Surface Tension Model for Concentrated Electrolyte Aqueous

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Ind. Eng. Chem. Res. 1999, 38, 1133-1139

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Surface Tension Model for Concentrated Electrolyte Aqueous Solutions by the Pitzer Equation Zhi-Bao Li, Yi-Gui Li,* and Jiu-Fang Lu Department of Chemical Engineering, Tsinghua University, Beijing 100084, People’s Republic of China

A new surface tension model based on a thermodynamic analysis on a vapor-liquid surface in an aqueous concentrated electrolyte solution has been proposed. The relation between the surface tensions and the osmotic coefficients of electrolyte solutions which are calculated by the Pitzer equation is established. The surface tensions of 46 single-electrolyte solutions are correlated with only one parameter and the overall average deviation is 1.22%. On the basis of the parameters obtained by correlating the surface tensions of single-electrolyte aqueous solutions, the surface tensions of mixed- and single-electrolyte solutions at different temperatures can also be predicted with deviations of 1.50%. In all cases investigated, good agreement is observed, even for the systems containing high concentrations of electrolytes. Introduction Knowledge about surface tensions of concentrated electrolyte aqueous solutions is of great importance for many processes and phenomena, such as distillation with salts, extraction separation, and liquid-liquid dispersions. In absorption refrigeration and heat pump systems, the surface tension data are also the basic properties to analyze heat and mass transfers of working fluids in the systems. The surface tension affects the transfer rates of vapor absorption where a vaporliquid interface exists. Therefore, the thermodynamic model of surface tensions of electrolyte solutions, especially concentrated electrolyte solutions, is needed for engineering use. In the past decades, many determinations of surface tension of aqueous electrolyte solutions were reported and a lot of calculation models were proposed in the published literature. According to Horvath, the calculation approaches can be divided into three main groups:1 (i) ion additivity method; (ii) conventional thermodynamic method by use of the Gibbs adsorption equation; (iii) calculations starting from a radial distribution for the ions based on statistical mechanical methods. The methods in the first group are most often used in engineering design. This is because no other physical properties are needed, especially when experimental data are not available. Ariyama first proposed the ion additivity method.2,3 According to this method the difference between the surface tension of electrolyte solutions and that of pure water at the same temperature is proportional to the number of respective ions. Another useful method was proposed by Lorenz who showed that the surface tensions of molar aqueous solutions of many inorganic salts are additive functions of the ion species in the system.4 At the same time, Lorenz tabulated the incremental values to the surface tensions of aqueous electrolytes of concentration 1 mol‚kg-1 at 298.15 K. With the aid of the Debye-Huckel theory, the quantitative limiting model of surface tension of a strong electrolyte was derived by Oka.5 Ariyama also proposed a theory of surface tension of Debye-Huckel electrolytes considering the various * To whom correspondence should be addressed. Tel.: 861062784540. Fax: 8610-62770304. E-mail: [email protected].

kinds of ions to be homogeneously distributed in the solutions and obtained a simpler and more practical equation for the increase in the surface tension.6,7 These direct calculation methods are very useful when experimental data are not available. In the second group, probably one of the most successful expressions for the calculation of the surface tension of aqueous electrolyte solutions was introduced by Onsager and Samaras.8 They consider that the increase of the surface tension caused by the addition of a strong electrolyte to water is mainly due to the repulsion of ions from the surface by the electrostatic image force. The Onsager-Samaras equation was later modified by Randles.9 Recently, Stairs modified the Onsager-Samaras expression to include ion-induced dipole terms in the image potential.10 The modified theory agrees well with the experiments. In the third group, one of the most important theories is the Buff-Stilbinger theory.11 While Onsager and Samaras used elegant analytical integration of Wagner theory, Buff and Stilbinger substituted the single-ion and ion-pair distribution function of very dilute ionic solutions into the statistical mechanical formula for surface tension. The model gave a better agreement with the experimental data than the Onsager-Samaras equation, in which the Gibbs adsorption equation was used. Nakamura et al. carried out a self-consistent treatment for the ion distribution near the surface of electrolyte solutions, by solving the Debye-Huckel equation in the Wagner-Kirkwood-Buff (WKB) approximation.12 The surface tension is calculated for electrolyte aqueous solutions with improvement of Onsager and Samaras results. More recently, the modified Poisson-Boltzmann (MPB) theory has been used to study the surface tension of aqueous electrolytes within the framework of the primitive model of the planar electric double layer.13 They show that the primitive model can mimic the surface properties of real systems to a remarkable degree. But the calculation procedure is rather complicated. From the treatment of experimental data of surface tension of strong electrolyte solutions, Abramzon et al. found that the surface tensions are determined by the nature of the anions and their content in the solution while the role of the cation is not significant.14 They gave

