Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice M

D o s s i e r

This paper is a part of the hereunder thematic dossier published in OGST Journal, Vol. 68, No. 2, pp. 187-396 and available online here Cet article fait partie du dossier thématique ci-dessous publié dans la revue OGST, Vol. 68, n°2, pp. 187-396 et téléchargeable ici © Photos: IFPEN, Fotolia, X. DOI: 10.2516/ogst/2012094

DOSSIER Edited by/Sous la direction de : Jean-Charles de Hemptinne InMoTher 2012: Industrial Use of Molecular Thermodynamics InMoTher 2012 : Application industrielle de la thermodynamique moléculaire

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2, pp. 187-396 Copyright © 2013, IFP Energies nouvelles 187 > Editorial électrostatique ab initio X. Rozanska, P. Ungerer, B. Leblanc and M. Yiannourakou 217 > Improving the Modeling of Hydrogen Solubility in Heavy Oil Cuts Using an Augmented Grayson Streed (AGS) Approach 309 > Improving Molecular Simulation Models of Adsorption in Porous Materials: Modélisation améliorée de la solubilité de l’hydrogène dans des Interdependence between Domains coupes lourdes par l’approche de Grayson Streed Augmenté (GSA) Amélioration des modèles d’adsorption dans les milieux poreux R. Torres, J.-C. de Hemptinne and I. Machin par simulation moléculaire : interdépendance entre les domaines J. Puibasset 235 > Improving Group Contribution Methods by Distance Weighting Amélioration de la méthode de contribution du groupe en pondérant 319 > Performance Analysis of Compositional and Modified Black-Oil Models la distance du groupe For a Gas Lift Process A. Zaitseva and V. Alopaeus Analyse des performances de modèles black-oil pour le procédé d’extraction par injection de gaz 249 > Numerical Investigation of an Absorption-Diffusion Cooling Machine Using M. Mahmudi and M. Taghi Sadeghi C3H8/C9H20 as Binary Working Fluid Étude numérique d’une machine frigorifique à absorption-diffusion 331 > Compositional Description of Three-Phase Flow Model in a Gas-Lifted Well with High Water-Cut utilisant le couple C3H8/C9H20 H. Dardour, P. Cézac, J.-M. Reneaume, M. Bourouis and A. Bellagi Description de la composition des trois phases du modèle de flux dans un puits utilisant la poussée de gaz avec des proportions d’eau élevées 255 > Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice M. Mahmudi and M. Taghi Sadeghi Equation of State Propriétés thermophysiques des solutions aqueuses de sels 1:1 avec l’équation 341 > Energy Equation Derivation of the Oil-Gas Flow in Pipelines d’état de réseau pour électrolytes Dérivation de l’équation d’énergie de l’écoulement huile-gaz dans A. Zuber, R.F. Checoni, R. Mathew, J.P.L. Santos, F.W. Tavares and M. Castier des pipelines J.M. Duan, W. Wang, Y. Zhang, L.J. Zheng, H.S. Liu and J. Gong 271 > Influence of the Periodic Boundary Conditions on the Fluid Structure and on the Thermodynamic Properties Computed from the Molecular Simulations 355 > The Effect of Hydrogen Sulfide Concentration on Gel as Water Shutoff Influence des conditions périodiques sur la structure et sur les propriétés Agent thermodynamiques calculées à partir des simulations moléculaires Effet de la concentration en sulfure d’hydrogène sur un gel utilisé en tant qu’agent de traitement des venues d'eaux J. Janeček Q. You, L. Mu, Y. Wang and F. Zhao 281 > Comparison of Predicted pKa Values for Some Amino-Acids, Dipeptides and Tripeptides, Using COSMO-RS, ChemAxon and ACD/Labs Methods 363 > Geology and Petroleum Systems of the Offshore Benin Basin (Benin) Comparaison des valeurs de pKa de quelques acides aminés, Géologie et système pétrolier du bassin offshore du Benin (Benin) dipeptides et tripeptides, prédites en utilisant les méthodes COSMO-RS, C. Kaki, G.A.F. d’Almeida, N. Yalo and S. Amelina ChemAxon et ACD/Labs O. Toure, C.-G. Dussap and A. Lebert 383 > Geopressure and Trap Integrity Predictions from 3-D Seismic Data: Case Study of the Greater Ughelli Depobelt, Niger Delta 299 > Isotherms of Fluids in Native and Defective Zeolite and Alumino-Phosphate Pressions de pores et prévisions de l’intégrité des couvertures à partir de Crystals: Monte-Carlo Simulations with “On-the-Fly”ab initio Electrostatic données sismiques 3D : le cas du grand sous-bassin d’Ughelli, Delta du Niger Potential A.I. Opara, K.M. Onuoha, C. Anowai, N.N. Onu and R.O. Mbah Isothermes d’adsorption de fluides dans des zéolithes silicées et dans des cristaux alumino-phosphatés : simulations de Monte-Carlo utilisant un potentiel Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2, pp. 255-270 Copyright Ó 2013, IFP Energies nouvelles DOI: 10.2516/ogst/2012088 Dossier

InMoTher 2012 - Industrial Use of Molecular Thermodynamics InMoTher 2012 - Application industrielle de la thermodynamique moléculaire

Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State

A. Zuber1,2, R.F. Checoni1, R. Mathew1, J.P.L. Santos3, F.W. Tavares4 and M. Castier1*

1 Chemical Engineering Program, Texas A&M University at Qatar, Doha - Qatar 2 Departamento de Engenharia Química, Universidade Estadual de Maringá, Maringá, Paraná - Brazil 3 Departamento de Engenharia Química, Universidade Federal do Sergipe, São Cristóvão, Sergipe - Brazil 4 Escola de Química, Universidade Federal do Rio de Janeiro, Rio de Janeiro - Brazil e-mail: [email protected] - [email protected] - [email protected] [email protected] - [email protected] - [email protected]

* Corresponding author, on leave from the Federal University of Rio de Janeiro, Brazil

Re´sume´— Proprie´te´s thermophysiques des solutions aqueuses de sels 1:1 avec l’e´quation d’e´tat de re´seau pour e´lectrolytes — L’e´quation d’e´tat, dite e´lectrolattice, est un mode` le qui e´tend l’e´quation d’e´tat de Mattedi-Tavares-Castier a` des syste` mes avec e´lectrolytes. Ce mode` le prend en compte l’effet de trois termes. Le premier terme est base´sur les trous dans le re´seau en conside´rant les effets de la composition locale, e´tude effectueé dans le cadre de la theórie geńe´raliseé de Van der Waals : l’e´quation d’e´tat de Mattedi-Tavares-Castier a e´tećhoisie pour ce premier terme. Les deuxie` me et troisie` me termes sont les contributions de Born et du MSA. Ils tiennent compte du chargement et du dećhargement des ions, et des interactions ioniques a` longue distance, respectivement. Le mode` le n’ayant besoin que de deux parame` tres d’interaction eńerge´tique, il mode´lise de manie` re satisfaisante la pression de vapeur et le coefficient d’activite´ ionique moyenne pour des solutions aqueuses simples contenant du LiCl, LiBr, LiI, NaCl, NaBr, NaI, KCl, KBr, KI, CsCl, CsBr, CsI, ou du RbCl. Deux me´thodes pour obtenir les parame` tres du mode` le sont pre´senteés et mises en contraste : une me´thode spećifique pour le sel en question et une autre baseé sur les ions. Par conse´quent, l’objectif de ce travail est de calculer les proprie´te´s thermophysiques qui sont largement utiliseés pour la conception, l’exploitation et l’optimisation de nombreux proce´de´s industriels, parmi eux le dessalement de l’eau.

