Non-Applicability of the Gibbs–Duhem Relation in ... - ACS Publications

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Non-Applicability of the Gibbs−Duhem Relation in Nonextensive Thermodynamics. Case of Micellar Solutions Pierre Letellier†,‡ and Mireille Turmine*,†,‡ †

Laboratoire Interfaces et Systèmes Electrochimiques, UPMC Université Paris 06, UMR 8235, Sorbonne Universités, F-75005, Paris, France ‡ CNRS, UMR 8235, LISE, F-75005, Paris, France ABSTRACT: This work is a discussion on the applicability of the Gibbs−Duhem relationship. We show that it does not exist in the nonextensive thermodynamic approaches. To illustrate this, we have chosen to present the properties of three surfactants (dodecyltrimethylammonium bromide (DTABr), sodium dodecyl sulfate (SDS), sodium decanoate (NaDec) in aqueous solutions at 298 K) whose behavior after the micellization threshold can be described by the rules of the nonextensive thermodynamics. We show that the Gibbs−Duhem relationship does not apply to these systems and we propose to formalize the deviation from this law. The consequences of this study are discussed for the different approaches that involve the use of Gibbs− Duhem relation in the case of micellized solutions. In particular, the applicability of the Gibbs relation which links surface tension and the chemical potentials of solutes is considered. More generally, this study warns against the blind application of the Gibbs− Duhem (or Gibbs) relation, to any system.

1. INTRODUCTION It is recognized that the relationships of thermodynamics, introduced by Gibbs in the 19th century, apply without restriction to solutions of ionic and molecular compounds in water or hydro-organic or organic solvents at different temperatures. They link, among others, changes in intensive magnitudes characteristic of the considered system, using the Gibbs−Duhem relationship. But, we must not forget that this relation is a mathematical consequence of the conventional choice to define the internal energy and entropy as homogeneous functions of the degree unit of the system mass (known as “extensive” state functions). The consequence is that the derivatives of these quantities with respect to the system mass are homogeneous functions of degree 0 of the mass (“intensive” variable). This is the case of the chemical potential, pressure, temperature, and so on. This convention on the internal energy and entropy is restrictive. It limits the generality of the thermodynamics proposed by Gibbs. It cannot pretend to describe the behavior of all systems. This is so, for example, for metal nanoparticles, whose melting points vary with the size. This feature means that the chemical potential of these compounds depends on the particle mass. In this case, the chemical potential is not an intensive magnitude and the extent from which it is derived is not extensive. It is partly for this reason that conventions other than those of Gibbs, have been proposed to build other thermodynamics in which the internal energy and entropy are not extensive state functions. In 1988, Tsallis1 introduced, from statistical considerations a thermodynamics in which the entropy is not an additive © 2015 American Chemical Society

function. Thus, when two systems A and B are joined, the entropy of the whole system A−B is related to those of A and B by Sq(A , B) = Sq(A) + Sq(B) +

1−q Sq(A)Sq(B) kT

(1)

The deviation from unity of the entropic index, q (positive), measures the degree of nonextensivity of the system. The value of kT is likely to vary with that of q. This thermodynamics is not extensive. Temperature and pressure can be not intensive magnitudes.2 This approach is very fruitful. However, its drawback for chemists and physical chemists is that it is formally inconvenient to describe the reactivity of the systems we are studying. This is why, in 2004, we introduced a “nonextensive” thermodynamics3 from a phenomenological approach without statistical reference, suitable for the description of the chemical reactivity. The application of the established relations allowed the description of the behavior of many systems, such as those with small size4−6 or others that involve significant interfacial energies.7 From that time, other approaches have been proposed based on Tsallis’ formalism to describe the thermodynamic potentials.8 In our approach, the geometric boundaries of the systems are not defined, and the interfaces are assumed to be “fuzzy”. The entropy and the internal energy are Euler’s functions of degree ω (we use in this paper the notation ω to denote the system Received: December 17, 2014 Revised: February 12, 2015 Published: February 22, 2015 4143

DOI: 10.1021/jp512576y J. Phys. Chem. B 2015, 119, 4143−4154

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The Journal of Physical Chemistry B dimension instead of “m” as in our previous works, in order to avoid any confusion with the molality, also denoted m) of the system mass, different from one. Thus, they are not extensive. The extents, as pressure and chemical potential, derived from the internal energy, vary with the system mass. Thus, they are not intensive. This means that the Gibbs−Duhem relation cannot be established. It does not apply. A number of systems are affected by this restriction. One can obviously mention the nanoparticles, but also the micellar aggregates of ionic surfactant. Indeed, we have shown in a previous publication,6 that their behavior in water and in electrolyte solutions can be described very precisely from the nonextensive thermodynamics relations. The logic should be that the Gibbs−Duhem relationship does not apply to the micellar media. This is a problem because this relationship is widely used, among other for • calculating values of the activity coefficients of the surfactant in solution from those of the activity solvent (osmotic coefficient). • analyze the values of surface tensions of micellar solutions from those of chemical potentials (or vice versa). The Gibbs relation expressing the surface excess is directly deducted from the Gibbs−Duhem relation. On a first step, the purpose of this article is to complete the formalism of the nonextensive thermodynamics by establishing for mixtures the relationship between extensities involved in the internal energy and the variables of tension. On a second step, in the light of the established relations, we will examine the behavior of three surfactants, dodecyltrimethylammonium bromide (DTABr), sodium dodecyl sulfate (SDS), and sodium decanoate (NaDec) in aqueous solutions at 298 K. We will discuss, for these three surfactants, the applicability of the Gibbs−Duhem relation to micellar solutions.

