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Representation of nonideality in concentrated electrolyte solutions using the Electrolyte NRTL model with concentration-dependent parameters
Fluid Phase Equilibria 150–151 Ž1998. 277–286
Representation of nonideality in concentrated electrolyte solutions using the Electrolyte NRTL model with concentration-dependent parameters V. Abovsky, Y. Liu ) , S. Watanasiri Aspen Technology, Ten Canal Park, Cambridge, MA 02141, USA
Abstract The Electrolyte NRTL model proposed by Chen et al. wC.-C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE Journal 28 Ž1982. 588–596.x and Chen and Evans wC.-C. Chen, L.B. Evans, AIChE Journal 32 Ž1986. 444–454.x has been modified to include concentration dependence of interaction parameters in order to enhance the capability of the model in representing the nonideality of concentrated electrolyte solutions. The concentration-dependent interaction parameter is based on the concept that the multi-body effect should be considered in determining the local composition when the electrolyte concentration is high. In this work, a linear concentration dependence is assumed and the activity-coefficient expressions for cations, anions and molecular species are derived from the excess Gibbs energy expression through the thermodynamic relationship. Therefore, the activity-coefficient expressions are consistent and satisfy the Gibbs–Duhem equation. The experimental mean-ionic-activity-coefficients of highly soluble aqueous electrolyte systems have been used to regress the concentration-dependent parameters. The calculated values and experimental data are in excellent agreement. The deviations are usually within experimental uncertainty and significantly smaller than those using the original model. The modified model is completely compatible with the original model because it reduces to the original model when the concentration-dependent term is zero. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Activity coefficient model; Electrolytes; Aqueous solutions; Data regression
1. Introduction There is a wide variety of important industrial processes involving electrolyte solutions. Waste-water treatment, seawater desalination, gas scrubbing, extractive distillation, normal or extractive crystalliza)
Corresponding author. Tel.: q1-617-949-1201; fax: q1-617-949-1030; e-mail: [email protected]
0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 3 2 7 - 6
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
278
tion and hydro-metallurgical processes are a few such examples. Modeling the thermodynamic properties of electrolyte solutions is essential for design and simulation of these processes. For some processes, property modeling in the dilute or intermediate concentration range of electrolyte is all that is required, while for others modeling in the high concentration range is of great importance. Although in the last few decades many thermodynamic models for electrolyte solutions have been proposed w3x, most of them have applicable concentration limit around 6 m for the aqueous electrolytes of uni–uni valence Žfor nonuni–uni valent electrolytes, the limit is even lower. . Only a few models claimed to be able to model the entire concentration range for those electrolytes which have solubility far above the usual applicable limit. Among the few, one is the Electrolyte NRTL model proposed by Chen et al. w1x and Chen and Evans w2x. Since its publication in 1982, the model has been successfully applied to model the phase equilibria of a broad range of electrolyte systems including high concentration electrolyte systems and is widely used in the chemical and other industries. However, for some high concentration electrolyte systems such as LiBr–H 2 O and CaCl 2 –H 2 O where nonideality in the high concentration range is large, the results using the Electrolyte NRTL model is not satisfactory. There is still a need for further improvement. The objective of the present work is to improve the capability of the Electrolyte NRTL model for high concentration electrolyte systems by introducing concentration-dependent interaction parameters. The interaction-parameter concentration dependency is introduced by considering the multi-body effect in determining the local composition. The experimental mean-ionic-activity-coefficient data of highly soluble aqueous electrolyte systems are fitted using the Electrolyte NRTL model with and without the interaction-parameter concentration dependency and the results are compared.
2. Electrolyte NRTL model The Electrolyte NRTL model proposed by Chen et al. w1x and Chen and Evans w2x assumes the excess Gibbs energy to be the sum of two interaction contributions, one long range and one short range. The long-range interaction contribution is represented by the Pitzer–Debye–Huckel equation ¨ w4x, which accounts for the contribution due to the electrostatic forces among all ions. The short-range interaction contribution is represented by the Electrolyte NRTL expression, which accounts for the contribution due to short-range interaction forces among all species. The Electrolyte NRTL expression was developed based on the non-random-two-liquid local-composition concept w5x, the like-ion repulsion assumption and the local electro-neutrality assumption. Later, Chen et al. w1x introduced a Born contribution to the model to extend its capacity to the mixed-solvent electrolyte systems. The Born contribution accounts for the change of Gibbs energy associated with moving the ionic species from a mixed-solvent reference state to an aqueous reference state. After proper consideration of unsymmetric convention, the excess Gibbs energy expression of the three-contribution Electrolyte NRTL model is given as follows: )
g ex s g ex
)
,PDH
q g ex
)
,Born
q g ex
)
,lc
Ž1.
For detailed information on the complete excess Gibbs energy expression of the three-contribution Electrolyte NRTL model, please refer to Ref. w6x.
