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Surface tension of aqueous electrolyte solutions at high concentrations — representation and prediction
Chemical Engineering Science 56 (2001) 2879}2888
Surface tension of aqueous electrolyte solutions at high concentrations * representation and prediction Zhibao Li, Benjamin C.-Y. Lu* Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 Received 27 April 2000; received in revised form 4 October 2000; accepted 7 November 2000
Abstract A calculation method was developed for representing and predicting surface tension of aqueous electrolyte solutions over a wide range of concentrations, up to 36 m. Based on the Gibbs dividing surface concept, the Langmuir adsorption equation was adopted for modeling the surface excess of electrolyte(s). The activities of electrolytes in the aqueous solutions were calculated by the Pitzer equation. The surface tensions for 45 aqueous single-electrolyte systems at a single temperature were used to correlate the model parameters, and the correlation yields an overall average absolute percentage deviation (AAPD) of 0.47. Surface tensions for 23 aqueous inorganic electrolyte systems are available at several temperatures. Application of these model parameters to extrapolate surface tensions of these systems to di!erent temperatures yields an overall AAPD of 0.91. The proposed method was successfully applied to predict surface tensions for an aqueous binary electrolyte system containing a free acid (HNO ) and its salt (KNO ), which have opposite e!ects on surface tension with an increase in their concentrations, with an overall AAPD of 1.87. The predicted surface tensions for additional 11 binary and "ve ternary mixed-electrolyte aqueous systems indicate an overall AAPD of 1.69. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Surface tension; Aqueous electrolyte solution; Computation; Adsorption; Interface; Isothermal
1. Introduction The surface tension of aqueous electrolyte solutions is an important physical chemical property in the study and analysis of hydro-metallurgical process, distillation in the presence of salt, and absorption refrigeration. There are many methods for calculating surface tension of dilute aqueous solutions of single electrolyte. Other than the empirical methods, there are theoretical models that involve a combination of thermodynamic equations with an adsorption model. Desnoyer, Masbernat, and Gourdon (1997) used this method for the liquid}liquid interfacial tension calculation of non-ideal electrolytic systems. A combination of the Gibbs chemical potential concept with the Langmuir adsorption model resulted in the von Szyszkowski (1908) equation that is widely used for surfactants. Because the concentrations of surfactants in the aqueous solutions are usually very low, there is no
* Corresponding author. Tel.: #1-613-562-5772; fax: #1-613-5625172.
need to take non-ideality into account. Aveyard and Saleem (1976) used the Schmutzer (1955) equation for the representation of the surface tension of electrolyte solutions of higher concentrations by introducing osmotic coe$cient of the solutions to account for the non-ideality. In their approach, the thickness of the surface of a salt-free layer was required in the calculation. Although the results are good, its applicable concentration range was limited to about 1 m (molality, mols per kg of water) and the method was limited to single-electrolyte aqueous systems. Most systems encountered in industrial processes are aqueous solutions of highly concentrated and/or mixed electrolytes. For such systems, there are few theoretical models for the surface tension calculation. It might be mentioned that prediction of surface tension for a mixture of a free acid and its salt, where the acid contributes a decrease in surface tension with concentration while the salt increases its contribution with concentration, encountered di$culties. For example, the recent e!ort of Li, Li, and Lu (1999) on surface tension for aqueous electrolyte solutions at high concentrations would yield an average absolute percent deviation (AAPD) of 15 for the HNO }KNO mixtures in water.
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 2 5 - X
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This kind of systems is of particular interest in hydrometallurgy industries. In this work, a theoretically based calculation method is proposed, using the Gibbs dividing surface, and the Langmuir adsorption equation (1918) with electrolyte activities calculated by the Pitzer (1973) equation. The model parameters are obtained from single-electrolyte aqueous solutions at a single temperature. Subsequently, these parameters are used to extrapolate the surface tensions of these systems at other temperatures. Finally, surface tensions for mixed-electrolyte aqueous solutions are predicted without introducing any additional parameters or coe$cients.
The quantity U is de"ned (Adamson & Gast, 1997) as +6 the excess of MX in a unit cross-sectional area of the surface region over the moles which would be present in the bulk liquid phase containing the same number of moles of water as does the section of surface region. The superscript w on indicates that the dividing surface is chosen so that "0. U The electrolyte adsorption on the surface of solution is considered to be behaving in the same manner as that in the Langmuir gas}solid adsorption (Langmuir, 1918). At equilibrium, the adsorption rate of electrolyte MX is equal to its desorption rate (Desnoyer et al., 1997), leading to k (1! )a "k , ? +6 +6 +6 B +6 +6
2. Proposed calculation method In a given aqueous solution of concentrated electrolytes, the reversible surface tension change occurring between the bulk vapor and liquid phases at constant temperature and pressure may be expressed by (Adamson & Gast, 1997) !d" dN # dN, U U G G G
(1)
where is the surface tension of the aqueous electrolyte solution. and are the surface excess of water and U G electrolyte i, respectively, and are based on an arbitrarily chosen dividing surface. N and N are the chemical U G potentials of water and electrolyte i in the interface between the vapor and liquid, respectively, and are identical to those in the bulk liquid. For a single electrolyte MX aqueous solution, Eq. (1) reduces to !d" dN # dN . U U +6 +6
(2)
N "* "*#RT ln a , U U U U
(3)
N "* "* #RT ln a . +6 +6 +6 +6
(4)
Thus, dN "d* "RT d ln a , U U U
(5)
dN "d* "RT d ln a . +6 +6 +6
(6)
Substituting Eqs. (5) and (6) into Eq. (2) yields (7)
Using the Gibbs dividing surface and choosing "0 U (Adamson & Gast, 1997), Eq. (7) simpli"es to !d"U RT d ln a . +6 +6
where, k and k are the adsorption and the de? +6 B +6 sorption rate constants, respectively. is the fraction +6 of surface covered by the electrolyte MX. Rearranging Eq. (9) leads to K a +6 +6 , " +6 1#K a +6 +6
(10)
where k K " ? +6 +6 k B +6
(11)
is the adsorption equilibrium constant. Alternatively, can be replaced by U /U , where U denotes +6 +6 +6 +6 the saturated surface excess. Hence, the "nal form of U is given by +6 K a +6 +6 . U "U +6 +6 1#K a +6 +6
(12)
Substituting Eq. (12) into Eq. (8) leads to
In terms of activities,
!d" RT d ln a # RT d ln a . U U +6 +6
(9)
(8)
K a +6 +6 RT d ln a . !d"U +6 +6 1#K a +6 +6 Integrating Eq. (13) from a
+6
"0 to a
1 " #RT U ln , U +6 1#K a +6 +6
+6
(13) yields (14)
where is the pure water surface tension at the system U temperature. The equations used in this work for calculating surface tensions are based on Eq. (14). The two model parameters of this equation, U and K , are +6 +6 obtained from correlating surface tensions of aqueous single-electrolyte solutions. An activity calculation is required for highly concentrated electrolyte solutions to take non-ideality into account. In this work, the Pitzer (1973) equation, which has been used extensively in the calculation of activities of
Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
electrolytes in aqueous solutions, was chosen for the calculation. 2.1. Single-electrolyte aqueous solutions
(15)
where " # . is the stoichiometric coe$cient of + 6 + the cation, is that of the anion, m is the molality of the 6 electrolyte MX, and is the mean ionic activity coef+6 "cient calculated by the Pitzer equation (Pitzer, 1973). Following the expressions of Kim and Frederick (1988), ln
+6
I 2 # ln(1#bI ) "!z z A( + 6 1#bI b #4m + 6
I B # B +6 2 +6
#6m + 6 z C , + + +6
(16)
where z and z are the charges of positive and negative + 6 ions, respectively, I" m z is the ionic strength (mols G G G per kg of water), and b is an empirical parameter with a value of 1.2. A(, the Debye}Huckel coe$cient, is given by
1 2 N d e U A(" , 3 1000 Dk¹
(17)
where N is Avogadro's number, d is the density of U water, D represents the dielectric constant of pure water, k is Boltzmann's constant; and e is the absolute electron charge. The parameters B and B describe the inter+6 +6 action of pairs of oppositely charged ions, which are de"ned as explicit functions of ionic strength by B " # f ( I )# f ( I ) +6 +6 +6 +6
(18)
and B " f ( I )/I# f ( I )/I. +6 +6 +6
considered with "2 (Pitzer, 1973). For higher valence type electrolytes, such as 2}2 electrolytes, the entire two equations are used with "1.4 and "12 (Pitzer, 1973). The C term of Eq. (16) is expressed by +6 C "C( /(2z z ). +6 +6 + 6
The activity of the electrolyte MX is de"ned by a "( )J+ ( )J6 ( m)J, +6 + 6 +6
2881
(19)
In these two expressions, f (x)"2[1!(1#x)e\V]/x,
(20)
f (x)"!2[1!(1#x#0.5x)e\V]/x.
