On the application of nonelectrolyte UNIQUAC-NRF

Fluid Phase Equilibria 417 (2016) 70e76

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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d

On the application of nonelectrolyte UNIQUAC-NRF model for strong aqueous electrolyte solutions Seyed Hossein Mazloumi* Chemical Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 November 2015 Received in revised form 6 January 2016 Accepted 16 February 2016 Available online 18 February 2016

An excess Gibbs free energy model composed of a short range term, expressed by the nonelectrolyte UNIQUAC-NRF model, and a long range contribution based on the Pitzer-Debye-Hückel equation is implemented for strong aqueous electrolyte solutions. The present model is applied for calculations of mean activity coefficient and osmotic coefficient of aqueous electrolyte solutions using two approaches: salt parameter model and ion specific parameter model. In salt parameter scheme, the present model has two adjustable parameters per each aqueous salt that are salt-water and water-salt energy parameters. In ion specific approach, the interaction energy parameters of each species with the other species are obtained. The experimental activity coefficients of aqueous electrolyte solutions are used to determine the energy parameters. The surface and volume parameters of ions are calculated using their radius. The results of the models are in good agreements with the experimental data. The extensions of the two approaches for prediction of solubility and osmotic coefficients of some mixed-salt systems have been carried out. © 2016 Elsevier B.V. All rights reserved.

Keywords: Electrolyte solutions Mean activity coefficient Osmotic coefficient Non-electrolyte UNIQUAC-NRF Mixed salt

1. Introduction Simulation and modeling of various oil, gas, petrochemical, biological and environmental processes cannot be properly carried out except the important role of electrolyte solutions in calculations of properties of the mixtures is taken into account sufficiently. Calculation of thermodynamic properties of electrolyte solutions, however, is not an easy subject [1]. Several attempts have been and will be done to thermodynamically model the electrolyte solutions. The electrolyte models either in excess Gibbs energy from Refs. [2e26] or in Helmholtz energy equations [27e30], are not a simple extension of nonelectrolyte models [1]. Some of the successful models in excess Gibbs energy category are the model of Meissner [2] Bromley [3] Pitzer [4] and those based on local composition theory such as models of Cruze and Renon [5], E-NRTL [6], E-Wilson [9], E-UNIQUAC [12], E-NRTL-NRF [7], E-Wilson-NRF [10], E-UNIQUAC-NRF [13,14]. In these models the excess Gibbs energy is summation of two parts, one for long range contribution and one for short range interaction of species. A version of Debye-Hückel equation is frequently implemented for

* Department of Chemical Engineering, Ferdowsi University of Mashhad, P.O. Box 9177948944, Iran. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fluid.2016.02.029 0378-3812/© 2016 Elsevier B.V. All rights reserved.

long range term. Also the Chen et al. assumptions that are like-ion repulsion and local electro-neutrality are often adopted in derivation of the short range term of electrolyte local composition models. On the other hand, some non-electrolyte models such as UNIQUAC model [19] N-Wilson-NRF [15] and N-NRTL-NRF [16] are applied for electrolyte solutions. In these models the original nonelectrolyte local composition is used as short range term by considering fixed suitable values for like-ion repulsion [19] or by viewing of the salt as a pseudo component [15]. A version of DebyeeHückel model is also implemented for long range interactions in non-electrolyte local composition models. In previous works, nonelectrolyte Wilson-NRF and nonelectrolyte NRTL-NRF have been successfully used to represent the properties of aqueous electrolytes with salt parameters [15,16] and ion specific parameters [23,24]. It should be noted that the salt parameter approach of nonelectrolyte Wilson-NRF and nonelectrolyte NRTL-NRF models [15,16] have been applied for binary strong aqueous electrolytes. The aim of this study is to use the nonelectrolyte UNIQUAC-NRF model [31] for strong aqueous electrolyte solutions. To do so, the Pitzer-Debye-Hückel equation for long range contribution and non-electrolyte UNIQUAC-NRF model [31] for short range term are used. Two scenarios are adopted salt parameter model and ion specific parameter approach. The model

S.H. Mazloumi / Fluid Phase Equilibria 417 (2016) 70e76

parameters in both cases are obtained using correlation of the activity coefficient experimental data of binary strong aqueous electrolyte solutions. In both approaches the surface and volume parameters of the combinatorial part of the model are calculated using ionic radii. The predictions of solubility and osmotic coefficients of several ternary systems have been done by the present model.

