research article thermodynamic modeling of surface tension

Research ArticleThermodynamic Modeling of Surface Tension ofAqueous Electrolyte Solution by Competitive Adsorption Model

Mohamad Javad Kamali1 Zakarya Kamali2 and Gholamhossein Vatankhah1

1Department of Chemical Engineering Islamic Azad University Bushehr Branch Bushehr 751961955 Iran2National Iranian Gas Company (NIGC) Fajr-e Jam Gas Company Bushehr Iran

Correspondence should be addressed to Mohamad Javad Kamali kamalimjgmailcom

Received 14 August 2015 Accepted 4 October 2015

Academic Editor Marc D Donohue

Copyright copy 2015 Mohamad Javad Kamali et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Thermodynamic modeling of surface tension of different electrolyte systems in presence of gas phase is studied Using thesolid-liquid equilibrium Langmuir gas-solid adsorption and ENRTL activity coefficient model the surface tension of electrolytesolutions is calculatedThe newmodel has two adjustable parameters which could be determined by fitting the experimental surfacetension of binary aqueous electrolyte solution in single temperature Then the values of surface tension for other temperatures inbinary and ternary system of aqueous electrolyte solution are predicted The average absolute deviations for calculation of surfacetension of binary and mixed electrolyte systems by new model are 198 and 170 respectively

1 Introduction

For studying the aqueous electrolyte solution in porousmedia distillation extraction process and liquid-liquiddispersion the calculation of surface tension of aqueoussolutions is required [1] So different equations have beendeveloped to calculate surface tension of mineral saltsAriyama [2] Lorenz [3] Young and Grinstead [4] andGleim and Shelomov [5] formulated useful equations asgroup contribution method for calculation of surface tensionof some limited binary electrolyte-water systems Oka [6]proposed an equation for calculation of surface tension basedon the concentration of solution electronic charge dielectricconstant of water ionic charge and Avogadrorsquos constantLater Hovarth [7] developed this equation by introducing theionic strength and degree of dissociation into Okarsquos model[6] Onsager and Samaras [8] obtained a relation based on thetemperature dielectric constant of water and concentrationof solution for calculation of surface tension of electrolytesolution Schmutzer [9] considered the osmotic coefficientas an important factor for calculation of surface tension ofelectrolyte solution Adding a proportional factor of anionconcentration to the surface tension of water the surfacetension of aqueous electrolyte solution was determined by

Abramazon and Gaukhberg [10] This parameter was con-sidered as a function of the inverse of square of ionic radiusand anion charge Li et al [1] developed a new model forcalculation of surface tension of single and mixed electrolytesolution In this model the surface layer is considered asa distinct phase where the electrolytes could be enteredinto it from other phases The surface tension was obtainedusing the proportion of molality of salt in surface layer toliquid bulk phase While this model had satisfactory resultsin low concentration of electrolytes in high concentrationthe calculated surface tension was not in good agreementwith experimental data Yu et al [11] combined Li et al [1]model with modified mean spherical approximation (MSA)as osmotic coefficient model The results showed that thecalculated surface tension in highly concentrated regions wasimproved Furthermore the Langmuir gas-solid adsorptionmodel was used at equilibrium condition for calculation ofsurface tension of mineral salts by Li and Lu [12] The resultsindicated the satisfactory agreement with experimental dataSadeghi et al [13] used the combination of MSA model [14]with the Ghotbi and Vera [15] and the Mansoori et al [16]equations of state for correlation of the surface tension ofsingle aqueous solution Also the surface tension of differentmixed aqueous solutions was predicted by this approachThe

Hindawi Publishing CorporationJournal of ermodynamicsVolume 2015 Article ID 319704 8 pageshttpdxdoiorg1011552015319704

2 Journal of Thermodynamics

Bulk vapor phase(V)

Surface phase

(S)

Bulk liquid phase

(L)

= H2O + electrolytes


Figure 1 Different phases in aqueous electrolyte solution-vaporsystem [1]

results indicate the satisfactory agreement between calculatedand experimental data [13]

In this paper a new model for calculation of surfacetension of the electrolyte systems is developed using theLangmuir adsorption equation and E-NRTL [17] modelThe adjustable parameters of this model are obtained byexperimental data of surface tension at single temperatureThen the model is verified by prediction of surface tension of65 binary electrolyte-water systems and 17 ternary electrolytesystems

2 Thermodynamic Modeling

For calculation of surface tension of electrolyte system theaqueous electrolyte solution-vapor system is supposed asthree different phases bulk vapor phase surface phase andbulk liquid phase (Figure 1) The surface phase is consideredas distinct layer for adsorption of electrolyte from liquidphase In this system the chemical potential of water in liquidbulk phase and surface would be defined as follows [1]

120583119871

119908= 1205831198710

119908+ 119877119879 ln 119886119871

119908

120583119878

119908= 1205831198780

119908+ 119877119879 ln 119886119878

119908minus 119860119908120590sol

(1)

where 120583 119886 119860 and 120590sol represent the chemical potentialactivity partial molar area and surface tension of solutionThe subscripts 119908 119871 119878 119871

0 and 119878

0refer to water liquid phase

surface phase reference state of liquid phase and referencestate of surface phase respectively

At equilibrium condition the chemical potential of waterin surface and liquid phase is equal So we have

1205831198780

119908minus 1205831198710

119908+ 119877119879 ln 119886119878

119908minus 119877119879 ln 119886119871

119908= 119860119908120590sol (2)

Rewriting the above equation for pure water it yields thefollowing equation

120590119908119860119908= 1205831198780

119908minus 1205831198710

119908 (3)