(21)
, and are the Pitzer parameters. For one or +6 +6 +6 both ions in univalent type electrolytes, only the "rst two terms of Eq. (18) and the "rst term of Eq. (19) are
(22)
2.2. Mixed-electrolyte aqueous solutions Eq. (14) together with the Pitzer equation is suitable for representing surface tension for single-electrolyte solution, using the two model parameters obtained for the individual electrolytes. In order to extend the calculation to mixed-electrolyte aqueous solutions, additional formulation should be considered for the adsorption of the mixed electrolytes on the interface of the solution. Two approaches, without inclusion of any additional parameters, are proposed for such an adsorption. The simplest approach is to assume that the adsorption of electrolyte i in a mixed-electrolyte aqueous solution still follows Eq. (12). Namely, Ka G G . U"U G G 1#K a G G
(23)
In other words, it is assumed that there is no interaction or competing adsorption between electrolytes at the interface, a similar assumption was used by Desnoyer et al. (1997) for the liquid}liquid interfacial tension calculation of non-ideal electrolytic systems. Combining Eq. (23) with Eq. (1), and using the Gibbs dividing surface with "0 leads to U ) Ka G G dN. !d" U G 1#K a G G G G
(24)
Integrating Eq. (24) from a "0 to a yields Model 1 for G G the mixed-electrolyte aqueous solution:
1 ) " #RT U ln . U G 1#K a G G G
(25)
It may be desirable to consider the interacting and competing adsorption between electrolytes in the interface for mixed-electrolyte aqueous systems at higher concentrations. Considering that the adsorption rate of electrolyte i in the mixed electrolyte is equal to its desorption rate leads to
k 1! a "k H G B G G ? G H
includes all electrolytes , H
(26)
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where k and k are the adsorption and the desorption ? G B G rate constants, respectively. is the fraction of surface H covered by the electrolyte j. Rearranging Eq. (26) yields Ka G G " , (27) G 1# K a H H H where K "k /k is the adsorption equilibrium conH H? HB stant of electrolyte j. Alternatively, can be replaced by G U/U where U denotes the saturated surface excess. G G G Hence, Ka G G U"U . (28) G G 1# K a H H H In a similar manner, substituting Eq. (28) into Eq. (1) yields another expression (Model 2),
)
Ka G G . (29) U ln 1! G 1# K a H H H G For single-electrolyte solutions, both Models 1 and 2 reduce to Eq. (14). In order to avoid using any additional parameters than those obtained from correlation of surface tension for single-electrolyte solutions, the activity of electrolyte i in mixed-electrolyte aqueous solutions is calculated by the simpli"ed Pitzer equation (Pitzer & Kim, 1974; Whit"eld, 1975). These authors demonstrated that interactions between ions of opposite charge signs are predominant, and to a "rst approximation, the activity of most electrolyte mixtures could be calculated from the properties of single-electrolyte solution. Hence, the adoption of the simpli"ed equation facilitates the calculation. Based on the presentation of Whit"eld (1975), " #RT U
ln
+6
"ln #(2 /) m X #(2 /) m X #* + ? ? 6 A A ? A # m m X , A ? A? A ?
(30)
where ln "!0.392z z [I /(1#1.2I ) #* + 6 #1.6667 ln(1#1.2I )],
X "B # m Z C , ? +? A A +? A
(31) (32)
(33) X "B # m Z C , A A A6 A A6 A X "z z B #(2 z /)C , (34) A? + 6 A? + + A? B " #(2 / I) f ( )#(2 / I) f ( ), +6 +6 +6 +6 (35) B "(2 / I) f ( )#(2 / I) f ( ), +6 +6 +6
(36)
f ( )"1!(1# I ) exp(! I ), L L L
(37)
f ( )"(1# I #0.5 I) exp(! I )!1. L L L L
(38)
In Eq. (30), m and m are the molalities of cation and A ? anion, respectively. All the other symbols are identical to those used in Eqs. (18) and (19). With the parameters obtained from correlating the surface tensions of single-electrolyte aqueous solutions, both models can be used to predict directly the surface tension of concentrated aqueous solutions of mixed electrolytes without any additional parameters.
3. Results and discussion 3.1. Single-electrolyte aqueous solutions Surface tensions for aqueous electrolyte solutions available in the literature were used to verify the validity of the proposed calculation methods. Recently, Weissenborn and Pugh (1996) reported the values of 36 single-electrolyte aqueous solutions containing simple inorganic electrolytes. However, the electrolyte concentrations are low, and limited to 1M (molarity, mols per liter of solution) or less. Abramzon and Gaukhberg (1993) tabulated surface tensions in aqueous solutions for 179 electrolytes in various combinations, covering a wide concentration range. However, some of the compiled systems include few data points, or without including densities of the solutions, which are required to change molarity to molality in the activity calculation by means of the Pitzer equation. Hence, the surface tensions of only 45 single-electrolyte aqueous systems covered in their compilation at a single temperature were used in this work to establish the model parameters of Eq. (14). The correlated results together with the parameter values, the number of experimental data points, and the maximum molalities are presented in Table 1. The overall average absolute percentage deviation (AAPD) of the correlation is 0.47. Three typical examples are shown in Figs. 1}3, representing an acid HNO , a salt NaNO and an alkali KOH, respectively. It is seen that the surface tension rapidly decreases with the concentrations of HNO while NaNO and KOH contribute an increase in the surface tension with concentration. A negative value of saturated surface excess U is +6 obtained when surface tensions increase with concentration, indicating that a negative adsorption occurs on the interface between the vapor and liquid phases. This applies to all aqueous single inorganic-salt solutions, sodium formate in water, and the aqueous H SO solution. All the other inorganic acids and organic salts in water yield positive values of U , corresponding to a decrease +6 in surface tension with the addition of electrolyte into the
Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
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Table 1 Correlated surface tensions of single-electrolyte aqueous solutions Electrolyte
AgNO BaCl CaCl CsCl CuSO HCl HClO HNO H SO KBr KCl KC H O K CrO K Fe(CN) KI KNO KNO KOH K SO LaCl LiBr LiCl MgCl MgSO MnCl NaBr NaCl NaCHO NaC H O NaC H O NaC H O Na CrO NaI NaNO NaOH Na SO NH Cl NH NO (NH ) SO RbCl SrCl Sr(NO ) UO SO Zn(NO ) ZnSO Overall AAPD
¹ (K)
373.15 303.15 303.15 298.15 303.15 293.15 298.15 303.15 303.15 293.15 293.15 303.15 298.15 298.15 298.15 293.15 298.15 291.15 298.15 298.15 303.15 298.15 293.15 283.15 291.15 293.15 298.15 303.15 303.15 303.15 303.15 303.15 298.15 291.15 291.15 303.15 293.15 293.15 303.15 298.15 293.15 291.15 293.15 313.15 298.15
Parameters U +6 (mol/m)
K +6 (dimensionless)
!3.24;10\ !5.84;10\ !4.88;10\ !1.30;10\ !4.50;10\ 4.12;10\ 5.69;10\ 8.05;10\ !6.75;10\ !9.70;10\ !7.31;10\ 8.57;10\ !3.67;10\ !1.85;10\ !1.13;10\ !1.02;10\ !2.38;10\ !5.44;10\ !7.05;10\ !1.82;10\ !4.84;10\ !6.01;10\ !2.49;10\ !1.25;10\ !5.15;10\ !6.54;10\ !1.05;10\ !8.56;10\ 8.74;10\ 9.58;10\ 2.09;10\ !8.06;10\ !4.80;10\ !1.66;10\ !1.13;10\ !8.37;10\ !1.01;10\ !3.08;10\ !8.79;10\ !1.22;10\ !4.00;10\ !8.48;10\ !2.68;10\ !2.89;10\ !1.67;10\
4.86;10\ 2.60;10 1.50;10 1.42;10 1.40;10 4.68;10\ 1.12;10 1.06;10\ 1.65;10 1.70;10 4.16;10 1.21;10\ 1.73;10 1.72;10 4.65;10\ 3.92;10\ 1.02;10 8.00;10 9.58;10 3.38;10 3.39;10 2.61;10 2.27;10 3.79;10 1.36;10 9.10;10 1.20;10 1.07;10 6.23;10\ 1.02;10 1.25;10 3.77;10 1.84;10 1.25;10 1.17;10 7.57;10 1.30;10 4.89;10\ 3.84;10 1.19;10 9.68;10 2.05;10 2.28;10 3.22;10 2.64;10
AAPD
N N
m
0.45 0.56 0.67 0.28 0.16 0.51 0.90 1.04 0.88 0.71 0.33 0.69 0.14 0.62 0.44 1.57 0.12 0.31 0.50 0.39 0.35 0.38 0.26 0.42 0.52 0.50 0.31 0.31 0.61 0.25 0.96 0.28 0.34 0.22 0.84 0.25 0.27 0.55 0.21 0.29 0.27 0.37 0.54 0.16 0.22
6 5 7 11 3 7 6 9 7 5 4 5 6 16 11 8 6 4 12 11 6 8 5 3 6 4 5 11 11 11 11 4 8 10 5 3 5 8 4 11 9 7 12 5 6
13.6 1.60 7.37 8.89 1.11 14.98 11.56 29.09 16.00 5.60 3.35 23.79 1.37 1.45 7.29 36.02 2.63 3.80 1.05 1.03 17.27 15.67 3.51 2.26 5.62 6.48 5.50 9.80 10.30 4.40 5.86 6.60 8.82 9.84 14.00 1.24 5.63 19.36 5.06 6.94 2.82 3.12 2.34 7.01 2.86
0.47
Data source: Abramzon and Gaukhberg (1993).
