- Email: [email protected]
On the dissociation degree of ionic solutions considering solvation effects
Electrochemistry Communications 92 (2018) 56–59
Contents lists available at ScienceDirect
Electrochemistry Communications journal homepage: www.elsevier.com/locate/elecom
On the dissociation degree of ionic solutions considering solvation effects
T
M. Landstorfer Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstraße 39, Berlin 10117, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Ion pair Dissociation degree Double layer Solvation shell Mixture theory
In this work the impact of solvation effects on the dissociation degree of strong electrolytes and salts is discussed. The investigation is based on a thermodynamic mixture theory which explicitly accounts for the solvation effect. Based on a remarkable relationship between differential capacity maxima and partial molar volume of ions in solution the solvation number of specific ions in solution is determined. A subsequent investigation of the electrolytic space charge layer shows that a saturated solution of 1 mol L−1 solvated ions forms near the electrode, and we point out some fundamental similarities of this state to a saturated bulk solution. This finding challenges the assumption of complete dissociation, even for moderate electrolyte concentrations, whereby we introduce an undissociated ion-pair in solution. We re-derive the equilibrium conditions for a two-step dissociation reaction, including solvation effects, which leads to a new relation to determine the dissociation degree. A comparison to Ostwald's dilution law clearly shows the shortcomings when solvation effects are neglected and we emphasize that complete dissociation is questionable beyond 0.5 mol L−1 for aqueous, monovalent electrolytes.
1. Introduction Strong electrolytes and salts are frequently assumed to completely dissociate into their respective ionic species, for all concentrations up to saturation [1-6]. After Arrhenius introduced the idea of dissociation (and also incomplete dissociation) at the end of the 19th century [7], Debye and Hückel concluded in 1923 [8] that strong electrolytes always completely dissociate in their respective ionic species [9]. From a thermodynamical point of view, this is a very strong a priori assumption and we show within this letter that solvation effects challenge this assumption, especially for concentrations beyond 0.5 mol L−1. The concept of incomplete dissociation, ion association, or formation of ionpairs in strong electrolytes was re-introduced several times [10-13] and is again of great scientific interest [14], especially investigated by MD simulations [15,16]. Our investigation presented here is based on a thermodynamic model [17] which is capable to predict qualitatively and quantitatively the double layer capacity of various electrolytes (see Fig. 1). It turns out that the capacity maxima are determined by the partial molar volume of the anion and the cation, respectively, whereby capacity measurements can be consulted to determine explicit values for different ions. For mono-valent ions in water we find that the partial molar volume of the ionic species is about 45 times larger than the solvent [17]. This suggests that the ionic species are strongly solvated, and based on a simple relation for the molar volume we can determine the solvation number from a single capacity measurement. An investigation of the
E-mail address: [email protected]. https://doi.org/10.1016/j.elecom.2018.05.011 Received 24 January 2018; Received in revised form 4 May 2018; Accepted 9 May 2018 Available online 01 June 2018 1388-2481/ © 2018 Elsevier B.V. All rights reserved.
double layer structure in the potential region beyond the capacity maximum shows the formation of an ionic saturation layer [19,20,21], which has some fundamental and remarkable similarities to a saturated bulk solution. This is then the starting point for our reflections on the dissociation degree, and it is shown that even for simple salts at concentrations of (0.5–1) mol L−1 the assumption of complete dissociation is questionable. 2. Theory Consider exemplarily a mono-valent salt AC of concentration c which completely dissociates in solvated anions A− and cations C+. Each ion A− and C+ is assumed to bind κA and κC solvent molecules S in its solvation shell, whereby the number density of free solvent molecules S in solution is
nS = nSR − κA⋅nA − κ C⋅nC .
(1)
nSR
corresponds to the mole density of the liquid The parameter solvent, i.e. for water nSR = 55.4 mol L−1. The number of mixing particles is then n = nS + nA + nC, and not the total number of molecules in solution, which is nT = nSR + nA + nC . 2.1. Entropy of mixing For the entropy of mixing this is extremely important. In a solvation
Electrochemistry Communications 92 (2018) 56–59
M. Landstorfer
Fig. 3. Entropy of mixing various solvation numbers κ = κA = κC and an ideal mixture.
n n n s = −kB ⎛nS ln ⎛ S ⎞ + nA ln ⎛ A ⎞ + nC ln ⎛ C ⎞ ⎞ . ⎝n⎠ ⎝n⎠ ⎝ n ⎠⎠ ⎝
(2)
For an ideal solution, however, the mixing entities are all solvent molecules in addition to the dissolved ions, which gives for the entropy of mixing
nR n n s ideal = −kB ⎛⎜nSR ln ⎛⎜ ST ⎞⎟ + nA ln ⎛ AT ⎞ + nC ln ⎛ CT ⎞ ⎞⎟ . n n n ⎠⎠ ⎝ ⎝ ⎠ ⎝ ⎠ ⎝
(3)
Fig. 3 displays the difference between the models and shows that the impact of the solvation effect is enormous, even for small solvation numbers. Fig. 1. Comparison between measured and computed double layer capacity for a non-adsorbing and completely dissociated salt KPF6. Source: Top: Fig. 2.a from [18], reprinted with permission of Elsevier.