It would be worth noting that the activity of pure water isunity and the partial molar area for pure water is equal to themolar area

Substituting the above equation into (2) results in thefollowing equation

120590sol = 120590119908 +119877119879

119860119908

ln119886119878

119908

119886119871119908

(4)

It is assumed that in the above equation the partial molar areaand molar area of water are equal The molar area of water iscalculated by [18]

119860119908= 119860119908= (119881119908)23

(119873119860)13

(5)

where 119881119908and 119873

119860are molar volume of water and Avogadro

numberUsing the osmotic definition (120601) instead of activity of

water (4) converts to the following equation

120590sol = 120590119908 +V119877119879

5551119860119908

(119898119871120593119871minus 119898119878120593119878) (6)

where 119898 is molality of electrolyte in aqueous solution and Vis stoichiometric coefficient For calculation of the molalityof electrolyte in surface phase (119898119878) the fraction of adsorbedelectrolyte (120579) on the interface between vapor and liquidphases could be related to the excess area (Γ) as [12]

120579 =Γ

Γ0 (7)

where superscript 0means saturated condition On the otherhand the excess area is defined as moles of electrolyte insurface per area of surface [19] So

120579 =119899119878119860119904

1198990119860119904

=119899119878

1198990=119898119878

1198980 (8)

where 119899 is mole number and superscript 119878 indicates thesurface phase The saturated molality of surface phase isconsidered as fraction of molality of liquid bulk phase or

1198980= 1198700119898119871 (9)

where 119871 is liquid bulk phase and1198700 is a constantUsing the Langmuir gas-solid adsorption [20] and equal-

ity of adsorption and desorption rate for an electrolyte insurface phase [21] the adsorbed fraction of electrolyte on thesurface phase is obtained as [12]

120579 =119870119886

1 + 119870119886 (10)

Combining (6)ndash(8) the molality of electrolyte in surfacephase would be obtained as

119898119878=

119870lowast119886

1 + 119870119886119898119871 (11)

where119870lowast = 1198700119870So substituting (11) into (6) the surface tension of pure

aqueous electrolyte solution is calculated It ismentioned thatthe two adjustable parameters 119870lowast and 119870 are obtained from

Journal of Thermodynamics 3

fitting the experimental surface tension data to the calculatedvalues (see (6)) at single temperature

For mixed electrolyte solution the surface tension wouldbe calculated by the following equation

120590sol = 120590119908 +119877119879

5551119860119908

(120593119871sum

119894

V119894119898119871

119894minus 120593119878sum

119894

V119894119898119878

119894) (12)

Assuming competitive adsorption between electrolytes insurface phase [12] the molality of surface phase is

119898119878

119894=

119870lowast

119894119886119894

1 + sum119895119870119895119886119895

119898119871

119894 (13)

3 Result and Discussion

For studying the surface tension of electrolyte solution 65binary electrolyte systems and 17 ternary electrolyte sys-tems are selected The surface tension of these electrolytesolutions is obtained by competitive adsorption model (see(6) or (12)) In this model for calculation of osmotic andactivity coefficients E-NRTL [17] model is used Using theregression of model with experimental surface tension ofbinary electrolyte-water system the adjustable parameters ofcompetitive adsorption model (119870lowast 119870) are optimized Theobjective function is defined as follows

AAD =100

119873119901

119873119901

sum

119894=1

10038161003816100381610038161003816120590calsol minus 120590

expsol10038161003816100381610038161003816

120590expsol

(14)

where AAD 119873119901 and superscripts cal and exp are average

absolute deviation number of data points and calculated andexperimental data of surface tension respectivelyThe resultsfor different binary electrolyte system are given in Table 1 Soat single temperature for each salt which is given in Table 1the surface tension is correlated

As it is shown in Table 1 when the first adjustableparameter (119870lowast) tends to zero or small positive value or thesecond adjustable parameter (119870) becomes a large numberthe molality of electrolyte system in surface phase is closeto zero Moreover when 119870

lowast is lower than unity and 119870 isnot moderately large number the molality of electrolyte insurface phase decreases with respect to the liquid bulk phaseIn these cases which point out to themineral salts the rate ofadsorption in surface layer would be lower than desorptionrate and surface tension of electrolyte solution is increasedwith respect to the surface tension of water Strong inorganicacids organic acetates and propionates which have highvapor pressure tend to escape from liquid phase to surfacephase and consequently the concentration of these compo-nents in surface phase would be greater than liquid phase [11]For these electrolytes the first adjustable parameter is higherthan unity and so the molality of electrolyte in surface phasebecomes greater than the liquid bulk phase and consequentlythe surface tension of aqueous electrolyte solution increaseswith respect to the liquid bulk phase

For other temperatures the value of surface tension ofelectrolyte aqueous solution is predicted and the results are

50

55

60

65

70

75

Surfa

ce te

nsio

n (m

Nm

)

2 4 6 8 10 12 140AgNO3 molality

Figure 2 Prediction of surface tension of AgNO3-water binary

system at 28315 K using the competitive adsorption model Experi-mental data are taken from [10]

60

62

64

66

68

70

72

74

76

78

80Su

rface

tens

ion

(mN

m)

02 04 06 08 1 12 14 16 180BaCl2 molality

Figure 3 Prediction of surface tension of BaCl2-water binary

system using the competitive adsorption model Experimental dataare taken from [10] (⧫ 119879 = 29815 K 998771 119879 = 32315 K and ◼ 119879 =35315 K)

given in Table 2 Using the competitive adsorption modelthe experimental and predicted values of surface tension ofaqueous binary electrolyte system are illustrated in Figures2ndash8 These figures indicate the agreement between predictedand experimental values of surface tension values for AgNO