At this stage of reasoning, the only relations between the tension magnitudes and the extensities result from the properties of the second derivatives that characterize the exact differentials (Schwarz’ theorem). In Gibbs thermodynamics, it is assumed that G is an extensive state function, that means a homogeneous function of degree one of the system mass. Thus, if the system mass is multiplied by a number λ, then U is multiplied by λ Uλ = U (λn1 , λn2) = λU (n1 , n2) = λU

Consequently, the properties of Euler’s functions allows writing the integral function of U = U(S,V,n1,n2,X...), as U = TS − PV + μ1n1 + μ2 n2 + YX

⎛ ∂U ⎞ ⎜ ⎟ ⎝ ∂S ⎠V , n , n 1

⎛ ∂U ⎞ +⎜ ⎟ ⎝ ∂n1 ⎠S , V , n

dS + 2,X

2,X

⎛ ∂U ⎞ ⎜ ⎟ ⎝ ∂V ⎠S , n , n 1

0 = S dT − V dP + n1 dμ1 + n2 dμ2 + X dY

1

dV

⎛ ∂U ⎞ dn1 + ⎜ dn 2 ⎟ ⎝ ∂n2 ⎠S , V , n , X 1

2

(2)

Conventionally, the tension magnitudes T, P, μ1, μ2, Y (temperature, pressure, chemical potential of 1, chemical potential of 2 and Y which is the variable of tension associated with X) are defined from the partial derivatives of U. Thus, the following relation can be established dU = T dS − P dV + μ1 dn1 + μ2 dn2 + Y dX

(6)

Variations of intensive magnitudes can be then linked, and especially in solution for which the chemical potential of a solute is linked to the chemical potential of the solvent or to the surface tension of the solution. As can be seen, all these properties ensue from the convention adopted for U. But, what happens when it is different? In 1997, Sorensen et al.10 suggested a debate on this subject, they particularly discussed the case of systems for which the variables of extensity can be Euler’s functions of different degrees. Then, they justified that the Gibbs−Duhem relation does not apply in the general case. In the nonextensive thermodynamic approach we propose, the situation is simpler than in those treated by Sorensen because we consider the case of systems where the products of the tension variables by the extensities are defined as homogeneous functions of degree ω of the mass, as well as for the internal energy. We will describe the behavior of these systems. Consider a system consisting of n1 moles of 1 and n2 moles of 2, but whose geometric boundaries between 1 and 2 are unclear or ill-known (fuzzy interface). In this situation, it is not possible to define the interfacial energy of the system as the product of an area by a surface tension. Nevertheless, this type of system must have an interfacial energy, Eσ, which must be defined. We considered that changes in Eσ are equal to the product of a tension variable, τ, by a magnitude of extensity, χ, whose property is to be a homogeneous function of degree ω of the mass, such as

2 ,X

⎛ ∂U ⎞ dX +⎜ ⎟ ⎝ ∂X ⎠S , V , n , n

(5)

This integral form of U is a mathematical property of homogeneous functions of degree one of the mass. It is checked only if S and X are extensive variables too. V and n are extensive by definition. The result is that T, P, μ, and Y, which derive from the internal energy with the mass are intensive magnitudes. They do not vary with the mass of the considered system (homogeneous functions of degree 0 of the mass). The Gibbs−Duhem relationship is an equation of consistency, which allows one to obtain eq 3 by deriving eq 5.

2. THE GIBBS−DUHEM RELATION DOES NOT EXIST IN NONEXTENSIVE THERMODYNAMICS 2.1. The Relations of the Nonextensive Thermodynamics. Consider a system made of n1 moles of component 1 and n2 moles of component 2 which can be molecular or ionic. The internal energy, U, is a function of variables of extensity9 as entropy, S, volume, V, of the system content (n1 and n2) or, to generalize, of an extent X, such as U = U(S,V,n1,n2,X...). The internal energy has an exact differential. dU =

(4)

χλ = χ (λn1 , λn2) = λ ωχ (n1 , n2) = λ ωχ

(7)

This contribution can be introduced in the expression of U = U(S,V,n1,n2,χ). The exact differential of U is written as dU = T dS − P dV + μ1 dn1 + μ2 dn2 + τ dχ

(3) 4144

(8)


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molality, m, of the surfactant and the critical micelle concentration expressed in molality, cmm.

It has the same form as eq 3, the Schwarz relations continue to apply, but U is a homogeneous function of the same degree of the mass that χ, i.e., ω which can be different from one. A number of other conventions are needed to continue the calculation. We supposed that • T and τ are intensive magnitudes. It follows that S and U are homogeneous functions of the same degree ω of the mass that χ, • The volume V and the numbers of moles, n1 and n2, being necessarily extensive variables, it follows that P and μ are tension magnitudes which are not intensive. They are homogeneous functions of degree (ω − 1) of the mass. The mathematical properties of Euler’s functions (see Appendix) allow writing the expression of U ωU = TS − PV + n1μ1 + n2μ2 + τχ

ln aM = ln aM(cmm) + AM (m − cmm)η ln aY = ln aY (cmm) + AY (m − cmm)η ln aMY = ln aMY (cmm) + (AM + AY )(m − cmm)η

The extent (m − cmm) is linked to the amount of micellized surfactant. aM(cmm), aY(cmm), and aMY(cmm) are respectively the activities of ions M, Y, and of the salt MY, at the critical micelle molality. Chemical potentials of the amphiphilic ions and the counterions are not intensive. As they derive from the internal energy with respect to an extensive quantity (number of moles), they are homogeneous functions of degree (ω − 1) of the amount of surfactant and then η = ω − 1. The parameters A contains constant terms characteristic of the system (positive) and the parameter τ (positive or negative). They are such as the quantities A(m − cmm)η are dimensionless. On this point, it is interesting to note that in all our analyses we have always found that the value of A for the amphiphilic ion is negative, implying that τ is also. For the counterions, the value A is positive and greater than that those related to the amphiphiles. This implies that AMY = AM + AY > 0. We were able to describe with great precision the variation in the activity of the ions constituting the micelle with their mass in the solution. Note: At this level, it is necessary to comment on the use of variable masses of micellar solutions at constant composition. The application of the above relations could fear that the values of tension magnitudes depend on the amount of micellar solution chosen for the measurement. This would imply that for different amounts of micellar solution, we would not find the same values for the parameters of tension (activities, surface tension, etc.). It is not the case because, the solution composition is fixed, and to choose a more or less important quantity comes to a simple operation of replication of the system. It does not change the ratio extensity/mass in this type of evolution. The extensity associated with the micellar phase varies linearly with the solution mass. The extensity is then an extensive quantity and the tension magnitude is intensive. In what follows, we will consider variations of the surfactant mass in solution at the constant number of moles of solvent in the molal system. 2.2.2. Relations between Surfactant Activity and Osmotic Coefficient in the Framework of Nonextensive Thermodynamics. Let consider a surfactant salt, MY, dissolved in water whose the values of activity experimentally determined by potentiometry and the osmotic coefficients obtained from the properties of water (osmometry, vapor pressure, melting point or boiling point of the solvent, etc.) are available. These both series of values come from independent experimental determinations. We want to check the validity of eq 11, in the molal scale, that means for 1 kg of water. The number of moles nMY contained in 1 kg of water corresponds to the molality, m. The number of moles of water contained in 1 kg of water is ne = (1/ M*e ), with M*e the molar mass of water expressed in kg mol−1. The relation of consistency is written in nonextensive thermodynamics