The excess Gibbs energy and activity coefficient of an electrolyte solution in local composition models can be expressed as sum of short and long range contributions. For long range contribution, the unsymmetrical PitzerDebyeeHückel [32] activity coefficients of ions and water can be respectively expressed as

1000 Mw

0:5

z2 I 0:5 2Ix1:5 þ ion x 1 þ rIx0:5

ln gLR;w ¼ Af

1000 0:5 Mw

ln

li ¼

(1)

2Ix1:5 1 þ rIx0:5

! (2)

The right hand side of the Eq. (12) is calculated by expansion of Eqs. (3) and (5) for binary system. To calculated the long range contribution, first the long range activity coefficients of anion and cation are calculated using PitzerDebyeeHückel model, Eq. (1), then the long range mean activity coefficient of the electrolyte is obtained using

ln g*1;LR ¼ ln g*±;LR ¼

va ðln ga Þ*LR þ vc ðln gc Þ*LR v

(13)

Finally, the mean activity coefficient of the electrolyte is computed as

(14)

2.2. Ion specific parameter model

(4) n X

qj

!3 n X qk ln tkj 5 Gij ln tij tji þ

j¼1

k¼1

(5) Gji ¼ P

In this approach, using ion pair assumption [15], the salt, e.g. NaCl, is considered as component 1 and water as component 2. Thus, the unsymmetrical short range mean activity coefficient of an electrolyte is given by

(3)

Z ðr qi Þ ðri 1Þ 2 i

ln gRi;SR ¼ qi 41 þ ln Gii

(11)

2.1. Salt parameter model

ln g*± ¼ ln g*1 ¼ ln g*±;SR þ ln g*±;LR

n f Z q f X ¼ ln i þ qi ln i þ li i xl xi 2 fi xi j¼1 j j

2

Where ∞ and PDH denote the infinite dilution state (i.e. x1 approaches to zero) and Pitzer-DebyeeHückel, respectively. The expression for activity coefficient of water is as

(12)

P Where Af, Mw, r and Ix (¼0.5 xionz2ion) are DebyeeHückel constants, molecular weight of solvent (kg mol1), the closest approach and ionic strength, respectively. z is the charge number of ionic species. The non-electrolyte UNIQUAC-NRF model [31] for short range, consisting of combinatorial (C) and residual part (R), is given by

gCi;SR

(10)

þ ln gR1 ln gR;∞ ln g*±;SR ¼ ln g*1;SR ¼ ln gC1 ln gC;∞ 1 1

2z2ion ln 1 þ rIx0:5 r !

þ ln gRion ln gR;∞ þ ln gPDH ln g*ion ¼ ln gCion ln gC;∞ ion ion ion

ln gw ¼ ln gCw þ ln gRw þ ln gPDH w

2. Thermodynamic framework

ln g*LR;ion ¼ Af

71

tji qk tkj

(6)

k

aji uji uii ¼ exp tji ¼ exp T T

(7)

qx qi ¼ Pm i i j¼1 qj xj

(8)

rx fi ¼ Pmi i j¼1 rj xj

(9)