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
279
3. Concentration-dependent interaction parameters In 1986, Sander et al. w7x introduced the concentration-dependent interaction parameters into an electrolyte UNIQUAC expression by an empirical way. In the current work, we introduce interaction parameter concentration dependency into the Electrolyte NRTL expression by considering the multi-body effect in the local composition determination. In deriving the Electrolyte NRTL expression with interaction-parameter concentration dependency, we considered not only the non-random-twoliquid local-composition concept, the like-ion repulsion assumption and the local electroneutrality assumption but also the following: 1. In the high concentration region, the mole ratio of solvents to ions is low. The probability of an ion being immediately surrounded by ions of opposite charge becomes significant. 2. The amount of the ions in the immediate surrounding of a central ion varies with the electrolyte concentration. 3. The interaction energy between a central ion and the ions in the immediate surrounding is affected by the amount of ions in the immediate surrounding. 4. For simplicity, this multi-ion effect in the immediate surrounding is assumed to be a linear function of a key solvent. 5. Since the high concentration electrolyte systems are basically the aqueous electrolyte systems, we assume the key solvent is water. Following the derivation steps outlined by Chen and Evans w2x, we obtained the Electrolyte NRTL excess-Gibbs-energy expression with interaction-parameter concentration dependency as follows: g ex ,lc RT
Ý X j Gjmt jm s Ý Xm
q Ý Xc Ý
Ý X k Gk m
m
X
c
a
X aX
0 Ý Xa
Y
Y
k
a
q Ý Xa Ý c
X
X cX
0 Ý Xc
Y
c
Y
where: Gjm s exp Ž ya jmt jm . Gjc, ac s exp Ž ya jc, act jc, ac . Gja,c a s exp Ž ya ja ,c at ja ,c a .
X
j
Ý X k Gk c, a c X
k
Ý X j Gja,c at ja ,c a X
a
Ý X j Gjc, a ct jc, a c X
j
X
j
Ý X k Gk a,c a
Ž2.
X
k
Ž 2a . Ž 2b. Ž 2c .
Ý X aGca, m Gcm s
a
Ý Xa
Ž 2d.
X
X
a
Ý X cGca, m Gam s
c
Ý Xc
X
c
X
Ž 2e .
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
280
Ý X a a ca, m a
a cm s
Ž 2f .
Ý Xa
X
X
a
Ý X c a ca, m c
a am s
Ž 2g.
Ý Xc
X
c
X
tmc, ac s tcm y tc a , m q tm ,c a tma,c a s ta m y tc a , m q tm ,c a Dca, m Tr Ž Tr y T . tca, m s Cc a , m q q Ec a , m q ln T T T tm,c a s Cm ,c a q
Dm,cX a T
q Em ,c a
Ž Tr y T . T
q ln
ž / ž / Tr T
Ž 2h. Ž 2i . q Fc a , m X w
Ž 2j .
q Fm ,c a X w
Ž 2k.
The variables tcm and tam are computed from Gcm and Gam according to Eqs. Ž2a. , Ž 2d. and Ž2e. . C, D, E and F are adjustable parameters where D and E are related to temperature dependency and F is related to concentration dependency. For simplicity, we assume there is no concentration dependency for the interaction parameters between ion pairs and non-water molecular species. Therefore, only when the molecular species, m, is water, F can be applied. Otherwise F is set to zero. The short-range NRTL activity-coefficient expressions for cations, anions and molecular species are obtained through the thermodynamic relation: E NT g ex ,lc s RT lng ilc Ž3. E Ni T , P , Ni/ j
ž
/
where Ni is the mole number of species i and NT is the total mole number for all species in the system. The effective mole fraction of species i is defined as X i s Ni A irNT . For ionic species, A i is equal to the absolute value of the charge for ion i. For molecular species, A i is equal to unity. For detailed information on the short-range NRTL activity-coefficient expressions with parameter-concentration dependency, please refer to Ref. w6x. As pointed out by Refs. w8,9x, the equation-of-state expressions or activity-coefficient expressions should be invariant when a component is divided into two or more identical subcomponents. We performed an invariant analysis on the newly derived expressions and found they satisfied the invariant condition. 4. Data correlation and discussion The mean-ionic-activity-coefficient data of aqueous electrolytes have been correlated using the Electrolyte NRTL model with and without parameter-concentration dependency. In this work the data correlation was carried out based on the least-squares analysis using the following objective function: n cal . exp. OF s Ý Ž lng ",i y lng ",i .
2
i
where n is the total number of experimental data points.
Ž4.
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
281
Table 1 Results of fit for molality mean-ionic-activity-coefficient data of concentrated aqueous electrolytes at 258C using the original and modified Electrolyte NRTL models Ždata from Lobo w10x. Maximum m
The original
The modified
Cm ,ca
Cca,m
sg "
Cm ,c a
Cc a ,m
Fm ,c a
Fc a ,m