10.1021/ie980465m CCC: $18.00 © 1999 American Chemical Society Published on Web 02/11/1999

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For the surface phase (S) of an aqueous electrolyte solution, the expressions of the chemical potentials are changed as follows:

Figure 1. Sketch illustrating the surface phase in an aqueous electrolyte system.

a simple linear relation between the surface tension of solutions and molality of electrolytes. This empirical relation is very useful in predicting the surface tension of concentrated electrolyte aqueous solutions. All of the thermodynamic methods and theories mentioned above are suitable for the dilute electrolyte solutions, but fail in the calculation of surface tension for concentrated electrolyte aqueous solutions. However, most electrolyte systems encountered in chemical industrial processes are concentrated electrolyte aqueous solutions. Therefore, the thermodynamic method of surface tensions for these systems are an industrial design requirement. For this purpose, in the present study we propose a thermodynamic predictive model to calculate the surface tensions of systems containing concentrated single and mixed electrolytes, based on Gibbs thermodynamics and Pitzer’s osmotic coefficient equation for electrolyte aqueous solutions. Thermodynamic Model The Gibbs thermodynamic method or the phenomenological surface-phase method15,16 which we used in this paper, is more empirical, yet applied more easily to complex systems such as the concentrated electrolyte aqueous solutions we have at hand. For an electrolyte aqueous solution-vapor system as shown in Figure 1, there exists a surface layer, located between the bulk liquid and vapor phases, called the surface phase which is regarded as a separate phase instead of a monolayer of molecules. This method differing from the earlier idea that the surface phase contains no electrolyte,17 we consider that it has a constant and uniform electrolyte concentration, but different from that of a bulk liquid phase. At the same time, we must keep in mind that the surface phase of an electrolyte aqueous solution is always electrically neutral. In an electrolyte aqueous solution containing single salt MX, the chemical potentials of water and ions are expressed as follows. For the bulk liquid phase (B), we have B µBw ) µB0 w + RT ln aw

(1)

B µBM ) µB0 M + RT ln aM

(2)

B µBX ) µB0 X + RT ln aX

(3)

B where µBi , µB0 i , and aw are the chemical potential, the standard state chemical potential of the component i, and activity of water in the bulk liquid phase of an electrolyte solution, respectively; R and T (K) are the universal gas constant and absolute temperature of the system.

S µSw ) µS0 h wσsolu w + RT ln aw - A

(4)

S h Mσsolu µSM ) µS0 M + RT ln aM - A

(5)

S h wσsolu µSX ) µS0 X + RT ln aX - A

(6)

S where µSi , µS0 i , and aw represent the chemical potential, the standard state chemical potential of the component i, and activity of water in the surface phase of an electrolyte solution, respectively. A h i is the partial molar area of component i while σsolu is the surface tension of an aqueous solution containing electrolyte MX. The derivation of eqs 4-6 has been made in Butler’s original paper using Gibbs equation as a starting point.18 At equilibrium, the chemical potentials of water in the surface and bulk phase are equal. This yields the following equation of state:

µBw ) µSw

(7)

With eq 7, eq 4 minus eq 1 gives the expression of surface tension for an aqueous electrolyte solution as

σsolu )

Aw A hw

σw +

S RT aw ln B A hw aw

(8)

Here, σw is the surface tension of pure water at system temperature. Assuming the partial molar surface area of water (A h w) in an electrolyte solution to be equal to the molar surface area of pure water (Aw), we have eq 9:

Aw ) A h w ) (Vw)2/3(NA)1/3

(9)

In eq 9, Vw is the molar volume of pure water and NA is Avogadro constant. With the assumption mentioned above, eq 8 is simplified as follows: S RT aw ln B σsolu ) σw + Aw a