2.2. Chemical potential Based on the entropy of mixing (2) it is possible to derive the chemical potential of the constituents in the liquid, incompressible electrolyte [17]. The chemical potential of the free solvent, the solvated anion and the solvated cation is
μα = gα + kB T ln (yα ) + vα⋅(p − pE )
α ∈ {S, A, C } ,
(4)
nα n
denotes the mole fraction with respect to the number where yα = density n = nS + nA + nC of mixing particles, gα the constant molar Gibbs energy, vα the partial molar volume, and pE the bulk pressure. 2.3. Partial molar volume of solvated ions While the solvation effect decreases the number of free solvent molecules in the mixture, it actually increases the molar mass and the partial molar volume of the solvated ions. The molar mass of a solvated ∼ + κ ⋅m , where m ∼ ion is clearly mA, C = m A, C A, C S A, C be the mass of the central ion and mS the molar mass of the solvent. A quite similar relation holds for the partial molar volume vA,C of a solvated ion. However, while the molecular mass is conserved during the solvation process, the volume is not necessarily. However, for the sake of this work it is sufficient to assume that the partial molar volume of a solvated ion is
Fig. 2. Sketch of solvation effect in a liquid mixture and the consequence on the entities for the entropy of mixing.
vA, C = v͠A, C + κA, C⋅vS mixture the mixing entities are now the free solvent molecules, the solvated anions and the solvated cations (see Fig. 2), leading to an entropy of mixing
with vS = (nSR )−1,
(5)
where v͠A, C is the molar volume of the central ion, vS the molar volume of the solvent and κA,C the solvation number. This relation allows us then to deduce the solvation number from a measured value of the partial molar volume. 57
Electrochemistry Communications 92 (2018) 56–59
M. Landstorfer
2.4. Determination of the partial molar volume and the solvation number In [17] it was found that the maxima of the differential capacity C of an electrolyte in contact to a metal are determined by the partial molar volume of the solvated anion and cation, respectively. The investigation is based on the chemical potential function (4), together with diffusional equilibrium, incompressibility and the Poisson-momentum equation system [17,22,23]. This yields an expression for C which can be compared to experimental data (see Fig. 1). The continuum model shows an exceptional agreement to experimental data, and we refer to [17] for the full derivation and validation. Surprisingly, the capacity is symmetric for many non-adsorbing salts [18,24,25], whereby the partial molar volumes of the anion and the cation are equal, i.e. vA ≈ vC. The ionic volume for mono-valent salts was found to be (40–50) times larger than the molar volume of the solvent, vA,C ≈(40–50) ⋅ vS, which gives a radius of 3
rA, C = 3 4π vA, C ≈ (6.6–7.1) Å for solvated ions. Based on the relation (5), the measured value of vA,C suggest with that each (mono-valent) ion solvates about κA,C ≈ 45 solvent molecules. This value captures all solvent molecules from the first and second solvation shell around the central ion. The first solvation shell is well understood and agglomerates (3–8) solvent molecules [26]. However, the second shell is rather poor understood but could cover far more solvent molecules simply from its geometrical arrangement (see Fig. 2). Bound solvent molecules may have a slightly smaller volume than bulk molecules due to microscopic charge-dipole interaction which decreases their thermal motion. Ab initio methods could probably predict precise relations between κA,C and vA,C based on a microscopic structure model. But the goal of this work is not to predict a precise value of κA,C, but rather show the general impact of the solvation effect on the dissociation degree, whereby we proceed the discussion with κA,C = 45.