sg "
11.0 16.0 16.0 10.0 28.0 17.5 20.0 20.0 20.0 12.0 11.0 20.0
11.615 11.343 12.625 11.500 9.246 10.115 12.703 11.721 9.756 10.244 8.641 11.242
y5.794 y5.629 y6.081 y5.807 y4.746 y4.999 y6.079 y5.719 y4.975 y5.164 y4.207 y5.441
0.213 0.211 0.560 0.179 0.051 0.112 0.648 0.303 0.072 0.099 0.090 0.291
20.990 16.016 15.659 20.315 17.254 15.769 19.749 22.313 15.060 16.742 22.492 18.696
y7.604 y6.790 y7.872 y7.723 y6.054 y6.143 y7.592 y7.262 y5.955 y6.557 y12.37 y6.858
y9.708 y5.869 y7.547 y9.448 y6.228 y5.488 y9.144 y10.50 y4.664 y6.373 y24.29 y8.068
1.739 1.491 3.434 1.991 0.387 0.909 2.062 1.923 0.585 1.199 13.48 1.412
0.022 0.005 0.007 0.006 0.005 0.007 0.045 0.024 0.013 0.003 0.007 0.022
2–1 BaŽClO4 . 2 5.0 CaBr2 9.2 CaCl 2 10.0 CaŽClO4 . 2 6.0 CoI 2 10.0 CuŽNO 3 . 2 7.8 MgBr2 5.0 MgCl 2 5.0 MgI 2 5.0 MgŽNO 2 . 2 6.5 MgŽNO 3 . 2 5.0 SrŽClO4 . 2 8.0 UO 2 Cl 2 3.0 UO 2 ŽClO4 . 2 5.5 ZnBr2 6.0 ZnCl 2 23.2 ZnŽClO4 . 2 4.0 ZnI 2 12.0 ZnŽNO 3 . 2 6.0
9.117 12.379 11.176 11.223 11.504 10.070 11.155 10.774 11.617 9.622 9.938 10.988 8.632 12.499 6.411 9.842 11.312 8.784 10.002
y4.788 y5.959 y5.510 y5.656 y5.800 y5.073 y5.629 y5.434 y5.840 y4.918 y5.108 y5.522 y4.685 y6.222 y3.772 y4.803 y5.756 y4.727 y5.123
0.061 0.911 0.312 0.251 0.216 0.130 0.219 0.182 0.288 0.134 0.111 0.232 0.035 0.437 0.107 0.463 0.190 0.299 0.131
24.692 25.212 29.316 26.063 40.676 21.739 22.230 22.833 23.478 14.597 21.861 23.598 29.769 30.772 19.902 19.056 28.886 17.854 22.374
y6.947 y8.159 y7.711 y7.897 y8.401 y6.917 y8.031 y7.776 y8.464 y7.362 y7.222 y7.383 y7.111 y8.791 y6.570 y6.462 y8.414 y6.613 y7.145
y12.73 y14.13 y16.93 y14.29 y26.80 y10.583 y11.576 y11.896 y12.783 y9.486 y10.847 y11.974 y16.774 y18.349 y9.935 y9.299 y16.707 y9.213 y11.163
0.793 2.368 1.348 1.769 1.247 1.163 2.425 2.086 2.792 4.180 1.503 1.370 0.448 2.288 1.245 1.370 2.077 1.714 1.334
0.007 0.063 0.036 0.015 0.077 0.010 0.012 0.012 0.020 0.013 0.007 0.007 0.008 0.015 0.008 0.026 0.013 0.022 0.012
3–1 SmCl 3 SmŽClO4 . 3 SmŽNO 3 . 3 TbŽClO4 . 3 TbŽNO 3 . 3 YbCl 3 YbŽClO4 . 3
10.348 12.374 8.761 12.468 8.908 10.828 12.586
y5.281 y6.060 y4.569 y6.103 y4.665 y5.465 y6.141
0.174 0.698 0.091 0.721 0.114 0.266 0.779
29.917 28.775 24.994 30.083 24.827 29.883 29.692
y7.796 y8.720 y6.935 y8.785 y7.012 y8.019 y8.848
y17.719 y17.385 y13.462 y18.444 y13.421 y17.890 y18.160
1.484 2.678 0.989 2.621 1.094 1.775 2.733
0.012 0.039 0.015 0.034 0.015 0.019 0.033
1– 1 HBr HCl HClO4 HI HNO 3 KF LiBr LiCl LiNO 3 NaI NaNO 3 NaOH
3.6 4.5 4.3 4.5 4.5 4.0 4.6
282
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
Table 1 shows the results of correlation for molality mean-ionic-activity-coefficient data of 38 concentrated aqueous electrolytes at 258C using the original model Žwithout parameter-concentration dependency. and the modified model Žwith parameter-concentration dependency. . The NRTL nonrandomness factor is set at the conventional value of 0.2. The root-mean-squares relative deviation, sg ", in the table is defined as:
It can be seen clearly that the modified model reduces the deviation by an order of magnitude in most cases. The correlation results for aqueous lithium bromide and nitric acid are also shown in Figs. 1 and 2. Plot Ža. in Fig. 1 displays the experimental and calculated mean ionic activity coefficient of aqueous lithium bromide at 0–20 m and 258C. Plot Žb. in Fig. 1 displays the experimental and calculated osmotic coefficient of aqueous lithium bromide at 0–20 m and 258C. The osmotic coefficient is calculated using the parameter values regressed solely from the mean-ionic-activitycoefficient data. The excellent representation of the osmotic coefficient evidently proves that the modified model satisfies Gibbs–Duhem thermodynamic relationship. Plots Ža. and Žb. in Fig. 2 display the mean ionic activity coefficient and osmotic coefficient, respectively, of aqueous nitric acid at 0–28 m and 258C. Although the concentration is very high, the results from the original model is reasonable. This is because the nonideality of aqueous nitric acid in the high concentration range is not large. Its maximum mean-ionic-activity-coefficient value is only about 2.81, which is much smaller than the maximum value of 485.0 for aqueous lithium bromide. For aqueous electrolytes of nonuni–uni valence, because of deficiency of the Debye–Huckel theory, ¨ there are also significant deviations at the low concentration range when using the original model. By
Fig. 1. Mean ionic activity coefficient Ža. and osmotic coefficient Žb. of aqueous lithium bromide at 258C. Ž`.: Experimental data; Ž- -.: calculated by the original model; Ž — .: calculated by the modified model.