Abstract — Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equa- tion of State — The electrolattice Equation of State (EOS) is a model that extends the Mattedi- Tavares-Castier EOS (MTC EOS) to systems with electrolytes. This model considers the effect of three terms. The first one is based on a lattice-hole model that considers local composition effects derived in the context of the generalized Van der Waals theory: the MTC EOS was chosen for this term. The second and the third terms are the Born and the MSA contributions, which take into account ion charging and discharging and long-range ionic interactions, respectively. Depending only on two energy interaction parameters, the model represents satisfactorily the vapor pressure and the mean ionic activity coefficient data of single aqueous solutions containing LiCl, LiBr, LiI, NaCl, NaBr, NaI, KCl, KBr, KI, CsCl, CsBr, CsI, or RbCl. Two methods are presented and contrasted: the salt-specific and the ion-specific approaches. Therefore, the aim of this work is to calculate thermodynamic properties that are extensively used to design, operate and optimize many industrial processes, including water desalination. 256 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2

NOMENCLATURE c(x) Mean ionic activity coefficient based on mole fraction scale List of Symbols c(m) Mean ionic activity coefficient based on molality scale A Helmholtz free energy C MSA shielding parameter D Dielectric constant of solution. e0 Permittivity of vacuum DS Dielectric constant of pure solvent Hma Exponential term in the expression of the most e Elementary charge probable distribution f (T) Function of temperature i j Debye screening parameter m Molality of the salt l Chemical potential M Number of cells of fixed volume m* m a Number of regions of type a in the molecule of M Molar mass of solvent i s type i nc Number of components Na Auxiliary symbol defined in Equation (20) ng Number of regions q Liquid density n Number of moles n” Function defined by Equation (29) N Avogadro’s number 3 a r Ionic diameter Np Number of experimental points t Total number of ionic charge OF Objective function given by Equation (44) / Volume fraction P Pressure W Characteristic constant of the lattice Qa Superficial area of a region of type a ri Volume parameter of component i R Universal gas constant Sm Area fraction of region of type m on a void-free Superscripts basis A Anion T Temperature Born Born contribution to EOS uma Molar interaction energy between regions of types m and a C Cation ma calc Calculated u0 Temperature independent molar interaction energy between regions of types m and a D Dispersive interactions v* Volume occupied by one mol of lattice cells exp Experimental v Molar volume of the solution at an given pres- IGM Ideal Gas Mixture sure and temperature m Molality scale ~v Reduced molar volume MSA Mean Spherical Approximation to EOS MTC Contribution of the Mattedi-Tavares-Castier vs Molar volume of the solution at infinite dilution at an given pressure and temperature EOS V Total volume pure Pure solvent val Any of the three thermodynamic properties R Residual property as defined in Equation (4) x Mole fraction S Solvent x Mole fraction scale xs Mole fraction of solvent Z Number of charges z Lattice coordination number zq Mean number of nearest neighbors Subscripts ± Mean ionic property Anion property Greek letters + Cation property a Quantity defined by Equation (27) 1 Infinite dilution a,b Electron-donor and electron-acceptor, respec- chg Charging process of ion in solution tively disc Discharging process of ion in vacuum c Activity coefficient mix Mixture properties A. Zuber et al. / Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State 257

INTRODUCTION two different equations to account for the cation- hydrated diameter, and correlated and predicted Electrolyte solutions are encountered in many chemical mean ionic activity coefficients insingle and mixed industrial processes such as water desalination, distilla- electrolyte solutions. tion, extraction, solution crystallization, mineral scale Haghtalab and Mazloumi (2009) assert gE models are formation in steam systems, gas scrubbing, hydrometal- easily applicable to engineering design and also have lurgy and biotechnology according to Pitzer (1973), good accuracy for different electrolyte solutions. How- Loehe and Donohue (1997) and Myers et al. (2002). ever, such models are not suitable for calculating solu- So, thermodynamic models for electrolyte systems have tion densities and activity coefficients as functions of been developed for evaluating the properties needed to pressure. Models for electrolyte solutions that express design and simulate processes involving electrolytes. the Helmholtz energy as function of temperature, vol- Compared to non-electrolyte solutions, electrolyte solu- ume and species amounts do not suffer such shortcom- tions are more difficult to model because of the presence ings and therefore, provide a more complete of dissolved ions. Myers et al. (2002) explain these ions description of the systems modeled. can interact with the solvent and with one another by In the case of models based on Helmholtz energy, Wu long-range or short-range electrostatic interactions. and Prausnitz (1998) extended the Peng-Robinson EOS Both interactions are always present in electrolyte solu- using different terms: the SAFT term to account hydro- tions. However, at low electrolyte concentrations, the gen bonding and a term to account the electric charging long-range forces are dominant whereas, at high concen- of ions and subsequent ion-ion and ion-solvent interac- trations, the short-range forces are dominant. Therefore, tions. Liu et al. (1999) proposed an equation that an accurate model applicable to electrolyte solutions includes 6 terms: interactions of ion-ion, ion-dipole, must consider all these interactions. dipole-dipole, Lenard-Jones dispersion, hydrogen bond Several reviews discuss in depth models for electrolyte association accounted by the MSA equation and hard solutions such as the publications of Friedman (1981), sphere repulsion. They investigated 30 aqueous electro- Loehe and Donohue (1997) and Anderko et al. (2002). lyte solutions, correlating mean ionic activity coefficient Generally, these models can be classified in two types using only one adjustable parameter for each salt. Den- of engineering-oriented approaches: excess Gibbs energy sity and mean ionic activity coefficient in mixtures of E E (g ) or Helmholtz energy. Some examples based on g salts were also predicted. Myers et al. (2002) determined expressions are the models of: mean ionic activity coefficients and osmotic coefficients – Debye and Hu¨ ckel (1923), who are pioneers in devel- at 298.15 K for 138 aqueous salt solutions by a three oping models to describe the behavior of electrolyte adjustable parameter equation containing a term based solutions. Their model considers the ions as charged on the Peng-Robinson EOS, a Born term and a MSA hard spheres present in a continuum dielectric med- term. Based on the perturbed-chain statistical associated ium. The ions has fixed diameters values, and the fluid theory, Cameretti et al. (2005) used the equation concentration of the solutions must be lower than named ePC-SAFT, which joins the PC-SAFT and the 0.1 molal (Guggenheim and Turgeon, 1955); Dubye-Hu¨ ckel term, to correlate the vapor pressure – Pitzer (1973), which can predict the behavior of single and the density of 12 salt solutions, considering two solvent solutions; adjustable parameters for each ion. Held et al. (2008) – Chen and Evans (1986), who extended the Non-Random extended the work done by Cameretti et al. correlating Two Liquid model (NRTL) to electrolyte solutions; densities and mean ionic activity coefficients at – Papaiconomou et al. (2002), who used the e-NRTL 298.15 K and predicting vapor pressures. Inchekel equation, combining the NRTL model with the Born et al. (2008) proposed an extension of the Cubic Plus term and the restricted primitive MSA term, in order Association (CPA) equation along with Born and to describe systems formed by electrolytes and multi- MSA terms, in order to calculate the apparent molar vol- ple solvents; ume, mean ionic activity coefficient and osmotic coeffi- – Nasirzadeh et al. (2005), who proposed an equation cient for 10 aqueous electrolyte solutions considering considering the sum of contributions of the Mean three ion-specific parameters. Held et al. (2008) studied Spherical Approximation (MSA) and NRTL, and the behavior of systems containing weak electrolytes, applied it to obtain the vapor-pressure and osmotic correlating densities and mean ionic activity coefficients, coefficients of aqueous solutions of lithium hydroxide; using ion-pairing parameters. Lee and Kim (2009) used – Salimi et al. (2005), who combined the Ghotbi-Vera the equation PC-SAFT along with the primitive MSA Mean Spherical Approximation (GV-MSA) with term and adjusted 4 ion-specific parameters for 258 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2