Fig. 4. Computed structure of the double layer for a completely dissociated salt AC. The solid lines display the molar concentration of solvated anions, solvated cations and the solvent near a metal electrode for an applied voltage of 0.5 V. The dashed line represents the electrostatic potential in the electrolytic space charge layer.
mixture. However, from an elementary perspective, the process actually occurs in two steps, initially the dissolution reaction1 (7)
AC|R ⇌ AC and subsequent the dissociation reaction
AC + (κA + κ C )S ⇌ A− + C+ ,
(8)
which accounts for the solvation effect. The reaction (8) implies that the constituent CE is actually a species of the liquid mixture and thus has a chemical potential in solution. Whether to term the constituent CE an ion pair, associated ion, Bjerrium pair or undissociated salt molecule in solution, is thermodynamically insignificant. What is significant, however, is the equilibrium condition the reaction (8) implies, namely
2.5. Saturation in the electrochemical double layer With the thermodynamic model of [17] it is possible to compute the structure of the double layer for arbitrary bulk concentrations and applied voltages. It is to emphasize that this structure is obtained from the very same model which predicts the validated capacity data (Fig. 1). Fig. 4 shows a representative computation for and applied voltage of 0.5 V and a bulk salt concentration of c = 0.0025 mol L−1. It turns out that a saturation layer of solvated ions forms near the electrode surface with a concentration of about 1 mol L−1. In this saturation layer the free water molecules are pushed out of the space charge layer in order to ensure the incompressibility of the liquid. What does this imply for an electrolytic solution of (1–2) mol L−1 bulk concentration ? A completely dissociated solution of c = 0.5 mol L−1 AC requires about (κA + κC) ⋅ c = 45 mol L−1 solvent molecules, which is almost the bulk value of water, nSR = 55.4 mol L−1. In this state there are not much more free solvent molecules left, which is a similar state as the saturation layer of the electrolytic space charge layer. Increasing the salt concentration further would even imply a negative value of the free solvent molecules, which certainly does not occur. In consequence one has to requisition the concept of complete dissociation, even at moderate concentrations of (0.1–1) mol L−1. Note that this effect occurs also when the solvation number is smaller, but then at little higher concentrations.
μAC + (κA + κ C ) μS = μA + μC .
(9)
With the representation (4) of the chemical potentials μα, this condition actually rewrites as Δg D yA ·yC = ekB T , κA+ κC yAC ·yS
(10)
with Δg = gAC + (κA + κC) ⋅ gS − gA − gC. Introducing the dissociation degree δ via D
nA = nC = δ⋅c
and nAC = (1 − δ ) c,
(11)
where c is the molar concentration of the salt, leads to
yA = yC =
yAC =
δ ⋅c , nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
(12)
(1 − δ )⋅c , nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
and yS =
(13)
nSR − 2κ δ⋅c . nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
(14)
The equilibrium condition (10) is thus an algebraic constraint 3. Discussion
δ2 c ⋅ (1 − δ ) nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
Requisitioning the concept of complete dissociation requires to state the actual bulk reactions occurring during the dissociation process. It is quite common to write the dissociation reaction of AC as
AC|R ⇌ A− + C+
2κ
⋅ ⎛⎜ ⎝
nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c ⎞ ⎟ nSR − 2κ δ⋅c ⎠
Δg D
− ekB T = 0
(6)
where R refers to the solid phase and A−, C+ are parts of the electrolytic
1
58
Note that this process could also require solvent molecules.
(15)
Electrochemistry Communications 92 (2018) 56–59
M. Landstorfer
or an electrolyte beyond a bulk concentration of 0.5 mol L−1 is questionable, and the degree of dissociation is determined by Eq. (15). Surprisingly, the double layer capacity maxima are correlated to the saturation maximum (or the degree of dissociation) and represent a well defined, experimentally accessible quantity to determine the crucial parameters of the dissociation degree. Due to the solvation effect incomplete dissociation (or the formation of ion pairs) is a necessary feature for a thermodynamic consistent theory of electrolytes. References [1] R. Haase, Elektrochemie I: Thermodynamik Elektrochemischer Systeme, Springer, 1972. [2] P.W. Atkins, J.D. Paula, Physical Chemistry, Oxford University Press, 2006. [3] C. Hamann, W. Vielstich, Elektrochemie, Wiley-VCH, 2005. [4] J. Newman, Electrochemical Systems, Prentice Hall, 1973. [5] K. Vetter, S. Bruckenstein, B. Howard, S. Technica, Electrochemical Kinetics, Academic Press, 1967. [6] I. Prigogine, R. Defay, Chemical Thermodynamics, Longmans, 1954. [7] S. Arrhenius, Über die dissociation der in Wasser gelösten stoffe, Z. Phys. Chem. 1 (1887) 631. [8] P. Debye, E. Hückel, Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen, Phys. Z. 24 (1923) 185–206. [9] O. Redlich, The dissociation of strong electrolytes, Chem. Rev. 39 (1946) 333–356. [10] Y. Marcus, G. Hefter, Ion pairing, Chem. Rev. 106 (2006) 4585–4621. [11] C.W. Davies, Incomplete dissociation. Introductory paper, Discuss. Faraday Soc. 24 (1957) 83–86. [12] R. Heyrovska, Physical electrochemistry of strong electrolytes based on partial dissociation and hydrationa, J. Electrochem. Soc. 143 (1996) 1789–1793. [13] H. Bian, X. Wen, J. Li, H. Chen, S. Han, X. Sun, J. Song, W. Zhuang, J. Zheng, Ion clustering in aqueous solutions probed with vibrational energy transfer, PNAS 108 (2011) 4737–4742. [14] A. Chen, R. Pappu, Quantitative characterization of ion pairing and cluster formation in strong 1:1 electrolytes, J. Phys. Chem. B 111 (2007) 6469–6478. [15] C.J. Fennell, A. Bizjak, V. Vlachy, K.A. Dill, Ion pairing in molecular simulations of aqueous alkali halide solutions, J. Phys. Chem. B 113 (2009) 6782–6791. [16] Y. Luo, W. Jiang, H. Yu, A.D. MacKerell, B. Roux, Simulation study of ion pairing in concentrated aqueous salt solutions with a polarizable force field, Faraday Discuss. 