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
283
Fig. 2. Mean ionic activity coefficient Ža. and osmotic coefficient Žb. of aqueous nitric acid at 258C. Ž`.: Experimental data; Ž- -.: calculated by the original model; Ž — .: calculated by the modified model.
introducing parameter-concentration dependency, the modified model is more flexible and is capable of providing adequate fit of data at both low and high concentration ranges for aqueous nonuni–uni electrolytes. An example is shown in Fig. 3. Fig. 4 illustrates the linear concentration dependency of interaction parameter tm ,ca and tca,m for aqueous lithium chloride using the modified model. It is a typical pattern of concentration dependency of interaction parameter for aqueous 1–1 electrolytes. For comparison, the corresponding tm ,ca and tca,m using the original model is also shown. Although the parameter-concentration dependency is assumed to be proportional to the mole fraction of water, for more direct analysis, we set the abscissa in Fig. 4 as mole fraction of the salt. From a close examination of the parameter values in Table 1, we
Fig. 3. Mean ionic activity coefficient Ža. and osmotic coefficient Žb. of aqueous samarium chloride at 258C. Ž`.: Experimental data; Ž- -.: calculated by the original model; Ž — .: calculated by the modified model.
284
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
Fig. 4. Concentration dependency of interaction parameter tm ,ca Ž1. and tca,m Ž2. for aqueous lithium chloride at 258C.
notice that the parameter-concentration dependency of aqueous sodium nitrate does not follow the general pattern of other 1–1 electrolytes. This needs further investigation. Table 2 and Fig. 5 show the results of fitting the mean-ionic-activity-coefficient data of aqueous potassium hydroxide over the temperature range of 0–808C and concentration range up to 17 m using the original and modified models. Six adjustable parameters are regressed for the original model while eight are used for the modified model. One order of magnitude improvement in deviation is also observed for the modified model. In this work, we assume the parameter concentration dependency is Table 2 Results of fit for molality mean-ionic-activity-coefficient data of aqueous potassium hydroxide over the temperature range of 0–808C and concentration range up to 17 m using the original and modified Electrolyte NRTL models Ždata from Harned and Owen w11x. Cm ,ca Cca,m Dm ,ca Dca,m Em ,ca Eca,m Fm ,ca Fca,m sg " Ž08C. sg " Ž108C. sg " Ž208C. sg " Ž308C. sg " Ž408C. sg " Ž508C. sg " Ž608C. sg " Ž708C. sg " Ž808C.
The original
The modified
8.054 y4.354 977.599 y360.448 y3.458 2.765
6.293 y4.568 1422.298 y541.833 2.417 2.582 y3.860 2.429 0.008 0.007 0.006 0.007 0.005 0.006 0.006 0.010 0.015
0.290 0.270 0.252 0.236 0.224 0.213 0.202 0.194 0.168
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
285
Fig. 5. Mean ionic activity coefficient of aqueous potassium hydroxide over temperature range of 0–808C.
the same at different temperatures. This appears to work well for the aqueous potassium hydroxide system. However, many more systems need to be studied before we can conclude that the parameter-concentration dependency is not a function of temperature.
5. Conclusion The Electrolyte NRTL model has been successfully modified by including a linear concentration dependency of interaction parameters. The parameter-concentration dependency is based on the concept that the multi-body effect should be considered in determining the local composition when the electrolyte concentration is high. The experimental mean-ionic-activity-coefficients of highly soluble aqueous electrolyte systems have been fitted using both the original and modified models. The results indicate that the parameter-concentration dependency can greatly enhance the capability of the Electrolyte NRTL model in representing the nonideality of the concentrated electrolyte solutions. The modified model is consistent and satisfies the Gibbs–Duhem thermodynamic relationship. Furthermore, it is completely compatible with the original model because it will reduce to the original model when the concentration dependent term is set to zero. Work is in progress to apply the modified model to vapor–liquid and liquid–liquid equilibria of industrial systems with multi-electrolytes and multi-solvents.
6. List of symbols C, D, E, F G N R T
Parameters defined in Eq. Ž 2. Quantity defined in Eq. Ž 2. Number of moles Gas constant Temperature ŽK.
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
286
X g m n
Effective mole fraction Gibbs energy Molality Number of data points
Greek letters a g s t
NRTL nonrandomness factor Activity coefficient Root mean squares relative deviation NRTL interaction parameter
Superscripts ) Born cal ex exp lc PDH
Unsymmetric convention Born contribution Calculated value Excess property Experimental value Local-composition NRTL equation Pitzer–Debye–Huckel equation ¨
Subscripts " a,aX ,aY c,cX ,cY ca i, j,k m r w
Mean ionic Anion Cation Electrolyte ca Any species Molecular species Reference Water
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x
C.-C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE Journal 28 Ž1982. 588–596. C.-C. Chen, L.B. Evans, AIChE Journal 32 Ž1986. 444–454. J.R. Loehe, M.D. Donohue, AIChE Journal 43 Ž1997. 180–195. K.S. Pitzer, J. Phys. Chem. 77 Ž1973. 268–277. H. Renon, J.M. Prausnitz, AIChE Journal 14 Ž1968. 135–144. Aspen Technology, ASPEN PLUS Electrolytes Manual, Cambridge, MA, U.S.A. Žin preparation.. B. Sander, P. Rusmussen, A. Fredenslund, Chem. Eng. Sci. 41 Ž1986. 1185–1195. M.L. Michelsen, H. Kistenmacher, Fluid Phase Equilibria 58 Ž1990. 229–230. P.M. Mathias, H.C. Klotz, J.M. Prausnitz, Fluid Phase Equilibria 67 Ž1991. 31–44. V.M.M. Lobo, Handbook of Electrolyte Solutions, Elsevier, New York, 1989. H.S. Harned, B.B. Owen, The Physical Chemistry of Electrolytic Solutions, 3rd edn., Reinhold Publishing, New York, 1958.