3

BaCl2 CaCl

2 KBr HNO

3 NaCl andUO

2SO4using E-NRTL

[17] modelFor ternary systems the surface tension of the aqueous

solutions is predicted in vast range of temperatures andmolalities The values of AAD for prediction of surfacetension of 16 ternary systems by competitive adsorptionmodel are shown in Table 3 The overall AAD for predictionof surface tension of mixed electrolyte solutions is 17


55

60

65

70

75

80

85

90

95

Surfa

ce te

nsio

n (m

Nm

)

2 4 6 80CaCl2 molality

Figure 4 Prediction of surface tension of CaCl2-water binary

system using the competitive adsorption model Experimental dataare taken from [10] (◼ 119879 = 28315 K 998771 119879 = 33315 K and ⧫ 119879 =37315 K)

40

45

50

55

60

65

70

75

Surfa

ce te

nsio

n (m

Nm

)

10 20 30 40 500HNO3 molality

Figure 5 Prediction of surface tension of HNO3-water binary


The predicted values of surface tension of NH4NO3-

KNO3-water KBr-KCl-water and NH

4Cl-(NH

4)2SO4-water

systems by new surface tension model and experimentalvalues are illustrated in Figures 9 10 and 11 respectivelyThe agreement between predicted and experimental valuesin these figures represents the satisfactory results of thecompetitive adsorption model

55

60

65

70

75

80

85

Surfa

ce te

nsio

n (m

Nm

)

1 2 3 4 5 60KBr molality

Figure 6 Prediction of surface tension of KBr-water binary systemusing the competitive adsorption model Experimental data aretaken from [10] (⧫119879=28315 K998771119879=32315 K and◼119879=36315 K)

4500

5000

5500

6000

6500

7000

7500

8000

8500

9000

Surfa

ce te

nsio

n (m

Nm

)

1 2 3 4 5 6 7 8 9 100NaCl molality

Figure 7 Prediction of surface tension of NaCl-water binary systemusing the competitive adsorption model Experimental data aretaken from [10] (998771 119879 = 29115 K ◼ 119879 = 31315 K e 119879 = 33315 K⧫ 119879 = 39315 K ◻ 119879 = 42315 K and I 119879 = 44315 K)

4 Conclusion

Based on the Gibbs thermodynamic and distinct area forsurface phase a new model for calculation of surface tensionof single and mixture electrolyte is developed The molalityin surface phase is calculated using Langmuir gas-solidadsorption theory for electrolytes The osmotic coefficientmodel in the competitive adsorption model is E-NRTLmodel The adjustable parameters of the model are obtained


60

62

64

66

68

70

72

74

76

78

Surfa

ce te

nsio

n (m

Nm

)

05 1 15 2 250UO2SO4 molality

Figure 8 Prediction of surface tension of UO2SO4-water binary

system using the competitive adsorption model Experimental dataare taken from [10] (⧫ 119879 = 29315 K ◼ 119879 = 31815 K 998771 119879 = 33315 Kand e 119879 = 34815 K)

70

72

74

76

78

80

82

84

86

88

90

Surfa

ce te

nsio

n (m

Nm

)

5 10 15 20 250NH4NO3 molality

E-NRTL modelExp data (T = 30

∘C)Exp data (T = 25


∘C)

Figure 9 Prediction of surface tension of NH4NO3-KNO

3-water

ternary system using the competitive adsorption model with non-competitive approach (KNO

3molality = 052) Experimental data

are taken from [10] (⧫ 119879 = 29115 K ◼ 119879 = 29815 K and 998771 119879 =30315 K)

by correlating the experimental values of surface tension ofbinary electrolyte solution in single temperature For othertemperatures and ternary systems competitive adsorptionmodel could predict the surface tension of aqueous solutionThe agreement between experimental and calculated values

73

74

75

76

77

78

79

80

Surfa

ce te

nsio

n (m

Nm

)

05 1 15 20KCl molality

Figure 10 Prediction of surface tension of KBr-KCl-water ternarysystem using the competitive adsorption model at 29115 K (KBrmolalityKCl molality = 1) Experimental data are taken from [10]

74

76

78

80

82

Surfa

ce te

nsio

n (m

Nm

)

1 2 3 4 5 60NH4Cl + (NH4)2SO4 molality

Figure 11 Prediction of surface tension of NH4Cl-(NH

4)2SO4-

water ternary system using the competitive adsorption model at29115 K (NH

4Cl molality(NH

4)2SO4molality = 15) Experimental

data are taken from [10]

of the competitive adsorption model could introduce thisnew model as effective one

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Table 1 The optimized values for adjustable parameters (119870lowast 119870) of competitive adsorption model Experimental data are from [10]

System 119879 (∘C) 119873119901

119870lowast

119870 AADAgNO

320 4 699 times 10minus2 616 times 1014 023

Al2(SO4)3

30 12 895 times 10minus1 393 times 1022 046BaCl2

30 5 0 453 times 104 061CaCl2

30 11 660 times 10minus1 427 times 104 081CdCl2

20 5 734 times 10minus1 219 times 101 008CdSO

420 2 747 times 10minus1 362 times 10minus2 915 times 10minus7

CoCl2


CsCl 25 15 130 times 10minus1 165 times 1016 043CsI 25 11 191 times 10minus6 191 times 10minus6 045CuSO


HBr 18 2 109 times 100 186 times 1016 007HCl 20 7 105 times 100 153 times 101 094HClO

