The Role of Concentration Dependent Static Permittivity of Electrolyte

Article pubs.acs.org/JPCB

The Role of Concentration Dependent Static Permittivity of Electrolyte Solutions in the Debye−Hü ckel Theory Ignat Yu. Shilov* Department of Chemistry, Lomonosov Moscow State University, 119991 Moscow, Russia

Andrey K. Lyashchenko† Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, 119991 Moscow, Russia ABSTRACT: The Debye−Hückel theory has been extended to allow for arbitrary concentration dependence of the electrolyte solution static permittivity. The theory follows the lines advanced by Erich Hückel (Hückel, E. Phys. Z. 1925, 26, 93) but gives rise to more general and lucid results. New theoretical expressions have been obtained for the excess free energy of solution, activity coefficient of water and mean ionic activity coefficient. The thermodynamic functions contain two terms representing interionic interactions and ion−water (solvation) interactions. The theory has been applied to calculate the activity coefficients of components in the aqueous solutions of alkali metal chlorides from LiCl to CsCl at ambient conditions making use of permittivities taken from experimental dielectric relaxation studies. Calculations without parameter adjustment have demonstrated a semiquantitative agreement with experimental data, reproducing both the nonmonotonic concentration dependence of the activity coefficients and the ordering of activity coefficients for the salts with different cations. A good agreement with experimental data is obtained for the aqueous solutions of LiCl in the concentration range up to 10 mol/kg. The nonmonotonic concentration dependence of activity coefficients is explained as a result of a balance between the effect of interionic interactions and the solvation contribution which appears quite naturally in the framework of the Debye−Hückel approach after incorporation of variable permittivity of solution.

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INTRODUCTION The static permittivity is important property of liquid matter characterizing its response to the electric field. It is employed as parameter to describe effective interionic interactions in those molecular theories of electrolyte solutions that treat ions as particles and solvent as continuum dielectric medium.1 According to the Friedman’s classification2 such theories are referred to as the McMillan-Mayer level Hamiltonian models. An eminent example of these models is the Debye−Hückel (DH) theory published in 1923.3 Although it was understood from the very beginning that the static permittivity of electrolyte solutions differ from that of the pure solvent, the reliable experimental data for solutions were not obtained until the middle of the 20th century.4−6 Thus, in most theoretical developments until the end of the 20th century, a common practice was to employ the experimental permittivity of pure solvent to describe the electrolyte solution medium.7 The models of the McMillan-Mayer level provide a simple and direct way to analyze the interrelationship between the dielectric and the thermodynamic properties of electrolyte solutions on a molecular footing. In the theories of the Born−Oppenheimer level,2 in which the solvent has explicit molecular structure, the permittivity and the activities of components are considered as separate macroscopic properties determined by particle parameters and interparticle interaction potentials. In the © XXXX American Chemical Society

framework of the integral equation theories, the thermodynamic properties were investigated, e.g., in refs 8−10 and the permittivity in refs 8, 11−14. The classical computer simulation methods were also employed to calculate activities15,16 and permittivity.17−19 The ab initio Molecular Dynamics simulations20−22 representing the Schrödinger level of theory, in which the matter consists of the nuclei and electrons, are presently focusing mainly on the structural issues of electrolyte solutions, attainable with the current computational resources. The experimental source of the static permittivity of electrolyte solutions is the dielectric relaxation spectroscopy methods.1 The dielectric relaxation spectroscopy of conducting medium provides data on the frequency dependent complex generalized permittivity,1 which includes both dielectric polarization and electric conductivity contributions. The static permittivity is then computed by extrapolation to zero frequency of the frequency dependent permittivity, which includes only polarization effect.1 This procedure is by no means straightforward because it requires selection of a relaxation model and estimation of its parameters. This problem is discussed in more detail in refs 23 and 24, and the references cited therein. Received: May 12, 2015 Revised: June 25, 2015

A

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solvation.48 The goal of the present work is thus to revise the Hückel’s extension of the DH theory and to test its applicability employing contemporarily available experimental values of permittivities.