In this approach, the binary electrolyte solution, e.g. NaCl þ Water, is considered as a ternary mixture containing Naþ, Cl and water. The short range activity coefficient of Naþ, Cl and water are calculated using expansion of Eqs. (3) and (5) for ternary system. Also the long range activity coefficients of Naþ, Cl are obtained using Pitzer-DebyeeHückel equation, Eq. (1), and water using Eq. (2). The long range mean activity coefficient of the electrolyte is calculated using Eq. (13). A similar equation to Eq. (13) is used to obtain short range mean activity coefficient of the electrolyte. Finally, the mean activity coefficient of the electrolyte is computed using Eq. (14). 3. Results and discussion 3.1. Salt parameter model

Where superscript C and R stand for combinatorial and residual part and aji, Z, q, r are interaction energy parameter, coordination number (Z ¼ 10), surface and volume parameters, respectively. The unsymmetrical activity coefficient can be calculated as

To test the capability of the present model based on salt parameter approach, a number of strong aqueous electrolyte solutions at 25 C and 1 bar is selected and given in Table 1. The adjustable parameters of the model appearing in short range part are a12 and a21 that are obtained for each electrolyte by correlation of the experimental mean activity coefficients of strong aqueous electrolytes. In calculations the values of 14.9 and 0.390947 have been considered for r and A4 in Pitzer-DebyeeHückel equation. Also the surface and volume parameters of ions, q and r, used in

72


Table 1 The adjusted parameters for salt parameter model approach and the results of the present model using salt parameter and ion specific approaches in correlating mean activity coefficients [32,33] and calculating osmotic coefficients [32,33] of binary strong aqueous electrolytes and comparisons with ion specific Nonelectrolyte NRTL-NRF (isNNN) [24], ion specific Nonelectrolyte Wilson-NRF (isNWN) [23], electrolyte NRTL-NRF [7], electrolyte UNIQUAC-NRF [13] and e-NRTL [6] models. Aq. sys.

max. m

RMSEa This work

isNNN

Salt Parameter

NaCl NaBr NaOH NaNO3 NaAc NaI Na2SO4 NaF LiCl LiBr LiOH LiNO3 LiAc LiI Li2SO4 KCl KBr KOH KNO3 KAc KI K2SO4 KF CaCl2 CaBr2 Ca(NO3)2 CaI2 MgCl2 MgBr2 Mg(NO3)2 MgAc2 MgI2 MgSO4 RbAc RbBr RbCl RbI RbNO3 Rb2SO4 CsAc CsBr CsCl CsI CsNO3 Cs2SO4 NH4C1 NH4NO3 (NH4)2SO4 HCI HBr HN03 HI BaCl2 BaBr2 Ba(NO3)2 BaAc2 BaI2 Al(Cl)3 Al2(SO4)3 CrCl3 Cr(NO3)2 Cr2(SO4)2 NaClO4 LiClO4 Ca(ClO4)2 Mg(ClO4)2 HClO4 Ba(ClO4)2

6.144 9 29 10.83 3.5 3.5 4 1 19.219 6 5 20 4 3 3 5 5.5 20 3.5 3.5 4.5 0.7 6 6 6 6 2 5 5 5 4 5 3 3.5 5 7.8 5 4.5 1.8 3.5 5 11 3 1.5 1.8 7.405 25.954 4 16 11 28 10 1.8 2 0.4 3.5 2 1.8 1 1.2 1.4 1.2 6 4.5 6 4 10 5

isNWN

e-NRTL-NRF

e-UNIQUAC-NRF

e-NRTL

Ion specific

a12/T

a21/T

g±

F

g±

F

g±

g±

g±

g±

g±

1.653 5.654 17.343 3.314 1.556 32.354 2.503 0.984 11.392 9.729 1.825 6.695 3.686 6.200 3.680 1.310 1.309 14.846 0.000 3.961 2.595 1.297 8.157 2.254 15.218 5.504 8.836 18.087 16.661 13.675 4.264 14.109 1.744 3.861 1.192 1.273 1.229 0.000 1.666 3.524 1.134 1.193 1.073 0.781 1.069 1.210 0.277 1.151 11.637 12.218 5.377 9.441 1.533 1.724 1.351 1.243 7.191 21.068 1.856 2.140 2.129 1.786 3.586 38.428 20.146 22.619 14.834 8.289