solutions. These electrolytes have the same e!ect on surface tension as surfactants. It is further observed that much higher values of K are obtained for H SO , HClO , K Fe(CN) and +6 LaCl in water, indicating that the surface tension of these electrolytes is mainly dependent on their activities in the solution.
Among the organic salts, the behavior of the aqueous sodium acetate (NaC H O ) system is unique. Not only that its K is much lower than the other three sodium +6 organic salts, its surface tension increases with concentration and reaches to a maximum value around a molality of 2, and then it decreases with further increase in concentration. As there are only two parameters in the proposed
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Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
Fig. 1. Surface tensions of aqueous HNO solutions as a function of molality at 303.15 K (Abramzon & Gaukhberg, 1993).
Fig. 2. Surface tensions of aqueous NaNO solutions as a function of molality at 291.15 K (Abramzon & Gaukhberg, 1993).
model, it cannot be expected to be capable of representing such a maximum. Nevertheless, the numerical di!erence between the calculated and the experimental surface tensions at the maximum is only about 1 (mN/m) out of 72 (mN/m), leading to a low AAPD value for the representation. 3.2. Extrapolation of surface tension of aqueous single-electrolyte solutions with temperature Among the 40 single inorganic electrolyte aqueous solutions, surface tensions for 23 of them are available at several temperatures. Application of the model parameters obtained at a single temperature as reported in Table 1 to predict surface tensions of these 23 systems at other temperatures was made with the application of the Pitzer equation. The temperatures at which the predictions were made are presented in Table 2. The calculated
Fig. 3. Surface tensions of aqueous KOH solutions as a function of molality at 293.15 K (Abramzon & Gaukhberg, 1993).
results are also summarized in the table, indicating an overall AAPD of 0.91. It should be mentioned, however, that such extrapolations are subject to certain limitations due to the application of the simpli"ed model. The parameters of the Pitzer equation were determined at 298.15 K. As indicated in Eq. (14), the activity a plays an important role. With+6 out including the temperature coe$cients of the Pitzer parameters in the calculation, prediction of surface tension at temperatures removed from 298.15 K is subject to error. The success reported in Tables 2 and 3 for some aqueous inorganic electrolyte solutions over relatively large temperature ranges is contributed to the small values of their activities in water. Among the "ve organic salts investigated, experimental surface tensions are only available for the aqueous potassium acetate (KC H O ) system. They are reported at six temperatures, from 303.15 to 353.15 K. Although the predicted values at 313.15 K is more or less acceptable with an AAPD of 2.65, much large AAPDs are obtained at higher temperatures. For example, the AAPD was found to be as high as 12.1 at 353.15 K. On the other hand, the AAPD obtained from predicting surface tension for aqueous KCl solutions at the same temperature was only 1.54. The reason for this is due to large di!erences in the activity between the two electrolytes in water. The activity of KCl in water at 353.15 K and at the highest experimental concentration (m"3.353) is less than 10% of the activity of KC H O in water at the same temperature but at the lowest experimental concentration (m"4.370). In the model proposed by Li et al. (1999), the surface tensions of aqueous electrolyte solutions were calculated using the osmotic coe$cients from the Pitzer equation. In that approach, the same model parameters were used for evaluating the osmotic coe$cients for the surface phase as well as the bulk phase. It is felt that at equilibrium, the equal chemical potential approach adopted in
Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
2885
Table 2 Predicted surface tensions of single inorganic electrolyte aqueous solutions at other temperatures Electrolyte
¹ (K)
BaCl CaCl CsCl CuSO HClO HNO H SO KBr
293.15, 293.15, 293.15, 293.15 288.15, 293.15, 283.15, 283.15, 333.15, 303.15, 343.15, 291.15, 293.15, 333.15 283.15, 333.15, 293.15, 333.15 283.15, 293.15, 293.15, 303.15, 343.15, 293.15, 343.15 298.15, 333.15 313.15 313.15, 293.15, 303.15,
KCl K CrO KI LiBr LiCl NaBr NaCl NaI NaNO Na SO NH Cl NH NO (NH ) SO RbCl UO SO
AAPD
N N
m
1.07 1.20 0.70 0.79 1.50 1.48 1.06 1.08
15 14 22 3 12 27 14 35
1.60 7.37 8.89 1.11 11.56 29.09 16.00 5.60
0.94
24
3.35
0.44 0.64
12 55
1.37 7.29
0.52
42
17.27
0.58
40
15.67
313.15, 323.15 323.15 313.15, 323.15 323.15, 333.15,
1.17 0.40 0.45 0.51
16 15 32 24
6.48 5.50 8.82 11.77
323.15, 333.15,
0.93
15
1.24
303.15, 313.15, 323.15,
0.33
25
5.63
323.15 303.15 318.15
2.55 0.98 0.38 1.30
8 8 22 24
19.36 5.06 6.94 2.34
313.15, 323.15 333.15 303.15 323.15 313.15, 298.15 303.15, 343.15, 313.15, 353.15 303.15 303.15,
323.15 313.15, 323.15, 353.15 323.15, 333.15,
313.15, 323.15,
293.15, 313.15, 323.15, 343.15, 353.15 303.15, 313.15, 323.15, 303.15, 313.15, 303.15, 313.15, 353.15 313.15,
Overall AAPD
0.91
Data source: Abramzon and Gaukhberg (1993).
this work is more reasonable. Furthermore, an empirical proportionality coe$cient was introduced in that model. Although its value was obtained from single-electrolyte solutions, it was used directly in the calculation of surface tensions for mixed-electrolyte solutions. These assumptions are avoided in this work. The overall AAPDs obtained in this work as reported in Tables 1 and 2 (0.47 and 0.91, respectively), are lower than the corresponding overall AAPDs reported by Li et al. (1.22 and 1.80, respectively), indicating the advantages of the theoretical consideration involved in the present work. In view of the high electrolyte concentrations involved, the modi"ed Pitzer parameters obtained by Kim and Frederick (1988) were applied in all the activity calculations in this work. 3.3. Prediction of surface tensions for mixed-electrolyte aqueous solutions The proposed two models, Model 1 represented by Eq. (25) and Model 2 represented by Eq. (29), can be used
directly to predict the surface tensions of mixed-electrolyte solutions by means of the model parameters reported in Table 1 together with the simpli"ed Pitzer equation. There is no need to introduce any additional parameters or empirical coe$cients. The predicted results for 11 binary and "ve ternary mixed-salt aqueous solutions are presented in Table 3. As no experimental values involving organic salts are available, only inorganic salts were considered. Model 1 yields an overall AAPD of 1.69, which is lower than that obtained by Model 2 (2.71), indicating that for these salt systems, Model 1 is more suitable. As expected, these deviations are slightly higher than those reported in Table 1, because all the predictions were based on the correlated model parameters for the single-electrolyte aqueous solutions reported in that table. The predicted surface tensions at "ve temperatures for the aqueous solution of HNO and KNO , an inorganic acid and one of its salts that contribute opposite e!ects on surface tensions from variation of concentration, are given in Table 4. The maximum ionic strength of
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Table 3 Predicted surface tensions of aqueous binary and ternary mixed-salt solutions Salt mixture
¹ (K)
(NH ) SO }NaNO Na SO }ZnSO NH Cl}(NH ) SO NH Cl}Sr(NO ) KNO }Sr(NO ) NaNO }Sr(NO ) KNO }NH Cl KBr}Sr(NO ) KNO }NH NO KBr}KCl KBr}NaBr KBr}KCl}NH Cl KBr}KNO }Sr(NO ) KBr}NH Cl}Sr(NO ) KNO }NH Cl}Sr(NO ) NH Cl}(NH ) SO }NaNO
AAPD
291.15 308.15 291.15 291.15 291.15 291.15 291.15 291.15 298.15 291.15 283.15 291.15 291.15 291.15 291.15 291.15
Overall AAPD
Model-1
Model-2
2.67 1.96 1.75 1.62 1.64 1.42 0.88 1.66 2.11 0.73 2.12 1.16 1.94 1.65 1.60 2.17
4.22 2.27 3.63 3.05 1.99 3.20 1.34 2.83 1.71 2.15 2.85 2.66 2.50 2.94 2.33 3.79
1.69
2.71
N N
I
9 13 10 10 10 9 10 10 45 10 9 10 10 10 10 10
13.44 18.10 11.18 7.94 5.79 9.60 4.83 7.45 17.95 4.90 5.83 5.55 5.70 7.18 6.13 10.79
Data source: Abramzon and Gaukhberg (1993).