425 10 106 times 100 286 times 1014 213

HNO3

20 7 142 times 100 603 times 1014 238KBr 20 5 236 times 10minus1 602 times 1014 021KC2H3O2

30 3 114 times 100 927 times 10minus2 040KCNS 25 12 936 times 10minus1 471 times 100 034KCl 20 10 107 times 10minus1 196 times 1016 033K2CrO4

30 15 249 times 10minus1 114 times 1018 035K3Fe(CN)

625 16 191 times 10minus6 191 times 10minus6 067

KI 25 12 191 times 10minus6 191 times 10minus6 317KNO


KOH 20 4 307 times 10minus1 846 times 1015 035K2SO4

25 12 191 times 10minus6 191 times 10minus6 072LiBr 30 4 640 times 10minus1 999 times 10minus2 034LiCl 25 7 470 times 10minus1 131 times 101 061LiI 18 2 654 times 10minus1 131 times 100 299 times 10minus6

LiOH 20 4 128 times 10minus1 596 times 1014 040Li2SO4

18 2 436 times 10minus1 357 times 10minus2 015MgCl


MgSO4

10 12 831 times 10minus1 543 times 10minus3 075MnCl


NH4Cl 25 6 290 times 10minus1 568 times 1014 010

NH4NO3

20 9 520 times 10minus1 642 times 10minus2 032(NH4)2SO4

30 4 577 times 10minus1 608 times 100 048NaBr 20 4 485 times 10minus1 177 times 10minus2 021NaCHO


NaC2H3O2

30 12 112 times 100 204 times 1016 104NaC3H5O2


30 12 328 times 100 194 times 1016 077NaCl 20 9 222 times 10minus1 205 times 1016 066NaClO

315 2 138 times 100 431 times 10minus1 003

NaClO4

25 3 861 times 10minus1 216 times 1016 017Na2CrO4

30 4 423 times 10minus1 159 times 10minus2 027NaI 25 8 615 times 10minus1 907 times 1014 016NaNO


NaOH 18 6 347 times 10minus1 246 times 10minus3 046Na2SO4

30 3 150 times 10minus1 563 times 1017 022Na2S2O3



Table 1 Continued

System 119879 (∘C) 119873119901

119870lowast

119870 AADNiSO


Pb(NO3)2

20 3 191 times 10minus6 191 times 10minus6 377RbCl 25 11 194 times 10minus1 965 times 1015 036SrCl2

20 9 595 times 10minus1 441 times 10minus3 049Sr(NO

3)2

18 8 339 times 10minus1 814 times 1014 050UO2SO4

30 12 870 times 10minus1 123 times 101 032Zn(NO

3)2

40 5 767 times 10minus1 127 times 10minus1 037Overall 404 055

Table 2The absolute average deviation percent (AAD) in prediction of surface tension using the competitive adsorption model for binarysystems Experimental data are from [10]

System 119879 (∘C) Molality 119873119901

AADAgNO

3100 103ndash1316 10 381

BaCl2

10ndash80 001ndash596 51 110CaCl2

10ndash100 01ndash737 122 138CsCl 20ndash30 006ndash888 28 052CuSO

410ndash80 033ndash111 21 212

HCl 25ndash90 03ndash3 18 382HClO

415ndash50 051ndash2592 23 256

HNO3

30ndash80 155ndash4686 66 371KBr 10ndash90 044ndash56 40 115KC2H3O2

0ndash80 05ndash2378 36 482KCl 25ndash80 071ndash516 56 103K3Fe(CN)

61235ndash208 029ndash062 4 076

KI 20ndash60 001ndash012 55 445KNO

318ndash100 01ndash263 27 128

KOH 30ndash95 477ndash1328 22 439LiBr 10ndash90 128ndash1727 48 152LiCl 10ndash90 119ndash1573 79 197MgCl

210ndash70 055ndash35 23 290

NH4Cl 19ndash60 102ndash721 39 101

NH4NO3

40ndash95 061ndash2943 29 121(NH4)2SO4

18ndash95 07ndash561 33 207NaBr 10ndash90 00007ndash648 58 314NaC2H3O2

0ndash25 050 6 094NaC4H7O2

0ndash50 050 10 163NaCl 10ndash200 071ndash942 81 173NaI 20ndash50 033ndash881 32 089NaNO

318ndash100 102ndash1177 42 159

NaOH 20ndash70 049ndash625 26 130Na2SO4

10ndash1934 02ndash124 62 287RbCl 20ndash30 011ndash693 22 059SrCl2

10ndash25 048ndash192 7 124UO2SO4

20ndash75 018ndash234 48 081Zn(NO

3)2

21 262 1 110Overall 1215 198


Table 3The absolute average deviation percent (AAD) in prediction of surface tension using the competitive adsorptionmodel for ternarysystems Experimental data are from [10]

Ternary system 1198981

1198982

119879 (∘C) 119873119901

AADBaCl2-HCl 045ndash113 010 25 3 047

CaCl2-HCl 037ndash148 010 25 4 192

LiCl-HCl 055ndash488 010 25 5 223SrCl2-HCl 048ndash192 010 25 4 113

KBr-KCl 035ndash208 034ndash189 18 10 065KBr-NaBr 044ndash357 051ndash42 101ndash908 45 259KNO