(9)

In order that eqs 9 and 8 are simultaneously true, it is necessary that (ω − 1) dU = S dT − V dP + n1 dμ1 + n2 dμ2 + χ dτ (10)

At constant T, P, and τ, we note that the Gibbs−Duhem relation is not valid for nonextensive functions. (ω − 1) dU = n1 dμ1 + n2 dμ2

(12)

(11)

The Gibbs−Duhem relation is recovered for ω = 1, corresponding to the case of extensive functions. In our approach, the Gibbs thermodynamics appears as a particular case of the nonextensive thermodynamics. It follows that the Gibbs−Duhem relation can be considered as a “criterion of extensivity”. It seems important to note, in conclusion of this section, that it can be risky to systematically apply the Gibbs−Duhem relation to systems without being sure that they are extensive. The results may have no meaning. Micellar solutions of ionic surfactants can be interesting examples to illustrate this situation. 2.2. Application to Micellar Solutions. 2.2.1. Surfactant Aggregates Considered as Nonextensive Phases. A large literature debate the status of micellar aggregates that can be considered as pseudophases (a series of historical references can be found in the publication of Elworthy et al. on the SDS11), or as complex in equilibrium with the free species in solution.12,13 Their behavior is sometimes simply described using empirical models.14 For our part,6 we proposed to consider these aggregates as “nonextensive phases” dispersed in an unlimited phase consisting of a surfactant solution at a concentration equal to the critical micelle concentration, cmm (critical micellar molality). Aqueous surfactant solutions are considered sufficiently dilute so that the values of concentrations and molalities can be identified. We therefore transposed into the molal system the relations demonstrated in concentration. From experimental values published in the literature, we showed for ionic surfactants that the aggregated amphiphiles and their counterions form two interpenetrated nonextensive phases of the same thermodynamic dimension, ω. Thus, for MY salt, the logarithms of the activities of each ion (aM and aY) and salt (aMY), in the micellar solution, can be written in the form of power laws of the difference between the 4145


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The Journal of Physical Chemistry B η dUr = m(dμMY ) +

1 (dμ ) Me* e

2(mϕ − cmmϕmmc) = AMY m(m − cmm)η

(13)

⎛ A ⎞ − ⎜ MY + ηβ ⎟(m − cmm)η+ 1 ⎝η + 1 ⎠

In this relation, dU is related to the variations of the internal energy of the system for 1 kg of water. For simplicity, the osmotic coefficient is usually defined by

ln ae = −2mMe*ϕ

This relation is general and includes the case where the Gibbs−Duhem relation applies with ηβ = 0. Let set

(14)

α=

The coefficient 2 corresponds to the number of particles generated by the salt MY in solution. The relation of consistency is written η dUr = md[ln aMY ] − 2d(mϕ) RT

= (m − cmm)η (AMY m − α(m − cmm))

(16)

Y = AMY m −

2(mϕ − cmmϕcmm) (m − cmm)η

= α(m − cmm)

(25)

If the above relations apply, we must obtain a straight line passing through the origin with a slope α. Once α known, we can determine the value of β and establish the deviations from the Gibbs−Duhem law as a function of molality. The values of the osmotic coefficient may then be recalculated by

(17)

The parameter β, assumed to be constant, characterizes the nature of the considered surfactant. The relation of consistency is written as ηβd(m − cmm)

(24)

When there are two independent sets of experimental data: one for the activities and one for the osmotic coefficient, this relationship is easy to check graphically by plotting Y against (m − cmm).

The particular case of this relation is that where ω − 1 = η = 0, i.e., ω = 1, which corresponds to the condition of extensity. In this case, Gibbs−Duhem’s relation is refound. dU is the derivative of a homogeneous function of degree η + 1 of the amount of micellized surfactant, so

η+1

(23)

2(mϕ − cmmϕcmm)

We replace the expressions of activities (eq 12)

dUr = βd(m − cmm)η+ 1 RT

AMY + ηβ η+1

Equation 22 then becomes

(15)

η dUr = AMY md(m − cmm)η − 2d(mϕ) RT

(22)

ϕ=

η

= AMY md(m − cmm) − 2d(mϕ)

⎛ (m − cmm) ⎞ 1 cmm ⎟ ϕ + (m − cmm)η ⎜AMY − α ⎝ ⎠ 2 m m cmm

(18)

(26)

The left-hand side of this equality represents deviations from the Gibbs−Duhem law. They vary with the solution composition. The product ηβ plays the role of “parameter of deviation to the extensivity” held by the index q in Tsallis’ approach. It is sufficient that one of the two terms is zero to find the properties of extensive functions. Thus, we can imagine nonextensive systems, η ≠ 0, but for which the nature of the surfactant is characterized by β = 0. In this case, the behavior of the system is extensive. Therefore, the analytical expression of the osmotic coefficient can be written as

We will assess the significance of these relations from experimental results concerning DTABr, SDS and NaDec.