Table 4 Predicted surface tensions for aqueous HNO }KNO solutions at "ve temperatures ¹ (K)
AAPD Model-1
Model-2
293.15 303.15 313.15 323.15 333.15
5.45 5.98 6.80 7.23 7.90
0.70 1.17 1.67 2.45 3.37
Overall AAPD
6.67
1.87
N N
I
6 6 6 6 6
25.11 25.11 25.11 25.11 25.11
Data source: Abramzon and Gaukhberg (1993).
the mixtures covered in the calculation is up to 25 (mol kg\). In this case, Model 2 yields a lower AAPD (1.87) than that obtained from Model 1 (6.67), indicating that the competing adsorption assumption introduced in Model 2 is more suitable for calculating surface tension for such a salt}acid system. The model of Li et al. (1999) would yield an AAPD of 15.80 for the system at 293.15 K. To illustrate the di!erent results obtained from these two models, the predicted results at 293.15 K are further compared, together with those calculated from the model of Li et al. (1999), in Table 5. The AAPD obtained by Model 2 is lower than that obtained by the model of Li et al. by an order of magnitude.
The accuracy of the calculated activities of electrolytes in the mixed-electrolyte aqueous solutions is essential for calculation. Since the simpli"ed Pitzer equation was used in the calculation of these activities, the -terms which summarize the interactions between the same charge ions, and the -terms which account for the higher-order terms in the Pitzer equation were not considered. As shown in the three "gures, as well as the quality of the predicted values reported in the tables, it appears that these simpli"cations did not contribute to any signi"cant deviations in the calculation, but facilitated the prediction.
4. Conclusions The proposed model based on the Gibbs dividing surface, the Langmuir-type adsorption model and the Pitzer equation for activity is found to be suitable for correlating the surface tension of single-electrolyte aqueous solutions. The two model parameters obtained at a single temperature are successfully used to extend surface tensions for these single-electrolyte aqueous solutions to other temperatures through the application of the Pitzer equation. Without introducing any additional parameters or empirical coe$cients, the generated model parameters are capable of predicting surface tensions of mixed-electrolyte aqueous solutions. The electrolytes studied in this work include various valence-types of salts, acids, and alkali. Model 1 was found to be suitable for predicting surface tensions for most of the tested mixed-salt aqueous solutions. However, for the aqueous
Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
2887
Table 5 A comparison of the predicted surface tensions with the Li et al. (1999) model for aqueous HNO }KNO solutions at ¹"293.15 K m (KNO )
m (HNO )
(mN m\)
1.157 2.442 3.879 1.301
19.397 19.397 19.397 23.806
63.80 64.20 65.30 62.10
Model 1
Model 2
Li et al.
calcd (mN m\)
APD
calcd (mN m\)
APD
calcd (mN m\)
APD
63.58 67.80 73.38 64.25
0.30 5.60 12.40 3.50
62.52 63.72 65.31 62.01
2.00 0.70 0.00 0.10
68.03 73.01 79.93 74.49
6.62 13.72 22.41 20.43
AAPD
5.45
0.70
15.80
Concentration exceeds the maximum molality of KNO in water (m "3.5) used in the correlation of the Pitzer parameters (Kim & Frederick,
1988).
HNO }KNO system, in which the two electrolytes ex hibit opposite e!ect on surface tension with concentration, the competing adsorption model (Model 2) appears to be more suitable.
Notation a AAPD A( b B +6 B +6 C +6 d U D e f f f and f I k k ? k B K m M MX N N N R ¹ X ,X ? A and X A? z ,z + 6
activity average absolute percentage deviation Debye}Huckel parameter for osmotic coef"cient 1.2, a Pitzer parameter for all electrolytes de"ned by Eqs. (18) and (35) de"ned by Eqs. (19) and (36) de"ned by Eq. (22) density of water, kg/m dielectric constant of pure water absolute electron charge de"ned by Eq. (20) de"ned by Eq. (21) de"ned by Eqs. (37) and (38), respectively ionic strength, mol kg\ Boltzmann's constant adsorption rate constant desorption rate constant adsorption equilibrium constant molality, mols per kilogram of water, mol kg\ molarity, mols per liter of solution, mol l\ electrolyte Avogadro's constant number of experimental data points gas constant temperature, K de"ned by Eqs. (32), (33) and (34), respectively charges of positive and negative ion, respectively
Greek letters , , , , +6 +6 C( +6 #* + 6
Pitzer universal constants Pitzer parameters activity coe$cient de"ned by Eq. (31) surface excess fraction of surface covered by electrolyte MX chemical potential # + 6 stoichiometric coe$cient of cation stoichiometric coe$cient of anion surface tension of aqueous electrolyte solution, mN m\
Superscripts 0 ¸ w
standard state or saturated bulk liquid water interface between vapor and liquid
Subscripts a c calcd d exptl i, j max. MX w
anion or adsorption cation calculated value desorption experimental value electrolytes maximum value electrolyte water
Acknowledgements The authors are indebted to the Natural Sciences and Engineering Research Council of Canada (NSERC) for "nancial support.
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References Abramzon, A. A., & Gaukhberg, R. D. (1993). Surface tension of aqueous solutions of inorganic salts. Russian Journal of Applied Chemistry, (Russian), 66(6), 1428}1438; 66(7), 1665}1674; 66(8), 1897}1906; and 66(9), 2145}2156. Adamson, A. W., & Gast, A. P. (1997). Physical chemistry of surfaces (3rd ed.). New York: Wiley Interscience. Aveyard, R., & Saleem, S. M. (1976). Interfacial tensions at alkaneaqueous electrolyte interfaces. Journal of Chemical Society, Faraday Transactions, I72(7), 1609}1617. Desnoyer, C., Masbernat, O., & Gourdon, C. (1997). Predictive model for the calculation of interfacial tension in nonideal electrolytic systems. Journal of Colloid and Interface Science, 191, 22}29. Kim, H. T., & Frederick Jr., W. J. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 253C. 1. Single salt parameters. Journal of Chemical and Engineering Data, 33, 177}184. Langmuir, I. (1918). The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of American Chemical Society, 40, 1361}1403. Li, Z.-B., Li, Y.-G., & Lu, J.-F. (1999). Surface tension model for concentrated electrolyte aqueous solutions by the Pitzer
equation. Industrial Engineering Chemistry Research, 38, 1133}1139. Pitzer, K. S. (1973). Thermodynamics of electrolytes I. Theoretical basis and general equations. Journal of Physical Chemistry, 77, 268}277. Pitzer, K. S., & Kim, J. J. (1974). Thermodynamics of electrolyte. IV. Activity and osmotic coe$cients for mixed electrolytes. Journal of American Chemical Society, 96, 5701}5707. Schmutzer, E. (1955). Zur Theorie der Ober#achenspannung von Losungen Bemerkungen uber den Zusammenhang zwischen radialer Verteilungsfunktion und den thermodynamischen Funktion. Zeitschrift fu( r Physikalische Chemie, 204, 131}156. von Szyszkowski, B. (1908). Experimentelle Studien uber kapillare Eigenschaften der wasserigen Losungen von Fettsauren. Zeitschrift fu( r Physikalische Chemie, 64, 385}414. Weissenborn, P. K., & Pugh, R. J. (1996). Surface tension of aqueous solutions of electrolytes: relationship with ion hydration, oxygen solubility, and bubble coalescence. Journal of Colloid and Interface Science, 184, 550}563. Whit"eld, M. (1975). The extension of chemical models for sea-water to include trace components at 25 and one atmosphere pressure. Geochimica et Cosmochimica Acta, 39, 1545}1557.