(10)

w

Substituting the relation between activity of water and the osmotic coefficient of an electrolyte solution into eq 10,19 we get the surface tension of an electrolyte aqueous solution as eq 11:

σsolu - σw )

RT (ln aSw - ln aBw) Aw

(

)

RT νmS S νmB B φ + φ Aw 55.51 55.51

)

RTν (mBφB - mSφS) Aw55.51

) (11)

Here, mB and mS are molality concentrations of an electrolyte solution in bulk liquid and surface phases; φB and φS represent osmotic coefficients of an electrolyte solution in both phases. Also, ν ) νM + νX, and νM is the stoichiometric coefficient of the cation, and νX is that of the anion.

Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1135

Up to present, we cannot calculate surface tension of an electrolyte aqueous system because the molality concentration of an electrolyte in the surface phase is unknown. For this, we propose two assumptions: (1) The molality of an electrolyte in the surface phase is proportional to that in bulk liquid, that is,

mS ) gmB

(12)

Here, g is a proportional constant, which can be obtained from correlation of the experimental data of the surface tension of a single-electrolyte aqueous solution. (2) The osmotic coefficients of bulk liquid and surface phases are all calculated by Pitzer equations by use of the molality concentrations of an electrolyte aqueous solution in bulk liquid and surface phases:20,21

νMνX (0) AφI0.5 + 2m [βMX + φ - 1 ) -|zMzX| ν (1 + bI0.5) 2(νMνX)3/2 φ 0.5 2 CMX (13) exp(-RI )] + m β(1) MX ν

In the case of a mixed-electrolyte aqueous solution, we consider that the proportional relation proposed in the first assumption is still tenable. Therefore, the expression of surface tension of a pure electrolyte aqueous solution is easily extended to such a mixedelectrolyte system as follows:

σsolu ) σw +

(

φ-1)

0.5

2 DkT

σsolu ) σw +

νRTmB B B [φ (m ) - gφS(gmB)] (14) 55.51Aw

This is an important equation because it provides us with a direct method to calculate the surface tension of a single-electrolyte aqueous solution. g is an empirical parameter, or called an interface parameter, which can be obtained by correlating surface tension data of a single-electrolyte aqueous solution.

(15)

{[If′(I) - f(I)] +

∑i mi 2∑∑mcma[Bca + IB′ca + 2(∑mz)Cca]} c a

(16)

∑mizi

∑mz ) ∑a ma|za| ) ∑c mczc f(I) ) f′(I) ) -2Aφ

1.5

Here, NA is Avogadro’s number; Fw is the density of pure water; D represents the dielectric constant of pure water; k is Boltzmann’s constant;  is the absolute electron charge; T is the absolute temperature (K). In this paper, the reason for choosing Pitzer equations is that probably it is the most widely used formula for the estimation of the osmotic coefficients for electrolyte aqueous solutions. Pitzer parameters for more than hundreds of electrolyte aqueous solutions can be found in the published papers, which is necessary for the calculation of surface tension of electrolyte aqueous systems. From the first assumption mentioned above, eq 11, the expression of surface tension of a single-electrolyte aqueous solution is changed to be

1

I)

)( )

1 2πNAFw 3 1000

∑νimBi - φS∑giνimBi )

Here, σsolu and σw represent the surface tension of a mixed-electrolyte aqueous solution and pure water at system temperature; mBi is the molality concentration of the electrolyte component i in a bulk liquid phase; νi ) νi+ + νi- is the number of ions resulting from one molecule of electrolyte; gi is the empirical parameter for electrolyte i, which is the same as that of the singleelectrolyte aqueous solution at the same temperature. φB and φS are osmotic coefficients of the bulk liquid phase and surface phase and calculated by the simplified Pitzer equations as the following:22

where φ is the osmotic coefficient of an electrolyte aqueous solution; ZM and ZX are charges of positive and negative ions; m is the concentration (molality) of the electrolyte; Aφ is Debye-Huckel parameter for osmotic coefficient, varies with temperature; b ) 1.2 for all electrolytes; I represents ionic strength (mol‚kg-1); (1) φ β(0) MX, βMX, and CMX are Pitzer parameters; R is 2.0 for 1:1, 1:2, and 1:3 electrolytes, but for 2:2 electrolytes, it is equal to 1.4. For 2:2 electrolytes and higher valence -R2I0.5 types, the right second term is added with β(2) MXe where R2 is 12.0. The Debye-Huckel parameter for the osmotic coefficient (Aφ) is given as

