The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions

Ionic association in electrolyte solutions is investigated in the framework of chemical models. The associative mean spherical approximation (AMSA) for electrolyte theory is reviewed. It is shown that AMSA in combination with the Ebeling association constant satisfactorily reproduces the thermody...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2000
Автори:	Barthel, J., Krienke, H., Holovko, M.F., Kapko, V.I., Protsykevich, I.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2000
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/121004
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions / J. Barthel, H. Krienke, M.F. Holovko, V.I. Kapko, I.A. Protsykevich // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 657-674. — Бібліогр.: 31 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-121004
record_format	dspace
spelling	irk-123456789-1210042017-06-14T03:06:03Z The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions Barthel, J. Krienke, H. Holovko, M.F. Kapko, V.I. Protsykevich, I.A. Ionic association in electrolyte solutions is investigated in the framework of chemical models. The associative mean spherical approximation (AMSA) for electrolyte theory is reviewed. It is shown that AMSA in combination with the Ebeling association constant satisfactorily reproduces the thermodynamic measurement data up to high concentrations for nonaqueous solutions of solvents of relative permittivities in the range 20 < ε < 36 . For ionic solutions of lower permittivity the AMSA is modiﬁed by including ion trimers and tetramers to obtain a correct description of osmotic coefﬁcients and ionic conductivities. Іонна асоціація в розчинах електролітів досліджується в рамках хімічних моделей. Дано огляд асоціативного середньосферичного наближення (АССН) в теорії електролітів. Показано, що АССН у комбінації з асоціативною константою Ебелінга задовільно відтворює дані термодинамічних вимірювань, включаючи область високих концентрацій, для неводних розчинів електролітів з розчинниками з діелектричною проникністю в області 20 < ε < 36 . Для іонних розчинів з низькою діелектричною проникністю АССН модифіковане включенням іонних тримерів і тетрамерів, щоб отримати правильний опис осмотичних коефіцієнтів та іонних провідностей. 2000 Article The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions / J. Barthel, H. Krienke, M.F. Holovko, V.I. Kapko, I.A. Protsykevich // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 657-674. — Бібліогр.: 31 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.657 PACS: 61.20.Qg, 66.10.Ed http://dspace.nbuv.gov.ua/handle/123456789/121004 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Ionic association in electrolyte solutions is investigated in the framework of chemical models. The associative mean spherical approximation (AMSA) for electrolyte theory is reviewed. It is shown that AMSA in combination with the Ebeling association constant satisfactorily reproduces the thermodynamic measurement data up to high concentrations for nonaqueous solutions of solvents of relative permittivities in the range 20 < ε < 36 . For ionic solutions of lower permittivity the AMSA is modiﬁed by including ion trimers and tetramers to obtain a correct description of osmotic coefﬁcients and ionic conductivities.
format	Article
author	Barthel, J. Krienke, H. Holovko, M.F. Kapko, V.I. Protsykevich, I.A.
spellingShingle	Barthel, J. Krienke, H. Holovko, M.F. Kapko, V.I. Protsykevich, I.A. The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions Condensed Matter Physics
author_facet	Barthel, J. Krienke, H. Holovko, M.F. Kapko, V.I. Protsykevich, I.A.
author_sort	Barthel, J.
title	The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions
title_short	The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions
title_full	The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions
title_fullStr	The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions
title_full_unstemmed	The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions
title_sort	application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2000
url	http://dspace.nbuv.gov.ua/handle/123456789/121004
citation_txt	The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions / J. Barthel, H. Krienke, M.F. Holovko, V.I. Kapko, I.A. Protsykevich // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 657-674. — Бібліогр.: 31 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT barthelj theapplicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT krienkeh theapplicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT holovkomf theapplicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT kapkovi theapplicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT protsykevichia theapplicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT barthelj applicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT krienkeh applicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT holovkomf applicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT kapkovi applicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions AT protsykevichia applicationoftheassociativemeansphericalapproximationinthetheoryofnonaqueouselectrolytesolutions