3-NH4NO3

052ndash243 012ndash1954 18ndash30 143 693KNO

3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2

023ndash13 019ndash117 18 10 092NH4Cl-(NH

4)2SO4

051ndash286 033ndash204 18 10 063NH4NO3-Pb(NO

3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2

071ndash348 025ndash117 18 9 124NaClO

4-HCl 05ndash134 010 25 3 070

KNO3-NH4Cl 023ndash13 049ndash286 18 10 106

NH4Cl-Sr(NO

3)2

051ndash286 019ndash117 18 10 091(NH4)2SO4-NaNO

3033ndash204 056ndash347 18 9 103

Average 314 170

References

[1] Z Li Y Li and J Lu ldquoSurface tension model for concentratedelectrolyte aqueous solutions by the pitzer equationrdquo Industrialamp Engineering Chemistry Research vol 38 no 3 pp 1133ndash11391999

[2] K Ariyama ldquoA theory of surface tension of aqueous solutionsof inorganic acidsrdquo Bulletin of the Chemical Society of Japan vol12 no 3 pp 109ndash113 1937

[3] P B Lorenz ldquoThe specific adsorption isotherms of thiocyanateand hydrogen ions at the free surface of aqueous solutionsrdquoJournal of Physical and Colloid Chemistry vol 54 no 5 pp 685ndash690 1950

[4] T F Young and S R Grinstead ldquoThe surface tensions of aque-ous sulfuric acid solutionsrdquo Annals of the New York Academy ofSciences vol 51 pp 765ndash780 1949

[5] V G Gleim and I K Shelomov ldquoCalculation of surfacetension of aqueous salt systemsrdquoThe Russian Journal of AppliedChemistry vol 30 pp 29ndash35 1957

[6] S Oka ldquoAdsorption and surface tension of strong electrolytesrdquoProceedings of the Physico-Mathematical Society of Japan vol 14pp 649ndash664 1932

[7] A LHorvathAqueous Electrolyte Solutions Physical PropertiesEstimation and Correlation Methods Halsted Press ChichesterUK 1985

[8] L Onsager andN N T Samaras ldquoThe surface tension of debye-huckel electrolytesrdquoThe Journal of Chemical Physics vol 2 no8 pp 528ndash536 1934

[9] E Schmutzer ldquoIons at liquidair and liquidliquid interfacesrdquoZeitschrift fur Physikalische Chemie vol 204 pp 131ndash156 1955

[10] A A Abramzon and R D Gaukhberg ldquoSurface tension of saltsolutionsrdquo Russian Journal of Applied Chemistry vol 66 no 6ndash9 pp 1428ndash2156 1993 (Russian)

[11] Y Yu G Gao and Y Li ldquoSurface tension for aqueous electrolytesolutions by themodifiedmean spherical approximationrdquo FluidPhase Equilibria vol 173 no 1 pp 23ndash28 2000

[12] Z Li and B C Lu ldquoSurface tension of aqueous electrolyte solu-tions at high concentrationsmdashrepresentation and predictionrdquoChemical Engineering Science vol 56 no 8 pp 2879ndash28882001

[13] M Sadeghi V Taghikhani and C Ghotbi ldquoApplication ofthe MSA-based models in correlating the surface tension forsingle and mixed electrolyte solutionsrdquo Journal of ChemicalThermodynamics vol 41 no 11 pp 1264ndash1271 2009

[14] L L LeeMolecularThermodynamics of Nonideal Fluids Butter-worth Publishers 1988

[15] C Ghotbi and J H Vera ldquoExtension to mixtures of two robusthard-sphere equations of state satisfying the ordered close-packed limitrdquo The Canadian Journal of Chemical Engineeringvol 79 no 4 pp 678ndash686 2001

[16] G AMansoori N F Carnahan K E Starling and TW LelandJr ldquoEquilibrium thermodynamic properties of the mixture ofhard spheresrdquo The Journal of Chemical Physics vol 54 no 4pp 1523ndash1526 1971

[17] C-C Chen and L B Evans ldquoLocal composition model forthe excess gibbs energy of aqueous electrolyte systemsrdquo AIChEJournal vol 32 no 3 pp 444ndash454 1986

[18] F B Sprow and J M Prausnitz ldquoSurface thermodynamics ofliquidmixturesrdquoTheCanadian Journal of Chemical Engineeringvol 45 no 1 pp 25ndash28 1967

[19] A W Adamson and A P Gast Physical Chemistry of SurfacesWiley Interscience New York NY USA 3rd edition 1997

[20] I Langmuir ldquoThe adsorption of gases on plane surfaces ofglassmica and platinumrdquoThe Journal of the AmericanChemicalSociety vol 40 no 9 pp 1361ndash1403 1918

[21] C Desnoyer OMasbernat and C Gourdon ldquoPredictivemodelfor the calculation of interfacial tension in nonideal electrolyticsystemsrdquo Journal of Colloid and Interface Science vol 191 no 1pp 22ndash29 1997

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ThermodynamicsJournal of


Bulk vapor phase(V)

Surface phase

(S)

Bulk liquid phase

(L)



Figure 1 Different phases in aqueous electrolyte solution-vaporsystem [1]

results indicate the satisfactory agreement between calculatedand experimental data [13]

In this paper a new model for calculation of surfacetension of the electrolyte systems is developed using theLangmuir adsorption equation and E-NRTL [17] modelThe adjustable parameters of this model are obtained byexperimental data of surface tension at single temperatureThen the model is verified by prediction of surface tension of65 binary electrolyte-water systems and 17 ternary electrolytesystems

2 Thermodynamic Modeling

For calculation of surface tension of electrolyte system theaqueous electrolyte solution-vapor system is supposed asthree different phases bulk vapor phase surface phase andbulk liquid phase (Figure 1) The surface phase is consideredas distinct layer for adsorption of electrolyte from liquidphase In this system the chemical potential of water in liquidbulk phase and surface would be defined as follows [1]