2.565 1.537 2.666 1.961 2.172 29.251 1.259 1.245 2.240 1.977 3.489 0.656 0.000 0.000 1.027 1.916 1.865 2.554 1.915 0.000 0.000 2.097 1.942 4.177 2.520 0.000 0.001 2.396 2.301 1.388 0.905 2.268 3.004 0.000 1.777 1.966 1.795 1.964 0.264 0.000 1.857 1.986 1.638 2.095 1.322 1.688 1.221 1.881 2.027 2.353 0 1.988 2.070 2.257 2.674 1.308 0.000 0.000 2.495 2.515 2.585 1.665 1.312 29.695 2.162 2.259 2.615 0.000

0.023 0.013 0.122 0.077 0.011 0.008 0.015 0.001 0.090 0.020 0.016 0.024 0.005 0.022 0.016 0.004 0.015 0.036 0.008 0.009 0.007 0.007 0.005 0.300 0.159 0.040 0.036 0.080 0.094 0.052 0.008 0.141 0.047 0.007 0.003 0.003 0.003 0.012 0.007 0.008 0.003 0.005 0.006 0.002 0.008 0.002 0.048 0.014 0.058 0.067 0.066 0.092 0.018 0.024 0.003 0.016 0.037 0.082 0.070 0.048 0.048 0.133 0.007 0.015 0.077 0.090 0.084 0.068

0.019 0.008 0.050 0.073 0.009 0.006 0.020 0.001 0.030 0.012 0.018 0.012 0.003 0.016 0.017 0.004 0.005 0.018 0.011 0.006 0.004 0.010 0.004 0.167 0.067 0.025 0.028 0.043 0.044 0.031 0.008 0.059 0.124 0.005 0.002 0.003 0.003 0.020 0.009 0.005 0.003 0.005 0.006 0.002 0.008 0.002 0.167 0.020 0.019 0.030 0.037 0.038 0.019 0.023 0.006 0.016 0.025 0.058 0.150 0.045 0.043 0.219 0.005 0.010 0.038 0.045 0.037 0.029

0.018 0.031 0.147 0.118 0.042 0.008 0.067 0.002 0.067 0.052 0.063 0.115 0.092 0.054 0.031 0.007 0.014 0.029 0.028 0.013 0.012 0.033 0.005 0.060 0.259 0.050 0.041 0.072 0.081 0.075 0.078 0.119 0.032 0.011 0.013 0.020 0.016 0.046 0.007 0.034 0.034 0.052 0.016 0.017 0.011 0.023 0.015 0.046 0.025 0.164 0.065 0.140 0.046 0.045 0.009 0.099 0.044 0.081 0.065 0.115 0.074 0.152 0.009 0.016 0.147 0.148 0.207 0.074

0.015 0.020 0.058 0.107 0.012 0.006 0.073 0.002 0.026 0.029 0.039 0.056 0.039 0.032 0.016 0.006 0.003 0.014 0.036 0.008 0.008 0.045 0.004 0.035 0.099 0.028 0.029 0.039 0.039 0.025 0.053 0.051 0.090 0.007 0.010 0.015 0.012 0.064 0.009 0.021 0.028 0.038 0.014 0.022 0.005 0.019 0.021 0.047 0.012 0.060 0.038 0.052 0.035 0.031 0.012 0.090 0.027 0.059 0.148 0.084 0.054 0.230 0.006 0.009 0.062 0.064 0.080 0.032

0.023 0.040 0.290 0.097 0.010 0.008 0.012 0.001 0.046 0.072 0.057 0.024 0.006 0.033 0.017 0.008 0.014 0.226 0.022 0.012 0.010 0.008 0.004 0.068 0.317 0.041 0.037 0.080 0.204 0.055 0.018 0.263 0.067 0.008 0.011 0.017 0.015 0.035 0.007 0.010 0.031 0.045 0.016 0.016 0.008 0.006 0.081 0.024 0.201 0.038 0.089 0.185 0.047 0.037 0.008 0.087 0.038 0.081 0.089 0.173 0.117 0.240 0.008 0.016 0.181 0.158 0.238 0.070

