface equilibrium between any two components (i and j) of the solution: σio

E E ωjo ωio RT ⎛ xs, i ⎞ ΔGs, i − ΔGi RT ⎛⎜ xs, j ⎞⎟ ln⎜ ⎟ + ln + = σjo + ωi ωi ⎝ xi ⎠ ωi ωj ωj ⎜⎝ xj ⎟⎠

+

ΔGs,Ej − ΔGjE ωj

(33)

(32)

The above equations can be used for practical calculations. For this purpose, a specific solution with its specific components 5800

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Langmuir mixing, the excess molar volume of mixing, the structural details of the surface region, the surface area occupied by one atom/molecule, and so forth. Finally, let us remind the reader that the thermodynamics of Gibbs is generally valid for all types of materials. Thus, the Gibbs adsorption equation is also generally valid for all types of interfaces. Thus, the renovated Butler equations, eqs 19 and 30−32, derived within the same framework, are also generally valid for all types of interfaces. This has already been demonstrated in my previous papers on coherent solid and liquid/ liquid interfaces,47 for the solid/liquid interfaces,48 and for the liquid/vapor interfaces of immiscible liquids, showing a surface phase transition.49−51 2.5. Simplified Analytical Solution of the Butler Equation. As follows from nonlinear eq 32, the Butler equation can be solved only numerically for real solutions. However, to understand its properties better, an analytical solution is needed, even if it leads to some simplifications. For this purpose, let us consider a binary A−B solution with the bulk mole fraction of component B denoted as x (dimensionless) and the mole fraction of component B in the surface region denoted as xs (dimensionless). Let this A−B solution be ideal in both its bulk and in its surface region, i.e., all activities will equal the corresponding mole fractions. Let us also take the molar surface areas of the components to be equal and therefore neglect their concentration dependence: ωA = ωoA = ωB = ωoB = ω. Then, eq 32 simplifies to σAo +

RT ⎛ 1 − xs ⎞ RT ⎛⎜ xs ⎞⎟ o ⎟ = σ + ln⎜ ln B ⎝ ⎠ 1−x ω ω ⎝x⎠

(34)

Equation 34 can be solved for the mole fraction of component B in the surface region as xs =

Kx 1 + Kx − x

(35)

where K (dimensionless) is the equilibrium adsorption constant of component B, defined as ⎡ ω(σAo − σBo) ⎤ K ≡ exp⎢ ⎥ ⎣ ⎦ RT

(36)

Substituting eq 35 back into the two sides of eq 34, two identical equations for the concentration dependence of the surface tension are obtained: σ=

σAo

RT − ln(1 + Kx − x) ω

σ = σBo +

⎞ RT ⎛ K ⎟ ln⎜ ω ⎝ 1 + Kx − x ⎠

Figure 2. Mole fraction of component B in the surface region (a), the surface tension (b), and the relative surface excess of component B (c) as a function of the bulk mole fraction of component B, calculated with eqs 35−38 and 45. General parameters: T = 298 K, ω = 5 × 104 m2/mol, and σoA = 0.072 J/m2. Particular parameters: σoB = 0.030 J/m2 (line 1); σoB = 0.072 J/m2 (line 2); and σoB = 0.11 J/m2 (line 3).

(37)

(38)

Let us analyze eqs 35−38 for the following boundary cases: (i) at x → 0, xs → 0 and σ → σoA; (ii) at x → 1, xs → 1 and σ → σoB. One can see that both of these boundary cases provide reasonable results (Figure 2a,b).

thermodynamic framework of Gibbs. There is one thing missing: to prove that the two equations are identical to each other. Although such proof has already been provided in the literature,13 it received little attention. Therefore, the way in which the Gibbs adsorption equation can be derived from the simplified Butler equation will be shown here. It is also understood in the literature that the Gibbs adsorption equation alone is not sufficient for performing model calculations. An additional equation (such as the Langmuir equation) is needed for this purpose, which is able to connect the surface excess with the bulk composition. These types of equations (including the Langmuir equation) do not follow from the Gibbs adsorption equation. Therefore, it is of

