Aqueous Salt Solutions


Extended UNIQUAC model

Sander et al. (ref. 1) presented in 1986 an extension of the UNIQUAC model by adding a Debye-Hückel term allowing this Extended UNIQUAC model to be used for electrolyte solutions. The model has since been modified and it has proven itself applicable for calculations of vapor-liquid-liquid-solid equilibria and of thermal properties in aqueous solutions containing electrolytes and non-electrolytes. The model is shown in its current form here as it is presented by Thomsen (1997) (ref 2). The extended UNIQUAC model consists of three terms: a combinatorial or entropic term, a residual or enthalpic term and an electrostatic term

Excess Gibbs energy function

The combinatorial and the residual terms are identical to the terms used in the traditional UNIQUAC equation (ref 3 and 4). The electrostatic term corresponds to the extended Debye-Hückel law. The combinatorial, entropic term is independent of temperature and only depends on the relative sizes of the species:

Debye-Hückel law

z = 10 is the co-ordination number. xi is the mole fraction,  is the volume fraction, and θi is the surface area fraction of component i:

The residual or enthalpic term in the UNIQUAC equation

The two model parameters ri and qi are the volume and surface area parameters for component i. In the classical application of the UNIQUAC model, these parameters are calculated from the properties of non electrolyte molecules (ref 3). In the Extended UNIQUAC application to multi component electrolyte solutions, this approach gave unsatisfactory results. The volume and surface area parameters were instead considered to be adjustable parameters. The values of these two parameters are determined by fitting to experimental data. Especially thermal property data such as heat of dilution and heat capacity data are efficient for determining the value of the surface area parameter q, because the UNIQUAC contribution to the excess enthalpy and excess heat capacity is proportional to the parameter q. The residual, enthalpic term is dependent on temperature through the parameter ψji:

The residual or enthalpic part of the UNIQUAC equation

the parameter ψji is given by:

UNIQUAC interaction energy parameter

uji and uii are interaction energy parameters. The interaction energy parameters are considered symmetrical and temperature dependent in this model  

UNIQUAC interaction parameters

The values of the interaction energy parameters  and  are determined by fitting to experimental data. By partial molar differentiation of the combinatorial and the residual UNIQUAC terms, the combinatorial and the residual parts of the rational, symmetrical activity coefficients are obtained:

combinatorial and residual parts of activity coefficients

The infinite dilution terms are obtained by setting xw = 1 in the above equation:  

Infinite dilution activity coefficients in UNIQUAC equation

The combinatorial and the residual terms of the UNIQUAC excess Gibbs energy function are based on the rational, symmetrical activity coefficient convention. The Debye-Hückel electrostatic term however is expressed in terms of the rational, symmetrical convention for water, and the rational, unsymmetrical convention for ions.

The electrostatic contributions to the water activity coefficients and the ionic activity coefficients are obtained by partial molar differentiation of the extended Debye-Hückel law excess Gibbs energy term. The term used for water is

Extended Debye-Hückel term for water activity coefficient

In this expression, b = 1.5 (kg/mol)½. The term used for ions is:   

Debye Hücckel expression of ionic activity coefficient

Based on table values of the density of pure water, and the relative permittivity of water, εr, the Debye-Hückel parameter A can be approximated in the temperature range 273.15 K < T < 383.15 K by:

Expression for Debye-Hückel parameter A

The activity coefficient for water is calculated in the Extended UNIQUAC model by summation of the three terms:

water activity coefficient

The activity coefficient for ion i is obtained as the rational, unsymmetrical activity coefficient according to the definition of rational unsymmetrical activity coefficients by adding the three contributions:

rational, ionic activity coefficient

The rational, unsymmetrical activity coefficient for ions calculated with the Extended UNIQUAC model can be converted to a molal activity coefficient by use of its definition which can be seen here.  This is relevant for comparison with experimental data.

The temperature dependency of the activity coefficients in the Extended UNIQUAC model is built into the model equations as outlined above. The temperature dependency of the equilibrium constants used in the Extended UNIQUAC model is calculated from the temperature dependency of the Gibbs energies of formation of the species Parameters for water and for the following ions can be found in Thomsen (1997) (ref 2) H+, Na+, K+, NH4+, Cl-, SO42-, HSO4-, NO3-, OH-, CO32-, HCO3-, S2O82-. Parameters and model modifications for gas solubility at pressures up to 100 bar in aqueous electrolyte systems have later been published (refs 5 and 6). Also phase equilibria for systems containing non-electrolytes are described by the model, including liquid-liquid equilibria (refs 7 and 8). Besides, parameters have been determined for systems containing heavy metal ions (ref 9).

The model has been applied by A.V. Garcia to include the pressure dependence of the solubility of salts. Two parameters for the pressure dependence of the solubility product of each salt were introduced in order to achieve this (refs 10 and 11). A significant advantage of the Extended UNIQUAC model compared to models like the Bromley model or the Pitzer model is that temperature dependence is built into the model. This enables the model to also describe thermodynamic properties that are temperature derivatives of the excess Gibbs function, such as heat of mixing and heat capacity. 

  1. B. Sander; P. Rasmussen and Aa. Fredenslund, “Calculation of Solid-Liquid Equilibria in Aqueous Solutions of Nitrate Salts Using an Extended UNIQUAC Equation”. Chemical Engineering Science, 41(1986)1197-1202
  2. Thomsen, K., Aqueous electrolytes: model parameters and process simulation, Ph.D. Thesis, Department of Chemical Engineering, Technical University of Denmark, 1997.
  3. Abrams D.S. and Prausnitz J.M., “Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems”, AIChE journal 21(1975)116-128
  4. Maurer G., Prausnitz J.M., ”On the derivation and extension of the UNIQUAC equation”, Fluid Phase Equilibria, 2(1978)91-99
  5. Thomsen K and Rasmussen P, “Modeling of Vapor-liquid-solid equilibrium in gas-aqueous electrolyte systems” Chemical Engineering Science 54(1999)1787-1802
  6. Pereda S, Thomsen K, Rasmussen P, “Vapor-Liquid-Solid Equilibria of Sulfur Dioxide in Aqueous Electrolyte Solutions” Chemical Engineering Science 55(2000)2663-2671
  7. Iliuta MC, Thomsen K, Rasmussen P, “Extended UNIQUAC model for correlation and prediction of vapor-liquid-solid equilibrium in aqueous salt systems containing non-electrolytes. Part A. Methanol – Water – Salt systems”  Chemical Engineering Science 55(2000)2673-2686
  8. Thomsen K, Iliuta MC, Rasmussen P, “Extended UNIQUAC model for correlation and prediction of vapor-liquid-liquid-solid equilibria in aqueous salt systems containing non-electrolytes. Part B. Alcohol (Ethanol, Propanols, Butanols) - water - salt systems“ Chemical Engineering Science 59(2004)3631-3647
  9. Iliuta MC, Thomsen K, Rasmussen P, “Modeling of heavy metal salt solubility using the extended UNIQUAC model” AIChE Journal, 48(2002)2664-2689
  10. Garcia AV, Thomsen K, Stenby EH, “Prediction of mineral scale formation in geothermal and oilfield operations using the extended UNIQUAC model Part I. Sulfate scaling minerals” Geothermics 34(2005)61-97
  11. Garcia AV, Thomsen K, Stenby EH, ”Prediction of mineral scale formation in geothermal and oilfield operations using the extended UNIQUAC model Part II. Carbonate scaling minerals”, Geothermics, 35(2006)239-284
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