0.007 0.008 0.073 0.067 0.012 0.01 0.014 0.001 0.05 0.055 0.046 0.012 0.008 0.025 0.016 0.01 0.021 0.065 0.014 0.008 0.024 0.014 0.006 0.078 0.278 0.069 0.04 0.119 0.175 0.059 0.088 0.262 0.1 0.006 0.007 0.02 0.006 0.024 0.008 0.008 0.017 0.031 0.011 0.014 0.01 0.034 0.099 0.098 0.116 0.306 0.085 0.215 0.047 0.047 0.009 0.104 0.045 0.080 0.090 0.223 0.105 0.229 0.170 0.015 0.100 0.157 0.214 0.094

0.011 0.061 0.057 0.072

0.008 0.008 0.155 0.072 0.007 0.007 0.014 0.001 0.070 0.014 0.020 0.012 0.003 0.021 0.016 0.002 0.014 0.022 0.006 0.007 0.005 0.007 0.003 0.061 0.141 0.037 0.034 0.070 0.080 0.050 0.009 0.117 0.051 0.006 0.004 0.007 0.005 0.008 0.007 0.006 0.007 0.017 0.007 0.002 0.008 0.003 0.017 0.021 0.019 0.034 0.024 0.055 0.021 0.024 0.003 0.013 0.029 0.081 0.067 0.063 0.060 0.150 0.007 0.015 0.065 0.084 0.056 0.046

0.012 0.033 0.324 0.083 0.003 0.006 0.011 0.001 0.261 0.041 0.020 0.039 0.003 0.019 0.016 0.003 0.013 0.269 0.006 0.004 0.003 0.007 0.009 0.205 0.322 0.033 0.030 0.202 0.241 0.104 0.013 0.287 0.088 0.006 0.003 0.002 0.003 0.008 0.005 0.003 0.005 0.006 0.007 0.002 0.005 0.001 0.012 0.016 0.198 0.179 0.024 0.180 0.006 0.012 0.003 0.035 0.021 0.100 0.066 0.039 0.042 0.129 0.007 0.022 0.260 0.193 0.182 0.041

0.028 0.026 0.002 0.052 0.045 0.021 0.016 0.005 0.020 0.016 0.021 0.004 0.039 0.004 0.008 0.005 0.015 0.006 0.279 0.034 0.036 0.179 0.211 0.048 0.010 0.276 0.074 0.008 0.001 0.003 0.002 0.007 0.007 0.008 0.005 0.013 0.005 0.000 0.008 0.002 0.010 0.056 0.024 0.040 0.018 0.062 0.023

0.029 0.080 0.051 0.073 0.070 0.132 0.009 0.017 0.182 0.180 0.060 0.042


73

Table 1 (continued ) Aq. sys.

max. m

RMSEa This work

isNNN

Salt Parameter a12/T

a

RMSE g ¼

ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P exp 2 1 ðln g1 ln gcal NP 1 Þ

a21/T

isNWN

e-NRTL-NRF

e-UNIQUAC-NRF

e-NRTL

Ion specific

g±

F

g±

0.040 0.031 0.059 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 1 ðln Fexp ln Fcal Þ2 . RMSE F ¼ NP