160 (2013) 135–224. [17] M. Landstorfer, C. Guhlke, W. Dreyer, Theory and structure of the metal-electrolyte interface incorporating adsorption and solvation effects, Electrochim. Acta 201 (2016) 187–219. [18] G. Valette, Double layer on silver single-crystal electrodes in contact with electrolytes having anions which present a slight specific adsorption: part I. The (110) face, J. Electroanal. Chem. 122 (1981) 285–297. [19] M. Kilic, M. Bazant, A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations, Phys. Rev. E 75 (2007) 021503. [20] I. Borukhov, D. Andelman, H. Orland, Steric effects in electrolytes: a modified Poisson-Boltzmann equation, Phys. Rev. Lett. 79 (1997) 435–438. [21] J. Bikerman, XXXIX. Structure and capacity of electrical double layer, Philos. Mag. 7 (33) (1942) 384–397. [22] W. Dreyer, C. Guhlke, R. Müller, Overcoming the shortcomings of the NernstPlanck model, Phys. Chem. Chem. Phys. 15 (2013) 7075–7086. [23] W. Dreyer, C. Guhlke, M. Landstorfer, A mixture theory of electrolytes containing solvation effects, Electrochem. Commun. 43 (2014) 75–78. [24] G. Valette, Double layer on silver single crystal electrodes in contact with electrolytes having anions which are slightly specifically adsorbed: part II. The (100) face, J. Electroanal. Chem. 138 (1982) 37–54. [25] G. Valette, Double layer on silver single crystal electrodes in contact with electrolytes having anions which are slightly specifically adsorbed: part III. The (111) face, J. Electroanal. Chem. 269 (1989) 191–203. [26] J.F. Hinton, E.S. Amis, Solvation numbers of ions, Chem. Rev. 71 (1971) 627–674.
Fig. 5. Comparison of the computed dissociation degree according to the solvation mixture model (15) with κ = 45 and Ostwald's dilution law (16).
between the dissociation degree δ and the salt concentration c. The only parameters of this relation are the dissociation energy ΔgD and the solvation number κ = κA,C. Since we know the solvation number κA,C from the capacity maximum, we can numerically solve Eq. (15) for various values of ΔgD in order to determine the dissociation degree δ = δ(c). Fig. 5 shows computations of the dissociation degree from very dilute solutions up to high concentrations. Note that Ostwald's dilution law, which is frequently used to compute or approximate the dissociation degree of acids, completely ignores the solvation effect and the concept of free solvent molecules. The corresponding constraint of Ostwald's dilution law reads D
Δg δ2 c · −ek B T = 0 1−δ nSR
(16)
and significantly underestimates the dissociation degree, especially for higher concentrations (see comparison in Fig. 5). 4. Summary For a large dissociation energy, i.e. ΔgD > 0.1 eV, which corresponds to a salt or a very strong electrolyte, Ostwald's dilution law predicts complete dissociation for arbitrary concentrations. In contrast, the solvation effect and its consistent incorporation in the thermodynamic theory requires incomplete dissociation beyond 0.5 mol L−1 (see Fig. 5). Origin of this effect is simply that not enough free solvent molecules are present anymore to shift the reaction equilibrium of Eq. (8) towards the ions in solution. For a solvation number of κA,C = 45 and a dissociation energy of ΔgD = 0.1 eV we find that dissociation degree is δ = 0.99 for 0.5 mol L−1, δ = 0.44 for 1 mol L−1 and δ = 0.19 for 2 mol L−1. Hence the assumption of complete dissociation of a salt
59
Contents lists available at ScienceDirect
Electrochemistry Communications journal homepage: www.elsevier.com/locate/elecom
On the dissociation degree of ionic solutions considering solvation effects
T
M. Landstorfer Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstraße 39, Berlin 10117, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Ion pair Dissociation degree Double layer Solvation shell Mixture theory
In this work the impact of solvation effects on the dissociation degree of strong electrolytes and salts is discussed. The investigation is based on a thermodynamic mixture theory which explicitly accounts for the solvation effect. Based on a remarkable relationship between differential capacity maxima and partial molar volume of ions in solution the solvation number of specific ions in solution is determined. A subsequent investigation of the electrolytic space charge layer shows that a saturated solution of 1 mol L−1 solvated ions forms near the electrode, and we point out some fundamental similarities of this state to a saturated bulk solution. This finding challenges the assumption of complete dissociation, even for moderate electrolyte concentrations, whereby we introduce an undissociated ion-pair in solution. We re-derive the equilibrium conditions for a two-step dissociation reaction, including solvation effects, which leads to a new relation to determine the dissociation degree. A comparison to Ostwald's dilution law clearly shows the shortcomings when solvation effects are neglected and we emphasize that complete dissociation is questionable beyond 0.5 mol L−1 for aqueous, monovalent electrolytes.