3. DISCUSSION The Gibbs adsorption equation is widely accepted in the literature as the basic equation for the surface tension of solutions. On the other hand, the Butler equation was usually treated in the past as one of the model equations, being much below in its ranking compared to the Gibbs adsorption equation. This article might change this view, as the general validity of the Butler equation was proven above within the extended 5801

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the relative surface excess is negative if the given component prefers the bulk to the surface region, i.e., if xs < x: this is in accordance with eq 45. (4) According to Gibbs, the relative surface excess is zero if the given component has the same concentration in the bulk and in the surface region, i.e., if xs = x: this is in accordance with eq 45. (5) In the limit of dilute solutions (x ≪ 1), eq 45 reduces to the following reasonable equation x −x ΓB(A) = s (46) ω

additional interest to underline here that one of the simplified Butler equations (eq 35) is identical to the Langmuir equation, as extended by Piñeiro et al.53 Furthermore, another form of the simplified Butler equation (eq 37) simplifies to the Szyszkowski equation9 for dilute solutions. As was shown by Langmuir,7 the Gibbs adsorption equation, the Langmuir equation and the Szyszkowski equation are interconnected. As two of them follow from the simplified Butler equation, the third one (the Gibbs adsorption equation) should also follow from the Butler equation. Let us show this in somewhat more detail. Let us take the derivative of the simplified Butler equation (eq 37) with respect to x and multiply both sides of the resulting equation by dx and its right-hand side by x/x: dσ = −

(K − 1)x dx RT (1 + Kx − x)ω x

(6) In the limit of dilute solutions (x ≪ 1) with high surface activity of component B (x ≪ xs), eq 45 reduces to the following reasonable equation x ΓB(A) = s (47) ω

(39)

(7) In the limit of concentrated solutions (x → 1), eq 43 reduces to the following equation

For an ideal A−B solution, the partial molar Gibbs energy of component B (GB, J/mol) is written as

G B = G Bo + RT ln x

ΓB(A) →

(40)

where GBo (J/mol) is the standard molar Gibbs energy (chemical potential) of component B. Let us take the derivative of eq 40 with respect to x and multiply both sides of the resulting equation by dx:

dx (41) x Now, let us substitute eq 41 into eq 39 to replace RT dx/x: (K − 1)x dG B (1 + Kx − x)ω

Γ oB(A) = (42)

(K − 1)x (1 + Kx − x)ω

(43)

Substituting eq 43 into eq 42, the following equation is obtained: dσ = −ΓB(A) dG B

1 ω

(49)

(9) Trial calculations by eq 45 show a reasonable dependence of ΓB(A) on x, in agreement with the concentration dependence of the surface concentration and surface tension calculated with eqs 35 and 37 (Figure 2a−c). On the basis of the above, we can conclude that eqs 43 and 45, which follow from the simplified Butler equations (eqs 35−37) are identical to the relative surface excess of component B, in accordance with the definition originally given by Gibbs. Thus, it is proven that the Gibbs adsorption isotherm follows from the Butler equation. Furthermore, the above discussion also proves that the physical quantity introduced above “the mole fraction of component i in the surface region” is a meaningful quantity, and it is in agreement with the thermodynamic framework of Gibbs. All this is true at least under the simplification introduced in section 2.5. The general proof of the equality of the Gibbs adsorption equation with the Butler equation is given in ref 13. Above it has been proven that both the Langmuir equation and the Gibbs adsorption equation can be derived from the

Let us define the following physical quantity: ΓB(A) ≡

(48)

As follows from eq 48, the relative surface excess of pure component B depends on the selection of component A because it depends on the positioning of the dividing surface of Gibbs, which is a function of component A. (8) In the limit of concentrated solutions (x → 1) with a high surface activity of component B (K ≫ 1), eq 43 reduces to the following meaningful equation:

dG B = RT

dσ = −

K−1 Kω

(44)