26 aqueous electrolyte solutions, correlating density and hypothetical ideal gas state at temperature T and vol- mean ionic activity coefficient data at 298.15 K and ume V. In the first step, the charges on all ions are 1 bar. removed. The change in Helmholtz energy is The aim of this work is to present an EOS for electro- accounted by the Born equation for ions in a vacuum, D Born lyte solutions, applicable to calculations of phase equi- Adisc ; librium and thermodynamic properties. This equation – step II. The short-range attractive dispersion and uses the Helmholtz approach, including three terms: repulsive forces due to excluded volume are turned one related to the Mattedi-Tavares-Castier EOS (MTC on. Also, self-association of solvent molecules can EOS) of Mattedi et al. (1998), based on the hole-lattice occur. The MTC EOS is used to calculate the change theory and whose partition function is obtained from in Helmholtz energy for this step, DAMTC; the generalized van der Waals theory, along with the – step III. The ions are recharged. The change in Born and MSA terms. Due to the combination of the Helmholtz energy is accounted by the Born equation D Born hole-lattice theory with long-range interactions typical for ions in a dielectric solvent, Achg ; of electrolyte solutions, it is named electrolattice EOS. – step IV. The long-range interactions among the ions in This model is used to calculate vapor pressures, mean solution are taken into account using the Mean Spher- ionic activity coefficients, densities and osmotic coeffi- ical Approximation (MSA), and the corresponding cients of 13 aqueous solutions of strong electrolytes con- change in the molar Helmholtz free energy is denoted taining salts of type 1:1 (LiCl, LiBr, LiI, NaCl, NaBr, by DAMSA. NaI, KCl, KBr, KI, CsCl, CsBr, CsI and RbCl). Two The residual Helmholtz energy for forming an electro- approaches are presented: one considering ion-specific lyte solution is given by: parameters, whose idea is to use parameters for the ions independently of the aqueous solutions they are inserted; ATðÞ¼; V; n AMTCðÞþT; V; n DABornðÞT; V; n the other considering salt-specific parameters, whose ð1Þ goal is to use ionic parameters dependent on the salt that þ DAMSAðÞT; V; n constitutes the aqueous solution. Both kinds of parameters are analyzed in terms of the accuracy of the calcu- wherein lated thermodynamic properties compared to their experimental data. AMTCðÞ¼T; V; n AIGM ðÞþT; V; n DAMTCðÞðT; V; n 2Þ

D Born ; ; n D Born ; ; n D Born ; ; n 1 ELECTROLATTICE EQUATION OF STATE A ðÞ¼T V Adisc ðÞþT V Achg ðÞT V ð3Þ According to the Helmholtz energy (A) approach, it is possible to develop an EOS applicable to electrolytes fol- Equation (1) can be arranged in order to obtain the lowing the suggestions of Myers et al. (2002). In general, residual Helmholtz energy in the following manner: they consider a path that starts from the reference state and passes through several intermediate states until ATðÞ; V; n AIGM ðÞ¼T; V; n ARðÞT; V; n reaching the final state that corresponds to the electro- MTC Born MSA lyte solution. This procedure allows the formulation of ¼ DA þ DA þ DA a model using a path of constant temperature and vol- ð4Þ ume. The change in Helmholtz energy between the final and initial states is equal to the residual Helmholtz In these equations, T is the temperature of the system, energy computed by adding the Helmholtz energy V is the volume, n is a vector of the number of moles of changes between consecutive states along the path. The the species, AIGM is the Helmholtz energy of the ideal gas transition from each state to the next corresponds to mixture that corresponds to the reference state and AR is adding another type of specific interaction to the system. the residual Helmholtz energy. The term AMTC is the These changes represent ion-solvent and solvent-solvent contribution of the MTC EOS to the Helmholtz energy physical interactions of short range, association or solva- that includes the repulsive and attractive effects, associ- tion and long-range ion-ion interactions. ation and the ideal gas contribution, whereas DAMTC is Santos (2010) developed the electrolattice EOS follow- the contribution of the MTC EOS to the Helmholtz ing a four step thermodynamic path (Fig. 1), as follows: energy without ideal gas contribution. Expressions for – step I. It is assumed that a reference mixture the pressure and chemical potential can be obtained constituted by charged ions and molecules is in a using standard techniques. According to Myers et al. A. Zuber et al. / Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State 259

IONS AND SOLVENT IONS AND SOLVENT MOLECULES MOLECULES (IDEAL GAS)

I. DISCHARGE IONS IV. TURN ON ION-ION ELECTROSTATIC INTERACTIONS Δ Born (T,V, n) Adisc ΔAMSA (T,V, n)

UNCHARGED IONS UNCHARGED IONS IONS AND SOLVENT AND SOLVENT AND SOLVENT MOLECULES (IDEAL GAS) MOLECULES

II. TURN ON SHORT- III. RECHARGE IONS RANGE INTERACTION Δ Born (T,V, n) Achg ΔAMTC (T,V, n)

Figure 1 Path to the formation of an electrolyte solution at constant temperature and volume proposed by Myers et al. (2002).