Aφ )

RT (φB 55.51Aw

[

Bca ) β(0) ca + B′ca )

4AφI ln(1 + bI0.5) b

I0.5 2 + ln(1 + bI0.5) 0.5 b 1 + bI

]

2β(1) 0.5 ca [1 - (1 + RI0.5)e-RI ] 2 RI

0.5 2β(1) 1 -1 + 1 + RI0.5 + R2I e-RI 2 2 2 RI

[

(

]

)

Cca ) Cφca/2{zcza}0.5 but for 2:2 electrolytes:

Bca ) β(0) ca +

2β(1) 0.5 ca [1 - (1 + R1I0.5)e-R1I ] + 2 R1 I 2β(2) 0.5 ca [1 - (1 + R2I0.5)e-R2I ] 2 R2 I

B′ca )

0.5 2β(1) 1 -1 + 1 + R1I0.5 + R12I e-R1I + 2 2 2 R1 I

[

(

)

]

0.5 2β(2) 1 -1 + 1 + R2I0.5 + R22I e-R2I 2 R22I2

[

(

)

]

where mi is the molality concentration of ion i; c and a (subscripts) are cation and anion, respectively; mc and ma are the molality of the cation and anion.

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Table 1. Calculated Results of Surface Tension and Parameters for Single-Electrolyte Aqueous Solutionsa

system HCl NaCl HNO3 KOH NaOH HClO4 KCl LiCl NaBr NaNO3 CsCl LiBr KI KNO2 NH4Cl NH4NO3 HCOONa NaAc C2H5COONa NaI RbCl KAc KSCN AgNO3 KNO3 KBr ZnCl2 H2SO4 BaCl2 CaCl2 CuSO4 K2CrO4 K2SO4 LaCl3 K3Fe(CN)6 K4Fe(CN)6 MgCl2 MgSO4 MnCl2 (NH4)2SO4 Na2CrO4 Na2SO4 SrCl2 UO2SO4 Zn(NO3)2 ZnSO4

T (K) 293.15 293.15 293.15 291.15 291.15 298.15 293.15 298.15 293.15 293.15 298.15 303.15 298.15 293.15 298.15 293.15 303.15 303.15 303.15 298.15 298.15 303.15 298.15 293.15 298.15 293.15 298.15 298.15 303.15 303.15 303.15 298.15 298.15 298.15 298.15 298.15 293.15 283.15 291.15 303.15 303.15 303.15 293.15 303.15 313.15 298.15

average deviation % a

maximum parameter % concentration g AAD (m) 1.069 35 0.108 98 1.237 25 0.011 99 0.536 82 1.006 47 0.057 12 0.316 07 0.203 03 0.062 80 0.164 98 0.848 35 0.398 99 0.210 82 0.193 10 0.072 58 0.461 71 1.070 64 1.832 75 0.708 73 0.103 32 1.076 34 0.955 74 0.178 69 0.000 14 0 0.241 09 0.844 11 0 0.602 96 0 0 0 0 0 0 0.609 67 0 0.401 18 0 0.358 85 0 0 0 0.766 58 0

0.35 0.52 0.96 0.73 2.51 2.69 0.09 0.56 0.22 0.16 0.57 4.08 0.17 1.41 0.21 0.60 0.59 1.14 0.31 1.17 0.23 4.47 0.04 1.61 0.41 0.13 0.19 0.31 1.15 2.43 2.51 0.89 0.56 0.30 0.49 0.26 1.84 1.64 1.48 1.27 3.49 1.46 0.65 2.04 2.51 2.11

14.98 6.00 19.08 3.80 14.00 25.92 4.39 3.82 2.90 7.00 8.89 17.27 7.29 30.06 5.63 19.36 9.80 10.31 4.40 8.82 6.93 23.79 6.83 6.02 2.18 1.05 2.15 5.16 1.60 6.00 1.10 1.37 1.05 1.03 1.45 0.54 3.50 2.32 5.62 5.06 6.59 1.24 2.81 2.34 7.01 2.86

1.22

All the surface tension data are taken from literature (ref 23).

