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Pdf 700.57 K Regular Article PHYSICAL CHEMISTRY RESEARCH Published by the Iranian Chemical Society www.physchemres.org [email protected] Phys. Chem. Res., Vol. 6, No. 4, 773-784, December 2018 DOI: 10.22036/pcr.2018.136748.1499 A Nonextensive Electrolyte UNIQUAC Model for Prediction of Mean Activity Coefficients of Binary Electrolyte Solutions H. Rezaeia, S. Abdolrahimib, G.R. Pazukib,* and A.A. Rohanic aChemical Engineering Department, Islamic Azad University, Shahrood Branch, Iran bChemical Engineering Department, Amirkabir University of Technology, (Tehran Polytechnic), Tehran, Iran cRefining Technology Development Division, Research Institute of Petroleum Industry, National Iranian Oil Company, Tehran, Iran (Received 22 June 2018, Accepted 10 September 2018) In this work, an electrolyte-UNIQUAC model was developed by replacement of Boltzmann weight binary interaction parameters by the nonextensive Tsallis weight. A summation of the long-range electrostatic term (Debye-Huckel equation) and a short-range interaction term were considered in the calculation of thermodynamic properties. A framework proposed by Chen et al. was employed for the derivation of the local mole fractions. Application of the nonextensive theory increased the degree of freedom of the present model (T-E-UNIQUAC). Furthermore, the strength of the model lies in its ability to calculate individual activity coefficients of ions. The applicability of the T-E- UNIQUAC model were tested using aqueous electrolyte solutions, and subsequently, results were compared with Messnaoui, Chen and Pitzer models. Keywords: Electrolyte solutions, Activity coefficient, Tsallis weight INTRODUCTION and k is a positive constant. The entropic index q is a measure of the degree of nonextensivity in the system The Boltzmann-Gibbs statistical mechanics (BG) is not explained by the following pseudoadditivity rule: applicable to systems that are not in the thermodynamic limit. This is due to the finite size of the system, the 1 q (2) SABSASBSASBq()()()()() q q q q existence of long-range interactions, and the presence of k dissipative structures [1]. Nonextensive statistical thermodynamics proposes a new formalism for In this equation systems A and B are independent variables, and S 0 . Consequently, superextensivity, extensivity, and characterisation of the systems which are not properly q described by Boltzmann-Gibbs statistical mechanics. Tsallis subextensivity are the cases related to q < 1, q = 1, and q > 1, respectively. In the limit q 1 of the Eqs. (1) and introduced the concept of nonextensive entropy as [2-7], (2), the Boltzmann-Gibbs entropy and extensivity rule are W q respectively recovered as: 1 pi W i1 (1) Sq k, p i 1 q 1 i1 W S S k pln p , (3) BG1 B i i i1 where pi is the probability of the microscopic state i , W is the total number of microstates, q is the entropic index, SABSASB( ) ( ) ( ), (4) *Corresponding author. E-mail: [email protected] where kB is Boltzmann’s constant. It is worthwhile Rezaei et al./Phys. Chem. Res., Vol. 6, No. 4, 773-784, December 2018. mentioning that the constant, k, in Eq. (1) differs from kB, Tsallis’ entropy has found its platform in modern statistical but in the limit q = 1, it reduces to kB = k. In the mechanics. microcanonical ensemble, the probability of all microstates Souza et al. replaced the ordinary product by a new are equal and, thus pi = 1/W. This concept simplifies Tsallis’ nonquadratic mixing rule between species by means of the entropy to q-product of mole fractions [18]. They applied q-product for generalization of van der Walls mixing rules, and validated (5) Sq kln q W their proposal by evaluating vapor-liquid equilibrium at different temperatures. x1q 1 In this study, the nonextensive Tsallis weight is applied where ln q is q-logarithm defined as ln x [8]. q 1 q rather than Boltzmann weight, and thus, nonextensive It is reasonable to introduce the inverse function of q- statistical mechanics propose a theoretical basis for exponential as derivation of the UNIQUAC model for electrolyte solutions. On this account, the Gibbs free energy of electrolyte 1/1q x [1 (1 q ) x ] if [1 (1 q ) x ] 0 solutions has been developed based on Tsallis entropy expq (x ) e q . 