1. Introduction Strong electrolytes and salts are frequently assumed to completely dissociate into their respective ionic species, for all concentrations up to saturation [1-6]. After Arrhenius introduced the idea of dissociation (and also incomplete dissociation) at the end of the 19th century [7], Debye and Hückel concluded in 1923 [8] that strong electrolytes always completely dissociate in their respective ionic species [9]. From a thermodynamical point of view, this is a very strong a priori assumption and we show within this letter that solvation effects challenge this assumption, especially for concentrations beyond 0.5 mol L−1. The concept of incomplete dissociation, ion association, or formation of ionpairs in strong electrolytes was re-introduced several times [10-13] and is again of great scientific interest [14], especially investigated by MD simulations [15,16]. Our investigation presented here is based on a thermodynamic model [17] which is capable to predict qualitatively and quantitatively the double layer capacity of various electrolytes (see Fig. 1). It turns out that the capacity maxima are determined by the partial molar volume of the anion and the cation, respectively, whereby capacity measurements can be consulted to determine explicit values for different ions. For mono-valent ions in water we find that the partial molar volume of the ionic species is about 45 times larger than the solvent [17]. This suggests that the ionic species are strongly solvated, and based on a simple relation for the molar volume we can determine the solvation number from a single capacity measurement. An investigation of the
E-mail address: [email protected]. https://doi.org/10.1016/j.elecom.2018.05.011 Received 24 January 2018; Received in revised form 4 May 2018; Accepted 9 May 2018 Available online 01 June 2018 1388-2481/ © 2018 Elsevier B.V. All rights reserved.
double layer structure in the potential region beyond the capacity maximum shows the formation of an ionic saturation layer [19,20,21], which has some fundamental and remarkable similarities to a saturated bulk solution. This is then the starting point for our reflections on the dissociation degree, and it is shown that even for simple salts at concentrations of (0.5–1) mol L−1 the assumption of complete dissociation is questionable. 2. Theory Consider exemplarily a mono-valent salt AC of concentration c which completely dissociates in solvated anions A− and cations C+. Each ion A− and C+ is assumed to bind κA and κC solvent molecules S in its solvation shell, whereby the number density of free solvent molecules S in solution is
nS = nSR − κA⋅nA − κ C⋅nC .
(1)
nSR
corresponds to the mole density of the liquid The parameter solvent, i.e. for water nSR = 55.4 mol L−1. The number of mixing particles is then n = nS + nA + nC, and not the total number of molecules in solution, which is nT = nSR + nA + nC . 2.1. Entropy of mixing For the entropy of mixing this is extremely important. In a solvation
Electrochemistry Communications 92 (2018) 56–59
M. Landstorfer
Fig. 3. Entropy of mixing various solvation numbers κ = κA = κC and an ideal mixture.
n n n s = −kB ⎛nS ln ⎛ S ⎞ + nA ln ⎛ A ⎞ + nC ln ⎛ C ⎞ ⎞ . ⎝n⎠ ⎝n⎠ ⎝ n ⎠⎠ ⎝
(2)
For an ideal solution, however, the mixing entities are all solvent molecules in addition to the dissolved ions, which gives for the entropy of mixing
nR n n s ideal = −kB ⎛⎜nSR ln ⎛⎜ ST ⎞⎟ + nA ln ⎛ AT ⎞ + nC ln ⎛ CT ⎞ ⎞⎟ . n n n ⎠⎠ ⎝ ⎝ ⎠ ⎝ ⎠ ⎝
(3)
Fig. 3 displays the difference between the models and shows that the impact of the solvation effect is enormous, even for small solvation numbers. Fig. 1. Comparison between measured and computed double layer capacity for a non-adsorbing and completely dissociated salt KPF6. Source: Top: Fig. 2.a from [18], reprinted with permission of Elsevier.
