uids 135 (2007) 1–4www.elsevier.com/locate/molliq

Journal of Molecular Liq

Excess surface tension and molecular interactions of Pb–Sn moltenmixture at elevated temperatures

R.K. Shukla ⁎, Ashish Narain Dubey, Piyush Awasthi

Department of Chemistry, V.S.S.D. College, Nawabganj, Kanpur 208002, India

Received 18 October 2005; accepted 12 July 2006Available online 7 March 2007

Abstract

Surface tension and excess surface tension of Pb–Sn molten mixtures have been evaluated theoretically over a wide range of temperature andcompositions with the use of the statistical mechanical theory of Flory. A reasonable agreement has been obtained between theory and experimentfor 10, 20, 30, 38, 1, 45, 60 and 80% Pb–Sn mixtures at 400, 600 and 700 °C temperature. A study of molecular interactions has also been madewhich agrees well both in sign and magnitude over a wide range of temperature and composition.© 2007 Published by Elsevier B.V.

Keywords: Thermodynamics; Surface tension; Excess surface tension; Interactions; Molten mixtures

1. Introduction

A better understanding of the surface thermodynamics of aliquid mixture [1–5] is of considerable physico-chemicalinterest and is essential in designing calculation involvingseparations, heat transfer, mass transfer and fluid flow. Asubstantial amount of work has been done on binary and ternary[6–12] liquid mixtures and is still in progress. The physico-chemical properties of a multicomponent system can be studiedusing the Bertrand–Acree–Bruchfield [13,14] (BAB) equationas well as the statistical mechanical concept of Flory [15–18]. InFlory's theory, the properties of a binary system are based on theproperties of pure component liquids. Pressure and temperaturedependent data on surface properties are very rare except for afew measurements of Takagi et al. [19–22]. It is of interest toemploy the Flory theory for a theoretical evaluation of thesurface tension of Pb–Sn molten binary liquid mixtures atvarious compositions and elevated temperature from morereadily available thermodynamic data. To our knowledge, verylittle effort has been made to study the surface tension of moltenbinary mixtures at elevated temperatures. In this paper wepresent the theoretical evaluation of surface tension and excesssurface tension for Pb–Sn molten binary liquids mixture at

⁎ Corresponding author.

0167-7322/$ - see front matter © 2007 Published by Elsevier B.V.doi:10.1016/j.molliq.2006.07.011

different compositions and elevated temperatures with the useof the statistical mechanical concept of Flory. We hope thataccurate and real information of surface thermodyanamics willultimately lead to a better understanding of the molecularconfiguration and molecular interactions of liquid mixtures.

2. Theoretical

The statistical concept of Flory [15,16] yields the followingexpression for the surface tension of a pure liquid and itsmixtures

r ¼ r⁎reðmeÞ ð1Þwhere σ, σ⁎, σ and ν are the surface tension, characteristicssurface tension, reduced surface tension and reduced volumerespectively. Flory's theory is closely related with the corre-sponding state theory [23] of Prigogine, which employs asimple cell model of the liquid state. Patterson and Rastogi [17]in their extension of the corresponding state theory, dealt withthe surface tension in terms of reduced parameters

r⁎ ¼ k1=3P⁎2=3T

⁎1=3 ð2Þwhere P⁎ is the characteristics pressure and T⁎ is thecharacteristics temperature.

Table 1Parameters of the pure component at various temperatures

Component Temperature(°C)

Thermalexpansioncoefficient(α×105) K

Isothermalcompressibility(βT×10

5) Kcm3 dyne−1

Molarvolume (v)cm3 mol−1

Reducedvolume (v~)cm3 mol−1

Characteristicpressure P⁎ Kbar

CharacteristictemperatureT⁎ K

Experimentalsurface tensionσexp dyne cm−1

Theoreticalsurfacetension σtheo

dyne cm−1

Sn 400 8.8600 2.7880 17.3097 1.0575 23.960 28561.8 523 527.23600 8.9986 3.0500 17.7079 1.0747 29.744 39578.8 507 505.71700 9.0997 3.1540 17.8910 1.0863 32.960 39940.3 501 495.2