Figure 2. Surface tension of a HCl aqueous solution versus the electrolyte concentration in the aqueous phase (T ) 293.15 K).

Figure 3. Surface tension of a NaCl aqueous solution versus the electrolyte concentration in the aqueous phase (T ) 293.15 K).

Figure 4. Surface tension of a MnCl2 aqueous solution versus the electrolyte concentration in the aqueous phase (T ) 291.15 K).

Finally, in the case of a mixture of electrolytes, the interface parameters of each electrolyte previously determined in the single-electrolyte aqueous solutions are kept constant in eq 15. By use of eq 15 and simplified Pitzer equations, it is therefore possible to predict the surface tension of concentrated aqueous solutions of mixed electrolytes without any additional parameters. Results and Discussion The solutions of inorganic salts in water have greater surface tension than that of the pure solvent, whereas the surface tension generally decreases with increasing inorganic acid concentration. Equation 14, together with the Pitzer equations, has been applied to 46 pure electrolyte aqueous solutions using modified leastsquares methods to obtain optimum values of the parameter g. The results are shown in Table 1 and Figures 2-5. In this table, the values of g and average

Figure 5. Surface tension of a SrCl2 aqueous solution versus the electrolyte concentration in the aqueous phase (T ) 293.15 K).

deviations of the calculated surface tension from the experimental values23 are presented. In all cases the agreement is quite good with average deviations of 1.22%. Figures 2-5 show the experimental points and

Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1137 Table 2. Predicted Results of Surface Tension for Single-Electrolyte Solutions at Different Temperaturesa T (K)

deviation %

maximum concentration (m)

CsCl

293.15 303.15

0.86 0.74

8.89 8.89

HClO4

288.15 323.15 303.15 313.15 323.15 333.15 343.15 353.15

1.36 2.54 1.08 1.31 1.46 1.45 1.47 1.23

303.15 313.15 323.15 333.15 343.15 353.15

0.70 0.70 0.37 1.25 0.41 0.37

283.15 293.15 313.15 323.15 333.15 343.15 353.15

3.50 3.82 4.28 4.52 4.79 5.02 5.33

293.15 303.15 313.15 323.15 333.15

0.16 0.16 0.41 0.70 1.06

7.29 7.29 7.29 7.29 7.29

293.15 303.15 313.15 323.15 333.15

0.13 0.33 0.23 0.75 1.06

5.63 5.63 5.63 5.63 5.63

NH4NO3

313.15

1.35

NaBr

283.15 303.15 313.15 323.15 333.15 343.15 353.15

1.95 1.60 1.57 1.55 1.73 1.87 2.06

system

KCl

LiBr

KI

NH4Cl

25.9 25.9 19.1 19.1 19.1 19.1 19.1 19.1 3.35 3.35 3.35 3.35 3.35 3.35 17.2 17.2 17.2 17.2 17.2 17.2 17.2

T (K)

deviation %

maximum concentration (m)

NaCl

298.15 303.15 313.15 323.15

0.29 0.32 0.44 0.65

5.49 5.49 5.49 5.49

NaI

293.15 303.15 313.15 323.15

1.08 1.29 1.53 1.66

8.81 8.81 8.81 8.81

RbCl

293.15 303.15

0.27 0.47

6.93 6.93

BaCl2

293.15 313.15 323.15 333.15 343.15 353.15

1.23 0.77 0.79 0.41 0.23 0.31

1.60 1.60 1.60 1.60 1.60 1.60

CaCl2

293.15 313.15 333.15 353.15

2.47 2.44 2.33 2.21

6.00 6.00 6.00 6.00

CuSO4

323.15 343.15 293.15

3.18 4.14 1.96

1.11 1.11 1.11

H2SO4

323.15

2.46

K2CrO4

291.15 303.15

0.32 0.95

1.37 1.37

MgCl2

313.15 323.15 343.15

2.33 2.79 3.33

3.50 3.50 3.50

(NH4)2SO4

313.15 323.15 333.15 343.15 353.15

1.43 1.63 1.52 1.47 1.56

5.06 5.06 5.06 5.06 5.06

Na2SO4

293.15 313.15 323.15 333.15 343.15 353.15

1.57 1.45 1.86 2.04 2.36 2.69

1.24 1.24 1.24 1.24 1.24 1.24

system

19.3 6.47 6.47 6.47 6.47 6.47 6.47 6.47

average deviation % a

24.0

1.80

All the surface tension data are taken from literature (ref 23).