2.2. Chemical potential Based on the entropy of mixing (2) it is possible to derive the chemical potential of the constituents in the liquid, incompressible electrolyte [17]. The chemical potential of the free solvent, the solvated anion and the solvated cation is
μα = gα + kB T ln (yα ) + vα⋅(p − pE )
α ∈ {S, A, C } ,
(4)
nα n
denotes the mole fraction with respect to the number where yα = density n = nS + nA + nC of mixing particles, gα the constant molar Gibbs energy, vα the partial molar volume, and pE the bulk pressure. 2.3. Partial molar volume of solvated ions While the solvation effect decreases the number of free solvent molecules in the mixture, it actually increases the molar mass and the partial molar volume of the solvated ions. The molar mass of a solvated ∼ + κ ⋅m , where m ∼ ion is clearly mA, C = m A, C A, C S A, C be the mass of the central ion and mS the molar mass of the solvent. A quite similar relation holds for the partial molar volume vA,C of a solvated ion. However, while the molecular mass is conserved during the solvation process, the volume is not necessarily. However, for the sake of this work it is sufficient to assume that the partial molar volume of a solvated ion is
Fig. 2. Sketch of solvation effect in a liquid mixture and the consequence on the entities for the entropy of mixing.
vA, C = v͠A, C + κA, C⋅vS mixture the mixing entities are now the free solvent molecules, the solvated anions and the solvated cations (see Fig. 2), leading to an entropy of mixing
with vS = (nSR )−1,
(5)
where v͠A, C is the molar volume of the central ion, vS the molar volume of the solvent and κA,C the solvation number. This relation allows us then to deduce the solvation number from a measured value of the partial molar volume. 57
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M. Landstorfer
2.4. Determination of the partial molar volume and the solvation number In [17] it was found that the maxima of the differential capacity C of an electrolyte in contact to a metal are determined by the partial molar volume of the solvated anion and cation, respectively. The investigation is based on the chemical potential function (4), together with diffusional equilibrium, incompressibility and the Poisson-momentum equation system [17,22,23]. This yields an expression for C which can be compared to experimental data (see Fig. 1). The continuum model shows an exceptional agreement to experimental data, and we refer to [17] for the full derivation and validation. Surprisingly, the capacity is symmetric for many non-adsorbing salts [18,24,25], whereby the partial molar volumes of the anion and the cation are equal, i.e. vA ≈ vC. The ionic volume for mono-valent salts was found to be (40–50) times larger than the molar volume of the solvent, vA,C ≈(40–50) ⋅ vS, which gives a radius of 3
rA, C = 3 4π vA, C ≈ (6.6–7.1) Å for solvated ions. Based on the relation (5), the measured value of vA,C suggest with that each (mono-valent) ion solvates about κA,C ≈ 45 solvent molecules. This value captures all solvent molecules from the first and second solvation shell around the central ion. The first solvation shell is well understood and agglomerates (3–8) solvent molecules [26]. However, the second shell is rather poor understood but could cover far more solvent molecules simply from its geometrical arrangement (see Fig. 2). Bound solvent molecules may have a slightly smaller volume than bulk molecules due to microscopic charge-dipole interaction which decreases their thermal motion. Ab initio methods could probably predict precise relations between κA,C and vA,C based on a microscopic structure model. But the goal of this work is not to predict a precise value of κA,C, but rather show the general impact of the solvation effect on the dissociation degree, whereby we proceed the discussion with κA,C = 45.