3. BEHAVIOR OF IONIC SURFACTANTS IN MICELLAR SOLUTIONS 3.1. Before the Critical Micellar Molality (for DTABr and SDS). In order to describe the behavior of DTABr and SDS, we adopted to account for the variations of activity coefficients of the two ions in solution, for molalities, m, lower than the critical micellar concentration, the equation used by Guntelberg,15

2d(mϕ) = AMY md(m − cmm)η − ηβd(m − cmm)η+ 1

ln γM = ln γY = ln γ± = −2.3

(19)

The integration can be realized between m and cmm. The first term of the right-hand side of this equality can be integrated by parts m

∫cmm (m − cmm)η dm m

− ηβ

∫cmm d(m − cmm)η+1

(20)

ϕ=1−

then 2(mϕ − cmmϕcmm) = AMY m(m − cmm)η − (m − cmm)η + 1 − ηβ(m − cmm)η + 1

(27)

Assuming that the premicellar solution meets the criteria of the extensivity, we can establish how the osmotic coefficient varies with the molality of the solution, which allows to determine the value of the osmotic coefficient at the cmm, essential for use eq 24. Calculating of values of osmotic coefficients request to integrate the logarithm of the mean activity coefficient between 0 to m. This calculation was presented by Hamer.16

2(mϕ − cmmϕcmm) = AMY m(m − cmm)η − AMY

0.5 m 1+ m

AMY η+1

2.3B ⎛ ⎜1 + m ⎝

m − 2 ln(1 +

m) −

⎞ 1 ⎟ 1+ m⎠ (28)

Regarding NaDec, another approach of the deviations from the infinitely dilute behavior will be justified. 3.2. After the cmm: Aggregates form Nonextensive Phases. There are in the literature independent values of

(21)

Terms of the same power are gathered 4146


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The Journal of Physical Chemistry B Table 1. Characteristic Parameters of Nonextensive Phases Made of Surfactants beyond the cmma DTABr SDS NaDec a

cmm/mol kg−1

η

A+

A−

AMY

β

ϕcmm

0.0136 0.0086 0.0114

0.5 0.69 0.65

−2.4171 7.02 1.122

3.2417 −5.82 −0.537

0.8246 1.2 0.585

0.192 0.0531 −0.249

0.962 15 0.968 89 0.945

The values of A for the amphiphiles are negative, whereas they are negative for the counterions.

activities and osmotic coefficients for DTABr, SDS, and NaDec. Therefore, these data will be used to illustrate and discuss previous relations. The different parameters characterizing the nonextensive phases were reported in Table 1. 3.2.1. Behavior of DTABr considered as nonextensive phase. Values of osmotic coefficients and mean activity coefficients, γ±, obtained by applying the Gibbs−Duhem relation have been published by Delisi.17 From eq 12 and the parameters given in Table 1, DTABr activity is calculated by assuming that micellar aggregates form nonextensive phases. These values are given in Table 2. Table 2. Values of Osmotic Coefficients and Logarithms of Mean Activity Coefficients Published by Delisi17 (a) and Values of DTABr Activities Calculated According to Our Approach6 (b) m/mol kg−1

ϕexp (a)

ln γ± (a)

ln aMY (b)

0.009006 0.011 46 0.014 02 0.020 86 0.028 42 0.031 99 0.035 87 0.047 33 0.050 04 0.057 74 0.070 14 0.079 93 0.099 64 0.1138 0.1484 0.1968 0.2260 0.2945 0.3900 0.4922 0.5908 0.7876

0.9655 0.9591 0.9342 0.7177 0.5296 0.4497 0.415 0.3177 0.3014 0.2781 0.2202 0.2038 0.177 0.1593 0.1415 0.129 0.1204 0.1061 0.1075 0.1037 0.1009 0.1004

−0.053 −0.0653 −0.1008 −0.3797 −0.6815 −0.8216 −0.9213 −1.1947 −1.2494 −1.3746 −1.579 −1.6986 −1.9049 −2.0334 −2.2791 −2.5387 −2.6687 −2.9203 −3.1731 −3.3869 −3.5545 −3.8171

−9.6191 −9.1602 −8.8187 −8.7653 −8.7352 −8.7238 −8.7125 −8.6841 −8.6782 −8.6623 −8.6395 −8.6232 −8.5937 −8.5746 −8.5328 −8.4826 −8.4555 −8.3985 −8.3297 −8.2651 −8.2091 −8.1101

Figure 1. Checking of eq 25 for DTABr. The value of α is equal to 0.6278.

Figure 2. Illustration of eq 25 for DTABr, which highlights the deviations from Gibbs−Duhem law vs the number of cmm.

Therefore, we can graphically check the validity of eq 25 (Figure 1). The plot of Y against the difference (m-cmm) leads to excellent straight line passing through the origin. The slope value α is equal to 0.6278. Then, the value of β = 0.156 is determined. Knowing these parameters allows calculating the deviations from the Gibbs−Duhem law as follows: η

dUr = ηβ(η + 1)(m − cmm)η dm

cmm for reaching 0.04 for a dozen cmm. For 60 cmm, the deviation from the Gibbs−Duhem law exceeds 0.1. Although these differences are small, they are significant. The values of DTABr activities calculated with the Gibbs−Duhem relation are different from those experimentally obtained by potentiometry as illustrated in Figure 3. The difference between the two sets of data is clear. 3.2.2. Behavior of SDS Considered as a Nonextensive Phase. The same development can be realized with the SDS. The values of osmotic coefficients are those published by Crisantino et al.18 (Table 3). In Table 3, the activity values are calculated from eq 12 with the parameters reported in Table 1, which were determined from the experimental values of Cutler et al.19 Verification of eq

(29)

Figure 2 presents the variation η dUr/dm vs the molality expressed in number of cmm. As shown on Figure 2, the deviation from the Gibbs−Duhem law increases rapidly for surfactant concentrations close to the 4147


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Figure 5. Illustration of eq 25 for SDS. This highlights the deviations from the Gibbs−Duhem law in respect to the number of cmm.