the calculated curves for HCl, NaCl, MnCl2, and SrCl2 aqueous systems. Because the concentration of many systems listed in Table 1 exceeds the applicable concentration range (0-6 m) of the Pitzer equations, the modified Pitzer parameters evaluated by Kim and Frederick24 are used in this paper to calculate the osmotic coefficients both for the bulk phase and surface phase instead of the original Pitzer parameters.20 As shown in Table 1, there are three cases for the values of the interface parameter g. First, g is greater than 1 for most strong inorganic acids, organic acetate, and propionate systems. This case indicates that the electrolyte concentration of the surface phase is larger than that of the bulk liquid phase and this corresponds to a positive adsorption of electrolyte in the surface phase. Second, the parameter g is zero for most non 1:1 salt solutions. In this case, There is no electrolyte in the surface phase, which is consistent

with the early idea of Harkins.25 According to Davies compilation,26 most of those electrolytes whose interface parameters g are zero are of large association constants in water. Those ions associating as ion pairs form dipoles and enter into the bulk phase by the strong attraction force of water dipoles. So these electrolytes are not adsorped on the surface phase. Finally, the model parameter g is less than 1 for all 1:1 salts and some non 1:1 salts. This indicates a deficiency of electrolyte in the surface phase corresponding to a negative adsorption of salts. Our g values are dependent not only on the cation species but also on the anion species because in our model we assume that the cations and anions ionized from an electrolyte are adsorped on the surface phase in the same ratios. So there are different assumptions between Abramzon et al.14 and ours. They supposed that the surface phase is adsorped with anion species and is negatively charged. So their

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Table 3. Predicted Results of Surface Tension for Mixed-Electrolytes Aqueous Solutions system NH4Cl-(NH4)2SO4 (NH4)2SO4-NaNO3 Na2SO4-ZnSO4 KNO3-SrNO3 NH4Cl-SrNO3 NaNO3-Sr(NO3)2 KBr-Sr(NO3)2 KNO3-NH4Cl KNO3-NH4NO3 KBr-KCl KBr-NaBr

LiCl-KCla NaCl-KCla LiCl-NaCla LiCl-NaCl-KCla KBr-KCl-NH4Cl KBr-KNO3-Sr(NO3)3 KBr-NH4Cl-Sr(NO3)2 KNO3-NH4Cl-Sr(NO3)2 NH4Cl-(NH4)2SO4-NaNO3

T (K)

deviation %

Imax (mol‚kg-1)

291.15 291.15 308.15 291.15 291.15 291.15 291.15 291.15 298.15 291.15 303.15 283.15 323.15 343.15 298.15 298.15 298.15 298.15 291.15 291.15 291.15 291.15 291.15

1.35 2.13 2.99 1.38 1.98 2.49 0.95 0.34 1.24 0.40 1.24 0.91 1.43 1.75 0.59 0.41 0.23 0.34 0.37 0.75 0.47 0.49 0.79

11.18 13.43 14.67 3.05 5.13 9.60 3.32 4.84 17.96 4.90 5.82 5.82 5.82 5.82 4.47 3.23 3.21 3.28 3.23 5.70 7.18 6.12 10.79

average deviation %

Figure 7. Surface tension of a KBr(1)-KCl(2) aqueous solution versus the total electrolyte concentration in the aqueous phase (T ) 291.15 K, m1/m2 ) 1).

1.06

a

The surface tension values for these 4 mixed-electrolyte systems were measured experimentally in our laboratory and have not been published yet. The surface tension data for all the other 16 mixed-electrolyte systems are taken from literature (ref 27).