Nevertheless, the experimental static permittivity is now available for tens of electrolytic systems,25−32 and more references to experimental data can be found elsewhere.23,33,34 Besides the ambiguity in experimental permittivity associated with the choice of a particular relaxation model, there is another problem that concerns its application in the theoretical models for electrolyte solutions. The static permittivities extracted from dielectric relaxation experiments correspond to a state with stationary ionic current and thus can consist in principle of both equilibrium and nonequilibrium contributions, the latter known as the kinetic depolarization effect.1 On the other hand, the theoretical equations for thermodynamic properties of solutions require only equilibrium permittivities. The theories of Hubbard and Onsager35−38 implied that the kinetic depolarization effect is responsible for the most of the permittivity decrement with increased concentration of electrolyte, however, subsequent theoretical calculations and computer simulations showed that it might well be negligible.39,40 Concerning the theoretical approximations utilized to estimate the role of kinetic depolarization effect, the question is still open for real solutions of electrolytes. The experimental static permittivities of electrolyte solutions were used in calculations of mean ionic activity coefficients of electrolyte solutions on the basis of the Mean Spherical Approximation,41−44 the corrected Debye−Hückel theory45 and a combined model and computer simulation approach.46,47 In the earlier studies,41−44 the incorporation of the variable permittivity modified the energetics of ion−ion interactions and improved agreement with experimental data on activity coefficients, though usually on a limited concentration scale and with the help of numerous fitting procedures, including, for example, adjustment of ionic sizes. The more recent works are focusing on the role of the variable permittivity in the solvation term, adopted from the Born theory of solvation,48 which is considered responsible for the nonmonotonic concentration behavior of the activity coefficients.46,47 Surprisingly, although the DH theory remains an important reference point for the molecular theories of electrolyte solutions,49 the role of the static permittivity concentration dependence in new theoretical approaches studied in refs 41−47 was not properly compared with its role in the theory of DH level. In fact, as early as in 1925, Hückel developed an extension of DH theory assuming linear concentration dependence of the static permittivity,50 but the lack of experimental permittivities prevented real acknowledgment of this advance at that time. Later in 1956, seemingly independently from Hückel, Teitler and Ginsburg51 published another extension of the DH theory also assuming a linear variation of permittivity. Though being mathematically more elegant, this work evidently was ignored by the experts of the field. Not long ago, a rather discouraging opinion was reported:41 “If one attempts to improve the extended Debye−Hückel model by changing the constants to reflect changes in solvent permittivity, no improvement in the concentration range over which the model describes experimental data is found.” Contrary to this opinion, we believe that despite a number of drawbacks criticized in the past52,53 the old Hückel’s work remains conceptually insightful and deserves reevaluation. Its important achievement is that not only the interionic interaction contribution considered in the original DH theory is modified by the variation of permittivity, but also there appears quite naturally, within the framework of classical electrostatics, a solvation contribution to the free energy of solution, uniting the DH theory with the Born theory of

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THEORY Basic Model. The general model is essentially the same as employed in the Hückel’s work.50 We consider a reference ion as a sphere of radius R bearing the charge q, uniformly spread on the surface. The ion is surrounded by continuum solvent containing excess charge density, i.e., ionic atmosphere, with the total charge −q. The charge density starts at a distance a from the center of the reference ion, so that the ion is surrounded by a spherical shell of pure dielectric with radius spanning the interval R < r < a. The parameter a is thus the distance of closest approach of ions. The static permittivity of the medium ε is independent of the distance r from the central ion. The static permittivity inside the ion is not important as it does not affect the result. The linearized Poisson−Boltzmann equation for the electrostatic potential ψ has a well-known form:7 Δψ = κ 2ψ

(1)

where κ is the inverse Debye screening length defined by κ2 =

4πe02 εkTV

∑ Nzi i2

(2)

i

where e0 is the elementary charge, k is the Boltzmann constant, T is the temperature, V is the volume of solution, Ni is the number of ions of ith type, zi is the valency (qi = zie0). The solution of eq 1 is qκ ⎧ q ⎪ εR − ε(1 + κa) , r ≤ R ⎪ qκ ⎪q , Ra ⎩ ε 1 + κa r

(3)

According to eq 3, the electrostatic potential on the surface of the ion can be considered as a sum of two components: one originates from the ionic atmosphere, ψatm = −(qκ/ε(1 + κa)), and the other is due to the central ion itself, ψion = q/εR. The electrostatic interaction contribution to the free energy (Gibbs energy) of solution can be calculated through the Debye charging process:53,54 Gel =