(2002), it is possible to obtain the following expression lMTC lIGM DlMTC ~v 1 i ¼ i þ i ¼ln f ðTÞr ln for the fugacity coefﬁcient: RT RT RT i i ~v 1 þ q=r 1 DlMTC DlBorn DlMSA x =r ~v / ðT; P; xÞ¼ exp i exp i exp i þ ln i l ln i Z RT RT RT ~v 1 þ q=r i ~v 1 þ q=r ð5Þ P~vvd Wðq=rÞðq r Þ In the next subsections, the terms that form the elec- i þ i i ð7Þ RT ð~v 1 þ q=rÞ trolattice EOS are presented and discussed. Xng ~v 1 þ q=r 1.1 Short-Range Contribution (MTC EOS) þ W maQa ln i ~v 1 q=r Na a¼1 þð Þ In the electrolattice EOS, the short-range interactions Pnc Png Png between species are accounted for by the MTC EOS, ma a mmHma a W xi i Q i Q ri whose Helmholtz energy expression was developed by i¼1 a¼1 m¼1 a Mattedi et al. (1998). This EOS can be used in three forms: r ð~v 1 þðq=rÞN Þ as a molecular model, as a conventional group-contribution model or, as done here, as a region-contribution ma In Equations (6) and (7), i is the number of regions of model, by splitting each molecule in regions. Each molec- type a in a molecule of type i, ng is the total number of ular region interacts differently with the regions of its regions, nc is the number of components, Qa is the sur- neighboring molecules (Santos et al., 2010). The expres- face area of a region of type a, z is the coordination num- sions for the pressure and chemical potential derived from ber (z = 10), W is a lattice constant (W = 1) and m*is this expression for the Helmholtz energy are: m 9 6 3 1 volume of one mol of cells ( *=5 10 m .mol ). PMTC 1 ~v z ðÞ~v 1 þ q=r l The number of external contacts (zq ), the bulkiness ¼ ln þ ln þ i RT v ðÞ~v 1 2v ~v r~vv factor (l ) and the reduced volume (v˜) are: i 1 ~W = Xnc Xng Na 1 : v ðÞq r ma a ln xi Q Xng r~vv ~v 1 þðq=rÞ i ~v 1 þ ðÞq=r Na i¼1 a¼1 ma a 8 zqi ¼ z i Q ð Þ ð6Þ a¼1 260 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2

z to connect the area and volume parameters for the sol- li ¼ ðÞri q ðÞri 1 ð9Þ 2 i vent. v ~v ¼ ð10Þ The average values of r and zq are: r:v riðv –vi Þ Xnc di ¼ ð11Þ r:v r ¼ rixi ð18Þ 1 Pnc i¼ wherein v is the molar volume, v / v with, Xnc ¼ i i 19 i¼1 zq ¼ zqixi ð Þ i¼1 / rixi 12 i ¼ Pnc ð Þ Other symbols present in previous equations are rjxj deﬁned as follows: j¼1 Pnc m m Xng vi xiQ 1 Na ¼ i¼ Hma ð20Þ The ions are assumed to be spherical and their bulki- q ness factors are calculated accordingly. We follow m¼1 Xnc Xng Abrams and Prausnitz (1975) in deﬁning the volume ma a 21 q ¼ xi i Q ð Þ (r) and surface area (q) parameters as ratios: i¼1 a¼1 ma Hma exp u 22 3 3 ¼ ð Þ V 4p R R RT r ¼ i ¼ 3 i ¼ i ð13Þ i V 4p 3 R 3 R ma 2 where u is the interaction energy between the m and A 4p R2 R q ¼ i ¼ i ¼ i ð14Þ the a regions, given as: i 4p 2 A R R ma ma ma u u0 B ¼ 1 þ ð23Þ R R T wherein Ri is the radius of a spherical ion i, Vi is its volume and Ai is its surface area. The corresponding sym- The chemical potential of component i given by Equa- bols with an asterisk denote the same properties for a tion (7) is the total contribution, including the ideal gas lattice cell, assumed to be spherical in Equations (13) contribution, which is equal to: and (14). It follows from these equations that, for a IGM l ðT; V; xÞ RT di spherical ion: i ¼ln f ðTÞln ð24Þ RT i v r 2=3 15 qi ¼ ri ð Þ which is consistent with the MTC model. The difference between Equations (7) and (24) is the residual chemical The bulkiness factor is obtained by substituting Equa- potential of component i in the MTC model with only tion (15) in Equation (9): repulsive and attractive forces contributions included DlMTC= ð i RTÞ. l z 2=3 1 16 i ¼ 2 ri ri ðÞri ðÞ

For determining the volume parameter, the ratio 1.2 Born Contribution between the ion volume and the cell volume is used as In the electrolattice EOS, the Born contribution term follows: accounts for the variation in the Helmholtz energy for discharging an ion in vacuum (ideal gas) and charging p N a 3 ri ¼ r ð17Þ it in a solvent. In it, the ions are considered as hard 6 v i spheres immersed in a continuum with uniform in which ri is the diameter of the ion and Na is the dielectric constant. The Born contribution term to the Avogadro’s number. Therefore, knowing the ion diame- pressure is: ter, it is possible to calculate the volume parameter, the DPBorn 1 N e2 oD X n Z2 bulkiness factor and the surface area parameter by ¼ a i i ð25Þ 4p e0 o r applying these three equations. RT D DRT V T;n i i Equations (13) to (17) are exclusively for calculating ion parameters, since the spherical shape is imposed. in which e is the unit of elementary charge, e0 is the per- For the solvent molecule, no physical structure is mittivity of vacuum, D is the relative dielectric constant assumed. Thus, no mathematical relation is proposed of the solution and Zi is the number of charge of species i. A. Zuber et al. / Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State 261