combinatorial part of UNIQUAC-NRF model is taken from work of Haghtalab and peyvandi [13] in which r and q were calculated based on ions radius. As mentioned in introduction part, the electrolyte model such as E-Wilson-NRF [10] and E-UNIQUAC-NRF model [13,14] are based on assumptions proposed by Chen et al. which are like-ion repulsion and local electro-neutrality. On the other word, in the salt parameter approach of N-UNIQUAC-NRF, the local cells are solvent and ionic pair, however in the E-UNIQUACNRF, solvent and ionic species are as central cells. Using assumptions proposed by Chen et al., more terms appear in the electrolyte model equations and are generally more complicate than nonelectrolyte models such as N-Wilson-NRF and N-UNIQUACNRF. The adjusted parameters for each electrolyte are given in Table 1. Also included is the standard deviation of the present model in correlating experimental mean activity coefficient [33,34]. The overall deviation of the present model in fitting mean activity coefficients is 0.040 and in calculating osmotic coefficients is 0.031 demonstrating successful representation of the experimental data. Included in Table 1 is also the results of ion specific Noelectrolyte NRTL-NRF (isNNN) [24], ion specific Noelectrolyte Wilson-NRF (isNWN) [23], E-NRTL-NRF [7], E-UNIQUAC-NRF [13] and E-NRTL [6] models. As one can see the results of the present salt specific model are in better agreements with the experiments than salt specific electrolyte models of E-NRTL and E-NRTL-NRF. Also it is better than ion specific models of isNNN and isNWN. It should be noted that the values r and q for electrolyte (e.g. NaCl) have an effect on the results of the salt parameter approach. In this study, these parameters are calculated as qsalt ¼ (q2anion þ q2cation)0.5 and rsalt ¼ (r3anion þ r3cation)1/3. As it can be seen the results of the simple present model are quite comparable with the electrolyte models. In fact, considering non-fitted surface and volume parameters, the results are in very good agreements with the experiments.

3.2. Ion specific parameter model In this approach, as mentioned, the binary strong aqueous electrolyte is considered as a ternary solution containing anion, cation and water. The activity coefficient of each species, i.e. anion, cation or water, is computed using nonelectrolyte UNIQUAC-NRF and Pitzer-DebyeeHückel equation. In this scheme, the like-ion repulsion in local cells is satisfied by adopting proper values for interaction energy parameters of cationecation and anioneanion [18]. In this work, the value of 100 is fixed for these parameters. Since the parameters are ion specific, so there are a number of common interaction parameters in different binary electrolytes. For example interaction energy parameters of Cl-water would be the same for each electrolyte in which anion is Cl such as NaCl, KCl, LiCl, CsCl and etc. This means the correlation should be carried out by taking in to account the experimental data simultaneously. So the optimization procedure is more difficult rather than later approach. In this work and in first step, the selected cations are Naþ, Liþ, Kþ, Csþ, Rbþ, Ca2þ and Mg2þ and anions are Cl, Br, OH, 2 NO 3 and Ac , F , I , SO4 . Considering these ions, the numbers of