Pb 400 12.4533 3.6500 19.7351 1.0794 26.762 28897.4 481 480.07600 12.7670 4.1950 20.1441 1.1037 32.781 29778.8 452 442.31700 12.8993 4.4900 20.4172 1.1158 34.810 308001.3 418 421.00

2 R.K. Shukla et al. / Journal of Molecular Liquids 135 (2007) 1–4

Starting from the work of Prigogine and Saraga [24], theyderived a reduced surface tension equation in the case of a vander Waals liquid, which can be written as

reðmÞ ¼ Mme−5=3− ðme1=3−1Þ=me2� �ln ðme1=3−0:5Þ=ðme1=3−1Þ� �h i

ð3ÞAll the notations in the above equation have their usual

significance as explained in our earlier papers [7–10]. The mostsuitable value [18] of M is 0.29 but we have taken the values ofM as 0.46 at 400 °C, 0.41 at 600 °C and 0.39 at 700 °C forcalculations in the present study.

The molecular element or segment is to be defined in corres-pondence with the two neighboring species such that r1 and r2 shallbe in the ratio of the respective molar core volumes ν1⁎ and ν2⁎.Similarly s1 and s2 shall be in the ratio of the molecular surface areof contact per segment. LetA11,A12 andA22f represent the numbersof contact pairs between the respective species and let η11/ν be theenergies associated with each, then the intermolecular energy of thebinary liquid mixture can be represented as

−E0 ¼ ðA11g11 þ A22g22 þ A12g12Þm

ð4Þ

It is assumed that random mixing of the two species takesplace and that a species of two neighbours at any given site isequal to its site fraction θ2 which is defined as

h2 ¼ 1−h1 ¼ s2r2N2

srNð5Þ

By adopting the Berthlot relationship, η12= (η11 η33)1/2 and

assuming X22=X21

we obtain

X12 ¼ P1⁎ 1−

P2⁎

P1⁎

� �1=2s1s2

� �1=2" #2

ð6Þ

The characteristics pressure and characteristics temperatureof binary mixture are obtained as

P⁎ ¼ ½w1P1⁎þ w2P2

⁎−w1h2X12� ð7Þ

T⁎ ¼ P⁎

w1P1⁎

T1⁎ þ w2P2

⁎

T2⁎

� � ð8Þ

where P⁎, Ψ, θ, and X12 are the characteristics pressure,segment fraction, site fraction and interaction parameterrespectively.

Assuming the volume reduction parameter of a binary mixtureto be linear in mole fractions of the component, we have

me ¼ m

ðx1m1⁎þ x2m2⁎Þð9Þ

3. Results and discussion

Parameters of pure components at elevated temperatures forPb–Snmolten binarymixtures are given in Table 1while values ofreduced volume (v~) characteristics pressure (P⁎), characteristicstemperature (T⁎), characteristics surface tension (σ⁎), excesssurface tension (σE), calculated and observed surface tension ofthe Pb–Sn system and their percentage deviations are presented inTable 2 over a wide range of temperatures and compositions. Theessential data required for the evaluation of these values have beentaken from the literature [25]. Surface tension of Pb–Sn mixturehas been computed with the help of the reduced surface tensionequation. In this equation the most suitable value of M variesindifferently from temperature to temperature. Prigogine andSarga [24] suggested the most suitable value of M forhydrocarbons ranging between 0.25 and 0.29. From hydrocarbonsto Pb–Sn molten mixtures structures of the liquid moleculeschange abruptly i.e. their lattice structure becomes so prominentthat it changes thewhole liquid geometry and deviation in themostsuited values has been observed. That is why, in the present study,we have taken the most suited value (M ) for the Pb–Sn system at400 °C as 0.46, at 600 °C as 0.41 and at 700 °C as 0.39. Theoreticaland experimental values are presented graphically in Figs. 1 and 2as a function of temperature for different mole fractions.