120583119871

119908= 1205831198710

119908+ 119877119879 ln 119886119871

119908

120583119878

119908= 1205831198780

119908+ 119877119879 ln 119886119878

119908minus 119860119908120590sol

(1)

where 120583 119886 119860 and 120590sol represent the chemical potentialactivity partial molar area and surface tension of solutionThe subscripts 119908 119871 119878 119871

0 and 119878

0refer to water liquid phase

surface phase reference state of liquid phase and referencestate of surface phase respectively

At equilibrium condition the chemical potential of waterin surface and liquid phase is equal So we have

1205831198780

119908minus 1205831198710

119908+ 119877119879 ln 119886119878

119908minus 119877119879 ln 119886119871

119908= 119860119908120590sol (2)

Rewriting the above equation for pure water it yields thefollowing equation

120590119908119860119908= 1205831198780

119908minus 1205831198710

119908 (3)

It would be worth noting that the activity of pure water isunity and the partial molar area for pure water is equal to themolar area

Substituting the above equation into (2) results in thefollowing equation

120590sol = 120590119908 +119877119879

119860119908

ln119886119878

119908

119886119871119908

(4)

It is assumed that in the above equation the partial molar areaand molar area of water are equal The molar area of water iscalculated by [18]

119860119908= 119860119908= (119881119908)23

(119873119860)13

(5)

where 119881119908and 119873

119860are molar volume of water and Avogadro

numberUsing the osmotic definition (120601) instead of activity of

water (4) converts to the following equation

120590sol = 120590119908 +V119877119879

5551119860119908

(119898119871120593119871minus 119898119878120593119878) (6)

where 119898 is molality of electrolyte in aqueous solution and Vis stoichiometric coefficient For calculation of the molalityof electrolyte in surface phase (119898119878) the fraction of adsorbedelectrolyte (120579) on the interface between vapor and liquidphases could be related to the excess area (Γ) as [12]

120579 =Γ

Γ0 (7)

where superscript 0means saturated condition On the otherhand the excess area is defined as moles of electrolyte insurface per area of surface [19] So

120579 =119899119878119860119904

1198990119860119904

=119899119878

1198990=119898119878

1198980 (8)

where 119899 is mole number and superscript 119878 indicates thesurface phase The saturated molality of surface phase isconsidered as fraction of molality of liquid bulk phase or

1198980= 1198700119898119871 (9)

where 119871 is liquid bulk phase and1198700 is a constantUsing the Langmuir gas-solid adsorption [20] and equal-

ity of adsorption and desorption rate for an electrolyte insurface phase [21] the adsorbed fraction of electrolyte on thesurface phase is obtained as [12]

120579 =119870119886

1 + 119870119886 (10)

Combining (6)ndash(8) the molality of electrolyte in surfacephase would be obtained as

119898119878=

119870lowast119886

1 + 119870119886119898119871 (11)

where119870lowast = 1198700119870So substituting (11) into (6) the surface tension of pure

aqueous electrolyte solution is calculated It ismentioned thatthe two adjustable parameters 119870lowast and 119870 are obtained from




120590sol = 120590119908 +119877119879

5551119860119908

(120593119871sum

119894

V119894119898119871

119894minus 120593119878sum

119894

V119894119898119878

119894) (12)


119898119878

119894=

119870lowast

119894119886119894

1 + sum119895119870119895119886119895

119898119871

119894 (13)



AAD =100

119873119901

119873119901

sum

119894=1

10038161003816100381610038161003816120590calsol minus 120590

expsol10038161003816100381610038161003816

120590expsol

(14)






50

55

60

65

70

75

Surfa

ce te

nsio

n (m

Nm

)

2 4 6 8 10 12 140AgNO3 molality



60

62

64

66

68

70

72

74

76

78

80Su

rface

tens

ion

(mN

m)

02 04 06 08 1 12 14 16 180BaCl2 molality




3

BaCl2 CaCl

2 KBr HNO

3 NaCl andUO

2SO4using E-NRTL




55

60

65

70

75

80

85

90

95

Surfa

ce te

nsio

n (m

Nm

)




40

45

50

55

60

65

70

75

Surfa

ce te

nsio

n (m

Nm

)

10 20 30 40 500HNO3 molality





4Cl-(NH

4)2SO4-water


55

60

65

70

75

80

85

Surfa

ce te

nsio

n (m

Nm

)



4500

5000

5500

6000

6500

7000

7500

8000

8500

9000

Surfa

ce te

nsio

n (m

Nm

)

1 2 3 4 5 6 7 8 9 100NaCl molality


4 Conclusion



60

62

64

66

68

70

72

74

76

78

Surfa

ce te

nsio

n (m

Nm

)




70

72

74

76

78

80

82

84

86

88

90

Surfa

ce te

nsio

n (m

Nm

)





∘C)


3-water





73

74

75

76

77

78

79

80

Surfa

ce te

nsio

n (m

Nm

)



74

76

78

80

82

Surfa

ce te

nsio

n (m

Nm

)