Surface tension of aqueous electrolyte solutions at high concentrations * representation and prediction Zhibao Li, Benjamin C.-Y. Lu* Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 Received 27 April 2000; received in revised form 4 October 2000; accepted 7 November 2000
Abstract A calculation method was developed for representing and predicting surface tension of aqueous electrolyte solutions over a wide range of concentrations, up to 36 m. Based on the Gibbs dividing surface concept, the Langmuir adsorption equation was adopted for modeling the surface excess of electrolyte(s). The activities of electrolytes in the aqueous solutions were calculated by the Pitzer equation. The surface tensions for 45 aqueous single-electrolyte systems at a single temperature were used to correlate the model parameters, and the correlation yields an overall average absolute percentage deviation (AAPD) of 0.47. Surface tensions for 23 aqueous inorganic electrolyte systems are available at several temperatures. Application of these model parameters to extrapolate surface tensions of these systems to di!erent temperatures yields an overall AAPD of 0.91. The proposed method was successfully applied to predict surface tensions for an aqueous binary electrolyte system containing a free acid (HNO ) and its salt (KNO ), which have opposite e!ects on surface tension with an increase in their concentrations, with an overall AAPD of 1.87. The predicted surface tensions for additional 11 binary and "ve ternary mixed-electrolyte aqueous systems indicate an overall AAPD of 1.69. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Surface tension; Aqueous electrolyte solution; Computation; Adsorption; Interface; Isothermal
1. Introduction The surface tension of aqueous electrolyte solutions is an important physical chemical property in the study and analysis of hydro-metallurgical process, distillation in the presence of salt, and absorption refrigeration. There are many methods for calculating surface tension of dilute aqueous solutions of single electrolyte. Other than the empirical methods, there are theoretical models that involve a combination of thermodynamic equations with an adsorption model. Desnoyer, Masbernat, and Gourdon (1997) used this method for the liquid}liquid interfacial tension calculation of non-ideal electrolytic systems. A combination of the Gibbs chemical potential concept with the Langmuir adsorption model resulted in the von Szyszkowski (1908) equation that is widely used for surfactants. Because the concentrations of surfactants in the aqueous solutions are usually very low, there is no
* Corresponding author. Tel.: #1-613-562-5772; fax: #1-613-5625172.
need to take non-ideality into account. Aveyard and Saleem (1976) used the Schmutzer (1955) equation for the representation of the surface tension of electrolyte solutions of higher concentrations by introducing osmotic coe$cient of the solutions to account for the non-ideality. In their approach, the thickness of the surface of a salt-free layer was required in the calculation. Although the results are good, its applicable concentration range was limited to about 1 m (molality, mols per kg of water) and the method was limited to single-electrolyte aqueous systems. Most systems encountered in industrial processes are aqueous solutions of highly concentrated and/or mixed electrolytes. For such systems, there are few theoretical models for the surface tension calculation. It might be mentioned that prediction of surface tension for a mixture of a free acid and its salt, where the acid contributes a decrease in surface tension with concentration while the salt increases its contribution with concentration, encountered di$culties. For example, the recent e!ort of Li, Li, and Lu (1999) on surface tension for aqueous electrolyte solutions at high concentrations would yield an average absolute percent deviation (AAPD) of 15 for the HNO }KNO mixtures in water.
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 2 5 - X
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This kind of systems is of particular interest in hydrometallurgy industries. In this work, a theoretically based calculation method is proposed, using the Gibbs dividing surface, and the Langmuir adsorption equation (1918) with electrolyte activities calculated by the Pitzer (1973) equation. The model parameters are obtained from single-electrolyte aqueous solutions at a single temperature. Subsequently, these parameters are used to extrapolate the surface tensions of these systems at other temperatures. Finally, surface tensions for mixed-electrolyte aqueous solutions are predicted without introducing any additional parameters or coe$cients.
The quantity U is de"ned (Adamson & Gast, 1997) as +6 the excess of MX in a unit cross-sectional area of the surface region over the moles which would be present in the bulk liquid phase containing the same number of moles of water as does the section of surface region. The superscript w on indicates that the dividing surface is chosen so that "0. U The electrolyte adsorption on the surface of solution is considered to be behaving in the same manner as that in the Langmuir gas}solid adsorption (Langmuir, 1918). At equilibrium, the adsorption rate of electrolyte MX is equal to its desorption rate (Desnoyer et al., 1997), leading to k (1! )a "k , ? +6 +6 +6 B +6 +6
2. Proposed calculation method In a given aqueous solution of concentrated electrolytes, the reversible surface tension change occurring between the bulk vapor and liquid phases at constant temperature and pressure may be expressed by (Adamson & Gast, 1997) !d" dN # dN, U U G G G
(1)
where is the surface tension of the aqueous electrolyte solution. and are the surface excess of water and U G electrolyte i, respectively, and are based on an arbitrarily chosen dividing surface. N and N are the chemical U G potentials of water and electrolyte i in the interface between the vapor and liquid, respectively, and are identical to those in the bulk liquid. For a single electrolyte MX aqueous solution, Eq. (1) reduces to !d" dN # dN . U U +6 +6
(2)
N "* "*#RT ln a , U U U U
(3)
N "* "* #RT ln a . +6 +6 +6 +6
(4)
Thus, dN "d* "RT d ln a , U U U
(5)
dN "d* "RT d ln a . +6 +6 +6
(6)
Substituting Eqs. (5) and (6) into Eq. (2) yields (7)
Using the Gibbs dividing surface and choosing "0 U (Adamson & Gast, 1997), Eq. (7) simpli"es to !d"U RT d ln a . +6 +6
where, k and k are the adsorption and the de? +6 B +6 sorption rate constants, respectively. is the fraction +6 of surface covered by the electrolyte MX. Rearranging Eq. (9) leads to K a +6 +6 , " +6 1#K a +6 +6
(10)
where k K " ? +6 +6 k B +6
(11)
is the adsorption equilibrium constant. Alternatively, can be replaced by U /U , where U denotes +6 +6 +6 +6 the saturated surface excess. Hence, the "nal form of U is given by +6 K a +6 +6 . U "U +6 +6 1#K a +6 +6
(12)
Substituting Eq. (12) into Eq. (8) leads to
In terms of activities,
!d" RT d ln a # RT d ln a . U U +6 +6
(9)
(8)
K a +6 +6 RT d ln a . !d"U +6 +6 1#K a +6 +6 Integrating Eq. (13) from a
+6
"0 to a
1 " #RT U ln , U +6 1#K a +6 +6
+6
(13) yields (14)
where is the pure water surface tension at the system U temperature. The equations used in this work for calculating surface tensions are based on Eq. (14). The two model parameters of this equation, U and K , are +6 +6 obtained from correlating surface tensions of aqueous single-electrolyte solutions. An activity calculation is required for highly concentrated electrolyte solutions to take non-ideality into account. In this work, the Pitzer (1973) equation, which has been used extensively in the calculation of activities of
Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
electrolytes in aqueous solutions, was chosen for the calculation. 2.1. Single-electrolyte aqueous solutions
(15)
where " # . is the stoichiometric coe$cient of + 6 + the cation, is that of the anion, m is the molality of the 6 electrolyte MX, and is the mean ionic activity coef+6 "cient calculated by the Pitzer equation (Pitzer, 1973). Following the expressions of Kim and Frederick (1988), ln
+6
I 2 # ln(1#bI ) "!z z A( + 6 1#bI b #4m + 6
I B # B +6 2 +6
#6m + 6 z C , + + +6
(16)
where z and z are the charges of positive and negative + 6 ions, respectively, I" m z is the ionic strength (mols G G G per kg of water), and b is an empirical parameter with a value of 1.2. A(, the Debye}Huckel coe$cient, is given by
1 2 N d e U A(" , 3 1000 Dk¹
(17)
where N is Avogadro's number, d is the density of U water, D represents the dielectric constant of pure water, k is Boltzmann's constant; and e is the absolute electron charge. The parameters B and B describe the inter+6 +6 action of pairs of oppositely charged ions, which are de"ned as explicit functions of ionic strength by B " # f ( I )# f ( I ) +6 +6 +6 +6
(18)
and B " f ( I )/I# f ( I )/I. +6 +6 +6
considered with "2 (Pitzer, 1973). For higher valence type electrolytes, such as 2}2 electrolytes, the entire two equations are used with "1.4 and "12 (Pitzer, 1973). The C term of Eq. (16) is expressed by +6 C "C( /(2z z ). +6 +6 + 6
The activity of the electrolyte MX is de"ned by a "( )J+ ( )J6 ( m)J, +6 + 6 +6
2881
(19)
In these two expressions, f (x)"2[1!(1#x)e\V]/x,
(20)
f (x)"!2[1!(1#x#0.5x)e\V]/x.