first_indexed	2025-07-08T19:00:24Z
last_indexed	2025-07-08T19:00:24Z
_version_	1837106438128795648
fulltext	Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 657–674 The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions J.Barthel 1 , H.Krienke 1 , M.F.Holovko 2 , V.I.Kapko 2 , I.A.Protsykevich 2 1 Institut für Physikalische und Theoretische Chemie, Universitat Regensburg, D-93040, Regensburg, Germany 2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received June 9, 2000 Ionic association in electrolyte solutions is investigated in the framework of chemical models. The associative mean spherical approximation (AMSA) for electrolyte theory is reviewed. It is shown that AMSA in combination with the Ebeling association constant satisfactorily reproduces the ther- modynamic measurement data up to high concentrations for nonaqueous solutions of solvents of relative permittivities in the range 20 < ε < 36 . For ionic solutions of lower permittivity the AMSA is modified by including ion trimers and tetramers to obtain a correct description of osmotic coefficients and ionic conductivities. Key words: electrolyte solutions, ionic association, associative mean spherical approximation, osmotic and activity coefficients, ionic conductivity PACS: 61.20.Qg, 66.10.Ed 1. Introduction It is our pleasure to dedicate this article to I.R. Yukhnovskii whose fundamental work in statistical physics yielded a considerable progress in the theory of electrolyte solutions. To his work belongs a general scheme of plasma-parameter expansions for pair distribution functions of ionic fluids [1], the application of the collective variables method to the correct treatment of long and short-range interionic interactions [2] and the development of the ion-molecular approach for the correct description of solvation phenomena in electrolyte solutions [3–6]. Another important aspect of electrolyte solutions is connected with the ionic association concept introduced by Bjerrum [7] and used in the chemical model ap- c© J.Barthel, H.Krienke, M.F.Holovko, V.I.Kapko, I.A.Protsykevich 657 J.Barthel et al. proach [8,9]. In the framework of this approach the electrolyte is considered to be a mixture of free ions and ion aggregates (usually ion pairs and sometimes trimers, tetramers etc.) which are assumed to take part in chemical equilibrium according to the corresponding mass action law (MAL). For the description of such mixtures the theory of ionic fluids is traditionally modified by simply correcting the ion concen- trations using the concentration of free ions obtained from the MAL [7–9]. However this approach neglects the contributions of electrostatic interaction from ionic ag- gregates and therefore is correct only in the regime of weak association. At the increasing association, the electrostatic contributions of the ion aggregates increase and are not negligible. Another route [10] starts from the associative mean spherical approximation (AMSA) [11] based on the theory of associating fluids [12–14]. It was shown that this approach coincides with the traditional approach of weak association, but also is correct in the regime of strong association. With Ebeling’s expression of the asso- ciation constant [15,16] the theory correctly reproduces both the high coupling limit of ion association and the low density limit. This theory reproduces satisfactorily the experimental osmotic coefficients of nonaqueous electrolyte solutions with solvents of relative permittivities in the range 20 < ε < 36 up to high ion concentrations [10]. For solvents with lower permittivities the theory was modified to take into account the effect of ion trimers and tetramers [17], and then satisfactorily reproduces the experimental data of low permittivity electrolyte solutions. This AMSA approach will be reviewed in the present paper. Applications used for the description of the concentration dependence of vapour pressures and conduc- tivities of nonaqueous solutions of associating electrolytes will be discussed. 2. Model and theory The description of thermodynamic and transport excess properties of nonaqueous electrolyte solutions is based on the ionic approach (McMillan-Mayer level) using a chemical model considering free ions and ion associates in equilibrium. The effect of solvent molecules is accounted for by introducing the permittivity ε into Coulomb’s interaction law and by appropriately choosing the short-range part of interionic interactions. We begin with the formation of electrically neutral ionic pairs in the framework of the restricted primitive model given by the following interactions Uab(r) = { ∞, r < R, ZaZbe 2 εr , r > R, (1) where Z+ = −Z− = 1, R is the diameter of ions, e is the elementary charge, ε is the solvent permittivity. After solving the corresponding AMSA for this model we will modify the results obtained to take into account the possibilities of trimer and tetramer formation. 