∑ Ni ∫

qi

0

i

(ψatm, i + ψion, i) dqi = G1 + G2

(4)

where G1 =

∑ Ni ∫ i

0

qi

ψatm, i dqi ,

G2 =

∑ Ni ∫ i

0

qi

ψion, i dqi (5)

The summation in eqs 4 and 5 is taken over all ions in solution. The first term in the right-hand side of eq 4, G1, can be interpreted as interionic interaction contribution, while the second term, G2, can be considered as a contribution due to ion− solvent interactions (solvation contribution). The solution of a fully dissociated single electrolyte Cνz++ Aνz−− will further be considered. All ions are assumed to have equal sizes with a B

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The Journal of Physical Chemistry B radius R± and a distance of closest approach a. The model being developed in this work can be thus considered as belonging to the class of restricted primitive models of the McMillan-Mayer level.45 Concentration Dependent Permittivity. The concentration dependence of the permittivity will be represented using the function f(κ0) defined by ε = ε(c) = ε(κ0) = ε0f (κ0)

where τ1(κ0 , a) = 3

∑ Nzi i2

f (κ0λ)

,

λ 2 dλ x2 ⎤ 3⎡ = 3 ⎢ln(1 + x) − x + ⎥ 0 1 + xλ 2⎦ x ⎣ 3 3 3 3 = 1 − x + x 2 − x 3 + x 4 − ... 4 5 6 7

τ (x ) = 3

∑ Ni ∫

0

i

i

= −∑ i

(12)

the free energy of solution, the calculation of excess chemical potential of water (or whatever solvent) is straightforward: E μw,1 = RT ln γw,1 =

ε(λ) = ε0f (κ0λ) (9)

∂G1 ∂n w

(13)

where nw is the number of moles of water and R is the universal gas constant. From eq 13, the expression for the first contribution to activity coefficient of water is obtained:

1

ψatm(a , λ)zie0 dλ

∫0

ln γw,1 =

1

κ(κ0λ) zie0λ zie0 dλ ε(κ0λ) 1 + κ(κ0λ)a 2 2 Nz kTV 3 i i e0 κ0 τ1(κ0 , a) = − κ0 τ1(κ0 , a) 3ε0 12π

= −∑ Ni

∫

1

Activity Coefficient of Water: First Contribution. Given

The relationships expressed by eqs 9 were assumed, though not explicitly written, in ref 51. Free Energy of Solution: First Contribution. According to eq 5, the first contribution to the free energy of solution is given by G1 =

(11)

theory:3,53

The charging process implied in eq 4 is carried out by changing the instant charges qi = zie0λ of all ions in solution letting the parameter λ increase from 0 to 1. During the charging process, the inverse Debye length and the permittivity are assumed to depend on λ in the following way: κ0λ

dλ

τ(κ0a), where τ(x) is a function used in the original DH

Comparing eqs 2 and 7, one can derive a relationship between the two types of inverse lengths: κ0 κ = κ(κ0) = f (κ0) (8)

κ (λ ) =

f (κ0λ)( f (κ0λ) + κ0aλ)

independent permittivity, the function τ1(κ0, a) reduces to

(7)

i

λ2

1

If f(κ0λ) = 1, which corresponds to the case of concentration

(6)

where ε0 is the permittivity of the pure solvent and κ0 is the usual inverse Debye length, defined by eq 2 with the use of the pure solvent permittivity ε0 instead of permittivity of solution: 4πe02 κ02 = ε0kTV

∫0

∫

σ1(κ0 , a) = 3

0

1

Vw̅ κ03σ1(κ0 , a) 24πNA

(14)

In eq 14, V̅ w is the partial molar volume of water and NA is Avogadro’s number. The function σ1(κ0, a) is defined by

(10)

f (κ0λ)3/2 −

3 2

f (κ0λ) f ′(κ0λ)κ0λ − af ′(κ0λ)κ02λ 2

f (κ0λ)2 ( f (κ0λ) + κ0aλ)2

where f ′(x) is the derivative of f(x). If f(κ0λ) = 1, then the function σ1(κ0, a) reduces to σ(κ0a), where σ(x) is a function defined in the original DH theory:3,53

λ 2 dλ (15)

μjE,1 = RT ln γj ,1 =

∂G1 ∂nj

(17)

where nj is the number of moles of jth ion. Eq 17 leads to the expression for the first contribution to the activity coefficient of ion:

⎤ 3⎡ 1 σ(x) = [xτ(x)]′ = 3 ⎢1 + x − − 2 ln(1 + x)⎥ ⎦ 1+x x ⎣ 2 3 2 4 3 5 4 = 1 − 3· x + 3· x − 3· x + 3· x − ... 4 5 6 7

ln γj ,1 = −

(16)

z j2e02κ0χ1 (κ0 , a) 3kTε0

+

Vj̅ κ03σ1(κ0 , a) 24πNA

(18)

where

Ionic Activity Coefficient: First Contribution. Calculation of the excess chemical potential of an ion from the free energy of solution is straightforward:

χ1 (κ0 , a) = τ1(κ0 , a) + C

σ1(κ0 , a) 2

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The Journal of Physical Chemistry B and V̅ j is the partial molar volume of jth ion. The expression for the mean ionic activity coefficient readily follows from eq 18: ln γ±,1 = −

|z+z −|e02κ0χ1 (κ0 , a) 3kTε0

+

For the infinitely diluted solution of ions chosen as the reference state for γj, i.e., γj → 1 at κ0 → 0, eq 27 should be recast in

V±̅ κ03σ1(κ0 , a) 24πNA

ln γj ,2 =

(20)

where V̅ ± = [(ν+V̅ + + ν−V̅ −)/ν] is the mean ionic partial molar volume, ν = ν+ + ν−. The second term in eq 20 is small and can be neglected, similarly to the conventional DH theory.53 If f(κ0λ) = 1, the function χ1(κ0, a) reduces to the expression (3/2)[1/(1 + κ0a)], reproducing the well-known DH result for the mean ionic activity coefficient.7,53 Free Energy of Solution: Second Contribution. The second contribution to the free energy of solution according to eq 5 equals G2 =

∑ Ni ∫

0

i

1

ψion, i(R i , λ)zie0 dλ =

∑ Ni i

zi2e02 2R±ε0

ln γ±,2 =

1

λ dλ f (κ0λ)

∫0

1

∂G2 ∂nj

z j2e02χ2 (κ0) 2R±ε0kT

−

(25)

χ1 (κ0 , a) =

τ2(κ0) = −

(26)

χ2 (κ0) =

(27)

where σ (κ ) χ2 (κ0) = τ2(κ0) + 2 0 2

⎞ ⎟ ⎟ 1 − b2κ02 ⎠ κ0a

(33)

⎞ ⎟ ⎟ 1 − b2κ02 ⎠ κ0a

(34)

3 2(1 −

b2κ02)(

1 − b2κ02 + κ0a)

(35)

ln(1 − b2κ02) b2κ02

(36)

⎤ ⎡ ln(1 − b2κ 2) 1 0 ⎥ σ2(κ0) = 2⎢ + 2 2 2 2 b κ0 1 − b κ0 ⎦ ⎣

Vj̅ κ02σ2(κ0) 16πNAR±

(31)

(32)

⎤ − a 2 ln(1 − b2κ02) − 2ab arcsin(bκ0)⎥ ⎥ ⎦ 3 + (1 − b2κ02)( 1 − b2κ02 + κ0a)

follows the corresponding contribution to the ionic activity coefficient: ln γj ,2 =

(30)

ln γ± = ln γ±,1 + ln γ±,2

⎡ ⎛ 3 ⎢2b2 ln⎜1 + σ1(κ0 , a) = − 2 2 ⎜ ab (a + b2)κ03 ⎢⎣ ⎝

Obviously, if f(κ0λ) = 1, then σ2(κ0) = 0 and the second contribution to ln γw becomes zero. Ionic Activity Coefficient: Second Contribution. From the second contribution to the excess chemical potential of the ion defined by μjE,2 = RT ln γj ,2 =

16πNAR±

⎤ − a 2 ln(1 − b2κ02) − 2ab arcsin(bκ0)⎥ ⎥ ⎦

f ′(κ0λ)λ 2 dλ f (κ0λ)2

V±̅ κ02σ2(κ0)

⎡ ⎛ 3 ⎢2b2 ln⎜1 + τ1(κ0 , a) = ⎜ 2ab2(a 2 + b2)κ03 ⎢⎣ ⎝

(24)