(21)
, and are the Pitzer parameters. For one or +6 +6 +6 both ions in univalent type electrolytes, only the "rst two terms of Eq. (18) and the "rst term of Eq. (19) are
(22)
2.2. Mixed-electrolyte aqueous solutions Eq. (14) together with the Pitzer equation is suitable for representing surface tension for single-electrolyte solution, using the two model parameters obtained for the individual electrolytes. In order to extend the calculation to mixed-electrolyte aqueous solutions, additional formulation should be considered for the adsorption of the mixed electrolytes on the interface of the solution. Two approaches, without inclusion of any additional parameters, are proposed for such an adsorption. The simplest approach is to assume that the adsorption of electrolyte i in a mixed-electrolyte aqueous solution still follows Eq. (12). Namely, Ka G G . U"U G G 1#K a G G
(23)
In other words, it is assumed that there is no interaction or competing adsorption between electrolytes at the interface, a similar assumption was used by Desnoyer et al. (1997) for the liquid}liquid interfacial tension calculation of non-ideal electrolytic systems. Combining Eq. (23) with Eq. (1), and using the Gibbs dividing surface with "0 leads to U ) Ka G G dN. !d" U G 1#K a G G G G
(24)
Integrating Eq. (24) from a "0 to a yields Model 1 for G G the mixed-electrolyte aqueous solution:
1 ) " #RT U ln . U G 1#K a G G G
(25)
It may be desirable to consider the interacting and competing adsorption between electrolytes in the interface for mixed-electrolyte aqueous systems at higher concentrations. Considering that the adsorption rate of electrolyte i in the mixed electrolyte is equal to its desorption rate leads to
k 1! a "k H G B G G ? G H
includes all electrolytes , H
(26)
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where k and k are the adsorption and the desorption ? G B G rate constants, respectively. is the fraction of surface H covered by the electrolyte j. Rearranging Eq. (26) yields Ka G G " , (27) G 1# K a H H H where K "k /k is the adsorption equilibrium conH H? HB stant of electrolyte j. Alternatively, can be replaced by G U/U where U denotes the saturated surface excess. G G G Hence, Ka G G U"U . (28) G G 1# K a H H H In a similar manner, substituting Eq. (28) into Eq. (1) yields another expression (Model 2),
)
Ka G G . (29) U ln 1! G 1# K a H H H G For single-electrolyte solutions, both Models 1 and 2 reduce to Eq. (14). In order to avoid using any additional parameters than those obtained from correlation of surface tension for single-electrolyte solutions, the activity of electrolyte i in mixed-electrolyte aqueous solutions is calculated by the simpli"ed Pitzer equation (Pitzer & Kim, 1974; Whit"eld, 1975). These authors demonstrated that interactions between ions of opposite charge signs are predominant, and to a "rst approximation, the activity of most electrolyte mixtures could be calculated from the properties of single-electrolyte solution. Hence, the adoption of the simpli"ed equation facilitates the calculation. Based on the presentation of Whit"eld (1975), " #RT U
ln
+6
"ln #(2 /) m X #(2 /) m X #* + ? ? 6 A A ? A # m m X , A ? A? A ?
(30)
where ln "!0.392z z [I /(1#1.2I ) #* + 6 #1.6667 ln(1#1.2I )],
X "B # m Z C , ? +? A A +? A
(31) (32)
(33) X "B # m Z C , A A A6 A A6 A X "z z B #(2 z /)C , (34) A? + 6 A? + + A? B " #(2 / I) f ( )#(2 / I) f ( ), +6 +6 +6 +6 (35) B "(2 / I) f ( )#(2 / I) f ( ), +6 +6 +6
(36)
f ( )"1!(1# I ) exp(! I ), L L L
(37)
f ( )"(1# I #0.5 I) exp(! I )!1. L L L L
(38)
In Eq. (30), m and m are the molalities of cation and A ? anion, respectively. All the other symbols are identical to those used in Eqs. (18) and (19). With the parameters obtained from correlating the surface tensions of single-electrolyte aqueous solutions, both models can be used to predict directly the surface tension of concentrated aqueous solutions of mixed electrolytes without any additional parameters.
3. Results and discussion 3.1. Single-electrolyte aqueous solutions Surface tensions for aqueous electrolyte solutions available in the literature were used to verify the validity of the proposed calculation methods. Recently, Weissenborn and Pugh (1996) reported the values of 36 single-electrolyte aqueous solutions containing simple inorganic electrolytes. However, the electrolyte concentrations are low, and limited to 1M (molarity, mols per liter of solution) or less. Abramzon and Gaukhberg (1993) tabulated surface tensions in aqueous solutions for 179 electrolytes in various combinations, covering a wide concentration range. However, some of the compiled systems include few data points, or without including densities of the solutions, which are required to change molarity to molality in the activity calculation by means of the Pitzer equation. Hence, the surface tensions of only 45 single-electrolyte aqueous systems covered in their compilation at a single temperature were used in this work to establish the model parameters of Eq. (14). The correlated results together with the parameter values, the number of experimental data points, and the maximum molalities are presented in Table 1. The overall average absolute percentage deviation (AAPD) of the correlation is 0.47. Three typical examples are shown in Figs. 1}3, representing an acid HNO , a salt NaNO and an alkali KOH, respectively. It is seen that the surface tension rapidly decreases with the concentrations of HNO while NaNO and KOH contribute an increase in the surface tension with concentration. A negative value of saturated surface excess U is +6 obtained when surface tensions increase with concentration, indicating that a negative adsorption occurs on the interface between the vapor and liquid phases. This applies to all aqueous single inorganic-salt solutions, sodium formate in water, and the aqueous H SO solution. All the other inorganic acids and organic salts in water yield positive values of U , corresponding to a decrease +6 in surface tension with the addition of electrolyte into the
Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
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Table 1 Correlated surface tensions of single-electrolyte aqueous solutions Electrolyte
AgNO BaCl CaCl CsCl CuSO HCl HClO HNO H SO KBr KCl KC H O K CrO K Fe(CN) KI KNO KNO KOH K SO LaCl LiBr LiCl MgCl MgSO MnCl NaBr NaCl NaCHO NaC H O NaC H O NaC H O Na CrO NaI NaNO NaOH Na SO NH Cl NH NO (NH ) SO RbCl SrCl Sr(NO ) UO SO Zn(NO ) ZnSO Overall AAPD
¹ (K)
373.15 303.15 303.15 298.15 303.15 293.15 298.15 303.15 303.15 293.15 293.15 303.15 298.15 298.15 298.15 293.15 298.15 291.15 298.15 298.15 303.15 298.15 293.15 283.15 291.15 293.15 298.15 303.15 303.15 303.15 303.15 303.15 298.15 291.15 291.15 303.15 293.15 293.15 303.15 298.15 293.15 291.15 293.15 313.15 298.15
Parameters U +6 (mol/m)
K +6 (dimensionless)
!3.24;10\ !5.84;10\ !4.88;10\ !1.30;10\ !4.50;10\ 4.12;10\ 5.69;10\ 8.05;10\ !6.75;10\ !9.70;10\ !7.31;10\ 8.57;10\ !3.67;10\ !1.85;10\ !1.13;10\ !1.02;10\ !2.38;10\ !5.44;10\ !7.05;10\ !1.82;10\ !4.84;10\ !6.01;10\ !2.49;10\ !1.25;10\ !5.15;10\ !6.54;10\ !1.05;10\ !8.56;10\ 8.74;10\ 9.58;10\ 2.09;10\ !8.06;10\ !4.80;10\ !1.66;10\ !1.13;10\ !8.37;10\ !1.01;10\ !3.08;10\ !8.79;10\ !1.22;10\ !4.00;10\ !8.48;10\ !2.68;10\ !2.89;10\ !1.67;10\
4.86;10\ 2.60;10 1.50;10 1.42;10 1.40;10 4.68;10\ 1.12;10 1.06;10\ 1.65;10 1.70;10 4.16;10 1.21;10\ 1.73;10 1.72;10 4.65;10\ 3.92;10\ 1.02;10 8.00;10 9.58;10 3.38;10 3.39;10 2.61;10 2.27;10 3.79;10 1.36;10 9.10;10 1.20;10 1.07;10 6.23;10\ 1.02;10 1.25;10 3.77;10 1.84;10 1.25;10 1.17;10 7.57;10 1.30;10 4.89;10\ 3.84;10 1.19;10 9.68;10 2.05;10 2.28;10 3.22;10 2.64;10
AAPD
N N
m
0.45 0.56 0.67 0.28 0.16 0.51 0.90 1.04 0.88 0.71 0.33 0.69 0.14 0.62 0.44 1.57 0.12 0.31 0.50 0.39 0.35 0.38 0.26 0.42 0.52 0.50 0.31 0.31 0.61 0.25 0.96 0.28 0.34 0.22 0.84 0.25 0.27 0.55 0.21 0.29 0.27 0.37 0.54 0.16 0.22
6 5 7 11 3 7 6 9 7 5 4 5 6 16 11 8 6 4 12 11 6 8 5 3 6 4 5 11 11 11 11 4 8 10 5 3 5 8 4 11 9 7 12 5 6
13.6 1.60 7.37 8.89 1.11 14.98 11.56 29.09 16.00 5.60 3.35 23.79 1.37 1.45 7.29 36.02 2.63 3.80 1.05 1.03 17.27 15.67 3.51 2.26 5.62 6.48 5.50 9.80 10.30 4.40 5.86 6.60 8.82 9.84 14.00 1.24 5.63 19.36 5.06 6.94 2.82 3.12 2.34 7.01 2.86
0.47
Data source: Abramzon and Gaukhberg (1993).