Figure 3. Logarithms of DTABr activities determined from the Gibbs−Duhem relation (blue diamonds) and from potentiometric measurements (black circles) plotted vs the number of cmm.

The differences are not significant; indeed at 50 times the critical micelle molality, they only reach 0.03. Nevertheless, the case of SDS is interesting because it corresponds to the situation where the behavior of the surfactant is perfectly described by the nonextensive thermodynamic approach, but with a relatively low deviation from the extensivity for all the studied range of molalities. This example shows the importance of the term β whose value depends on the surfactant nature. However, these differences are significant even if they are small. The values of SDS activities calculated by Gibbs−Duhem equation are different from those experimentally determined by potentiometry. For comparison, we plotted in Figure 6, the logarithms of activities found according to each of these two approaches versus the SDS molality.

Table 3. Values of Osmotic Coefficients and Logarithms of the Mean Activities from18 (a) and Values of DTABr Activities Calculated from our Approach6 (b) m/mol kg−1

ϕexp (a)

ln γ± (a)

ln aMY (b)

0.009 92 0.015 23 0.025 19 0.045 03 0.073 89 0.1001 0.1456 0.1936 0.2463 0.2979 0.3315 0.3856

0.877 0.536 0.409 0.261 0.147 0.16 0.144 0.133 0.124 0.126 0.125 0.13

−0.2 −0.66 −1.05 −1.6 −2.11 −2.36 −2.7 −2.96 −3.18 −3.36 −3.44 −3.57

−9.6948 −9.6695 −9.6362 −9.5851 −9.5246 −9.4767 −9.4027 −9.3326 −9.2619 −9.1972 −9.1571 −9.0950

25 leads to a straight line with a slope of 0.7467 which allows determining β = 0.0531 (Figure 4). These values are lower than those obtained for SDS. Variations of the deviation from the extensivity according to the SDS molality expressed in number of cmm is given in Figure 5.

Figure 6. Logarithms of SDS activities determined from the Gibbs− Duhem relation (blue diamonds) and from potentiometric measurements (black circles) plotted vs the number of cmm.

3.2.3. Behavior of Sodium Decanoate, NaDec, Considered as Nonextensive Phase. This case is probably the most interesting among the discussed examples. In 1979, Vikingstad20 published a potentiometric study on the activities of sodium and decanoate ions in water at 25 °C. Two years later, Delisi21 measured the osmotic coefficients of

Figure 4. Checking of eq 25 for SDS. The value of α is equal to 0.7467. 4148


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The Journal of Physical Chemistry B this same surfactant. Apparently, the two sets of results are not consistent. Indeed, the values of the osmotic coefficients show two breaks when plotted as a function of the molality, suggesting the existence of two successive forms of the surfactant aggregation. However, results from the potentiometric study highlight only one break due to the micellar aggregation. Another source of amazement concerns the significant differences between the experimental mean activity coefficients determined by potentiometry and those calculated using the Gibbs−Duhem relation. Delisi discussed this divergence primarily in terms of validity of the techniques whether for electrodes or for the determination of osmotic coefficients. He assumed that in both cases, the experimental values can be spoilt by mistakes, which is an indisputable fact. More recently, Sharma22 indicated the “unusual behavior” of NaDec, in water and in the presence of NaCl, from measurements of osmotic coefficients. The values of the surfactant activity coefficients are calculated from the GibbsDuhem relation, using an equilibrium model for the micelle. The calculated values seem surprisingly small. Experimentally, it should be noted that the values of osmotic coefficients found by Sharma perfectly match up with those of Delisi and extend over a larger concentration range. We can believe in the validity of experimental determinations proposed by Delisi. We took all these data and we first extracted the potentiometry values of Vikingstad from the graphs of its publication (Table 4). We have exploited them according to the same procedures used in our study in 2008. For NaDec, the activity values of both ions for the lower concentrations show that eq 27 used for DTABr and SDS does not properly describe the experimental values. The behavior of

these ions markedly deviates from the model of diluted solution as shown in Figure 9 where we have plotted the logarithm of the mean activity of the salt according to the logarithm of the molality. Without trying to explain this phenomenon, we have tried to formalize the experimental values and for this we used the expression of the mean activity coefficient proposed by Hamer16 with terms of higher order, usually reserved for concentrated salt solutions (most often greater than 1 mol kg−1 for 1−1 electrolytes). ln γ± = −2.3

0.5 m + bm + cm2 1+ m

(30)

Values of b and c can be determined by plotting W (eq 31) against the molality. W=

0.5 m m

ln γ± + 2.3 1 + m

= b + cm

(31)

A fair straight-line is obtained (Figure 7) with b = 4.8335 and c = −41.002.

Table 4. Molalities and Activities of Na+ and Dec− from the Graphs Published by Vikingstad.20 mNa+ (/mol kg−1)

aNa+

mDec‑ (/mol kg−1)

aDec‑

0 0.044 0.052 0.058 0.066 0.0765 0.0865 0.096 0.107 0.122 0.139 0.153 0.178 0.201 0.23 0.2545 0.283 0.318 0.3515 0.384 0.417 0.453 0.478

0 0.0425 0.048 0.054 0.06 0.0675 0.075 0.08 0.087 0.092 0.096 0.099 0.105 0.109 0.116 0.12 0.125 0.131 0.1355 0.14 0.144 0.149 0.152

0 0.044 0.052 0.058 0.066 0.0765 0.0865 0.095 0.107 0.126 0.144 0.165 0.193 0.22 0.252 0.28 0.315 0.345 0.382 0.413 0.452 0.493

0 0.0425 0.048 0.054 0.06 0.0675 0.075 0.079 0.081 0.084 0.083 0.081 0.08 0.078 0.076 0.074 0.0715 0.0705 0.069 0.0678 0.065 0.064