658 AMSA in the theory of nonaqueous electrolyte solutions 2.1. AMSA theory AMSA theory represents the two-density version of the traditional mean spherical approximation [18,19] for an ionic fluid of associative particles. To treat the ion pairs we introduce in the chemical picture in addition to the charged hard sphere interaction, equation (1), a formal short range cation-anion interaction, U as +− (r), which is responsible for the formation of electrically neutral ion pairs [C +A−]0 fixed by the value of the second ionic virial coefficient of the system in Ebeling’s physical picture of complete dissociation [15,16]. The formation of ion pairs determines the concentration of free ions c+ = c− = αc according to the MAL 1− α α2 = cKA (y ′ ± )2 y ′ 0 , (2) where α is the degree of dissociation, c = ρ/2NA is the analytical electrolyte concen- tration, ρ = ρ+ + ρ− is the total number density, NA is the Avogadro constant, KA is the equilibrium constant of ion-pair formation, y ′ ± is the mean activity coefficient of the free ions in solution, and y ′ 0 is that of the ion pairs. The MAL (2) is written in the form [12–14] 1− α α2 = 4πcNA ∫ ∞ 0 f as +− (r)g00+− (r)r2dr, (3) where f as +− (r) = exp(−βUas +− (r))− 1 is the Mayer function for the associative inter- action, β = 1/kT , T is the temperature, k is the Boltzmann constant. At the sticky limit follows f as +− (r) = Bδ(r − R) (4) and the equation (3) can be rewritten in the form 1− α α2 = c4πNABR2g00+− (R), (5) where g00+− (R) is the contact value at r = R of the pair-distribution function of the unbound oppositely charged ions. From the comparison of equations (5) and (2) for charged hard spheres follows KA = 4πNABR2eb, (y ′ ± )2 y ′ 0 = e−bg00+− (R), (6) where b = e2/(εkTR) is the Bjerrum parameter characterising the Coulomb inter- actions between ions at contact distance, and one has [15] KA = 8πNAR 3 ∑ m>2 b2m (2m)!(2m− 3) . (7) 659 J.Barthel et al. The total pair correlation function hij(r) between two ions i and j is represented as a sum of four terms [10–12] hij(r) = h00 ij (r) + h01 ij (r) + h10 ij (r) + h11 ij (r), (8) where the upper index 0 announces that the corresponding ion is free, and the index 1 indicates, when it is bounded in an ion pair, the function g 00 ij (r) = h00 ij (r) + 1. The functions hαβ ij (r) satisfy the system of Wertheim-Ornstein-Zernike (WOZ) equations [11,12] hij(r12) = Cij(r12 + ∑ l ρl ∫ Cil(r13)xhlj(r32)dr̄3, (9) where the matrices hij(r) and Cij(r) have the elements hαβ ij (r) and Cαβ ij (r), and x = ( 1 1 1 0 ) . (10) As usual, due to the symmetry of the RPM it is possible to define the sum and the difference functions hαβ s (r) = 1 2 ( hαβ ++(r) + hαβ +−(r) ) , hαβ D (r) = 1 2 ( hαβ ++(r)− hαβ +−(r) ) (11) and corresponding expressions for the C-functions. Then the WOZ equation (9) decouples into a set of two matrix equations hs(r12) = Cs(r12) + ρ ∫ Cs(r13)xhs(r32)dr̄3, (12) hD(r12) = CD(r12) + ρ ∫ CD(r13)xhD(r32)dr̄3. (13) The AMSA closures for the electroneutral sum problem (index s) are the same as for the associative hard sphere Percus-Yevick (PY) approximation [20] h00 s (r) = −1, h00 s (r) = h10 s (r) = h11 s (r) = 0, r < R; C00 s (r) = C01 s (r) = C10 s (r) = 0, C11 s (r) = (1− α) 4πρ δ(r − R), r > R. (14) Similarly, this follows for the difference case (index D) h00 DD(r) = h01 D (r) = h10 D (r) = h11 D (r) = 0, r < R; C00 D (r) = −bR r , C01 D (r) = C10 D (r) = 0, C11 D (r) = −(1 − α) 4πρ δ(r −R), r > R. (15) The function g00+− (r), introduced into the MAL, equation (5), is defined by g00+− (R) = 1 + h00 +− (R) = 1 + h00 s (R)− h00 D (R) (16) 660 AMSA in the theory of nonaqueous electrolyte solutions or in its exponential approximation [21] g00+− (R) = [1 + h00 s (R)] exp[−h00 D (R)]. (17) The solution for hs(r) formally coincides with Wertheim’s solution [20] for dimer- izing hard spheres with α calculated from equation (5). The solution for hD(r) was also obtained [10,11] and here we briefly repeat the general features of this solution. Using the Wertheim-Baxter factorization technique, equation (13) is transformed into two equations: SD(r) = Q(r)− ∫ ∞ 0 dtQ(r + t)xQT (t), (18) JD(r) = Q(r) + ∫ ∞ 0 dtJD(\|r − t\|)xQ(t), (19) where Jαβ D (r) = 2πρ ∫ ∞ r shαβ D (s)ds, Sαβ D (r) = 2πρ ∫ ∞ r sCαβ D (s)ds. (20) After differentiation equations (18)–(19) read 2πρrCD(r) = −Q′(r) + ∫ ∞ 0 dtQ′(r + t)xQT (t), (21) 2πρrhD(r) = −Q′(r) + 2πρ ∫ ∞ 0 dt (r − t)hD(\|r − t\|)xQ(t). (22) From the AMSA closures Q(r) has the form Q(r) = q(r)− lim µ→0+ Ae−µr, (23) where q(r) = q0 + q1(r −R), r 6 R, q(r) = 0, r > R. (24) The matrix A has nonzero values only in the first row Aαβ = aβδα0. (25) Taking into account the structure of hαβ D (r) for r 6 R according to equation (15) we have from equation (19) for the matrix elements of q1 qαβ1 = Jαaβ , (26) where Jα = 2πρ ∫ ∞ 0 r dr ∑ β hαβ D (r). (27) From the boundary conditions one has q000 = q010 = q100 = 0, q110 = −1− α 2 . (28) 661 J.Barthel et al. From equation (21) in the asymptotic cases r → ∞ and r → 0 four independent equations are obtained (κR)2 = a20 + 2a0a1 , 2a1 + (κR)2J1R + a0(1− α) = 0, (κR)2(1 + J0R)2 + 2a0J0R = 0, a0J1R + a1J0R + (1 + J0R) ( κ2R2J1R + 1 2 a0(1− α) ) = 0, (29) which make up a system of equations for the four parameters J0, J1, a0 and a1 depending on α and κ2R2, where κ2 = 4πe2βρe2/ε is the Debye-Huckel screening parameter. After some algebra this system can be presented as a single nonlinear equation