∂τ2(κ0) = −2κ0 ∂κ0

−

where b is a constant having dimension of length, one obtains the following expressions for the functions τ1(κ0, a), σ1(κ0, a), χ1(κ0, a), τ2(κ0), σ2(κ0), and χ2(κ0):

where σ2(κ0) = κ0

(29)

f (κ0) = 1 − b2κ02

leads to an equation for the corresponding contribution to the activity coefficient: ln γw,2

2R±ε0kT

(23)

V̅ κ 2σ (κ ) =− w 0 2 0 16πNAR±

16πNAR±

The Case of Linear Variation of Permittivity. The theory developed above covers any type of concentration dependence of permittivity. In case of linear variation of permittivity, the integrals in eqs 11, 15, 22, and 25 can be evaluated analytically. Assuming a linear dependence51

τ2(κ0)

(22)

∂G2 ∂n w

|z+z −|e02[χ2 (κ0) − 1]

ln γw = ln γw,1 + ln γw,2 ,

If f(κ0λ) = 1, then the function τ2(κ0) = 1 and we arrive at the expression similar to that obtained in the Born theory of solvation.48 Activity Coefficient of Water: Second Contribution. The definition of the second contribution to the excess chemical potential of water E μw,2 = RT ln γw,2 =

−

where the second term is small and can be neglected. Finally, the rational activity coefficient of water and mean ionic activity coefficient can be calculated summing up the corresponding contributions:

where

∫0

2R±ε0kT

Vj̅ κ02σ2(κ0)

Passing on to the mean ionic activity coefficient one obtains:

(21)

τ2(κ0) = 2

z j2e02[χ2 (κ0) − 1]

1 1 − b2κ02

(37)

(38)

It follows easily from eqs 33−38 that the DH limiting laws ln γw ∼ κ03 ,

(28) D

ln γ± ∼ κ0

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The Journal of Physical Chemistry B remain valid after inclusion of the concentration dependence of permittivity, as was predicted by Debye and Pauling.55 The expression for G1 (eq 10) with τ1(κ0, a) defined by eq 33 and the first term in ln γ±,1 (eq 20) with χ1(κ0, a) defined by eq 35 coincide with the results of Teitler and Ginsburg.51 The DH theory has thus been extended taking into account an arbitrary concentration dependence of the permittivity of electrolyte solution. The thermodynamic functions consist of two terms representing ion−ion and ion−water interactions. There is no explicit term for the water−water interactions in view of the implicit model of the solvent. The new theory reduces to the conventional DH theory if the permittivity of solution is assumed to be equal to that of pure solvent. The results obtained in case of linear variation of permittivity differ from rather cumbersome Hü ckel’s expressions50 because of different implementation of the charging process and other mathematical approximations and details. In case of the linear variation of permittivity, our expressions reproduce the results obtained by Teitler and Ginsburg,51 but only in the part regarding ion−ion interactions because the solvation contribution is totally missing in their work. Our solvation contribution to the ionic activity coefficient is different from a plain Born-like expression used by Vincze, Valiskó, and Boda.46,47

Figure 1. Static permittivity of aqueous solutions of alkali metal chlorides from LiCl to CsCl at 298 K calculated using interpolation formula (eq 40) with parameters taken from ref 43; ms is molality of salt.

where R+ and R− are the radii of cation and anion. The Pauling crystal ionic radii57 were taken to estimate the size of the ions (Table 2).

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APPLICATION OF THE THEORY TO AQUEOUS SOLUTIONS OF ALKALI METAL CHLORIDES The applicability of the new theory was examined considering the activity coefficients of components in the aqueous solutions of alkali metal chlorides from LiCl to CsCl at 298 K. The experimental static permittivity of solutions was used in the polynomial form ε = ε0 + aεcs + bεcs3/2 + gεcs2 + hεcs5/2

Table 2. Pauling Crystal Ionic Radii57

(40)

where cs is molar concentration of salt. The permittivity of pure water ε0 was taken equal to 78.4; 29 other parameters in eq 40 were adopted from ref 43 and are listed in Table 1. For all

ε(cs)