Figure 7. Graphical determination of parameters b and c of eq 31

Variations in mean activity are properly described for low NaDec concentrations, as shown in Figure 8. The dashed curve corresponds to the behavior of an infinitely dilute solution, (γ± = 1). The calculated values perfectly fit the experimental values. This preliminary study allows exploiting the results of Vikingstad according to eq 12. By adopting a critical micellar molality equal to 0.1149 mol kg−1, close to the value adopted by Sharma22 (0.113 mol kg−1) at 298 K, it is shown that the behaviors of Na+ and Dec− ions perfectly follow the model established for nonextensive phases (Figure 9). The dimension η = 0.652, common to both ions, is close to that determined for the SDS. As with other charged surfactants, the coefficient relating to the amphiphiles (A− = −0.537) is negative and the one corresponding to the counterions (A+ = 1.122) is positive. The coefficient ANaDec = 0.585 for salt is positive. We plotted in Figure 10, the experimental and calculated curves corresponding to studied molality range. The agreement is satisfactory. It does not show a “second break” on the curves beyond the critical micelle concentration. 4149


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We conclude from this preliminary study that the micellar solutions of NaDec behave like nonextensive phases, which justifies the questions the applicability of the Gibbs−Duhem relation for micellar solutions. We therefore examined the experimental values of osmotic coefficients and sought to verify eq 25, by plotting Y against the difference (m−cmm). But for this, the value of the osmotic coefficient at the critical micelle molality is needed. This meant that we had to formalize changes in osmotic coefficients in premicellar. Using Gibbs−Duhem relation, Hamer sets for the mean activity coefficient of eq 30 ϕ=1− + Figure 8. logarithm of the mean activity of NaDec vs the logarithm of the molality. The dots represent the experimental data. The open square correspond to the values calculated from eq 30. The dashed line shows the infinitely dilute behavior.

2.3B ⎛ ⎜1 + m ⎝

b 2c m + m2 2 3

m − 2 ln(1 +

m) −

⎞ 1 ⎟ 1+ m⎠ (32)

We calculated the values of osmotic coefficients by means of this relation and compared them with the experimental data. Plainly, the experimental and calculated data are different. It may be due, as suggested by Delisi, to a poor experimental determination of osmotic coefficients or activity coefficients, but also the result of an application of the Gibbs−Duhem relation in illicit conditions. Campbell,23 in 1965, stated the unusual behavior of sodium decanoate at low concentration in conductimetry and he evoked possible interactions between the ions of the electrolyte to form highly conductive structures. He determined a critical micellar molality of about 0.094 mol kg−1. So it seems that the association between Na+ and Dec− ions occurs for dilute solutions and the resulting medium cannot be described by an extensive thermodynamics. This feature poses the problem of the choice of the osmotic coefficient value at the critical micelle molality. From the experimental curve, we adopted a likely value enabling us to achieve consistent results for the nonextensive processing of data. We chose ϕcmm = 0.945. We can then calculate Y. The plot of this function according to the molality leads to a proper line through the origin of slope α = 0.2945 (Figure 11). We deduce the value of β = −0.2486. This allows calculating the deviations from Gibbs−Duhem relation (Figure 12).

Figure 9. experimental values for activities of Na+ (+) and Dec− (●) and the calculated values according ot the model of the nonextensive phases of Na+ (○) and Dec− (□).

Figure 10. Values of experimental osmotic coefficients (colored dots) and calculated (open squares) from eq 32 for NaDec. Figure 11. Checking of eq 25 for NaDec. The value of α is 0.2945. 4150


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exist in the nonextensive thermodynamic approaches, whether for that we develop or other derived from the conventions proposed by Tsallis.1 This first result is important because it shows that the Gibbs−Duhem relation is not universal and that its application depends on the considered system and the chosen thermodynamic conventions to describe the behavior. On this point we fully agree with the conclusions of Sørensen and Compan.10 In this work, we are particularly focused on micellar environments. Our thinking starts from the fact that we had previously shown, using results of potentiometric studies of DTABr and SDS, that the behavior of ionic micelles can be precisely described when aggregates of surfactants are considered as interpenetrating nonextensive phases, amphiphiles and counterions having the same thermodynamic dimension. It was therefore appropriate to consider the applicability of the Gibbs−Duhem relation for systems consisting of micellar solutions. This is the second aspect of our study. Our first task was to adjust the relations of nonextensive thermodynamics to this situation and to express the deviations from the Gibbs−Duhem equation (eq 24). To specify the values according to the surfactant molality, we propose a graphical exploitation of the osmotic coefficients according to eq 25. The latter does not leave much room for coincidence (straight line passing through the origin) and its verification cannot be fortuitous. For the three surfactants, we attempted this exploitation from experimental values of the literature, while being aware of their fragility. As Delisi et al. has noted,21 these are necessarily marred by mistakes and even a small change in values can lead to erroneous interpretations or hasty conclusions. However, for the three studied surfactants, graphic exploitation of data from the literature by eq 25 has led to the plot of straight-line passing through the origin, whose slope was used to determine the value of the parameter β which characterizes how the system deviates from the extensivity (eq 17). It depends on the surfactant nature, it may be positive or negative and more or less large. In the case of SDS, the β value is small but positive. Although SDS micelles have all the characteristics of nonextensive phases, deviations from the Gibbs−Duhem equation remain small. However, in the case of DTABr, β > 0, and still more in that of NaDec, β < 0, the deviations from the Gibbs−Duhem equation are important which excludes a valid application of this relation. The result is that a micellar system is determined, at given temperature, by five parameters. • The critical micellar molality, cmm. • ω (or η), which is the common dimension nonextensive phases of amphiphiles and counterions. The parameter ω qualifies the nonextensivity of the system. • Aamphiphile and Acounterion, which are parameters directly related to the tension magnitude τ (intensive) of each nonextensive phase. For amphiphile ions, the value of τ was always negative. • β, which characterizes the deviation from the Gibbs− Duhem relation. The value of this parameter depends on the surfactant nature. To determine it, we must have a series of independent data of osmotic coefficients and activity coefficients.

Figure 12. For NaDec, illustration of eq 25 which shows the deviation from Gibbs−Duhem relation according to the number of cmm.