Representation of nonideality in concentrated electrolyte solutions using the Electrolyte NRTL model with concentration-dependent parameters V. Abovsky, Y. Liu ) , S. Watanasiri Aspen Technology, Ten Canal Park, Cambridge, MA 02141, USA
Abstract The Electrolyte NRTL model proposed by Chen et al. wC.-C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE Journal 28 Ž1982. 588–596.x and Chen and Evans wC.-C. Chen, L.B. Evans, AIChE Journal 32 Ž1986. 444–454.x has been modified to include concentration dependence of interaction parameters in order to enhance the capability of the model in representing the nonideality of concentrated electrolyte solutions. The concentration-dependent interaction parameter is based on the concept that the multi-body effect should be considered in determining the local composition when the electrolyte concentration is high. In this work, a linear concentration dependence is assumed and the activity-coefficient expressions for cations, anions and molecular species are derived from the excess Gibbs energy expression through the thermodynamic relationship. Therefore, the activity-coefficient expressions are consistent and satisfy the Gibbs–Duhem equation. The experimental mean-ionic-activity-coefficients of highly soluble aqueous electrolyte systems have been used to regress the concentration-dependent parameters. The calculated values and experimental data are in excellent agreement. The deviations are usually within experimental uncertainty and significantly smaller than those using the original model. The modified model is completely compatible with the original model because it reduces to the original model when the concentration-dependent term is zero. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Activity coefficient model; Electrolytes; Aqueous solutions; Data regression
1. Introduction There is a wide variety of important industrial processes involving electrolyte solutions. Waste-water treatment, seawater desalination, gas scrubbing, extractive distillation, normal or extractive crystalliza)
Corresponding author. Tel.: q1-617-949-1201; fax: q1-617-949-1030; e-mail: [email protected]
0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 3 2 7 - 6
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
278
tion and hydro-metallurgical processes are a few such examples. Modeling the thermodynamic properties of electrolyte solutions is essential for design and simulation of these processes. For some processes, property modeling in the dilute or intermediate concentration range of electrolyte is all that is required, while for others modeling in the high concentration range is of great importance. Although in the last few decades many thermodynamic models for electrolyte solutions have been proposed w3x, most of them have applicable concentration limit around 6 m for the aqueous electrolytes of uni–uni valence Žfor nonuni–uni valent electrolytes, the limit is even lower. . Only a few models claimed to be able to model the entire concentration range for those electrolytes which have solubility far above the usual applicable limit. Among the few, one is the Electrolyte NRTL model proposed by Chen et al. w1x and Chen and Evans w2x. Since its publication in 1982, the model has been successfully applied to model the phase equilibria of a broad range of electrolyte systems including high concentration electrolyte systems and is widely used in the chemical and other industries. However, for some high concentration electrolyte systems such as LiBr–H 2 O and CaCl 2 –H 2 O where nonideality in the high concentration range is large, the results using the Electrolyte NRTL model is not satisfactory. There is still a need for further improvement. The objective of the present work is to improve the capability of the Electrolyte NRTL model for high concentration electrolyte systems by introducing concentration-dependent interaction parameters. The interaction-parameter concentration dependency is introduced by considering the multi-body effect in determining the local composition. The experimental mean-ionic-activity-coefficient data of highly soluble aqueous electrolyte systems are fitted using the Electrolyte NRTL model with and without the interaction-parameter concentration dependency and the results are compared.
2. Electrolyte NRTL model The Electrolyte NRTL model proposed by Chen et al. w1x and Chen and Evans w2x assumes the excess Gibbs energy to be the sum of two interaction contributions, one long range and one short range. The long-range interaction contribution is represented by the Pitzer–Debye–Huckel equation ¨ w4x, which accounts for the contribution due to the electrostatic forces among all ions. The short-range interaction contribution is represented by the Electrolyte NRTL expression, which accounts for the contribution due to short-range interaction forces among all species. The Electrolyte NRTL expression was developed based on the non-random-two-liquid local-composition concept w5x, the like-ion repulsion assumption and the local electro-neutrality assumption. Later, Chen et al. w1x introduced a Born contribution to the model to extend its capacity to the mixed-solvent electrolyte systems. The Born contribution accounts for the change of Gibbs energy associated with moving the ionic species from a mixed-solvent reference state to an aqueous reference state. After proper consideration of unsymmetric convention, the excess Gibbs energy expression of the three-contribution Electrolyte NRTL model is given as follows: )
g ex s g ex
)
,PDH
q g ex
)
,Born
q g ex
)
,lc
Ž1.
For detailed information on the complete excess Gibbs energy expression of the three-contribution Electrolyte NRTL model, please refer to Ref. w6x.