of the parameter J0 (κR)2(1 + J0R)4[α− J0R(1− α)] = 4(J0R)2. (30) This equation can be written in another form, if Blum’s screening parameter ΓB instead of J0 is introduced J0R = − ΓBR 1 + ΓBR . (31) Then a simple nonlinear equation is derived for the scaling parameter ΓB first given by Bernard and Blum [21] 4(ΓBR)2(1 + ΓBR)2 = (κR)2 α + ΓBR 1 + ΓBR . (32) The equations (32) and (30) containing the degree of dissociation α are consid- ered together with the MAL equation (5). For the contact value hD(r) needed for the calculation α from equation (22) we have h00 d (R) = − q001 2πρR = −b(1 + J0R)2 = − b (1 + ΓBR)2 (33) and using the exponential form (17) for g+−(R), the equation (5) can be written as 1− α α2 = c4πNABR2[1 + h00 s (R)] exp [ b (1 + ΓBR)2 ] . (34) In the AMSA the contact value of 1+h00 s (R) is equal to the Percus-Yevick value for the hard-sphere model [5,9] 1 + h00 s (R) = 1 + 0.5η (1− η)2 , (35) where η = 1 6 πρR3. The analysis of equation (30) for J0 or equation (32) for ΓB suggests the exis- tence of two different regimes, namely the weak (α → 1) and the strong (α → 0) 662 AMSA in the theory of nonaqueous electrolyte solutions association regimes. In the regime of weak association equation (32) for ΓB reduces to 4(ΓBR)2(1 + ΓBR)2 = (κR)2α (36) and in this regime only the electrostatic contribution of free ions is important; contri- butions from ion pairs can be neglected. The regime of weak association corresponds to the traditional description of electrostatic ion association [7–9] and is realized for 1 > α ≫ ΓBR (1 + 2ΓBR) . (37) In the strong association regime, equation (32) for ΓB reduces to 4ΓBR(1 + ΓBR)3 = (κR)2(1− α) (38) and only the electrostatic contribution from ion pairs is important; the contribution from free ions can be neglected. This regime is realized for 0 6 α ≪ ΓBR 1 + 2ΓBR . (39) In the AMSA the thermodynamic properties contain three different contribu- tions: the hard sphere contribution (HS), the contribution from the mass action law (MAL) and a contribution from electrostatic ionic interaction (el.) [10,21,22]. The osmotic coefficient is Φ = posm ρkT = pHS ρkT + pMAL ρkT + pel ρkT , (40) where the hard sphere contribution may be calculated by the Carnahan-Starling expression for hard spheres [23] PHS ρkT = 1 + η + η2 − η3 (1− η)3 . (41) The effect of ion pair formation on the hard sphere contributions can be calcu- lated with the help of thermodynamic perturbation theory [20] and is included into the MAL-terms pMAL ρkT = −1 2 (1− α) [ 1 + ρ ∂ lnhs(R) ∂ρ ] = −1 2 (1− α) 1 + η − 0.5η2 (1− η) (1− 0.5η) . (42) The electrostatic contribution can be obtained by the direct integration of the internal energy as shown by Bernard and Blum [21] pel ρkT = −(ΓBR)3 3πρR3 . (43) For the activity coefficient of the free ions one has ln y± = ln yHS ± + ln yMAL ± + ln yel ± , (44) 663 J.Barthel et al. where in the same approximation as for osmotic coefficient ln yHS ± = (1 + 2η)2 (1− η)4 , (45) ln yMAL ± = lnα− 1 4 (1− α)ρ ∂ lnhs(R) ∂ρ = lnα− 1 4 (1− α) 5η − 2η2 (1− η)(1− 0.5η) , (46) ln yel ± = −b ΓBR 1 + ΓBR . (47) From the comparison of equations (5), (6), (34) and (47) the electrostatic part of the activity coefficient of ionic pairs is obtained ln yel0 = −b (ΓBR)2 (1 + ΓBR)2 . (48) 2.2. Modified AMSA including the effect of ion trimers and tet ramers The formation of ion aggregates which are more complex than ion pairs was first introduced to explain the conductivity of low permittivity electrolyte solutions [24]. Later on it was shown that this concept is also important for the explanation of other phenomena [9]. We suppose that in the modified AMSA the formation of trimers and tetramers follows the reaction (C+A−)0 + C+(A−) ↔ [(C+A−)0C+(A−)], (49) where the notation C+(A−) means that the cation C+ can either be a free ion or is bound in another pair. In the first case we have the formation of ion trimers and in second case we have the formation of tetramers. A relation corresponding to (49) holds for the anion A−. As a simplification we consider bilateral triplet formation [9], assuming that the formation of a trimer is characterized by only one association constant K2. This means that the concentration of negative trimers (ACA)− equals that of positive trimers (CAC)+. Introducing γc as the concentration of ions of a species which is not bound either in a trimer or in a tetramer. Correspondingly (1 − γ)c will be the concentration of ions of each species bound in trimers and tetramers. Then the ionic solute is assumed to exist in the form of single ions of concentration αγc, pairs, (1 − α)γc, trimers, α(1− γ)c, and tetramers, (1− α)(1− γ)c αγc+ (1− α)γc = γc, α(1− γ)c+ (1− α)(1− γ)c = (1− γ)c. (50) Correspondingly, αc is the solute concentration of the charged particles (mono- mers and trimers) and (1−α)c is the solute concentration of the noncharged particles (pairs and tetramers) αγc+ α(1− γ)c = αc, (1− α)γc+ (1− α)(1− γ)c = (1− α)c. (51) 664 AMSA in the theory of nonaqueous electrolyte solutions Note that αc is only the solute concentration. The concentration of charged particles of one kind ( + or – ) is αγc+ 1 3 α(1− γ)c = 2γ + 1 3 αc. (52) For the concentration of the noncharged particles one