solutions. These electrolytes have the same e!ect on surface tension as surfactants. It is further observed that much higher values of K are obtained for H SO , HClO , K Fe(CN) and +6 LaCl in water, indicating that the surface tension of these electrolytes is mainly dependent on their activities in the solution.
Among the organic salts, the behavior of the aqueous sodium acetate (NaC H O ) system is unique. Not only that its K is much lower than the other three sodium +6 organic salts, its surface tension increases with concentration and reaches to a maximum value around a molality of 2, and then it decreases with further increase in concentration. As there are only two parameters in the proposed
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Fig. 1. Surface tensions of aqueous HNO solutions as a function of molality at 303.15 K (Abramzon & Gaukhberg, 1993).
Fig. 2. Surface tensions of aqueous NaNO solutions as a function of molality at 291.15 K (Abramzon & Gaukhberg, 1993).
model, it cannot be expected to be capable of representing such a maximum. Nevertheless, the numerical di!erence between the calculated and the experimental surface tensions at the maximum is only about 1 (mN/m) out of 72 (mN/m), leading to a low AAPD value for the representation. 3.2. Extrapolation of surface tension of aqueous single-electrolyte solutions with temperature Among the 40 single inorganic electrolyte aqueous solutions, surface tensions for 23 of them are available at several temperatures. Application of the model parameters obtained at a single temperature as reported in Table 1 to predict surface tensions of these 23 systems at other temperatures was made with the application of the Pitzer equation. The temperatures at which the predictions were made are presented in Table 2. The calculated
Fig. 3. Surface tensions of aqueous KOH solutions as a function of molality at 293.15 K (Abramzon & Gaukhberg, 1993).
results are also summarized in the table, indicating an overall AAPD of 0.91. It should be mentioned, however, that such extrapolations are subject to certain limitations due to the application of the simpli"ed model. The parameters of the Pitzer equation were determined at 298.15 K. As indicated in Eq. (14), the activity a plays an important role. With+6 out including the temperature coe$cients of the Pitzer parameters in the calculation, prediction of surface tension at temperatures removed from 298.15 K is subject to error. The success reported in Tables 2 and 3 for some aqueous inorganic electrolyte solutions over relatively large temperature ranges is contributed to the small values of their activities in water. Among the "ve organic salts investigated, experimental surface tensions are only available for the aqueous potassium acetate (KC H O ) system. They are reported at six temperatures, from 303.15 to 353.15 K. Although the predicted values at 313.15 K is more or less acceptable with an AAPD of 2.65, much large AAPDs are obtained at higher temperatures. For example, the AAPD was found to be as high as 12.1 at 353.15 K. On the other hand, the AAPD obtained from predicting surface tension for aqueous KCl solutions at the same temperature was only 1.54. The reason for this is due to large di!erences in the activity between the two electrolytes in water. The activity of KCl in water at 353.15 K and at the highest experimental concentration (m"3.353) is less than 10% of the activity of KC H O in water at the same temperature but at the lowest experimental concentration (m"4.370). In the model proposed by Li et al. (1999), the surface tensions of aqueous electrolyte solutions were calculated using the osmotic coe$cients from the Pitzer equation. In that approach, the same model parameters were used for evaluating the osmotic coe$cients for the surface phase as well as the bulk phase. It is felt that at equilibrium, the equal chemical potential approach adopted in
Z. Li, B. C.-Y. Lu / Chemical Engineering Science 56 (2001) 2879}2888
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Table 2 Predicted surface tensions of single inorganic electrolyte aqueous solutions at other temperatures Electrolyte
¹ (K)
BaCl CaCl CsCl CuSO HClO HNO H SO KBr
293.15, 293.15, 293.15, 293.15 288.15, 293.15, 283.15, 283.15, 333.15, 303.15, 343.15, 291.15, 293.15, 333.15 283.15, 333.15, 293.15, 333.15 283.15, 293.15, 293.15, 303.15, 343.15, 293.15, 343.15 298.15, 333.15 313.15 313.15, 293.15, 303.15,
KCl K CrO KI LiBr LiCl NaBr NaCl NaI NaNO Na SO NH Cl NH NO (NH ) SO RbCl UO SO
AAPD
N N
m
1.07 1.20 0.70 0.79 1.50 1.48 1.06 1.08
15 14 22 3 12 27 14 35
1.60 7.37 8.89 1.11 11.56 29.09 16.00 5.60
0.94
24
3.35
0.44 0.64
12 55
1.37 7.29
0.52
42
17.27
0.58
40
15.67
313.15, 323.15 323.15 313.15, 323.15 323.15, 333.15,
1.17 0.40 0.45 0.51
16 15 32 24
6.48 5.50 8.82 11.77
323.15, 333.15,
0.93
15
1.24
303.15, 313.15, 323.15,
0.33
25
5.63
323.15 303.15 318.15
2.55 0.98 0.38 1.30
8 8 22 24
19.36 5.06 6.94 2.34
313.15, 323.15 333.15 303.15 323.15 313.15, 298.15 303.15, 343.15, 313.15, 353.15 303.15 303.15,
323.15 313.15, 323.15, 353.15 323.15, 333.15,
313.15, 323.15,
293.15, 313.15, 323.15, 343.15, 353.15 303.15, 313.15, 323.15, 303.15, 313.15, 303.15, 313.15, 353.15 313.15,
Overall AAPD
0.91
Data source: Abramzon and Gaukhberg (1993).
this work is more reasonable. Furthermore, an empirical proportionality coe$cient was introduced in that model. Although its value was obtained from single-electrolyte solutions, it was used directly in the calculation of surface tensions for mixed-electrolyte solutions. These assumptions are avoided in this work. The overall AAPDs obtained in this work as reported in Tables 1 and 2 (0.47 and 0.91, respectively), are lower than the corresponding overall AAPDs reported by Li et al. (1.22 and 1.80, respectively), indicating the advantages of the theoretical consideration involved in the present work. In view of the high electrolyte concentrations involved, the modi"ed Pitzer parameters obtained by Kim and Frederick (1988) were applied in all the activity calculations in this work. 3.3. Prediction of surface tensions for mixed-electrolyte aqueous solutions The proposed two models, Model 1 represented by Eq. (25) and Model 2 represented by Eq. (29), can be used
directly to predict the surface tensions of mixed-electrolyte solutions by means of the model parameters reported in Table 1 together with the simpli"ed Pitzer equation. There is no need to introduce any additional parameters or empirical coe$cients. The predicted results for 11 binary and "ve ternary mixed-salt aqueous solutions are presented in Table 3. As no experimental values involving organic salts are available, only inorganic salts were considered. Model 1 yields an overall AAPD of 1.69, which is lower than that obtained by Model 2 (2.71), indicating that for these salt systems, Model 1 is more suitable. As expected, these deviations are slightly higher than those reported in Table 1, because all the predictions were based on the correlated model parameters for the single-electrolyte aqueous solutions reported in that table. The predicted surface tensions at "ve temperatures for the aqueous solution of HNO and KNO , an inorganic acid and one of its salts that contribute opposite e!ects on surface tensions from variation of concentration, are given in Table 4. The maximum ionic strength of
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Table 3 Predicted surface tensions of aqueous binary and ternary mixed-salt solutions Salt mixture
¹ (K)
(NH ) SO }NaNO Na SO }ZnSO NH Cl}(NH ) SO NH Cl}Sr(NO ) KNO }Sr(NO ) NaNO }Sr(NO ) KNO }NH Cl KBr}Sr(NO ) KNO }NH NO KBr}KCl KBr}NaBr KBr}KCl}NH Cl KBr}KNO }Sr(NO ) KBr}NH Cl}Sr(NO ) KNO }NH Cl}Sr(NO ) NH Cl}(NH ) SO }NaNO
AAPD
291.15 308.15 291.15 291.15 291.15 291.15 291.15 291.15 298.15 291.15 283.15 291.15 291.15 291.15 291.15 291.15
Overall AAPD
Model-1
Model-2
2.67 1.96 1.75 1.62 1.64 1.42 0.88 1.66 2.11 0.73 2.12 1.16 1.94 1.65 1.60 2.17
4.22 2.27 3.63 3.05 1.99 3.20 1.34 2.83 1.71 2.15 2.85 2.66 2.50 2.94 2.33 3.79
1.69
2.71
N N
I
9 13 10 10 10 9 10 10 45 10 9 10 10 10 10 10
13.44 18.10 11.18 7.94 5.79 9.60 4.83 7.45 17.95 4.90 5.83 5.55 5.70 7.18 6.13 10.79
Data source: Abramzon and Gaukhberg (1993).