The chemical potential of component i due to the Born wherein, model is: P 2 Born 2 2 niriZ Dl ðT; V; xÞ e N a Z 1 i i ¼ i 1 r iP 32 4pe r mix ¼ 2 ð Þ RT 0RT i D n Z 2 X Z2 i i oD a nj j i ð26Þ 2 2 X on ; ; 4pD r 2 e N 2 i T V ni6¼j j j j a Z 33 ¼ ni i ð Þ e0 DRTV i ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which: 1 2r j 1 2 C þ mix 34 2 e N a ¼ ð Þ a ¼ ð27Þ 2rmix e0DRT in which D is the dielectric constant presented in Equa- The Born term depends on the relative dielectric con- tion (28). In these equations, j is the Debye screening stant (D). We use the Pottel model for electrolyte solu- parameter and U is the MSA shielding parameter. The tions, which is presented by Zuo and Fu¨ rst (1997) as: "# contribution of this MSA term to the chemical potential 1 n00 is: 1 1 3 28 D ¼ þðDS Þ 00 ð Þ 1 þðn3=2Þ DlMSA ; ; x 2 oC 3 i ðT V Þ C2 1 r C wherein: ¼p V o þ 2 mix 3 RT ni ; ; p Xions r T V ni6¼j 00 N a niðÞi "#() n3 ¼ ð29Þ 6 V 2C3 3 or oC i C mix r 3p 2 V o þ mix o N a ni T;V;n ni T;V;n Here, D is the pure solvent dielectric constant. To eval- i6¼j i6¼j s 35 uate it for pure water, we used the following relation, ð Þ reported by Zuo and Fu¨ rst (1997), for temperatures in Kelvin: 2:98145 104 2 THERMODYNAMIC PROPERTIES 2 Ds ¼19:2905 þ 1:9678 10 T T The relation between the fugacity coefficient and the 4 2 7 3 þ 1:3189 10 T 3:0 10 T activity coefficient for an ion of species i and for solvent ð30Þ s can be expressed, respectively, as:

1.3 Long-Range Electrostatic Contribution – Mean u ; ; x cðxÞ ; ; x iðT P Þ 36 Spherical Approximation (MSA) i ðT P Þ¼u ; ; 1 ð Þ iðT P xs ! Þ Many published models try to describe the long-range elec- u ; ; x c ; ; x sðT P Þ 37 trostatic contributions of charged species in solution. The sðT P Þ¼ u ; ð Þ sðT PÞ MSA has been extensively used to describe such interac- cðÞx ; ; x tions, with different levels of approximation. Instead of wherein i ðÞT P is the unsymmetrical activity coeffi- solving implicit forms of the MSA iteratively, we opted cient of ion i on the mole fraction scale; ui(T, P, x) is the for the explicit approximation with single effective ion fugacity coefficient of ion I in the solution; ui(T, P, xs ? 1) diameter of Harvey et al. (1988), with diameter values is the fugacity coefficient of ion i at infinite dilution; for cations and anions obtained by Marcus (1988). This us(T, P, x) is the fugacity coefficient of solvent s in the form of MSA was successfully used by Myers et al. solution and us(T, P) is the fugacity coefficient of pure (2002) along with the Peng-Robinson EOS for obtaining solvent s. mean ionic activity coefficient of many salt solutions. The activity coefficient of ion i on the molality scale is The contribution of this MSA term to the pressure is: given by:

DPMSA 2RT ¼ 3p ðmÞ / ðT; P; xÞ RT "#N a c ðT; P; xÞ¼x i i s / T; P; x 1 2 oC 3 3 ið s ! Þ 3C V fg2Cr þ 1 þ C 1 þ r C oV mix 2 mix 1 / ; ; x T;n iðT P Þ 38 ¼ 1 / ; ; 1 ð Þ ð31Þ þ vmM s iðT P xs ! Þ 262 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2

wherein xs is the molar fraction of the solvent; v is the supposed to follow Equation (23) and there would be sum of stoichiometric coefficients of the ions in the salt 18 adjustable parameters in such a solution: 9 parame- ma ma molecules, in this work, v = 2 since the employed salts ters (u0 /R) and 9 parameters B . This is inconvenient are 1:1; m is the molality; and Ms is the molecular weight in an engineering-oriented model and we adopted simpli- of the solvent in kg.mol1. Expressions for the mean fications to reduce the number of adjustable parameters. ionic activity coefficient and for the osmotic coefficient We assumed the interactions between water regions and are available in several references (e.g., Myers et al., each charged species are equal, i.e.: 2002).

aC bC DC solventC u0 u u0 u0 ¼ 0 ¼ ¼ ð39Þ 3 PARAMETER FITTING R R R R aA bA DA solventA u0 u u0 u0 Solutions containing aqueous single strong 1:1 electro- ¼ 0 ¼ ¼ ð40Þ lytes are formed by three species: solvent, cation (C) R R R R and anion (A). Since the MTC EOS term is present These interactions are assumed to be temperature in the electrolattice EOS, according to Ehlker and independent; therefore, the Bma term of Equation (23) Pfenning (2002) and Mattedi et al. (1998), it is conve- is neglected as follows: nient to split the solvent molecule in three regions: an electron-donor (a), an electron-acceptor (b) and a dis- aC bC DC aA bA DA persion region (D). B ¼ B ¼ B ¼ B ¼ B ¼ B ¼ 0 ð41Þ Once the aim of this work is to determine thermodynamic properties of electrolyte solutions, the first step Two assumptions are made about the interactions toward this goal is to obtain the parameters for pure between ionic species. Following Zuo et al. (2000), the water. As in the MTC EOS, the electrolattice EOS uses short-range interaction between ions with the same seven parameters to represent pure polar solvents. These charge is neglected once the repulsion between the spe- parameters were obtained by simultaneous fit of liquid cies is strong, therefore: density data and pressure at saturation, obtained from the DIPPR correlation (American Institute of Chemical uAA uCC Engineers, 2007), in the temperature range from 292.84 ¼ ¼ 0 ð42Þ R R to 489.67 K. To reduce the number of adjustable solvent parameters, all region-region interactions, except that Also, according to Lee and Kim (2009), it is possible a b between and regions, are assumed to be dispersion to neglect the short-range interactions between ions of D–D interactions, numerically equal (u0 /R). These interac- different charges in solution, i.e.: tions are assumed to be temperature dependent, with a-a a-D b-b b-D D-D B = B = B = B = B . Also for reducing uAC uCA ¼ ¼ 0 ð43Þ the number of adjustable parameters, the interaction R R between a and b regions, which refers to the hydrogen bond, is considered to be temperature independent With these simplifications, each ion has only one a b (B - = 0). Table 1 shows the parameters for water used adjustable parameter and yet, the electrolattice EOS in this paper. gives good results. When an electrolyte solution of a single salt and a sol- To calculate the vapor pressure of electrolyte solu- vent is modeled by the electrolattice EOS, there are five tions, it is assumed the vapor phase only contains water. regions that interact with one another (three for the sol- Also, all the salts are strong electrolytes, which fully dis- vent (a, b,D), one for the cation (C) and one for the sociate in solution into cations and anions. Then, for anion (A)). Each of these interactions is, in principle, each data point, there is only one phase equilibrium