has (1− α)γc+ 1 2 (1− α)(1− γ)c = γ + 1 2 (1− α)c. (53) Now we can write the MAL of the ion-pair formation as 1− α α2 = cγKA(1 + h00 s (R)) (yel ± )2 yel0 , (54) which differs from equation (2) by the factor γ. As before KA is the association constant of ion pairs, for which we can use equation (7). The hard sphere contribution 1 + h00 s (R) is given by equation (35). We use the expressions (47) and (49) for the electrostatic contributions of the activity coefficients of free ions, y ′ ± , and of the ion pairs, y ′ 0, with correction of equation (32) for ΓB. For this purpose, we can neglect the electrostatic contribution from ion trimers and tetramers in y el ± and yel0 as for equation (36) in the regime of weak association, α → 1. For γ → 1 we generalize the equation for ΓB in the following way 4(ΓBR)2(1 + ΓBR)2 = γ(κR)2 α+ ΓBR 1 + ΓBR . (55) The second MAL relation for the ion trimer and tetramer formation can be written in two different forms. For the regime of weak ion pairing (α → 1) the trimer formation process is the dominating process and 1− γ γ2 = (1− α)yHS 2 cK2 yel ± yel0 yel3 . (56) For the regime of strong ionic pairing (α → 0) the tetramer formation process is the dominating process and 1− γ γ2 = (1− α)cyHS 2 K2 (yel0 ) 2 yel4 . (57) The association constant K2 is related to the formation of ion trimers and tetramers and by analogy with the Ebeling association constant it can be given as part of the third ionic virial coefficient in the case of equation (56) or as part of the fourth ionic virial coefficient in the case of equation (57). Third and fourth ionic virial coefficients can be approximated as electrostatic parts of the second virial co- efficient for ion-dipole and dipole-dipole interactions, correspondingly [17]. For the case of simplicity we consider K2 as an adjustable constant and neglect the hard 665 J.Barthel et al. sphere contribution of equations (56)–(57) assuming that yHS 2 = 1. For the electro- static part of the activity coefficient of the ion trimers, y el 3 , and ion tetramers, yel 4 , we assume that by analogy with the relations (47) and (49) we can write ln yel3 = −b (ΓBR)3 (1 + ΓBR)3 , ln yel4 = −b (ΓBR)4 (1 + ΓBR)4 . (58) The formation of trimers and tetramers is a process following the ion-pair for- mation. Usually KA ≫ K2 [9]. This leads to the following order of the quantities α and γ 0 6 α 6 γ 6 1. (59) Due to this fact equation (57) is preferable. The equations (54), (55) and (57) for α, γ and (ΓBR) form a nonlinear system which in the general case can be solved numerically, e.g. by a Newton-Raphson technique. After including trimers and tetramers the osmotic coefficient Φ has a form similar to equation (40), where the hard sphere contribution PHS/ρkT is again given by equation (41). For the MAL-contribution we take into account only the ideal terms from free ions, ion pairs, trimers and tetramers. As a result instead of equation (42) we have PMAL ρkT = αγ + 1 2 (1− α)γ + 1 3 α(1− γ) + 1 4 (1− α)(1− γ)− 1. (60) The electrostatic excess contribution is given by equation (43) where ΓB is cal- culated from equation (55). We note that due to the approximation made for the calculation of ΓB in equation (55) we neglect in this calculation electrostatic contri- butions from trimers and tetramers for the osmotic coefficient because with γ → 0 also ΓB → 0. The activity coefficient of the free ions has the form (44) with a similar modifi- cation for yMAL ± and yel ± . The results obtained are also useful for the calculation of the ionic conductivity of nonaqueous electrolyte solutions. Several attempts exist for the calculation of the molar conductivity of associating electrolytes beyond the limiting law in the chemical model at the level of the mean spherical approximation [9,25,26], where, however, only ion pairs were taken into account. Ion pairs and tetramers are electrically neutral, non-conducting species in the solution, by contrast to the ion trimers. The total concentration of charged particles is given by equation (52) and due to this we modify the result obtained by Turq et al. [25,26], changing α to 1 3 α(2γ + 1) and Γ to ΓB. Then we have the following expression for molar conductivity Λ=α (2γ+1) 3 Λ0 [ [ 1+Srel ( α 2γ+1 3 )][ 1+Sel ( α 2γ+1 3 )] + 3 2 [ Srel ( α 2γ+1 3 )]2 ] , (61) where Λ0 is the limiting molar conductivity of the salt in the form of simple ions at infinite dilution. 666 AMSA in the theory of nonaqueous electrolyte solutions The first order relaxation contribution S rel and the first order electrophoretic term Sel are given by Srel = − b 6 1− exp(− √ 2κ′R) (1 + ΓBR)2 + √ 2(1 + ΓBR) + 1− exp(κ′R/ √ 2) , (62) Sel = − Nae 2 3πη0Λ0 2ΓB 1 + ΓBR , (63) where (κ′)2 = κ2α 2γ+1 3 is the Debye-Huckel screening parameter for charged parti- cles, η0 is the solvent viscosity. 