Table 4 Predicted surface tensions for aqueous HNO }KNO solutions at "ve temperatures ¹ (K)
AAPD Model-1
Model-2
293.15 303.15 313.15 323.15 333.15
5.45 5.98 6.80 7.23 7.90
0.70 1.17 1.67 2.45 3.37
Overall AAPD
6.67
1.87
N N
I
6 6 6 6 6
25.11 25.11 25.11 25.11 25.11
Data source: Abramzon and Gaukhberg (1993).
the mixtures covered in the calculation is up to 25 (mol kg\). In this case, Model 2 yields a lower AAPD (1.87) than that obtained from Model 1 (6.67), indicating that the competing adsorption assumption introduced in Model 2 is more suitable for calculating surface tension for such a salt}acid system. The model of Li et al. (1999) would yield an AAPD of 15.80 for the system at 293.15 K. To illustrate the di!erent results obtained from these two models, the predicted results at 293.15 K are further compared, together with those calculated from the model of Li et al. (1999), in Table 5. The AAPD obtained by Model 2 is lower than that obtained by the model of Li et al. by an order of magnitude.
The accuracy of the calculated activities of electrolytes in the mixed-electrolyte aqueous solutions is essential for calculation. Since the simpli"ed Pitzer equation was used in the calculation of these activities, the -terms which summarize the interactions between the same charge ions, and the -terms which account for the higher-order terms in the Pitzer equation were not considered. As shown in the three "gures, as well as the quality of the predicted values reported in the tables, it appears that these simpli"cations did not contribute to any signi"cant deviations in the calculation, but facilitated the prediction.
4. Conclusions The proposed model based on the Gibbs dividing surface, the Langmuir-type adsorption model and the Pitzer equation for activity is found to be suitable for correlating the surface tension of single-electrolyte aqueous solutions. The two model parameters obtained at a single temperature are successfully used to extend surface tensions for these single-electrolyte aqueous solutions to other temperatures through the application of the Pitzer equation. Without introducing any additional parameters or empirical coe$cients, the generated model parameters are capable of predicting surface tensions of mixed-electrolyte aqueous solutions. The electrolytes studied in this work include various valence-types of salts, acids, and alkali. Model 1 was found to be suitable for predicting surface tensions for most of the tested mixed-salt aqueous solutions. However, for the aqueous
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Table 5 A comparison of the predicted surface tensions with the Li et al. (1999) model for aqueous HNO }KNO solutions at ¹"293.15 K m (KNO )
m (HNO )
(mN m\)
1.157 2.442 3.879 1.301
19.397 19.397 19.397 23.806
63.80 64.20 65.30 62.10
Model 1
Model 2
Li et al.
calcd (mN m\)
APD
calcd (mN m\)
APD
calcd (mN m\)
APD
63.58 67.80 73.38 64.25
0.30 5.60 12.40 3.50
62.52 63.72 65.31 62.01
2.00 0.70 0.00 0.10
68.03 73.01 79.93 74.49
6.62 13.72 22.41 20.43
AAPD
5.45
0.70
15.80
Concentration exceeds the maximum molality of KNO in water (m "3.5) used in the correlation of the Pitzer parameters (Kim & Frederick,
1988).
HNO }KNO system, in which the two electrolytes ex hibit opposite e!ect on surface tension with concentration, the competing adsorption model (Model 2) appears to be more suitable.
Notation a AAPD A( b B +6 B +6 C +6 d U D e f f f and f I k k ? k B K m M MX N N N R ¹ X ,X ? A and X A? z ,z + 6
activity average absolute percentage deviation Debye}Huckel parameter for osmotic coef"cient 1.2, a Pitzer parameter for all electrolytes de"ned by Eqs. (18) and (35) de"ned by Eqs. (19) and (36) de"ned by Eq. (22) density of water, kg/m dielectric constant of pure water absolute electron charge de"ned by Eq. (20) de"ned by Eq. (21) de"ned by Eqs. (37) and (38), respectively ionic strength, mol kg\ Boltzmann's constant adsorption rate constant desorption rate constant adsorption equilibrium constant molality, mols per kilogram of water, mol kg\ molarity, mols per liter of solution, mol l\ electrolyte Avogadro's constant number of experimental data points gas constant temperature, K de"ned by Eqs. (32), (33) and (34), respectively charges of positive and negative ion, respectively
Greek letters , , , , +6 +6 C( +6 #* + 6
Pitzer universal constants Pitzer parameters activity coe$cient de"ned by Eq. (31) surface excess fraction of surface covered by electrolyte MX chemical potential # + 6 stoichiometric coe$cient of cation stoichiometric coe$cient of anion surface tension of aqueous electrolyte solution, mN m\
Superscripts 0 ¸ w
standard state or saturated bulk liquid water interface between vapor and liquid
Subscripts a c calcd d exptl i, j max. MX w
anion or adsorption cation calculated value desorption experimental value electrolytes maximum value electrolyte water
Acknowledgements The authors are indebted to the Natural Sciences and Engineering Research Council of Canada (NSERC) for "nancial support.
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References Abramzon, A. A., & Gaukhberg, R. D. (1993). Surface tension of aqueous solutions of inorganic salts. Russian Journal of Applied Chemistry, (Russian), 66(6), 1428}1438; 66(7), 1665}1674; 66(8), 1897}1906; and 66(9), 2145}2156. Adamson, A. W., & Gast, A. P. (1997). Physical chemistry of surfaces (3rd ed.). New York: Wiley Interscience. Aveyard, R., & Saleem, S. M. (1976). Interfacial tensions at alkaneaqueous electrolyte interfaces. Journal of Chemical Society, Faraday Transactions, I72(7), 1609}1617. Desnoyer, C., Masbernat, O., & Gourdon, C. (1997). Predictive model for the calculation of interfacial tension in nonideal electrolytic systems. Journal of Colloid and Interface Science, 191, 22}29. Kim, H. T., & Frederick Jr., W. J. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 253C. 1. Single salt parameters. Journal of Chemical and Engineering Data, 33, 177}184. Langmuir, I. (1918). The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of American Chemical Society, 40, 1361}1403. Li, Z.-B., Li, Y.-G., & Lu, J.-F. (1999). Surface tension model for concentrated electrolyte aqueous solutions by the Pitzer
equation. Industrial Engineering Chemistry Research, 38, 1133}1139. Pitzer, K. S. (1973). Thermodynamics of electrolytes I. Theoretical basis and general equations. Journal of Physical Chemistry, 77, 268}277. Pitzer, K. S., & Kim, J. J. (1974). Thermodynamics of electrolyte. IV. Activity and osmotic coe$cients for mixed electrolytes. Journal of American Chemical Society, 96, 5701}5707. Schmutzer, E. (1955). Zur Theorie der Ober#achenspannung von Losungen Bemerkungen uber den Zusammenhang zwischen radialer Verteilungsfunktion und den thermodynamischen Funktion. Zeitschrift fu( r Physikalische Chemie, 204, 131}156. von Szyszkowski, B. (1908). Experimentelle Studien uber kapillare Eigenschaften der wasserigen Losungen von Fettsauren. Zeitschrift fu( r Physikalische Chemie, 64, 385}414. Weissenborn, P. K., & Pugh, R. J. (1996). Surface tension of aqueous solutions of electrolytes: relationship with ion hydration, oxygen solubility, and bubble coalescence. Journal of Colloid and Interface Science, 184, 550}563. Whit"eld, M. (1975). The extension of chemical models for sea-water to include trace components at 25 and one atmosphere pressure. Geochimica et Cosmochimica Acta, 39, 1545}1557.