4)2SO4-


4Cl molality(NH








System 119879 (∘C) 119873119901

119870lowast

119870 AADAgNO


Al2(SO4)3


30 5 0 453 times 104 061CaCl2




CoCl2





425 10 106 times 100 286 times 1014 213

HNO3












MgSO4




NH4NO3




NaC2H3O2





NaClO4








Table 1 Continued

System 119879 (∘C) 119873119901

119870lowast

119870 AADNiSO


Pb(NO3)2



3)2



3)2




AADAgNO

3100 103ndash1316 10 381

BaCl2



410ndash80 033ndash111 21 212


415ndash50 051ndash2592 23 256

HNO3



61235ndash208 029ndash062 4 076


318ndash100 01ndash263 27 128


210ndash70 055ndash35 23 290


NH4NO3

40ndash95 061ndash2943 29 121(NH4)2SO4


0ndash25 050 6 094NaC4H7O2


318ndash100 102ndash1177 42 159





3)2

21 262 1 110Overall 1215 198




1198982

119879 (∘C) 119873119901


CaCl2-HCl 037ndash148 010 25 4 192



3-NH4NO3


3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2


4)2SO4


3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2


4-HCl 05ndash134 010 25 3 070


NH4Cl-Sr(NO

3)2


3033ndash204 056ndash347 18 9 103

Average 314 170

References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120590sol = 120590119908 +119877119879

5551119860119908

(120593119871sum

119894

V119894119898119871

119894minus 120593119878sum

119894

V119894119898119878

119894) (12)


119898119878

119894=

119870lowast

119894119886119894

1 + sum119895119870119895119886119895

119898119871

119894 (13)



AAD =100

119873119901

119873119901

sum

119894=1

10038161003816100381610038161003816120590calsol minus 120590

expsol10038161003816100381610038161003816

120590expsol

(14)






50

55

60

65

70

75

Surfa

ce te

nsio

n (m

Nm

)

2 4 6 8 10 12 140AgNO3 molality



60

62

64

66

68

70

72

74

76

78

80Su

rface

tens

ion

(mN

m)

02 04 06 08 1 12 14 16 180BaCl2 molality




3

BaCl2 CaCl

2 KBr HNO

3 NaCl andUO

2SO4using E-NRTL




55

60

65

70

75

80

85

90

95

Surfa

ce te

nsio

n (m

Nm

)




40

45

50

55

60

65

70

75

Surfa

ce te

nsio

n (m

Nm

)

10 20 30 40 500HNO3 molality





4Cl-(NH

4)2SO4-water


55

60

65

70

75

80

85

Surfa

ce te

nsio

n (m

Nm

)



4500

5000

5500

6000

6500

7000

7500

8000

8500

9000

Surfa

ce te

nsio

n (m

Nm

)

1 2 3 4 5 6 7 8 9 100NaCl molality


4 Conclusion



60

62

64

66

68

70

72

74

76

78

Surfa

ce te

nsio

n (m

Nm

)




70

72

74

76

78

80

82

84

86

88

90

Surfa

ce te

nsio

n (m

Nm

)





∘C)


3-water





73

74

75

76

77

78

79

80

Surfa

ce te

nsio

n (m

Nm

)



74

76

78

80

82

Surfa

ce te

nsio

n (m

Nm

)



4)2SO4-


4Cl molality(NH








System 119879 (∘C) 119873119901

119870lowast

119870 AADAgNO


Al2(SO4)3


30 5 0 453 times 104 061CaCl2




CoCl2





425 10 106 times 100 286 times 1014 213

HNO3












MgSO4




NH4NO3




NaC2H3O2





NaClO4








Table 1 Continued

System 119879 (∘C) 119873119901

119870lowast

119870 AADNiSO


Pb(NO3)2



3)2



3)2




AADAgNO

3100 103ndash1316 10 381

BaCl2



410ndash80 033ndash111 21 212


415ndash50 051ndash2592 23 256

HNO3



61235ndash208 029ndash062 4 076


318ndash100 01ndash263 27 128


210ndash70 055ndash35 23 290


NH4NO3

40ndash95 061ndash2943 29 121(NH4)2SO4


0ndash25 050 6 094NaC4H7O2


318ndash100 102ndash1177 42 159





3)2

21 262 1 110Overall 1215 198




1198982

119879 (∘C) 119873119901


CaCl2-HCl 037ndash148 010 25 4 192



3-NH4NO3


3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2


4)2SO4


3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2


4-HCl 05ndash134 010 25 3 070


NH4Cl-Sr(NO

3)2


3033ndash204 056ndash347 18 9 103

Average 314 170

References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




55

60

65

70

75

80

85

90

95

Surfa

ce te

nsio

n (m

Nm

)




40

45

50

55

60

65

70

75

Surfa

ce te

nsio

n (m

Nm

)

10 20 30 40 500HNO3 molality





4Cl-(NH

4)2SO4-water


55

60

65

70

75

80

85

Surfa

ce te

nsio

n (m

Nm

)



4500

5000

5500

6000

6500

7000

7500

8000

8500

9000

Surfa

ce te

nsio

n (m

Nm

)

1 2 3 4 5 6 7 8 9 100NaCl molality


4 Conclusion



60

62

64

66

68

70

72

74

76

78

Surfa

ce te

nsio

n (m

Nm

)




70

72

74

76

78

80

82

84

86

88

90

Surfa

ce te

nsio

n (m

Nm

)





∘C)


3-water





73

74

75

76

77

78

79

80

Surfa

ce te

nsio

n (m

Nm

)



74

76

78

80

82

Surfa

ce te

nsio

n (m

Nm

)