TABLE 1 Water parameters

DD ab Compound QD Qa Qb r u0 BDD (K) u0 R (K) R (K) Water 0.857715 0.712207 0.172331 2.14949 535.864 700.671 2 625.69 A. Zuber et al. / Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State 263 equation, the isofugacity equation for water, which is 4 RESULTS AND DISCUSSION solved to find the vapor pressure at a given temperature. In this work, the experimental data set for vapor pres- The volume, the bulkiness and the surface area parame- sure is composed by: Hubert et al. (1995), who obtained ters for ions are calculated using Equations (15-17). data for NaCl in a temperature range from 293.15 to Marcus (1988) determined the ionic diameter of several 363.15 K; for LiCl, LiBr and LiI, the work of Patil et al. cations and anions in aqueous solutions, which are used (1990) was used and, for other salts, the work of Patil instead of Pauling-type crystal ionic diameters (Pauling, et al. (1991) was used, both in the temperature range from 1927). The diameter values used in our model are pre- 303.15 K to 343.15 K. For mean ionic activity coefficients sented in Table 2. and osmotic coefficients, the data were obtained from The parameters fitted by the salt-specific and ion- Robinson and Stokes (1949) and Lobo and Quaresma specific approaches are shown in Tables 3 and 4, (1989) at atmospheric pressure and 298.15 K. Considering the parameters obtained for the pure sol- TABLE 2 vent and the set of simplifications for the ions, the num- Ionic diameters according to Marcus (1988) ber of adjustable parameters in a single-salt aqueous r ˚ solventC Ion i (A) solution is reduced to two: ðu0 =RÞ and solventA + ðu0 =RÞ. Different approaches are used to deter- Li 1.42 mine these energy interactions: the first considers the Na+ 1.94 parameters as salt-specific, in which, these two parameters are simultaneously obtained for each salt; the second K+ 2.82 approach considers the parameters as ion-specific, which Rb+ 3.00 means that the energy interaction between water and an + ion must always be the same, independently of the salt it Cs 3.46 takes part. The ion-specific approach is more complex Cl 3.60 compared to the salt-specific approach since more parameters need to be fitted at the same time. Following Br 3.96 Held et al. (2008), we first determined the parameters for I 4.50 a basic set of salts, composed by three cations (Na+, K+,Li+) and three anions (Br,Cl,I), corresponding to nine single-salt aqueous solutions. In this initial fitting, 6 parameters were simultaneously obtained using TABLE 3 368 data points for vapor pressure and 208 data points Salt-specific parameters solventC solventA for mean ionic activity coefficient. Then, the anion Salt u0 u0 (K) (K) parameters obtained previously were fixed to determine R R + the Cs parameters, using experimental data of CsBr, LiCl 2 787.85 1 808.60 CsCl and CsI aqueous solutions. Likewise, the Rb+ LiBr 2 731.22 1 926.85 parameters were determined by correlating the experimental data of RbCl. LiI 2 725.82 1 799.22 For both approaches previously mentioned, the NaCl 808.08 2 110.77 adjustable parameters were determined using the Excess Gibbs Energy Models and Equations of State (XSEOS) NaBr 2 482.99 988.07 Ò package, an Excel add-in developed by Castier and NaI 2 448.56 1 416.51 Amer (2011), by minimizing the Objective Function (OF): KCl 612.10 613.18

KBr 97.54 1 189.34 XNp exp 2 XNp exp 2 Pcalc P ccalc c OF ¼ i i þ i i ð44Þ KI 482.85 459.96 calc ccalc i 1 Pi i 1 i ¼ ¼ CsCl 2 134.57 3 635.93

calc ccalc CsBr 2 171.67 5 990.57 in which Pi and i denote calculated values for vapor pressure and mean ionic activity coefficient, respectively; CsI 2 233.42 4 487.87 Pexp and cexp are the corresponding experimental values; i i RbCl 1 916.47 142.84 Np is the number of points. 264 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2 respectively. In Table 5, the values for the Average Rela- for vapor pressure, mean ionic activity coefficient and tive Deviation (ARD(%)), calculated by Equation (45), osmotic coefficient of each salt solution are presented. These outcomes are shown in Table 5 for both fitting approaches, as SSP for salt-specific parameters, and TABLE 4 ISP for ion-specific parameters. Ion-specific parameters solvention exp Ion u0 100 XNp calc R (K) vali vali ARDðÞ¼% exp ð45Þ Np val Li+ 2831.67 i¼1 i

Na+ 2448.74 calc exp In Equation (45), vali and vali represent the calcu- K+ 24.74 lated and experimental values of vapor pressure, mean ionic activity coefficient or osmotic coefficient. Table 5 + Rb 27.14 shows the electrolattice EOS with salt-specific and ion- Cs+ 626.51 specific parameters can correlate vapor pressure satisfactorily, with average of ARD values equal to 1.74% and Cl 1461.32 1.84%, respectively. A comparison of our results with Br 1488.42 those obtained by Held et al. (2008) for the same salts shows that their average ARD for vapor pressure is I 1321.96 4.48%.

TABLE 5 Average Relative Deviation (ARD(%)) of 1:1 salt thermodynamic properties considering two parameter ﬁtting approaches

Salt Vapor pressure ARD(%) Mean ionic activity coefﬁcient ARD(%) Osmotic coefﬁcient ARD(%)

Np mmax SSP ISP Np mmax SSP ISP Np mmax SSP ISP (mol/kg) (mol/kg) (mol/kg)