Unlike previous surfactants, deviations from the Gibbs− Duhem law are negative, and of large amplitude. At three times the cmm, the gap reaches -0.1 (60 times the cmm for DTABr reaching +0.1). This difference is largely responsible, in our view, of the divergence observed between the values of activities determined by potentiometry and those calculated from the values of osmotic coefficients by applying the Gibbs−Duhem law (Figure 13).

Figure 13. Values of osmotic coefficient of NaDec. Experimental (dots) and calculated (open squares) from eq 26, with β = −0.2486.

These parameters are used to properly describe the variation of osmotic coefficients for micellar solutions.

4. DISCUSSION This work presents several aspects. The first, purely formal, attempts to complete the equations of nonextensive thermodynamics, we introduced in 2004, to establish the relationship between the chemical potentials of the components when the considered system is a mixture. By considering the state functions as nonhomogeneous functions of degree one of the mass, the write of the integral form of the internal energy is forbidden (eq 5). The consequence is that the Gibbs−Duhem relation does not 4151


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The Journal of Physical Chemistry B The conclusion of this discussion is that the Gibbs−Duhem relation does not apply to micellar solutions (except if ηβ product would be equal to 0), which has the effect of calling into question the validity of a large number of approaches conventionally used for linking together the extents of tension in the case of micellar solutions. We can give some examples. 4.1. Relations between Osmotic Coefficients and Activity Coefficients. This is the case we have just discussed. It is not certain that we can calculate the activities of charged and nonionic surfactants from osmotic coefficients,24 even from a micelle equilibrium model as suggested by MacNeil et al.25,26 We fully share their opinion when they wrote that the osmometry offers many interesting and unexplored possibilities for studies of mixed surfactant thermodynamics, but we are not persuaded that the application of the Gibbs−Duhem relation is allowed for these systems. It follows that the obtained values of the activity coefficients of the surfactants may be uncertain. For charged surfactants, the values of activities can always be determined independently by potentiometry, provided to prepare the relevant ion-selective electrodes. This is not possible for nonionic surfactants.25 Calling into question the applicability of the Gibbs−Duhem relation for micellar solutions, is to abandon the main access to the activities of these compounds, which poses some problems. 4.2. Mixtures of Surfactants. The behavior of mixtures of ionic/ionic, ionic/nonionic and nonionic/nonionic surfactants was often described using the regular solution theory, RST,27 considering the mixed micelle as a pseudophase. The composition of the micelle is determined assuming the applicability of the Gibbs−Duhem relation. In a nonextensive thermodynamic approach, the use of RST has no meaning, like all the approaches using the Gibbs−Duhem relation for determining the compositions of micelles.28 For our part, we must also call into question our own work29,30 published in this field. 4.3. Surface Tensions and Gibbs’ Law. Another consequence of the nonapplicability of the Gibbs−Duhem relation for micellar solutions is challenging the validity of the Gibbs’ equation. For a mixture of the components 1 (solvent) and 2 (solute), Gibbs’ relation binds the variations in surface tension, γ, of the solution to the variation in chemical potential of the solute through the excess surface, Γ2

Γ2 = −

dγ dμ2

For our part, as micelles are supposed to behave as nonextensive phases, Gibbs’ equation is not justified in a formal level. A deviation term is missing. The consequence is that the surface excess values can be different according to the approaches, for a given concentration. Determinations from the application of the Gibbs’ equation seem to us less reliable. Regarding surfactant solutions of concentration below the cmm, some studies32 also question the applicability of the Gibbs’ relation and show by neutron reflection the formation of surfactant multilayers in surface in the premicellar domain.38 This observation is similar to what we report in this work, for NaDec. Even below the critical micelle molality, the Gibbs− Duhem relation does not seem to apply. Is it due to insufficient accuracy of the experimental values, to the exploitation that we have made, or to the nature of premicellar systems that would lead the system to follow no more the Gibbs extensive thermodynamics? It is difficult to answer with certainty. But, this phenomenon deserves to be mentioned.

5. CONCLUSION We know that the consequences of this idea are disturbing. We do not want to unnecessarily cast doubt, but it seems important that we debate this topic as it relates to systems widely used and often treated as simple solutions. They are not. The work we present is meant to be a basis for reflection on the applicability of the Gibbs−Duhem relation. We show that it does not exist in nonextensive thermodynamic approaches, but it does not mean that the extents of tension of the system cannot be linked together. For that, we have to introduce a term of deviation from Gibbs−Duhem law and to formalize it. But the most important is to show that seemingly insignificant systems may have behaviors that cannot be treated by Gibbs’ thermodynamics and that relations, assumed universal, are not. This study cautions against the indiscriminate application of the Gibbs−Duhem relation or Gibbs equation, to any system. This can be audacious. This is therein the paradox of the Gibbs−Duhem relation. Before applying, we should first check that the behavior of the system follows the requirements of extensive thermodynamics. And for that, the best way is to check from sets of independent tension extent values that the Gibbs−Duhem relation is justified. This is why it must be primarily considered as a criterion of extensivity and not necessarily as a practical relationship to calculate the variations of a variable of tension from another variable whose values are experimentally determined. We think that Gibbs−Duhem relation should be used carefully and, whenever possible, it would be suitable to determine in independent way the values of the tension extents by dedicated techniques. Gibbs−Duhem relation is not universal. It is totally related to the Gibbs thermodynamics. It may not apply to certain systems.