Figure 6. Surface tension of KNO3(1)-NH4Cl(2) aqueous solution versus the total electrolyte concentration in the aqueous phase (T ) 291.15 K, m1/m2 ) 1/2.165).

conclusion is that the surface tension is essentially independent of the cation species for a given anion. Our equation indicates that the surface tension is dependent not only on the interface parameter g but also on the original Pitzer parameters for a specific electrolyte in an aqueous solution. If the interface parameter g is zero, that is, the activity of water in the surface phase (aSw) is equal to 1, the expression of surface tension for electrolyte aqueous solutions, eq 11, is simplified as follows:

σsolu ) σw -

RT ln aBw Aw

(17)

This means that the surface tension is a linear function of the logarithm of the activity of water in the bulk liquid phase. The addition of inorganic salts to water decreases the activity of water (always aBw e 1) and increases the surface tension monotonically, which is consistent with eq 17. Here, eq 17 is a pure prediction model without any model parameters and accurate if g ) 0.

Figure 8. Surface tension of a NH4Cl(1)-(NH4)2SO4(2)-NaNO3(3) aqueous solution versus the total electrolyte concentration in the aqueous phase (T ) 291.15 K, m1:m2:m3 ) 1.54:1.0:1.66).

Figure 9. Surface tension of a KBr(1)-NH4Cl(2)-Sr(NO3)2(3) aqueous solution versus the total electrolyte concentration in the aqueous phase (T ) 291.15 K, m1:m2:m3 ) 1.77:2.6:1).

In the determination of the interface parameter in eq 14, the surface tension data of electrolyte aqueous solutions at only one temperature are used. But with eq 14 and these parameters, we can assume the interface parameters g as constants and predict the surface tension of single-electrolyte solutions at other temperatures with the average deviation of 1.80%. All predicted results compared with the experimental values23 are listed in Table 2. The accuracy is satisfactory to the engineering requirement over a wide range of temperatures (293.15-353.15 K). The interface parameters being identified, the surface tension of solutions composed of mixed electrolytes can be calculated on the basis of eq 15 and the simplified Pitzer equations (eq 16). The surface tensions of 20 mixed-electrolyte aqueous solutions are predicted by use of our model with the AADs of 1.06% compared with the experimental values.27 All results are listed in Table 3 and Figures 6-9 with good accuracy and show that our model is successful for the prediction of surface tension of both single- and mixed-electrolyte solutions.

Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1139

Conclusions

Literature Cited

On the basis of Gibbs thermodynamic analysis to electrolyte aqueous solution containing an interface, a new model has been developed for the calculation of the surface tension of single- and mixed-electrolyte aqueous solutions. With the interface parameters evaluated from correlating the surface tension data of single-electrolyte aqueous solutions only at one temperature, our model can successfully predict the surface tension of systems composed of single electrolytes at other temperatures (with the average deviation of 1.8%) and mixed electrolytes (with the average deviation of 1.06%). It provides a good result for the calculation of the surface tension for the systems studied.

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Acknowledgment Financial support of this work by National Natural Science Foundation of China (No. 62960702) is gratefully appreciated. Nomenclature A ) molar surface area Aφ ) Debye-Huckel parameter for the osmotic coefficient a ) activity B ) Pitzer parameter b ) 1.2, Pitzer parameter for all electrolytes C ) Pitzer parameter D ) dielectric constant of pure water g ) interface parameter I ) ionic strength k ) Boltzmann constant m ) molality concentration NA ) Avogadro’s constant R ) gas constant T ) absolute temperature V ) molar volume ZM, ZX ) charges of positive and negative ions Greek Letters R, R1, R2 ) Pitzer universal constants (0) (1) β(MX) , β(MX) , CφMX ) Pitzer parameters  ) absolute electron charge µ ) chemical potential ν ) ν M + γX νM ) stoichiometric coefficient of a cation νX ) stoichiometric coefficient of an anion F ) density σ ) surface tension φ ) osmotic coefficient Superscripts B ) bulk liquid phase S ) surface phase 0 ) standard state Subscripts a ) anion c ) cation i ) component MX ) electrolyte solu ) electrolyte aqueous solution w ) water

Received for review July 20, 1998 Revised manuscript received December 10, 1998 Accepted December 15, 1998 IE980465M