LiCl 20(2) 7.8 2.76 2.72 22(4) 6.0 4.56 4.44 22(5) 6.0 2.97 2.92

LiBr 20(2) 7.0 2.29 2.39 23(4) 6.0 6.98 6.92 23(5) 6.0 4.03 4.02

LiI 25(2) 7.5 2.12 2.12 17(4) 3.0 2.49 3.51 17(5) 3.0 1.67 2.30

NaCl 133(1) 5.4 2.16 1.97 35(4) 6.0 1.70 1.77 35(5) 6.0 1.29 1.37

NaBr 30(3) 8.0 1.71 2.07 19(4) 4.0 1.24 1.62 19(5) 4.0 0.80 1.12

NaI 40(3) 8.4 1.48 1.33 31(4) 8.0 2.32 2.24 31(5) 8.0 1.50 1.44

KCl 30(3) 4.3 1.72 1.51 20(4) 4.5 0.56 3.49 20(5) 4.5 0.42 1.54

KBr 30(3) 4.3 1.02 1.04 21(4) 5.0 0.49 0.53 21(5) 5.0 0.41 0.36

KI 40(3) 5.6 1.62 2.15 20(4) 4.5 0.43 2.56 20(5) 4.5 0.31 1.52

CsCl 40(3) 8.6 1.21 3.19 21(4) 5.0 2.10 5.83 21(5) 5.0 1.40 4.73

CsBr 30(3) 5.9 2.16 1.11 21(4) 5.0 1.83 3.24 21(5) 5.0 1.29 2.24

CsI 25(3) 2.6 1.20 1.21 17(4) 3.0 0.62 10.88 17(5) 3.0 0.62 4.57

RbCl 30(3) 6.5 1.21 1.13 21(4) 5.0 1.44 1.23 21(5) 5.0 1.01 0.87

Average - - 1.74 1.84 - - 2.06 3.71 - - 1.36 2.23

(1) Hubert et al. (1995) in temperature range 293.15-363.15 K. (2) Patil et al. (1991) in temperature range 303.15-343.15 K. (3) Patil et al. (1990) in temperature range 303.15-343.15 K. (4) Robinson and Stokes (1949) at temperature 298.15 K. (5) Lobo and Quaresma (1989) at temperature 298.15 K. SSP: Salt-Speciﬁc Parameters. ISP: Ion-Speciﬁc Parameters. A. Zuber et al. / Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State 265

35 000 1.0

30 000 0.9

0.8 25 000

0.7 20 000 0.6 15 000

Pressure (Pa) 0.5 10 000 0.4 5 000 Mean ionic activity coefficient 0.3 0 01234 56 01234567 Molality (mol/kg) Molality (mol/kg)

Figure 4 Figure 2 Mean ionic activity coefficient of NaCl (h), KCl (D) and Vapor pressure of KI in water at 303.15 K (h), 313.15 K CsCl (s) in water at 1 atm and 298.15 K, obtained from (D), 323.15 K (s), 333.15 K (9) and 343.15 K (e), obtained Lobo and Quaresma (1989). The lines represent the electro- by Patil et al. (1991). The full line (—) represents the elec- lattice model considering salt-specific parameters () trolattice model considering ion-specific parameters. and ion-specific parameters (—).

35 000

30 000 experimental data more adequately than the ion-specific parameters, except for RbCl. The average ARD for such 25 000 approaches are 2.06 % and 3.71 %, respectively. The 20 000 results obtained for LiBr and CsI are higher compared to other salts modeled by ion-specific parameters, and 15 000 their ARD values are 6.92% and 10.88 %, respectively. Pressure (Pa) 10 000 In Figure 4, it is possible to observe that the mean ionic activity coefficient is underestimated for KCl and CsCl 5 000 at 298.15 K and at atmospheric pressure, when ion-spe-

0 cific parameters are used. Also in Figure 4, it is possible 12345to observe that for salts whose anion is Cl, the larger is Molality (mol/kg) the diameter of the cation, the lower is the mean ionic activity coefficient in all the molality range. In Figure 5, Figure 3 salts containing Br as the anion are presented. Also in Vapor pressure of KBr in water at 303.15 K (h), 313.15 K this set of salts, the mean ionic activity coefficient (D), 323.15 K (s), 333.15 K (9) and 343.15 K (e), obtained decreases as the cation size increases, in the sequence by Patil et al. (1991). The full line (—) represents the elec- NaBr > KBr > CsBr. Figure 5 also shows that the electrolattice model considering ion-specific parameters. trolattice EOS can match the experimental mean ionic activity coefficient satisfactorily with the parameters fitted by both approaches. The osmotic coefficient data of aqueous solutions, Figures 2 and 3 present the vapor pressures of aque- using the ion-specific and the salt-specific methods, are ous solutions containing KI and KBr, respectively, as presented in Figure 6. The average ARD is 1.36% for function of molality in the temperature range from salt-specific approach and 2.23% for ion-specific 303.15 K to 343.15 K, using ion-specific parameters. approach. This thermodynamic property was not The results with the salt-specific parameters, not shown directly used for parameter fitting but the mean ionic in these figures, are similar. With both fitting activity coefficient is analytically related to the osmotic approaches, the results of the model are in good agree- coefficient via the Gibbs-Duhem equation (a complete ment with the experimental data. formulation about this relation can be found in For the mean ionic activity coefficient data (Tab. 5), Prausnitz et al., 1999). For aqueous solutions containing in general, the salt-specific parameters correlate the NaBr, KBr and RbCl, results using parameters fitted 266 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2 with the salt-specific and ion-specific approaches are in It is not possible to observe a trend in the salt-specific good agreement with experimental data. For RbCl at parameters of Table 3. Generally, all the parameters high concentrations, the parameters adjusted by both have negative values, except the ones obtained for the approaches cause underestimation of the osmotic coeffi- salts whose cation is Cs+, in which the energy interac- cients compared to the experimental data. tion between the solvent and the anion is positive.

1.0 1.25 1.20 0.9 1.15 0.8 1.10 0.7 1.05 0.6 1.00 0.5

Osmotic coefficient 0.95

Mean ionic activity coefficient 0.4 0.90

0.3 0.85 0123456 0123456 Molality (mol/kg) Molality (mol/kg)

Figure 5 Figure 6 h D s Mean ionic activity coefficient of NaBr (h), KBr (D) and Osmotic coefficient of NaBr ( ), KBr ( ) and RbCl ( )in CsBr (s) in water at 1 atm and 298.15 K, obtained from water at 1 atm and 298.15 K, obtained from Lobo and Lobo and Quaresma (1989). The lines represent the electro- Quaresma (1989). The lines represent the electrolattice lattice model considering salt-specific parameters () model considering salt-specific parameters () and ion- and ion-specific parameters (—). specific parameters (—).