V. AboÕsky et al.r Fluid Phase Equilibria 150–151 (1998) 277–286
279
3. Concentration-dependent interaction parameters In 1986, Sander et al. w7x introduced the concentration-dependent interaction parameters into an electrolyte UNIQUAC expression by an empirical way. In the current work, we introduce interaction parameter concentration dependency into the Electrolyte NRTL expression by considering the multi-body effect in the local composition determination. In deriving the Electrolyte NRTL expression with interaction-parameter concentration dependency, we considered not only the non-random-twoliquid local-composition concept, the like-ion repulsion assumption and the local electroneutrality assumption but also the following: 1. In the high concentration region, the mole ratio of solvents to ions is low. The probability of an ion being immediately surrounded by ions of opposite charge becomes significant. 2. The amount of the ions in the immediate surrounding of a central ion varies with the electrolyte concentration. 3. The interaction energy between a central ion and the ions in the immediate surrounding is affected by the amount of ions in the immediate surrounding. 4. For simplicity, this multi-ion effect in the immediate surrounding is assumed to be a linear function of a key solvent. 5. Since the high concentration electrolyte systems are basically the aqueous electrolyte systems, we assume the key solvent is water. Following the derivation steps outlined by Chen and Evans w2x, we obtained the Electrolyte NRTL excess-Gibbs-energy expression with interaction-parameter concentration dependency as follows: g ex ,lc RT
Ý X j Gjmt jm s Ý Xm
q Ý Xc Ý
Ý X k Gk m
m
X
c
a
X aX
0 Ý Xa
Y
Y
k
a
q Ý Xa Ý c
X
X cX
0 Ý Xc
Y
c
Y
where: Gjm s exp Ž ya jmt jm . Gjc, ac s exp Ž ya jc, act jc, ac . Gja,c a s exp Ž ya ja ,c at ja ,c a .
X
j
Ý X k Gk c, a c X
k
Ý X j Gja,c at ja ,c a X
a
Ý X j Gjc, a ct jc, a c X
j
X
j
Ý X k Gk a,c a
Ž2.
X
k
Ž 2a . Ž 2b. Ž 2c .
Ý X aGca, m Gcm s
a
Ý Xa
Ž 2d.
X
X
a
Ý X cGca, m Gam s
c
Ý Xc
X
c
X
Ž 2e .
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280
Ý X a a ca, m a
a cm s
Ž 2f .
Ý Xa
X
X
a
Ý X c a ca, m c
a am s
Ž 2g.
Ý Xc
X
c
X
tmc, ac s tcm y tc a , m q tm ,c a tma,c a s ta m y tc a , m q tm ,c a Dca, m Tr Ž Tr y T . tca, m s Cc a , m q q Ec a , m q ln T T T tm,c a s Cm ,c a q
Dm,cX a T
q Em ,c a
Ž Tr y T . T
q ln
ž / ž / Tr T
Ž 2h. Ž 2i . q Fc a , m X w
Ž 2j .
q Fm ,c a X w
Ž 2k.
The variables tcm and tam are computed from Gcm and Gam according to Eqs. Ž2a. , Ž 2d. and Ž2e. . C, D, E and F are adjustable parameters where D and E are related to temperature dependency and F is related to concentration dependency. For simplicity, we assume there is no concentration dependency for the interaction parameters between ion pairs and non-water molecular species. Therefore, only when the molecular species, m, is water, F can be applied. Otherwise F is set to zero. The short-range NRTL activity-coefficient expressions for cations, anions and molecular species are obtained through the thermodynamic relation: E NT g ex ,lc s RT lng ilc Ž3. E Ni T , P , Ni/ j
ž
/
where Ni is the mole number of species i and NT is the total mole number for all species in the system. The effective mole fraction of species i is defined as X i s Ni A irNT . For ionic species, A i is equal to the absolute value of the charge for ion i. For molecular species, A i is equal to unity. For detailed information on the short-range NRTL activity-coefficient expressions with parameter-concentration dependency, please refer to Ref. w6x. As pointed out by Refs. w8,9x, the equation-of-state expressions or activity-coefficient expressions should be invariant when a component is divided into two or more identical subcomponents. We performed an invariant analysis on the newly derived expressions and found they satisfied the invariant condition. 4. Data correlation and discussion The mean-ionic-activity-coefficient data of aqueous electrolytes have been correlated using the Electrolyte NRTL model with and without parameter-concentration dependency. In this work the data correlation was carried out based on the least-squares analysis using the following objective function: n cal . exp. OF s Ý Ž lng ",i y lng ",i .
2
i
where n is the total number of experimental data points.
Ž4.
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Table 1 Results of fit for molality mean-ionic-activity-coefficient data of concentrated aqueous electrolytes at 258C using the original and modified Electrolyte NRTL models Ždata from Lobo w10x. Maximum m
The original
The modified
Cm ,ca
Cca,m
sg "
Cm ,c a
Cc a ,m
Fm ,c a
Fc a ,m