F

g±

g±

g±

g±

g±

0.039

0.068

0.069

0.046

0.032

0.067

possible binary aqueous electrolytes and ion specific interaction parameters are 45 and 61, respectively. The adjusted ion specific interaction parameters obtained using objective function given in Eq. (19), are reported in Table 2. Table 1 also gives the standard deviations of the present model with ion specific parameters in correlating mean activity coefficients and calculating osmotic coefficients of the experimental data of strong aqueous electrolyte solutions [32,33]. The overall deviations of the present model for the mean activity coefficient correlation and osmotic coefficient calculation for these 45 systems are 0.049 and 0.031, respectively. As one can see the results are satisfactory because they are comparable and even better than E-NRTL model with salt specific parameter. It is worthy to note that in the present salt specific model, E-NRTL models the number of adjustable parameters for these 45 electrolytes is 90. So considering reduction of 29 adjustable parameters, the results of the ion specific nonelectrolyte UNIQUAC-NRF model are very good. Fig. 1 shows the comparisons of the correlation of the mean activity coefficient of the present models with salt parameter and ion specific parameter and experimental data for LiCl þ water, KOH þ water and LiNO3 þ water systems at 298 K. Fig. 2 shows the comparisons of the osmotic coefficient for LiCl þ water and LiNO3 þ water systems. The agreements are good even in high molalities. þ In the second step of this part, some cations including NHþ 4, H , Ba2þ, Al3þ and Cr3þ and an anion, ClO have been chosen to vali4 date the reliability and usability of the fitted ion specific parameþ 2þ ters. For the electrolytes in which the cations are NHþ 4 , H , Ba , Al3þ and Cr3þ, the fitted interaction parameters of waterewater and water-anions of the first step are fixed and used. So the þ 2þ remaining parameters that are water-NHþ 4 , water-H , water-Ba , water-Al3þ, water-Cr3þ and possible cationeanion are adjusted. Also For the electrolytes in which the anion is ClO 4 , the fitted interaction parameters of waterewater and water-cations of the first step are fixed and used. Then the remaining parameters that are water-ClO 4 and possible cationeanion are adjusted. The fitted parameters and the results are shown in Tables 1 and 2, respectively. The results are still in good agreements with the experiments and comparable with E-NRTL model. The overall deviations of the present model for the mean activity coefficient correlation and osmotic coefficient calculations for the 68 aqueous systems are respectively 0.059 and 0.039. It can be seen the results of the present ion specific model are in better agreements with the experiments than salt specific electrolyte models of E-NRTL. Also the accuracy of the present ion specific model is better than ion specific models of isNNN and isNWN.

3.3. Ternary systems To show the potential of the model with ion specific and salt parameter for ternary systems, several aqueous mixed salt solutions have been selected. In salt parameter approach, the activity coefficient of water (used for osmotic coefficient calculation) in a mixed salt solutions, can be calculated using expansion of Eqs. (3)

74


Table 2 The adjusted interaction energy parameters, uij/T for ion specific parameter model.

Water Naþ Liþ Kþ Ca2þ Mg2þ Rbþ Csþ NHþ 4 Hþ 2þ Ba Al3þ Cr3þ

Water

Cl

Br

OH

NO 3

Ac

I

(SO4)2

F

ClO 4

0.738 23.814 26.076 33.792 15.108 58.678 22.915 26.234 103.039 5.959 16.072 5.940 5.845

64.511 107.740 59.803 110.075 59.913 59.074 88.181 85.550 217.033 58.086 95.494 58.963 58.294

63.235 107.425 104.215 111.451 84.038 57.850 92.191 89.698 e 162.953 95.559 e e

81.757 77.537 224.634 77.589 e e e e e e e e e

57.104 94.044 53.344 108.196 83.133 58.139 84.687 83.331 207.330 240.102 111.324 e 82.715

50.181 49.543 49.567 65.720 e 53.278 48.954 47.324 e 6.273 48.261 e e

61.103 109.234 56.126 111.390 103.339 55.599 94.430 92.925 1.000 220.024 92.133 e e

50.845 46.407 50.207 46.208 e 51.762 48.754 50.149 195.894 1.000 48.754 52.569 46.097

57.715 53.662 e 54.282 e e e e e e e e e

56.655 103.689 88.566 e 56.455 51.752 e e e 151.769 84.929 e e

and (5) for ternary system, i.e. n ¼ 3, including salt 1, salt 2 and water. In the ion specific approach, the Eqs. (3) and (5) should be expanded using n ¼ 4 for common ion ternary systems. The prediction results for osmotic coefficients of some ternary systems are given in Table 3. The overall relative standard deviations are 0.062 and 0.035 using ion specific and salt parameter approaches, respectively, demonstrating good agreements with the experiments especially for salt parameter model. It should be noted that the results given in Table 3 is only based on the adjusted parameters of the binary electrolytes. To build the phase diagram of mixed salt system using ion specific approach the following equation is applied

Mean activity coefficient

100

10

LiCl 1

KOH

y c ya ma gm;* Ksp ¼ mc gm;* c a

LiNO3 Salt parameter Ion specific parameter

In the salt parameter approach, the following equivalent equation is used

0.1 0

5

10

15

20

(15)

25

Molality Fig. 1. The comparison of experimental [32] (symbol) and correlated mean activity coefficients for aqueous LiCl, KOH and LiNO3 at 298 K.