LiCl

ε = ε0 − 15.5cs + 1.96cs2 − 0.306cs5/2

NaCl

ε = ε0 − 16.2cs + 3.1cs3/2

KCl

ε = ε0 − 14.7cs + 3.0cs3/2

RbCl

ε = ε0 − 15.3cs + 3.7cs3/2

CsCl

ε = ε0 − 13.1cs + 2.9cs3/2

0.60 0.95 1.33 1.48 1.69 1.81

The mean partial molar volume of ions was estimated using the formula 4 V±̅ = πNAR±3 (43) 3 The quality of estimation defined by eq 43 is not crucial because the relative contribution of the terms depending on V̅ ± in eqs 20 and 30 increases with growing concentration starting from zero but not exceeding 5% in the most concentrated solutions. With the estimations defined by eqs 41−43, the theory allows one to evaluate the thermodynamic functions without parameter adjustment. The experimental ionic activity coefficients taken from ref 58 for comparison with the theoretical predictions were converted to rational scale using equation59

chlorides except LiCl, gε = 0 and hε = 0. The interpolation given by eq 40 is consistent with the more recent experimental determinations of permittivities for the systems under consideration.27−32 No correction of the permittivity data for the kinetic depolarization effect has been attempted. The permittivities of the solutions calculated using eq 40 are plotted in Figure 1. The densities of solutions were taken from ref 56. The mean ionic radii were calculated as arithmetic means R + R− R± = + 2

R (Å)

The distance of closest approach was assumed to be the sum of the radii: a = R+ + R − (42)

Table 1. Concentration Dependence of Static Permittivity of Electrolyte Solutions,43 ε0 = 78.4, cs Is Molar Concentration of Salt (mol/L) salt

ion Li+ Na+ K+ Rb+ Cs+ Cl−

γ±(x) = γ±(m)(1 + νmsM w )

(44)

where ms is the molality of salt and Mw is the molar weight of water.

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RESULTS AND DISCUSSION The results of calculations of water activity coefficients are presented in Figure 2 (left) in comparison with the experimental

(41) E

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Figure 2. Activity coefficient of water in aqueous solutions of alkali metal chlorides at T = 298 K calculated using the extended Debye−Hückel theory developed in this work (left) compared with experimental data58 (right).

Figure 3. Mean ionic activity coefficient (rational) in aqueous solutions of alkali metal chlorides at T = 298 K calculated using the extended Debye− Hückel theory developed in this work (left) compared with experimental data58 (right).

Figure 4. Activity coefficient of water in aqueous solutions of LiCl at T = 298 K shown in full (left) and limited (right) concentration scale. Open circles are experimental data,58 the solid line is calculated using the extended Debye−Hückel theory developed in this work, line 1 is the first contribution due to ion−ion interactions, line 2 is the second (solvation) contribution, line DH is calculated using the conventional Debye−Hückel theory, line DH,lim is the Debye−Hückel limiting law.

data58 given in Figure 2 (right). The calculated and experimental values for the mean ionic activity coefficients are presented in Figure 3 (left) and 3 (right), respectively. No parameter fitting

was employed at this stage. The nonmonotonic concentration dependence of activity coefficient of water and mean ionic coefficient is qualitatively reproduced for all the examined F

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Figure 5. Mean ionic activity coefficient (rational) in aqueous solutions of LiCl at T = 298 K shown in full (left) and limited (right) concentration scale. Other symbols and notations are the same as in Figure 4.

Figure 6. Effect of the ionic radius to the distance of closest approach ratio on activity coefficient of water (left) and mean ionic activity coefficient (rational) (right) in aqueous solutions of NaCl at T = 298 K. Open circles are experimental data,58 line 1 is calculated using the extended Debye−Hückel theory developed in this work with relationship R± = 0.5a, line 2 is calculated with R± = 0.7a.

was advocated in recent works,46,47 has been achieved in this study using a rather simple electrostatic framework. It ought to be noted that the new extension of the DH theory conserves the inherent limitations of the linearized Poisson−Boltzmann equation, which were usually thought responsible for the failure of the DH theory in the concentrated solutions, confirming the Debye’s perspective on his theory: “It applies much better than it should”.7 The experimental activity coefficients demonstrate a more pronounced dependence on the mean ionic radius in the series LiCl−CsCl than can be captured by the new theory, resulting in poorer agreement with experiment for the alkali metal chlorides starting with NaCl. In view of the well-known deficiencies of the Poisson−Boltzmann equation and limitations of the primitive model,1,7 the lack of the quantitative agreement in the whole concentration range seems inevitable, especially if calculations are performed without fitting. In general, the sources of discrepancy between the predictions of this theory and experiment can be subdivided into two categories: first, the difference between the DH level and the exact solution for the restricted primitive model,49 that is a system with equal-sized charged hard spheres in continuum