(33)

This relationship is directly derived from the Gibbs−Duhem equation. Let us suppose that the conditions are such that the nonextensive thermodynamics applies, this relation has no more meaning and there is no simple relation between changes in surface tension and in the chemical potential of the solute as assumed in various studies.22,31,32 These last few years, several studies have reported differences in excess surface value, for a given surfactant concentration, when determined by two different ways, in tensiometry and by a physical method (neutron reflection (NR). Radiotracers, etc.). The applicability of Gibbs’ equation has been called into question by several authors for micellar solutions. It was proposed to limit its significance in certain concentration ranges,33 or to reexamine it.34−37

■

APPENDIX: EULER’S FUNCTIONS AND THEIR USE IN THERMODYNAMICS We have summarized below some properties of homogeneous functions (Euler’s functions), directly related to the text of the publication. For a function F of two variables x and y, F(x,y), with an exact differential, the derivative of F is written as 4152


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A′n1 and A′n2 are the partial areas of components 1 and 2, respectively. They are also linked by the Schwarz equation. Let suppose that the system content is multiplied by λ. For the volume, we will have

d F = d F (x , y ) =

⎛ ∂F ⎞ ⎛ ∂F ⎞ ⎜ ⎟ dx + ⎜ ⎟ dy ⎝ ∂x ⎠ y ⎝ ∂y ⎠x

= F ′x dx + F ′ y dY

V (λn1 , λn2) = λV (n1 , n2)

(34)

The volume is a homogeneous function of degree 1 of the system mass (p = 1). It is an extensive magnitude. The derivatives of V with n1 and n2 are of homogeneous functions of degree 0 of the mass, they are intensive. They do not vary with the system mass. For the volumes, the partial derivatives are linked together by a relation similar to Gibbs−Duhem equation

Schwartz’ relations linked together the partial derivatives F′x and F′y by ⎛ ∂F ′ y ⎞ ⎛ ∂F ′x ⎞ ⎟ ⎜ ⎟ =⎜ ⎝ ∂y ⎠x ⎝ ∂x ⎠ y

(35)

0 = dV ′n1 n1 + dV ′n2 n2

Among the functions F(x,y) which have an exact differential, some are such that if we multiply by a number λ the variable values, then the function Fλ = (λx,λy) is related to F(x,y) by the equation Fλ = F(λx , λy) = λ pF(x , y)

A(λn1 , λn2) = λ 2/3A(n1 , n2)

1 − dA = dA′n1 n1 + dA′n2 n2 3

(37)

Equations 34 and 37 are simultaneous true if

⎛ ∂F(λx , λy) ⎞ F ′ λx = ⎜ ⎟ = λ p − 1F ′x ⎝ ⎠ λy ∂λx

■

(39)

*(M.T.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Tsallis, C. Possible Generalization of Boltzmann-Gibbs Statistics. J. Stat. Phys. 1988, 52, 479−487. (2) Abe, S. Y.; Martinez, S.; Pennini, F.; Plastino, A. Nonextensive Thermodynamic Relations. Phys. Lett. A 2001, 281, 126−130. (3) Turmine, M.; Mayaffre, A.; Letellier, P. Nonextensive Approach to Thermodynamics: Analysis and Suggestions, and Application to Chemical Reactivity. J. Phys. Chem. B 2004, 108, 18980−18987. (4) Letellier, P.; Mayaffre, A.; Turmine, M. Melting Point Depression of Nanosolids: Nonextensive Thermodynamics Approach. Phys. Rev. B 2007, 76 (045428), 1−9. (5) Letellier, P.; Mayaffre, A.; Turmine, M. Redox Behavior of Nanoparticules: Nonextensive Thermodynamics Approach. J. Phys. Chem. C 2008, 112, 12116−12121. (6) Letellier, P.; Mayaffre, A.; Turmine, M. Micellar Aggregation for Ionic Surfactant in Pure Solvent and Electrolyte Solution: Nonextensive Thermodynamics Approach. J. Colloid Interface Sci. 2008, 321, 195−204.

⎛ ∂V ⎞ ⎛ ∂V ⎞ dV = ⎜ ⎟ dn2 = V ′n1 dn1 + V ′n2 dn2 ⎟ dn1 + ⎜ ⎝ ∂n2 ⎠n ⎝ ∂n1 ⎠n 1

(40)

V′n1 and V′n2are the partial volumes of components 1 and 2, respectively. They are linked by the Schwarz’ equations. In the same way, the area of the droplet A = A(n1,n2) has an exact differential ⎛ ∂A ⎞ ⎛ ∂A ⎞ dA = ⎜ ⎟ dn2 = A′n1 dn1 + A′n2 dn2 ⎟ dn1 + ⎜ ⎝ ∂n2 ⎠n ⎝ ∂n1 ⎠n 2

AUTHOR INFORMATION

Corresponding Author

Similarly, when a homogeneous function of degree p is divided by one of the variables, a homogeneous function of order p-1 is obtained. These relations can be illustrated considering the volume, V, and the area, A, of a spherical drop of a liquid mixture consisting of n1 moles of 1 and n2 moles of 2. The volume V = V(n1,n2) of the droplet has an exact differential,

2

(45)

In the case of a drop which swells when its content increases, the Gibbs relation does not apply. However, if the considered system replicates, i.e., whose number of drops is multiplied by λn when content is multiplied by λ, the total area of the system is then that of a drop multiplied by λ. In this case, the area is an extensive magnitude. The extensive or nonextensive character of a magnitude depends on how the transformation is envisaged. For some of them, the considered function may not be a homogeneous function, for example, the total area of a cylinder of base area Ao of which the height is increased with the content.

(38)

Another important property of these functions is that the homogeneity is conserved during derivations and divisions. The derivatives of homogeneous functions of degree p are homogeneous functions of degree p − 1 of the same variables.

⎛ ∂F(λx , λy) ⎞ F ′ λy = ⎜ ⎟ = λ p − 1F ′ y ∂ λ y ⎝ ⎠ λx

(44)

In the selected case (a sphere) the area is a homogeneous function of degree p = 2/3 of the system volume, thus of its mass. The area is not an extensive extent. Its partial derivatives with n1 and n2 are of degree −1/3 of the mass. They are not intensive and vary with the system mass. For the area, the partial derivatives are linked together by

These functions are homogeneous of degree p of the variables x and y. They are called Euler’s functions. They have different mathematical properties invoked in thermodynamics and physical chemistry. Thus, the integral function F = F(x,y) is simply written from the partial derivatives

(p − 1) dF = dF ′x x + dF ′ y y

(43)

However, for the area, if the system content is multiplied by l, we will obtain

(36)

pF = F ′x x + F ′ y y

(42)

1

(41) 4153


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