sg "
11.0 16.0 16.0 10.0 28.0 17.5 20.0 20.0 20.0 12.0 11.0 20.0
11.615 11.343 12.625 11.500 9.246 10.115 12.703 11.721 9.756 10.244 8.641 11.242
y5.794 y5.629 y6.081 y5.807 y4.746 y4.999 y6.079 y5.719 y4.975 y5.164 y4.207 y5.441
0.213 0.211 0.560 0.179 0.051 0.112 0.648 0.303 0.072 0.099 0.090 0.291
20.990 16.016 15.659 20.315 17.254 15.769 19.749 22.313 15.060 16.742 22.492 18.696
y7.604 y6.790 y7.872 y7.723 y6.054 y6.143 y7.592 y7.262 y5.955 y6.557 y12.37 y6.858
y9.708 y5.869 y7.547 y9.448 y6.228 y5.488 y9.144 y10.50 y4.664 y6.373 y24.29 y8.068
1.739 1.491 3.434 1.991 0.387 0.909 2.062 1.923 0.585 1.199 13.48 1.412
0.022 0.005 0.007 0.006 0.005 0.007 0.045 0.024 0.013 0.003 0.007 0.022
2–1 BaŽClO4 . 2 5.0 CaBr2 9.2 CaCl 2 10.0 CaŽClO4 . 2 6.0 CoI 2 10.0 CuŽNO 3 . 2 7.8 MgBr2 5.0 MgCl 2 5.0 MgI 2 5.0 MgŽNO 2 . 2 6.5 MgŽNO 3 . 2 5.0 SrŽClO4 . 2 8.0 UO 2 Cl 2 3.0 UO 2 ŽClO4 . 2 5.5 ZnBr2 6.0 ZnCl 2 23.2 ZnŽClO4 . 2 4.0 ZnI 2 12.0 ZnŽNO 3 . 2 6.0
9.117 12.379 11.176 11.223 11.504 10.070 11.155 10.774 11.617 9.622 9.938 10.988 8.632 12.499 6.411 9.842 11.312 8.784 10.002
y4.788 y5.959 y5.510 y5.656 y5.800 y5.073 y5.629 y5.434 y5.840 y4.918 y5.108 y5.522 y4.685 y6.222 y3.772 y4.803 y5.756 y4.727 y5.123
0.061 0.911 0.312 0.251 0.216 0.130 0.219 0.182 0.288 0.134 0.111 0.232 0.035 0.437 0.107 0.463 0.190 0.299 0.131
24.692 25.212 29.316 26.063 40.676 21.739 22.230 22.833 23.478 14.597 21.861 23.598 29.769 30.772 19.902 19.056 28.886 17.854 22.374
y6.947 y8.159 y7.711 y7.897 y8.401 y6.917 y8.031 y7.776 y8.464 y7.362 y7.222 y7.383 y7.111 y8.791 y6.570 y6.462 y8.414 y6.613 y7.145
y12.73 y14.13 y16.93 y14.29 y26.80 y10.583 y11.576 y11.896 y12.783 y9.486 y10.847 y11.974 y16.774 y18.349 y9.935 y9.299 y16.707 y9.213 y11.163
0.793 2.368 1.348 1.769 1.247 1.163 2.425 2.086 2.792 4.180 1.503 1.370 0.448 2.288 1.245 1.370 2.077 1.714 1.334
0.007 0.063 0.036 0.015 0.077 0.010 0.012 0.012 0.020 0.013 0.007 0.007 0.008 0.015 0.008 0.026 0.013 0.022 0.012
3–1 SmCl 3 SmŽClO4 . 3 SmŽNO 3 . 3 TbŽClO4 . 3 TbŽNO 3 . 3 YbCl 3 YbŽClO4 . 3
10.348 12.374 8.761 12.468 8.908 10.828 12.586
y5.281 y6.060 y4.569 y6.103 y4.665 y5.465 y6.141
0.174 0.698 0.091 0.721 0.114 0.266 0.779
29.917 28.775 24.994 30.083 24.827 29.883 29.692
y7.796 y8.720 y6.935 y8.785 y7.012 y8.019 y8.848
y17.719 y17.385 y13.462 y18.444 y13.421 y17.890 y18.160
1.484 2.678 0.989 2.621 1.094 1.775 2.733
0.012 0.039 0.015 0.034 0.015 0.019 0.033
1– 1 HBr HCl HClO4 HI HNO 3 KF LiBr LiCl LiNO 3 NaI NaNO 3 NaOH
3.6 4.5 4.3 4.5 4.5 4.0 4.6
282
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Table 1 shows the results of correlation for molality mean-ionic-activity-coefficient data of 38 concentrated aqueous electrolytes at 258C using the original model Žwithout parameter-concentration dependency. and the modified model Žwith parameter-concentration dependency. . The NRTL nonrandomness factor is set at the conventional value of 0.2. The root-mean-squares relative deviation, sg ", in the table is defined as:
It can be seen clearly that the modified model reduces the deviation by an order of magnitude in most cases. The correlation results for aqueous lithium bromide and nitric acid are also shown in Figs. 1 and 2. Plot Ža. in Fig. 1 displays the experimental and calculated mean ionic activity coefficient of aqueous lithium bromide at 0–20 m and 258C. Plot Žb. in Fig. 1 displays the experimental and calculated osmotic coefficient of aqueous lithium bromide at 0–20 m and 258C. The osmotic coefficient is calculated using the parameter values regressed solely from the mean-ionic-activitycoefficient data. The excellent representation of the osmotic coefficient evidently proves that the modified model satisfies Gibbs–Duhem thermodynamic relationship. Plots Ža. and Žb. in Fig. 2 display the mean ionic activity coefficient and osmotic coefficient, respectively, of aqueous nitric acid at 0–28 m and 258C. Although the concentration is very high, the results from the original model is reasonable. This is because the nonideality of aqueous nitric acid in the high concentration range is not large. Its maximum mean-ionic-activity-coefficient value is only about 2.81, which is much smaller than the maximum value of 485.0 for aqueous lithium bromide. For aqueous electrolytes of nonuni–uni valence, because of deficiency of the Debye–Huckel theory, ¨ there are also significant deviations at the low concentration range when using the original model. By
Fig. 1. Mean ionic activity coefficient Ža. and osmotic coefficient Žb. of aqueous lithium bromide at 258C. Ž`.: Experimental data; Ž- -.: calculated by the original model; Ž — .: calculated by the modified model.