Fig. 4. Computed structure of the double layer for a completely dissociated salt AC. The solid lines display the molar concentration of solvated anions, solvated cations and the solvent near a metal electrode for an applied voltage of 0.5 V. The dashed line represents the electrostatic potential in the electrolytic space charge layer.
mixture. However, from an elementary perspective, the process actually occurs in two steps, initially the dissolution reaction1 (7)
AC|R ⇌ AC and subsequent the dissociation reaction
AC + (κA + κ C )S ⇌ A− + C+ ,
(8)
which accounts for the solvation effect. The reaction (8) implies that the constituent CE is actually a species of the liquid mixture and thus has a chemical potential in solution. Whether to term the constituent CE an ion pair, associated ion, Bjerrium pair or undissociated salt molecule in solution, is thermodynamically insignificant. What is significant, however, is the equilibrium condition the reaction (8) implies, namely
2.5. Saturation in the electrochemical double layer With the thermodynamic model of [17] it is possible to compute the structure of the double layer for arbitrary bulk concentrations and applied voltages. It is to emphasize that this structure is obtained from the very same model which predicts the validated capacity data (Fig. 1). Fig. 4 shows a representative computation for and applied voltage of 0.5 V and a bulk salt concentration of c = 0.0025 mol L−1. It turns out that a saturation layer of solvated ions forms near the electrode surface with a concentration of about 1 mol L−1. In this saturation layer the free water molecules are pushed out of the space charge layer in order to ensure the incompressibility of the liquid. What does this imply for an electrolytic solution of (1–2) mol L−1 bulk concentration ? A completely dissociated solution of c = 0.5 mol L−1 AC requires about (κA + κC) ⋅ c = 45 mol L−1 solvent molecules, which is almost the bulk value of water, nSR = 55.4 mol L−1. In this state there are not much more free solvent molecules left, which is a similar state as the saturation layer of the electrolytic space charge layer. Increasing the salt concentration further would even imply a negative value of the free solvent molecules, which certainly does not occur. In consequence one has to requisition the concept of complete dissociation, even at moderate concentrations of (0.1–1) mol L−1. Note that this effect occurs also when the solvation number is smaller, but then at little higher concentrations.
μAC + (κA + κ C ) μS = μA + μC .
(9)
With the representation (4) of the chemical potentials μα, this condition actually rewrites as Δg D yA ·yC = ekB T , κA+ κC yAC ·yS
(10)
with Δg = gAC + (κA + κC) ⋅ gS − gA − gC. Introducing the dissociation degree δ via D
nA = nC = δ⋅c
and nAC = (1 − δ ) c,
(11)
where c is the molar concentration of the salt, leads to
yA = yC =
yAC =
δ ⋅c , nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
(12)
(1 − δ )⋅c , nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
and yS =
(13)
nSR − 2κ δ⋅c . nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
(14)
The equilibrium condition (10) is thus an algebraic constraint 3. Discussion
δ2 c ⋅ (1 − δ ) nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c
Requisitioning the concept of complete dissociation requires to state the actual bulk reactions occurring during the dissociation process. It is quite common to write the dissociation reaction of AC as
AC|R ⇌ A− + C+
2κ
⋅ ⎛⎜ ⎝
nSR + 2(1 − κ ) δ⋅c + (1 − δ )⋅c ⎞ ⎟ nSR − 2κ δ⋅c ⎠
Δg D
− ekB T = 0
(6)
where R refers to the solid phase and A−, C+ are parts of the electrolytic
1
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Note that this process could also require solvent molecules.
(15)
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M. Landstorfer
or an electrolyte beyond a bulk concentration of 0.5 mol L−1 is questionable, and the degree of dissociation is determined by Eq. (15). Surprisingly, the double layer capacity maxima are correlated to the saturation maximum (or the degree of dissociation) and represent a well defined, experimentally accessible quantity to determine the crucial parameters of the dissociation degree. Due to the solvation effect incomplete dissociation (or the formation of ion pairs) is a necessary feature for a thermodynamic consistent theory of electrolytes. References [1] R. Haase, Elektrochemie I: Thermodynamik Elektrochemischer Systeme, Springer, 1972. [2] P.W. Atkins, J.D. Paula, Physical Chemistry, Oxford University Press, 2006. [3] C. Hamann, W. Vielstich, Elektrochemie, Wiley-VCH, 2005. [4] J. Newman, Electrochemical Systems, Prentice Hall, 1973. [5] K. Vetter, S. Bruckenstein, B. Howard, S. Technica, Electrochemical Kinetics, Academic Press, 1967. [6] I. Prigogine, R. Defay, Chemical