One can see that eq 44 is identical to the Gibbs adsorption equation6 written for a binary A−B solution, assuming that the physical quantity defined by eq 43 is identical to the relative surface excess of component B if the dividing surface is selected such that the surface excess of component A is zero (as defined by Gibbs6). Let us express K from eq 35; then, let us substitute this new equation into eq 43 to replace K. After simplification, an alternative equation for the relative surface excess of component B is obtained, in agreement with the Butler equation: xs − x ΓB(A) = ω(1 − x) (45) Equation 45 connects the relative surface excess of Gibbs with our surface mole fraction in the case of an ideal solution and equal molar surface areas of the components. Because eqs 43 and 45 are new equations, let us see whether they are meaningful. Their identity with respect to the relative surface excess of component B in a Gibbsian sense is supported by the following: (1) It has the right units of mol/m2. (2) According to Gibbs, the relative surface excess is positive if the given component preferably adsorbs to the surface region, i.e., if xs > x: this is in accordance with eq 45. (3) According to Gibbs,

Figure 3. Logic of different equations for the surface tension and adsorption of solutions. (Note that the Butler equation is upgraded to the root equation in this article.) 5802

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Langmuir

carried out in the framework of the Center of Excellence of Applied Materials Science and Nano-Technology at the University of Miskolc. This work was carried out in the project TÁ MOP-4.2.2.A-11/1/KONV-2012-0019, the New Széchenyi Plan, supported by the European Union and co-financed by the European Social Fund.

Butler equation whereas the Langmuir equation and the Gibbs adsorption equations do not follow from each other. Thus, the renovated Butler equation is a root equation for the surface tension of solutions (Figure 3).

4. CONCLUSIONS The partial surface tension of any component in a multicomponent solution was thermodynamically defined in this article, following the definition of surface tension by Gibbs, extending the limits of his thermodynamic framework. It was shown by a formal thermodynamic derivation that at equilibrium the partial surface tension values for all components in a solution must equal each other and the surface tension of the solution. Furthermore, the partial surface tension of a component was written as a function of the bulk and surface concentrations of the same component. This result coincides with the Butler equation, although the derivation is absolutely different; our new derivation is not based on the Gibbs adsorption equation, and it does not make any assumptions about the thickness or structure of the surface region, thus it has general validity. Furthermore, it was shown here that the renovated Butler equation is the root equation for both the Gibbs adsorption equation (1878) and the Langmuir equation (1918) as well as all of their derivatives, including the semiempirical Szyszkowski equation (1908). On the other hand, the Langmuir equation and the Gibbs adsorption equation do not follow from each other, so the renovated Butler equation is a more fundamental equation of the surface tension of solutions than any other. The renovated Butler equation provides both the surface composition and surface tension of binary and multicomponent solutions as a function of their bulk composition, temperature, and pressure. The Butler equation is flexible: it can take any solution model to describe the excess thermodynamic functions (activity coefficients) of the bulk solution and that of the surface region of the solution; also, any value/concentration dependence of the molar surface areas can be used. The renovated Butler equation is also easy to teach, especially if compared to the definition of the relative surface excess, the dividing surface, and the derivation of the Gibbs adsorption isotherm. In my opinion, the renovated Butler equation provides similarly deep insight into the nature of solution interfaces, as was the insight of Gibbs into the condition of heterogeneous equilibrium of multicomponent systems. The renovated Butler equation should be used together with the equation of Gibbs for bulk heterogeneous equilibria to calculate the equilibrium of phases and interfaces between them in multicomponent and multiphase systems for both macro- and nanosizes.33,37,52

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the TÁ MOP-4.2.2.A-11/1/ KONV-2012-0027 project. The project is cofinanced by the European Union and the European Social Fund. I also acknowledge financial support from the Hungarian Academy of Sciences under grant K101781. This research was partially 5803

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