3. Experimental data Ionic association occurring as a result of strong Coulomb interactions increases at decreasing solvent permittivity ε. However, ionic association also occurs as a result of chemical interactions due to specific solvation forces [9]. In the following discussions on solutions of LiClO4 we focus our attention only on the first case. LiClO4 is known for its remarkable solubility in low permittivity solvents. To compare theoretical and experimental data, experimental osmotic coefficients were chosen for acetonitrile and aceton [27], 2-propanol [28], 1,2 dimethoxyethane (DME) and dimethylcarbonate (DMC) [29] solutions of LiClO4 with solvents of relative permittivities ε = 35.95 for acetonitrile, ε = 20.70 for acetone, ε = 19.39 for 2-propanol, covering the range 20 < KA[mol−1dm3] < 2000 of the association constant. For electrolytes with solvents of significantly lower permittivities, ε = 7.02 for DME and ε = 3.09 for DMC, the association constant is much higher (KA > 106mol−1dm3). Osmotic coefficients ΦLR were obtained by measurements of vapour pressures p at molalitiesm and temperature T from vapour pressure lowering ∆p = p∗−p, with p∗ – the vapour pressure of the pure solvent, using the relationship ΦLR = − 1 2mMs [ ln p∗ −∆p p∗ − ∆p(B − V ∗(l) s RT ] , (64) where Ms is the molar mass of the solvent, B is the second virial coefficient of the gas phase solvent, V ∗(l) s is the molar volume of the liquid solvent, R = kNA is the gas constant. To compare with theoretical data, the experimental osmotic coefficients in the Lewis-Randall (LR) system must be converted to the McMillan-Mayer (MM) system and molarity c Φ = ΦLR(1 + 10−3mME) d∗ d , Φc = md (1 + 10−3mME) , (65) where d and d∗ are the densities of solution and solvent, ME = 105.39 g/mol is the molar mass of salt LiClO4. The densities are available in the form of polynomials d = d∗ + 3 ∑ i=1 Aim i. (66) 667 J.Barthel et al. Table 1. Solvent properties. Solvent acetonitrile acetone 2-propanol DME DMC Ms[g/mol] 41.053 58.08 60.069 90.123 90.079 B[cm3/mol] –6190 –2132 –3424 –2000 –1950 d∗[g/cm3] 0.77675 0.78421 0.780716 0.86135 1.06335 V ∗(l) s [cm3/mol] 52.852 70.995 76.971 104.63 84.664 Pi[Pa] 11745 30803 5777 9136 7226 ε 35.95 20.70 19.39 7.08 3.108 η0[cP] 0.34 0.303 2.08 0.41 0.5902 Table 2. Coefficients of the density polynomials, equation (66) for the various electrolyte solutions of LiClO4. Solvent acetone acetonitrile 2-propanol DME DMC A1 ∗ 102 8.2131 7.1513 7.0936 7.5609 8.087 A2 ∗ 102 –1.5952 –0.64268 –0.8510 5.3787 –0.4516 A3 ∗ 103 10.844 0.8780 1.972 –0.4516 –0.0958 The data used to calculate the osmotic and the activity coefficients for the so- lutions studied in this paper are taken from [27–29] and are given in table 1 and table 2. The experimental ∆p data used to calculate the osmotic coefficients are repre- sented in the cited literature by polynomials of the type ∆p = 5 ∑ i=1 Bim i−1. (67) The coefficients Bi are summarized in table 3. The experimental data for the ionic conductivity of LiClO4 solution in DME and DMC were taken from the references [30] and [31]. Table 3. Vapour pressure lowering ∆p = p∗ − p of various electrolyte solutions of LiClO4. Solvent Conc. range B1 B2 B3 B4 B5 Acetone 0.02–0.58 0.030530 17.7091 –12.3816 22.8147 –14.5306 Acetonitrile 0.06–1.15 0.027440 5.67914 –1.43816 1.36384 –0.433865 2-propanol 0.05–1.48 0.015177 2.74737 –1.01672 1.24942 –0.232199 DME 0.02–0.35 0.003686 6.57103 –19.2202 47.1899 –49.6216 DMC 0.03–1.77 0.052101 3.50040 –4.02156 3.40816 –0.682103 668 AMSA in the theory of nonaqueous electrolyte solutions 4. Description of the experimental data with the AMSA As shown previously [10] the regime of weak ion pairing with ΓB calculated from equation (36) correctly describes ionic solutions with electrostatic ion association for solvents of dielectric permittivity ε > 36. For electrolyte solutions with solvent permittivities in the range 20 < ε < 36 the correct description is obtained in the framework of the usual AMSA theory. This result is illustrated in figure 1, where the osmotic coefficients of LiClO4 in different solvents, derived from vapour pressure measurements, are compared with the predictions of the AMSA. For the association constant, equation (7) is used yielding KA which is connected with the association parameter B according to equation (6). ΓB and α are calculated from equations (32) and (34). The osmotic and activity coefficients are calculated using equations (40)– (43) and (44)–(47) respectively. Figure 1 shows the satisfactory agreement of one- parameter AMSA calculations with the experimental results. Distance parameters between 0.42 nm and 0.48 nm in protic solvents are reasonable, they are larger than the sum of ionic radii, indicating contributions of contact and solvent shared ion pairs. 0.0 0.2 0.4 0.6 0.8 1.0 c (mol/l) 0.0 0.2 0.4 0.6 0.8 1.0 Φ , y (1) (2) (3) (2) (1) (3) y Figure 1. Osmotic coefficients Φ and mean ionic activity coefficients y± of LiClO4 solutions of different solvents at 25◦: Experimental Φ-values: (circles: acetone; squares: acetonitrile; diamonds: 2-propanol). Calculations from AMSA theory (solid lines) (1): acetone, R = 0.48 nm; (2): acetonitrile, R = 0.42 nm; (3): 2-propanol, R = 0.43 nm. Mean ionic activity coefficients: (broken lines). For ionic solutions of very low permittivity, here for DME and DMC solutions of LiClO4, the occurrence of ion trimers and tetramers should be considered as it is known from conductivity measurements [30,31]. For the description of such ionic solutions we take into account that the permittivity of LiClO4 solutions increases with