Thermodynamics, Longmans, 1954. [7] S. Arrhenius, Über die dissociation der in Wasser gelösten stoffe, Z. Phys. Chem. 1 (1887) 631. [8] P. Debye, E. Hückel, Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen, Phys. Z. 24 (1923) 185–206. [9] O. Redlich, The dissociation of strong electrolytes, Chem. Rev. 39 (1946) 333–356. [10] Y. Marcus, G. Hefter, Ion pairing, Chem. Rev. 106 (2006) 4585–4621. [11] C.W. Davies, Incomplete dissociation. Introductory paper, Discuss. Faraday Soc. 24 (1957) 83–86. [12] R. Heyrovska, Physical electrochemistry of strong electrolytes based on partial dissociation and hydrationa, J. Electrochem. Soc. 143 (1996) 1789–1793. [13] H. Bian, X. Wen, J. Li, H. Chen, S. Han, X. Sun, J. Song, W. Zhuang, J. Zheng, Ion clustering in aqueous solutions probed with vibrational energy transfer, PNAS 108 (2011) 4737–4742. [14] A. Chen, R. Pappu, Quantitative characterization of ion pairing and cluster formation in strong 1:1 electrolytes, J. Phys. Chem. B 111 (2007) 6469–6478. [15] C.J. Fennell, A. Bizjak, V. Vlachy, K.A. Dill, Ion pairing in molecular simulations of aqueous alkali halide solutions, J. Phys. Chem. B 113 (2009) 6782–6791. [16] Y. Luo, W. Jiang, H. Yu, A.D. MacKerell, B. Roux, Simulation study of ion pairing in concentrated aqueous salt solutions with a polarizable force field, Faraday Discuss. 160 (2013) 135–224. [17] M. Landstorfer, C. Guhlke, W. Dreyer, Theory and structure of the metal-electrolyte interface incorporating adsorption and solvation effects, Electrochim. Acta 201 (2016) 187–219. [18] G. Valette, Double layer on silver single-crystal electrodes in contact with electrolytes having anions which present a slight specific adsorption: part I. The (110) face, J. Electroanal. Chem. 122 (1981) 285–297. [19] M. Kilic, M. Bazant, A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations, Phys. Rev. E 75 (2007) 021503. [20] I. Borukhov, D. Andelman, H. Orland, Steric effects in electrolytes: a modified Poisson-Boltzmann equation, Phys. Rev. Lett. 79 (1997) 435–438. [21] J. Bikerman, XXXIX. Structure and capacity of electrical double layer, Philos. Mag. 7 (33) (1942) 384–397. [22] W. Dreyer, C. Guhlke, R. Müller, Overcoming the shortcomings of the NernstPlanck model, Phys. Chem. Chem. Phys. 15 (2013) 7075–7086. [23] W. Dreyer, C. Guhlke, M. Landstorfer, A mixture theory of electrolytes containing solvation effects, Electrochem. Commun. 43 (2014) 75–78. [24] G. Valette, Double layer on silver single crystal electrodes in contact with electrolytes having anions which are slightly specifically adsorbed: part II. The (100) face, J. Electroanal. Chem. 138 (1982) 37–54. [25] G. Valette, Double layer on silver single crystal electrodes in contact with electrolytes having anions which are slightly specifically adsorbed: part III. The (111) face, J. Electroanal. Chem. 269 (1989) 191–203. [26] J.F. Hinton, E.S. Amis, Solvation numbers of ions, Chem. Rev. 71 (1971) 627–674.
Fig. 5. Comparison of the computed dissociation degree according to the solvation mixture model (15) with κ = 45 and Ostwald's dilution law (16).
between the dissociation degree δ and the salt concentration c. The only parameters of this relation are the dissociation energy ΔgD and the solvation number κ = κA,C. Since we know the solvation number κA,C from the capacity maximum, we can numerically solve Eq. (15) for various values of ΔgD in order to determine the dissociation degree δ = δ(c). Fig. 5 shows computations of the dissociation degree from very dilute solutions up to high concentrations. Note that Ostwald's dilution law, which is frequently used to compute or approximate the dissociation degree of acids, completely ignores the solvation effect and the concept of free solvent molecules. The corresponding constraint of Ostwald's dilution law reads D
Δg δ2 c · −ek B T = 0 1−δ nSR
(16)
and significantly underestimates the dissociation degree, especially for higher concentrations (see comparison in Fig. 5). 4. Summary For a large dissociation energy, i.e. ΔgD > 0.1 eV, which corresponds to a salt or a very strong electrolyte, Ostwald's dilution law predicts complete dissociation for arbitrary concentrations. In contrast, the solvation effect and its consistent incorporation in the thermodynamic theory requires incomplete dissociation beyond 0.5 mol L−1 (see Fig. 5). Origin of this effect is simply that not enough free solvent molecules are present anymore to shift the reaction equilibrium of Eq. (8) towards the ions in solution. For a solvation number of κA,C = 45 and a dissociation energy of ΔgD = 0.1 eV we find that dissociation degree is δ = 0.99 for 0.5 mol L−1, δ = 0.44 for 1 mol L−1 and δ = 0.19 for 2 mol L−1. Hence the assumption of complete dissociation of a salt
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