TABLE 6 Average Relative Deviation (ARD (%)) of predicted density for 1:1 salts considering the parameters of ion-speciﬁc approach

(1) Salt Np mmax (mol/kg) T range (K) Density ARD(%)

LiCl 23 13.80 298.15-373.15 7.43

LiBr 16 13.18 298.15-343.34 7.52

LiI 15 8.37 298.15-343.34 7.14

NaCl 45 0.96 298.15-328.15 2.16

NaBr 37 8.00 298.15-328.15 4.06

NaI 20 11.29 298.15-313.15 6.00

KCl 28 3.61 298.15-423.15 3.92

KBr 37 1.20 298.15-368.15 4.06

KI 36 1.00 298.15-368.15 1.63

CsCl 27 8.00 298.15-328.15 6.84

CsBr 18 4.18 298.15-343.34 8.36

CsI 15 2.90 298.15-343.34 5.84

RbCl 25 6.00 298.15-328.15 5.16

(1) Experimental data obtained from Lobo and Quaresma (1989). A. Zuber et al. / Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State 267

On the other hand, the values of ion-specific parameters for parameter fitting. Their inclusion in the objective of Table 4 tend to increase with cation diameter. Potas- function might have improved the correlation of density sium and rubidium do not follow this trend but have data. However, it is important to note that even predict- similar parameters, inside the range set by the sodium ing such results, the electrolattice EOS can describe the and cesium parameters. For the anion parameters, no correct tendency for the density in all the electrolyte trend appears in Table 4. The interaction parameter solutions. for cesium in Table 4 is positive, suggesting short range For the predictions of mean ionic activity and osmotic repulsion. coefficients at different temperatures, whose results are With the ion-specific parameters of Table 4, the elec- presented in Table 7, 267 and 192 data points, respec- trolattice EOS was also used to predict density, mean tively, were obtained from Lobo and Quaresma (1989). ionic activity coefficient and osmotic coefficient at tem- The average of the ARD values is 4.64% for mean ionic peratures other than 298.15 K and vapor pressures of activity coefficient and 3.73% for osmotic coefficient. binary salt solutions. The best prediction was obtained for NaBr with the Density values were predicted for 13 salts considering ARD values of 2.10% at 273.15 K and 2.45% at 342 experimental data points, obtained from Lobo and 313.15 K. For other salts, such as NaCl, KCl and KBr, Quaresma (1989). The results are presented in Table 6. the higher the temperature, the larger are the ARD val- The lowest values for ARD were obtained for KI, NaCl ues. For LiCl and CsCl, at any temperature the deviation and KCl, as 1.63%, 2.16% and 3.92%, respectively. obtained is higher than 5.0%, indicating that the model Compared to Cameretti et al. (2005) and Held et al. cannot predict the thermodynamic property accurately (2008), our results for density are not as accurate as for these solutions. For the osmotic coefficient all the theirs, which are generally lower than 1.0%. A possible ARD values are below 5.0%, except for CsCl, whose reason is that, unlike them, we did not use density data ARD is 7.66% at 323.15 K. For this thermodynamic

TABLE 7 Average Relative Deviation (ARD (%)) of predicted mean ionic activity and osmotic coefﬁcients at different temperatures considering ion-speciﬁc parameters

(A) Salt Np mmax (mol/kg) T (K) Mean ionic activity Osmotic coefﬁcient coefﬁcient ARD(%) ARD(%)

LiCl 22 6.0 273.15 11.06 4.55

22 6.0 323.15 8.81 3.74

LiBr 17 1.1 283.15 2.95 NA

NaCl 23 6.0 273.15 2.27 2.01

23 6.0 323.15 5.88 4.56

NaBr 12 4.0 273.15 2.10 NA

12 4.0 313.15 2.45 NA

KCl 22 6.0 273.15 2.03 1.37

17 1.1 283.15 0.46 NA

22 6.0 323.15 5.17 3.62

KBr 17 1.1 283.15 1.28 NA

14 4.0 333.15 4.62 1.90

CsCl 22 6.0 273.15 5.09 4.15

22 6.0 323.15 10.79 7.66

Average - - - 4.64 3.73

(A) The number of experimental data points is the same for both mean ionic activity and osmotic coefﬁcients, except when the data for osmotic coefﬁcient is Not Available (NA). All experimental data were obtained from Lobo and Quaresma (1989). 268 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2

TABLE 8 Prediction of vapor pressure of mixtures containing two salts using the ion-speciﬁc parameters

Salts Np T range (K) Vapor pressure ARD(%)

NaCl + KCl 10(1) 278.15-323.15 3.96

KBr + KCl 10(1) 278.15-323.15 1.93

KI + KCl 10(1) 278.15-323.15 5.06

NaBr + KCl 25(2) 303.15-333.15 0.78

NaCl + KBr 25(2) 303.15-333.15 0.74

(1) Apelblat and Korin (2009); (2) Hsu et al. (2003).

20 000 10 000 18 000 9 000 8 000 16 000 7 000 14 000

6 000 12 000 5 000 10 000 4 000

Pressure (Pa) 8 000 3 000 Vapor pressure (Pa) 2 000 6 000 1 000 4 000

0 2 000 275 285 295 305 315 325 012345 Temperature (K) Molality (mol/kg)

Figure 7 Figure 8 Vapor pressure prediction of saturated solutions of KCl + Vapor pressure prediction of NaBr + KCl in water at KBr (h), NaCl + KCl (D) and KCl + KI (s) in water, 303.15 K (h), 308.15 K (D), 313.15 K (s), 318.25 K (9), obtained from Apelblat and Korin (2009). The line repre- 323.15 K (e), 328.15 K (*) and 333.15 K (+), obtained sents the electrolattice model considering ion-speciﬁc from Hsu et al. (2003). The line represents the electrolattice parameters (—). model considering ion-speciﬁc parameters (—).

property, the prediction is acceptable in the temperature the predictions are very good in the temperature range range of each salt. from 303.15 to 333.15 K. It should be noted that no param- Predictions of vapor pressure considering binary salt eter fitting is needed when applying the electrolattice EOS solutions were also analyzed. For solutions composed by to predict vapor pressures of aqueous solutions of multiple NaCl + KCl, KBr + KCl and KI + KCl the experimental salts. data were from Apelblat and Korin (2009), obtained at saturation. For NaBr + KCl and NaCl + KBr, the data were obtained from Hsu et al. (2003) at a molality range from 1 CONCLUSIONS to 4 molal and temperature range from 303.15 K to 333.15 K. Table 8 sums up the results for such predictions In this work, the electrolattice equation of state was used and Figures 7-9 display the corresponding plots. For the to model the thermodynamic properties of aqueous solu- three first solutions, the ARD values are slightly higher tions of strong 1:1 electrolytes. The ion volume and sur- than for the other two since the molality is high throughout face area parameters used in this EOS were evaluated the temperature range and the model tends to be not so from ionic diameter data. Experimental data for vapor accurate at such condition. For the other two solutions, pressure and mean ionic activity coefficients were used A. Zuber et al. / Thermodynamic Properties of 1:1 Salt Aqueous Solutions with the Electrolattice Equation of State 269

20 000 and motivate additional investigations of this model’s

18 000 performance in future work.

16 000

14 000 ACKNOWLEDGMENTS 12 000

10 000 The participation of R. Mathew in this work was made possible by UREP grant # 10-124-2-039 from the Qatar Pressure (Pa) 8 000 National Research Fund (a member of Qatar Founda- 6 000 tion). The statements made herein are solely the respon- 4 000 sibility of the authors.

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