systems. The ordering of the curves for activity coefficients corresponding to the salts with different cations follows the experimentally observed trend despite a common belief that such a behavior cannot be described without resort to modeling with explicit solvent.10 For all the examined systems except the LiCl− water solutions, the calculated water activity coefficients are underestimated while the mean ionic activity coefficients are overestimated. For the LiCl solutions, a surprisingly satisfactory agreement with experiment is observed in the concentration range up to ms = 10 mol/kg as shown in Figures 4 and 5. The predictions from the conventional DH theory and the DH limiting law are also shown in Figure 4 and 5 for comparison, demonstrating that incorporation of the concentration dependent permittivity of solution extends the applicability of the DH approach from ms ≈ 0.01 mol/kg to the concentrations of 2 orders of magnitude larger. It follows from the theory and calculations that the nonmonotonic concentration dependence of the activity coefficients results from a balance between the contributions from the ion−ion and ion−water interactions which demonstrate opposite deviations from unity (Figures 4 and 5). This explanation of the nonmonotonic concentration behavior of activity coefficients, which goes back to the Hückel’s work50 and G

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framework of DH electrostatics in a quite natural way after incorporation of variable permittivity.

dielectric, and second, the difference between the restricted primitive model and a real electrolyte solution. The exact solution for the restricted primitive model can be approached by computer simulation methods (Monte Carlo or Molecular Dynamics) and comparison with the results of simulation would reveal the approximations made in the theoretical expressions. The simulations for the restricted primitive model were made in many works,60−62 but only with constant permittivity. In our case, the calculations with concentration dependent permittivity are required, but such simulations have not been reported yet, to our knowledge. The second opposition includes among a number of very complicated issues some rather simpler ones, for instance, the difference of ionic sizes between cation and anion, whose role was investigated on DH level in a recent work.63 A detailed analysis of all these issues is beyond the scope of this work and can become a subject for further studies. To illustrate the flexibility of the model, we can show that much better agreement with experiment can be obtained using a quite modest parameter adjustment. For instance, one may keep constant the mean ionic diameter a defined by eq 42, but vary the ionic radius R± as an arbitrary fraction of a: R± = ηa (compare with equality R± = a assumed in the Hückel’s work50). The optimal values of the factor η fitted to activity coefficients are 0.7 for NaCl and 0.8 for KCl. The effect of ionic radius adjustment is shown in Figure 6. For the other salts, the radius adjustment also improves the description but over a more limited concentration range. However, we do not attach much physical sense to this fitting and it is not our goal to develop in a systematic manner correlation relationships for activity coefficients using the new extension of the DH theory, because more fundamental issues mentioned in this discussion have first to be clarified, which can well improve the prediction power of the theory without resort to parameter fitting. What seems important in this study is that new theoretical arguments have been obtained to demonstrate the crucial role of the concentration dependent static permittivity of electrolyte solutions in understanding of their thermodynamic properties in the whole concentration range.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. † E-mail: [email protected].

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REFERENCES

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CONCLUSIONS The Debye−Hückel theory has been extended taking into account an arbitrary variation of the static permittivity of electrolyte solution with concentration. New theoretical expressions have been obtained for the excess free energy of solution, activity coefficient of water and mean ionic activity coefficient. The relationships for the thermodynamic functions consist of two terms representing interionic interactions and ion−water (solvation) interactions. The theory was applied to calculate the activity coefficients in the aqueous solutions of alkali metal chlorides from LiCl to CsCl at 298 K using permittivities derived from experimental dielectric relaxation studies. The calculations without any parameter adjustment show semiquantitative agreement with experimental data, reproducing the nonmonotonic concentration dependence of activity coefficient of water and mean ionic activity coefficient which can be interpreted as result of a balance between interionic interactions and solvation. The observed trend in activity coefficients for the salts with different cations is also reproduced taking into account the variation of crystal ion radii and solution permittivity. A surprisingly good agreement with experimental data is obtained for the aqueous solutions of LiCl in the concentration range up to 10 mol/kg. The important factor in the success of the theory is the inclusion of the solvation contribution which appears in the H

DOI: 10.1021/acs.jpcb.5b04555 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

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