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Fig. 2. Mean ionic activity coefficient Ža. and osmotic coefficient Žb. of aqueous nitric acid at 258C. Ž`.: Experimental data; Ž- -.: calculated by the original model; Ž — .: calculated by the modified model.
introducing parameter-concentration dependency, the modified model is more flexible and is capable of providing adequate fit of data at both low and high concentration ranges for aqueous nonuni–uni electrolytes. An example is shown in Fig. 3. Fig. 4 illustrates the linear concentration dependency of interaction parameter tm ,ca and tca,m for aqueous lithium chloride using the modified model. It is a typical pattern of concentration dependency of interaction parameter for aqueous 1–1 electrolytes. For comparison, the corresponding tm ,ca and tca,m using the original model is also shown. Although the parameter-concentration dependency is assumed to be proportional to the mole fraction of water, for more direct analysis, we set the abscissa in Fig. 4 as mole fraction of the salt. From a close examination of the parameter values in Table 1, we
Fig. 3. Mean ionic activity coefficient Ža. and osmotic coefficient Žb. of aqueous samarium chloride at 258C. Ž`.: Experimental data; Ž- -.: calculated by the original model; Ž — .: calculated by the modified model.
284
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Fig. 4. Concentration dependency of interaction parameter tm ,ca Ž1. and tca,m Ž2. for aqueous lithium chloride at 258C.
notice that the parameter-concentration dependency of aqueous sodium nitrate does not follow the general pattern of other 1–1 electrolytes. This needs further investigation. Table 2 and Fig. 5 show the results of fitting the mean-ionic-activity-coefficient data of aqueous potassium hydroxide over the temperature range of 0–808C and concentration range up to 17 m using the original and modified models. Six adjustable parameters are regressed for the original model while eight are used for the modified model. One order of magnitude improvement in deviation is also observed for the modified model. In this work, we assume the parameter concentration dependency is Table 2 Results of fit for molality mean-ionic-activity-coefficient data of aqueous potassium hydroxide over the temperature range of 0–808C and concentration range up to 17 m using the original and modified Electrolyte NRTL models Ždata from Harned and Owen w11x. Cm ,ca Cca,m Dm ,ca Dca,m Em ,ca Eca,m Fm ,ca Fca,m sg " Ž08C. sg " Ž108C. sg " Ž208C. sg " Ž308C. sg " Ž408C. sg " Ž508C. sg " Ž608C. sg " Ž708C. sg " Ž808C.
The original
The modified
8.054 y4.354 977.599 y360.448 y3.458 2.765
6.293 y4.568 1422.298 y541.833 2.417 2.582 y3.860 2.429 0.008 0.007 0.006 0.007 0.005 0.006 0.006 0.010 0.015
0.290 0.270 0.252 0.236 0.224 0.213 0.202 0.194 0.168
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Fig. 5. Mean ionic activity coefficient of aqueous potassium hydroxide over temperature range of 0–808C.
the same at different temperatures. This appears to work well for the aqueous potassium hydroxide system. However, many more systems need to be studied before we can conclude that the parameter-concentration dependency is not a function of temperature.
5. Conclusion The Electrolyte NRTL model has been successfully modified by including a linear concentration dependency of interaction parameters. The parameter-concentration dependency is based on the concept that the multi-body effect should be considered in determining the local composition when the electrolyte concentration is high. The experimental mean-ionic-activity-coefficients of highly soluble aqueous electrolyte systems have been fitted using both the original and modified models. The results indicate that the parameter-concentration dependency can greatly enhance the capability of the Electrolyte NRTL model in representing the nonideality of the concentrated electrolyte solutions. The modified model is consistent and satisfies the Gibbs–Duhem thermodynamic relationship. Furthermore, it is completely compatible with the original model because it will reduce to the original model when the concentration dependent term is set to zero. Work is in progress to apply the modified model to vapor–liquid and liquid–liquid equilibria of industrial systems with multi-electrolytes and multi-solvents.
6. List of symbols C, D, E, F G N R T
Parameters defined in Eq. Ž 2. Quantity defined in Eq. Ž 2. Number of moles Gas constant Temperature ŽK.
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X g m n
Effective mole fraction Gibbs energy Molality Number of data points
Greek letters a g s t
NRTL nonrandomness factor Activity coefficient Root mean squares relative deviation NRTL interaction parameter
Superscripts ) Born cal ex exp lc PDH
Unsymmetric convention Born contribution Calculated value Excess property Experimental value Local-composition NRTL equation Pitzer–Debye–Huckel equation ¨
Subscripts " a,aX ,aY c,cX ,cY ca i, j,k m r w
Mean ionic Anion Cation Electrolyte ca Any species Molecular species Reference Water
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x
C.-C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE Journal 28 Ž1982. 588–596. C.-C. Chen, L.B. Evans, AIChE Journal 32 Ž1986. 444–454. J.R. Loehe, M.D. Donohue, AIChE Journal 43 Ž1997. 180–195. K.S. Pitzer, J. Phys. Chem. 77 Ž1973. 268–277. H. Renon, J.M. Prausnitz, AIChE Journal 14 Ž1968. 135–144. Aspen Technology, ASPEN PLUS Electrolytes Manual, Cambridge, MA, U.S.A. Žin preparation.. B. Sander, P. Rusmussen, A. Fredenslund, Chem. Eng. Sci. 41 Ž1986. 1185–1195. M.L. Michelsen, H. Kistenmacher, Fluid Phase Equilibria 58 Ž1990. 229–230. P.M. Mathias, H.C. Klotz, J.M. Prausnitz, Fluid Phase Equilibria 67 Ž1991. 31–44. V.M.M. Lobo, Handbook of Electrolyte Solutions, Elsevier, New York, 1989. H.S. Harned, B.B. Owen, The Physical Chemistry of Electrolytic Solutions, 3rd edn., Reinhold Publishing, New York, 1958.