10

ðya þyc Þ Ksp ¼ ðmc Þyc ðma Þya gm;* ±

(16)

Where Ksp is solubility product, mc and ma are cation and anion molalities, respectively. Figs. 3 and 4, respectively, show the predicted phase diagrams of NaCl þ KCl þ water and NaCl þ CaCl2 þ water systems using both approaches. As one can see both approaches are capable to construct the phase diagrams relatively good. In these calculations, no extra fitted parameters are considered and the parameters of Table 1 are only used.

Osmotic coefficient

4. Conclusion The combination of nonelectrolyte UNIQUAC-NRF and PitzerDebyeeHückel has been successfully applied for aqueous strong electrolyte solutions at 298 K and 1 bar. Two scenarios have been studied salt parameter model and ion specific parameter model. 1 Table 3 The results of the salt and ion specific approaches in predicting osmotic coefficients of aqueous ternary solutions, experimental data were taken from Refs. [35e40].

LiCl LiNO3

Aqueous sys.

RMSE Salt parameter

Ion-specific

NaCl þ KCl LiCl þ CsCl NaCl þ CsCl NaCl þ LiCl NH4Cl þ NaCl (NH4)2SO4 þ K2SO4 LiCl þ MgCl2 Ave.

0.021 0.044 0.025 0.063 0.017 0.012 0.061 0.035

0.020 0.071 0.041 0.026 0.126 0.098 0.050 0.062

Salt parameter Ion specific parameter 0.1 0

5

10

15

20

25

Molality Fig. 2. The comparison of experimental [32] (symbol) and calculated osmotic coefficients for aqueous LiCl and LiNO3 at 298 K.


7

and osmotic coefficients of several aqueous ternary systems have been successfully carried out.

Experiment Salt parameter

6

List of symbols

Ion specific parameter

Molality NaCl

5 4 3 2 1 0 0

1

2

3

75

4

5

6

Molality KCl Fig. 3. The solubility prediction of aqueous NaCl þ KCl at 298 K, experimental data from Ref. [41].

7

A Af Ix m Mw NP r R q T u x z Z

energy parameter DebyeeHückel constant ionic strength in mole fraction based molality molecular weight of water number of experimental points volume parameter universal gas constant surface parameter absolute temperature interaction energy parameter mole fraction charge number of ionic species coordination number

Greek symbols g activity coefficient n stoichiometric number r closest approach parameter Gij nonrandom factor F osmotic coefficient

Experiment Salt parameter

6

Superscripts C combinatorial part cal calculated exp experimental E excess R residual part * unsymmetrical convention ∞ infinite dilution

Ion specific parameter

Molality NaCl

5 4 3 2 1 0 0

2

4

6

8

10

Molality CaCl2 Fig. 4. The solubility prediction of aqueous NaCl þ CaCl2 at 298 K, experimental data from Ref. [41].

While in salt parameter scheme, the energy parameters of the model is obtained for each binary aqueous electrolyte separately, however simultaneous fitting of the experimental data is needed in ion specific parameter model. In both approaches, good results were obtained. It is worthy note that in ion specific model, the number of adjusted parameters reduces from 90 to 61 without significant lake of precision. The overall deviation of the salt parameter model was 0.040 in correlating activity coefficient data of 68 aqueous strong electrolytes and was 0.031 in calculating the osmotic coefficients. The deviations were 0.059 and 0.039, respectively, for ion specific parameter model. In both approaches, the surface and volume parameters were those calculated using ionic radius. The comparisons of the results with E-NRTL model demonstrate the good capability of the present models. The predictions of phase diagrams of aqueous KCl þ NaCl and NaCl þ CaCl2

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