ionic concentration in both solvents. This effect can be described up to c = 0.3mol/dm3 by quadratic polynomials in c [9] ε = 7.08 + 24c− 21.3c2 (68) 669 J.Barthel et al. for DME and ε = 3.11 + 13.3c− 7.7c2 (69) for DMC. Experimental data for the osmotic coefficient of LiClO4 solutions in DMC is known up to 1.77mol/dm3 (see table 3). For higher concentrations the relation (69) has been altered to a kind of Pade-approximant, to include the saturation effects ε = 3.11 + 13.3c ( 1 + 7.7c 13.3 )−1 . (70) The increase of ε with ionic concentration produces an increase of α. This is im- portant to understand the concentration dependence of the electrolyte conductivity. For the calculations, the association constant KA of equation (7) was used. ΓB, α and γ were calculated from equations (54), (55) and (57). The osmotic coefficient is calculated from equations (40), (41), (43) and (60). To calculate the molar conduc- tivity we use equations (61)–(63) with Λ0 = 176.4 cm2Ω−1mol−1 for LiClO4 solution in DME [30] and Λ0 = 115.7 cm2Ω−1mol−1 for LiClO4 solution in DMC [31]. Now our approach contains two adjustable parameters, namely the contact dis- tance R and the second association constant K2. The calculations show that the osmotic coefficients can be described up to 0.4mol/dm3 with fixed K2 = 1dm3/mol and again with the contact distance as fitting parameter, namely R = 0.318 nm for 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 Φ,α,γ, ΓR c, [ mol l ] γ ΓR Φ α ✸✸✸✸✸✸ ✸ ✸✸✸✸✸✸ ✸✸✸ ✸✸✸✸✸✸✸✸✸ ✸✸✸✸✸✸✸✸✸ ✸ ✸ Figure 2. Osmotic coefficients Φ, of LiClO4 solu- tions in DME. ( ✸ ) experimental values, (solid line) theoretical values. Also shown: γ – the de- gree of particles not bound in a trimer or a tetramer, and α, the degree of charged particles in the system., and the dimensionless screening parameter ΓBRi See the text for details. LiClO4 solutions in DME and R = 0.1 nm for LiClO4 solu- tions in DMC. Distances R up to which two ions are counted as a pair which are much smaller than the ion contact distance, lead to high values of the as- sociation constant KA, when calculated from equation (7); α is practically zero and ac- cording to equation (61) Λ ≈ 0. To avoid this drawback we modify the theory in such a way that we introduce two fit- ting distances, one for the ionic contact distance, R, which we use to calculate the associa- tion constant KA and for the hard-sphere contribution, equa- tion (41), and the other one, Ri, to calculate the electrostatic contributions connected with the dimensionless screening pa- rameter ΓBR, equations (55) 670 AMSA in the theory of nonaqueous electrolyte solutions such that a modified ΓBRi term corrects some of the simplifications introduced with the chosen forms of the MAL, equations (55) to (57) and the expressions for the activity coefficients of the higher associates, y el 3 and yel4 , equation (58). Figure 2 shows the satisfactory agreement of the modified AMSA with two ionic distances R = 0.42 nm and Ri = 0.22 nm with the experimental results for the osmotic coefficient of LiClO4 solutions in DME. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Φ,α,γ, ΓR c, [ mol l ] γ ΓR Φ α ✸✸✸✸✸✸✸✸✸✸✸ ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ ✸✸ ✸✸ Figure 3. Osmotic coefficients Φ of LiClO4 solu- tions in DMC and related quantities (as in fig- ure 2). See the text for details. Also, the agreement (up to 0.8mol/dm3) of the modi- fied AMSA theory with R = 0.55 nm and Ri = 0.13 nm with experimental results of the os- motic coefficient of LiClO4 so- lutions in DMC is presented in figure 3. The second association constant for both cases is fixed as K2 = 1dm3/mol. In figure 2 and figure 3 the concentration dependences of α, γ and ΓBRi are also presented. For both cases the γ fraction decreases with the ionic concentration in the limit of the applicability of the ap- proximation of equation (55) for ΓB. The fraction α strongly decreases, and, beginning from concentrations near 10−2mol/dm3 slowly increases again. In general α < 0.15 which justifies the application of the form (57) for the second MAL. Such a behaviour of α and γ explains the specific features of the concentration dependence of the conductivity of the considered solutions as shown in the next figures. A comparison of the experimental data and the theoretical results for the equiv- alent conductivity obtained from equations (61)–(63) with the parameters used for the description of the osmotic coefficients is given in figure 4 for LiClO4 solutions in DME and in figure 5 for LiClO4 solutions in DMC. In both cases the theoretical curves satisfactorily reproduce the sharp decrease of conductivity for small ionic concentrations due to ion-pair formation, the existence of a conductivity minimum at a concentration near 10−2mol/dm3 and the increase of conductivity due to ionic trimer formation. For higher concentrations we observe for LiClO4 in DMC (fig- ure 5) a calculable maximum at a concentration near 1mol/dm3 and the decrease of conductivity at an increasing ionic concentration required by tetramer formation. 