0 if [1 (1 q ) x ] 0 function. (6) Thermodynamic Model Boltzmann weight displays exponential tails while Tsallis The basis of the present thermodynamic model is the weight presents power law tails which is long-lasting and modification of the original UNIQUAC model [19] by vanishing for q > 1 and q < 1, respectively. A new concept addition of Debye-Huckel term and replacement of of nondistributive q-deformed algebra has been introduced Boltzmann weight by the nonextensive Tsallis weight. based on the q-exponential and q-logarithm functions [9, Consequently, the proposed model has three fundamental 10]. Furthermore, reading for comprehension of some terms: a combinatorial or entropic term, a residual or properties of q-functions and q-algebra is recommended at enthalpic term, and an electrostatic term. As a result, the [11-14], and references therein. excess Gibbs energy consists of a long-range electrostatic In other words, the extensivity (additivity) and term along with a short-range interaction term. The intensivity concepts in thermodynamics are applied to the electrostatic interaction between ions is regarded in long- homogeneous systems with weak interactions, and the range term while non-electrostatic interactions are thermodynamic limit of (,)NV or NV/ finite . The accounted in the short-range term. Moreover, this paper notion of nonextensive statistical mechanics was introduced deals with a single completely dissociated liquid electrolyte to overcome the limitations of Boltzmann-Gibbs statistical and a single solvent. It is evident that the procedure can be mechanics. This formalism is suitable for the study of the generalized for multicomponent electrolyte system. The systems with long-range interactions as well as small UNIQUAC model is employed to describe the short-range systems, or systems with complex behavior, including term, while Debye-Huckel equation depicts the long-range nanosystems [15]. term, Some research studies have validated the application of Tsallis' entropy to complex systems. Cohen has successfully g E g E gEE g DH c res . (7) related the microscopic dynamics to the entropic index [16]. RT RT RT RT In another research, conducted by García-Morales and Pellicer [17], fractal phase space systems have been studied, Thus, i can be derived as: and the entropic index has been referred to the fractal dimensions of the available phase space. Consequently, ln ln DH ln c ln res . (8) i i i i 774 A Nonextensive Electrolyte UNIQUAC Model/Phys. Chem. Res., Vol. 6, No. 4, 773-784, December 2018. X r X q (14) The Long-range Contribution i i i i i , and i , Xj r j X j q j The Debye-Huckel model [20] is the fundamental model j j in electrolyte solutions that accounts for the long-range ion- ion interactions. The excess Gibbs energy in this model can where ri and qi are volume and surface area parameters for be written as: component i , respectively. Therefore, by partial molar differentiation of gE 4A b2 I combinatorial excess Gibbs energy, the activity coefficients DH x M[ln(1 b I ) b I ], (9) RTw w b3 2 are derived as [19] c i iz i i (15) where xw and Mw are mole fraction and molar mass of water, ln i ln( ) 1 ( ) q i [ln( ) 1 ( )]. XXi i2 i i respectively. A and b are the Debye-Huckel parameters, and I is the ionic strength defined as: Residual term Chen et al. [214,22] stated the existence of three types 1 2 (10) I zi m i. of cells. The first type has a solvent molecule (m) as its 2 i center, while other solvent molecules, anions (a) and cations (c) are in its immediate neighborhood. For this type of cell, Consequently, partial molar differentiation of g E leads to DH the local electroneutrality assumption is valid. The other activity coefficient of water, two types of cells either have a central cation or anion with solvent molecules and ions of opposite charge in the 2A 1 ln DH M (1 b I 2ln(1 b I )). (11) immediate neighborhood. For these two cells the like-ion w w b3 1b I repulsion assumption is valid; thus no ions of like charge exist near each other (i.e. ). Thus, local mole cc aa 0 Moreover, the activity coefficients for the ions are: fractions can be written as: DH 2 AI (12) cm am mm 1 (central solvents cells), ln i z i . (16) 1 b I ma ca 1 (central anions cells), mc ac 1 (central cations cells), The Short-range Contribution where Ɵ is the local surface fraction of component i Combinatorial term. Although we are dealing with an ij around the central j molecule. electrolyte solution, the combinatorial term is identical to The effective local surface areas with component i as the one used in the traditional UNIQUAC thermodynamic the center are given by model. The combinatorial or entropic term is defined as [1]: ji j H , (17) g E z ji, ki cXln( i ) q X ln( i ), (13) ki k RTi x 2 i i ii i i where Ɵji and Ɵki are the effective local surface area of where , for ions, and for solvent, z is Xi C i x i CZi i Ci 1 species j and k around the central component i, the coordination number and is equal to 10, and are respectively. H , is defined as: i i ji ki volume fraction and surface area fraction of component i, u Hexp ( ji, ki ) , and u h h , (18) respectively, ji,, ki qRT ji ki ji ki 775 Rezaei et al./Phys.
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