4)2SO4-


4Cl molality(NH








System 119879 (∘C) 119873119901

119870lowast

119870 AADAgNO


Al2(SO4)3


30 5 0 453 times 104 061CaCl2




CoCl2





425 10 106 times 100 286 times 1014 213

HNO3












MgSO4




NH4NO3




NaC2H3O2





NaClO4








Table 1 Continued

System 119879 (∘C) 119873119901

119870lowast

119870 AADNiSO


Pb(NO3)2



3)2



3)2




AADAgNO

3100 103ndash1316 10 381

BaCl2



410ndash80 033ndash111 21 212


415ndash50 051ndash2592 23 256

HNO3



61235ndash208 029ndash062 4 076


318ndash100 01ndash263 27 128


210ndash70 055ndash35 23 290


NH4NO3

40ndash95 061ndash2943 29 121(NH4)2SO4


0ndash25 050 6 094NaC4H7O2


318ndash100 102ndash1177 42 159





3)2

21 262 1 110Overall 1215 198




1198982

119879 (∘C) 119873119901


CaCl2-HCl 037ndash148 010 25 4 192



3-NH4NO3


3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2


4)2SO4


3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2


4-HCl 05ndash134 010 25 3 070


NH4Cl-Sr(NO

3)2


3033ndash204 056ndash347 18 9 103

Average 314 170

References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




60

62

64

66

68

70

72

74

76

78

Surfa

ce te

nsio

n (m

Nm

)




70

72

74

76

78

80

82

84

86

88

90

Surfa

ce te

nsio

n (m

Nm

)





∘C)


3-water





73

74

75

76

77

78

79

80

Surfa

ce te

nsio

n (m

Nm

)



74

76

78

80

82

Surfa

ce te

nsio

n (m

Nm

)



4)2SO4-


4Cl molality(NH








System 119879 (∘C) 119873119901

119870lowast

119870 AADAgNO


Al2(SO4)3


30 5 0 453 times 104 061CaCl2




CoCl2





425 10 106 times 100 286 times 1014 213

HNO3












MgSO4




NH4NO3




NaC2H3O2





NaClO4








Table 1 Continued

System 119879 (∘C) 119873119901

119870lowast

119870 AADNiSO


Pb(NO3)2



3)2



3)2




AADAgNO

3100 103ndash1316 10 381

BaCl2



410ndash80 033ndash111 21 212


415ndash50 051ndash2592 23 256

HNO3



61235ndash208 029ndash062 4 076


318ndash100 01ndash263 27 128


210ndash70 055ndash35 23 290


NH4NO3

40ndash95 061ndash2943 29 121(NH4)2SO4


0ndash25 050 6 094NaC4H7O2


318ndash100 102ndash1177 42 159





3)2

21 262 1 110Overall 1215 198




1198982

119879 (∘C) 119873119901


CaCl2-HCl 037ndash148 010 25 4 192



3-NH4NO3


3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2


4)2SO4


3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2


4-HCl 05ndash134 010 25 3 070


NH4Cl-Sr(NO

3)2


3033ndash204 056ndash347 18 9 103

Average 314 170

References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





System 119879 (∘C) 119873119901

119870lowast

119870 AADAgNO


Al2(SO4)3


30 5 0 453 times 104 061CaCl2




CoCl2





425 10 106 times 100 286 times 1014 213

HNO3












MgSO4




NH4NO3




NaC2H3O2





NaClO4








Table 1 Continued

System 119879 (∘C) 119873119901

119870lowast

119870 AADNiSO


Pb(NO3)2



3)2



3)2




AADAgNO

3100 103ndash1316 10 381

BaCl2



410ndash80 033ndash111 21 212


415ndash50 051ndash2592 23 256

HNO3



61235ndash208 029ndash062 4 076


318ndash100 01ndash263 27 128


210ndash70 055ndash35 23 290


NH4NO3

40ndash95 061ndash2943 29 121(NH4)2SO4


0ndash25 050 6 094NaC4H7O2


318ndash100 102ndash1177 42 159





3)2

21 262 1 110Overall 1215 198




1198982

119879 (∘C) 119873119901


CaCl2-HCl 037ndash148 010 25 4 192



3-NH4NO3


3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2


4)2SO4


3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2


4-HCl 05ndash134 010 25 3 070


NH4Cl-Sr(NO

3)2


3033ndash204 056ndash347 18 9 103

Average 314 170

References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Table 1 Continued

System 119879 (∘C) 119873119901

119870lowast

119870 AADNiSO


Pb(NO3)2



3)2



3)2




AADAgNO

3100 103ndash1316 10 381

BaCl2



410ndash80 033ndash111 21 212


415ndash50 051ndash2592 23 256

HNO3



61235ndash208 029ndash062 4 076


318ndash100 01ndash263 27 128


210ndash70 055ndash35 23 290


NH4NO3

40ndash95 061ndash2943 29 121(NH4)2SO4


0ndash25 050 6 094NaC4H7O2


318ndash100 102ndash1177 42 159





3)2

21 262 1 110Overall 1215 198




1198982

119879 (∘C) 119873119901


CaCl2-HCl 037ndash148 010 25 4 192



3-NH4NO3


3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2


4)2SO4


3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2


4-HCl 05ndash134 010 25 3 070


NH4Cl-Sr(NO

3)2


3033ndash204 056ndash347 18 9 103

Average 314 170

References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1198982

119879 (∘C) 119873119901


CaCl2-HCl 037ndash148 010 25 4 192



3-NH4NO3


3-Pb(NO

3)2

0-1 0-1 20 33 266KNO

3-Sr(NO

3)2


4)2SO4


3)2

033 0ndash0333 35 6 207NaNO

3-Sr(NO

3)2


4-HCl 05ndash134 010 25 3 070


NH4Cl-Sr(NO

3)2


3033ndash204 056ndash347 18 9 103

Average 314 170

References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article thermodynamic modeling of surface tension

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