671 J.Barthel et al. 2 4 6 8 10 -4 -3 -2 -1 Λ, [ cm2 molOm ] lg(c), [ mol l ] ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Figure 4. Equivalent conductivity Λ of LiClO4 solutions in DME with the pa- rameters used to calculate the osmotic coefficient. See the text for details. -3 -2 -1 0 1 -4 -3 -2 -1 0 lg(Λ), [ cm2 molOm ] lg(c), [ mol l ] ✸ ✸✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Figure 5. Equivalent conductivity Λ of LiClO4 solutions in DMC. See the text for details. 672 AMSA in the theory of nonaqueous electrolyte solutions 5. Conclusions An unified approach to the description of the thermodynamic and transport excess functions of associating and nonassociating electrolytes is proposed which leads to numerically simple final formulae capable of presenting experimental data with a minimum of adjustable parameters. These parameters show realistic values in their definition range. 6. Acknowledgment This work was completed in the framework of a cooperation between our in- stitutes at Regensburg and in Lviv, registered under UKR–028–96 at the “Inter- nationales Büro des Bundesministeriums für Bildung, Wissenschaft, Forschung und Technologie (BMBF) bei der Deutschen Luft- und Raumfahrtgesellschaft (DLR)”, Germany, and Ministry on Science and Technology of Ukraine. The financial support of this contract is gratefully acknowledged. References 1. Yukhnovskij I.R. // Zhurn. Eksp. and Teoret. Fiz., 1954, vol. 27, p. 690 (in Russian). 2. Yukhnovskij I.R. // Zhurn. Eksp. and Teoret. Fiz., 1958, vol. 34, p. 379 (in Russian). 3. Yukhnovskij I.R. // Dokl. Akad. Nauk SSSR, 1961, vol. 136, p. 1317 (in Russian). 4. Yukhnovskij I.R. // Ukr. Fiz. Zhurn., 1962, vol. 7, p. 267 (in Russian). 5. Yukhnovskij I.R., Holovko M.F. The Statistical Theory of the Classical Equilibrium Systems. Naukova dumka, Kiev, 1980. 6. Golovko M.F., Yukhnovskij I.R. – In: “The Chemical Physics of Solvation”, Elsevier, Amsterdam, 1985, vol. A, p. 207–262. 7. Bjerrum N. // Mat.-Fys. Medd. K. Dan. Vidensk. Selsk., 1926, vol. 7, p. 1. 8. Barthel J. // Ber. Bunsenges. Phys. Chem., 1979, vol. 83, p. 252. 9. Barthel J., Krienke H., KunzW. Physical Chemistry of Electrolyte Solutions – Modern Aspects. New York, Steinkopff, Darmstadt and Springer, 1998. 10. Krienke H., Barthel J., Holovko M., Protsykevich I., Kalyuzhnyi Yu. // J. Mol. Liq- uids, 2000 (in press). 11. Holovko M.F., Kalyuzhnyi Yu.V. // Mol. Phys., 1991, vol. 73, p. 1145. 12. Wertheim M.S. // J. Stat. Phys., 1984, vol. 35, p. 19; p. 35. 13. Wertheim M.S. // J. Stat. Phys., 1986, vol. 42, p. 495; p. 477. 14. Holovko M.F. // Cond. Matt. Phys., 1999, vol. 2(18), p. 205. 15. Ebeling W. // Z. Phys. Chem. (Leipzig), 1968, vol. 238, p. 400. 16. Ebeling W., Grigo M. // Ann. Physik (Leipzig), 1980, vol. 37, p. 21. 17. Holovko M., Protsykevich I., Kapko V., Krienke H., Barthel J. // J. Mol. Liquids, 2000 (in preparation). 18. Waisman E., Lebowitz L. // J. Chem. Phys., 1970, vol. 52, p. 4307. 19. Blum L. // Mol. Phys., 1975, vol. 30, p. 1529. 20. Wertheim M.S. // J. Chem. Phys., 1986, vol. 85, p. 2929. 21. Bernard O., Blum L. // J. Chem. Phys., 1996, vol. 104, p. 4746. 22. Kalyuzhnyi Yu., Holovko M.F. // J. Chem. Phys., 1998, vol. 108, p. 3709. 673 J.Barthel et al. 23. Carnahan M.F., Starling K.E. // J. Chem. Phys., 1969, vol. 51, p. 635. 24. Fuoss R.M., Kraus C.A. // J. Amer. Chem. Soc., 1933, vol. 55, p. 2387. 25. Chhih A., Turq P., Bernard O., Barthel J., Blum L. // Ber. Bunsenges Phys. Chem., 1994, vol. 98, p. 1516. 26. Turq P., Blum L., Bernard O., Kunz W. // J. Phys. Chem., 1995, vol. 99, p. 822. 27. Barthel J., Neuder R., Poepke H., Wittmann H. // J. Solution Chemistry, 1999 (in press). 28. Barthel J., Neuder R., Poepke H., Wittmann H. // J. Solution Chemitry, 1998, vol. 27, p. 1055. 29. Barthel J., Neuder R., Poepke H., Wittmann H. // J. Solution Chemitry, 1999 (in press). 30. Cachet H., Fekir M., Lestrade J.-C. // Can. J. Chem., 1981, vol. 59, p. 1051. 31. Delsignore M., Farber H., Petrucci S. // J. Phys. Chem., 1985, vol. 89, p. 4968. Застосування асоціативного середньосферичного наближення в теорії неводних розчинів електролітів Й.Бартель 1 , Г.Крінке 1 , М.Ф.Головко 2 , В.І.Капко 2 , І.А.Процикевич 2 1 Інститут фізичної і теоретичної хімії, Регенсбурзький університет, D-93040, Регенсбург, Німеччина 2 Інститут фізики конденсованих систем НАН Укpаїни, 79011 Львів, вул. Свєнціцького, 1 Отримано 9 липня 2000 р. Іонна асоціація в розчинах електролітів досліджується в рамках хіміч- них моделей. Дано огляд асоціативного середньосферичного на- ближення (АССН) в теорії електролітів. Показано, що АССН у комбі- нації з асоціативною константою Ебелінга задовільно відтворює дані термодинамічних вимірювань, включаючи область високих концен- трацій, для неводних розчинів електролітів з розчинниками з діелек- тричною проникністю в області 20 < ε < 36 . Для іонних розчинів з низькою діелектричною проникністю АССН модифіковане включен- ням іонних тримерів і тетрамерів, щоб отримати правильний опис ос- мотичних коефіцієнтів та іонних провідностей. Ключові слова: розчини електролітів, асоціативне середньосферичне наближення, осмотичний і активності коефіцієнти, іонна провідність PACS: 61.20.Qg, 66.10.Ed 674

The application of the associative mean spherical approximation in the theory of nonaqueous electrolyte solutions

Репозитарії

Схожі ресурси