A careful perusal of Table 2 reveals that all the parametersi.e. reduced volume, characteristic pressure, characteristic tem-perature and characteristic surface tension are increasing as thetemperature and composition of Pb are increasing whereas thevalues of calculated surface tension of the Pb–Sn system aredecreasing. Percent deviations between experimental and theoret-ical values of surface tension of the Pb–Sn binary mixture havebeen depicted in the last column of Table 2. Minimum andmaximum percentage deviation have been observed as 0.24 and8.23 for 10%Pb at 600 °C and 45%Pb at 400 °C, respectively. Theoverall average percent deviation for the Pb–Sn system over a

Table 2Parameters of the Pb–Sn molten binary mixture

ComponentX1 (Sn)

Temperature(°C)

ReducedVolume (v~)cm3 mol−1

Characteristicpressure (P⁎)K bar

Characteristictemperature(T⁎) K

Characteristicsurface tensionσ⁎ dyne cm−1

Experimentalsurface tension(σexp) dyne cm−1

Theoreticalsurface tension(σtheo) dyne cm−1

σtheoE

dynecm−1

Percentagedeviation(σexp−σtheo)×100 /σexp%Δ

Δσ

10% 400 1.0631 25.629 35451.61 1475.23 519 525.87 +41.08 −1.32 −0.132600 1.0838 32.502 35748.29 1744.60 507 505.79 +57.14 −0.24 +0.024700 1.0930 35.765 36441.88 1860.05 502 491.25 +62.83 −2.14 −0.0214

20% 400 1.0659 26.473 34073.82 1488.28 501 525.70 +36.2 −4.93 −0.0493600 1.0856 33.018 35085.65 1741.38 497 51.81 +46.82 −0.96 −0.0096700 1.0955 36.149 35604.77 1858.89 495 486.76 +50.92 −1.66 −0.0166

30% 400 1.0698 26.956 32368.20 1480.77 482 52.89 +8.67 −4.33 −0.0433600 1.0005 33.621 33434.81 1734.42 478 491.78 +30.45 −2.88 −0.0288700 1.1011 36.437 33911.03 1838.61 476 472.55 +29.29 −0.72 −0.0072

38.1% 400 1.0694 26.685 32521.43 1473.15 510 514.39 +16.35 − .86 −0.0086600 1.0905 31.947 33506.25 1670.63 464 474.05 +7.585 −2.16 −0.0216700 1.1006 35.897 34063.56 1823.42 462 469.39 +20.12 −1.59 −0.0159

45% 400 1.7011 27.379 31835.94 1488.61 475 514.12 +12.84 −8.23 −0.0823600 1.0925 33.639 32821.24 1724.36 467 470.24 −0.6 − .69 −0.0069700 1.1033 36.649 33286.73 1834.34 463 467.99 +13.6 −1.078 −0.01078

60% 400 1.0813 30.031 28285.8 1521.42 460 511.52 +3.15 −11.2 −0.112600 1.1080 37.146 28779.78 1765.29 451 472.84 −7.50 −4.84 −0.0484700 1.1215 40.68 29091.81 1878.24 447 452.19 −13.33 −1.16 −0.0116

80% 400 1.0800 28.815 28681.85 1486.95 490 501.94 −15.85 −2.2 −0.220600 1.1050 35.231 29450.81 1715.28 460 464.20 −28.83 −0.9 +0.009700 1.1168 37.807 29922.36 1744.89 430 426.42 −53.94 −0.83 +0.083700 1.0955 36.149 35604.77 1858.89 495 486.76 +50.92 −1.66 −0.0166

3R.K. Shukla et al. / Journal of Molecular Liquids 135 (2007) 1–4

wide range of temperatures and a high degree of coulombic inter-action would be expected in the near and next to near neighbours,resulting in ordering effects.

Fig. 1. Values of surface tension (σexp and σtheo) of molten Pb–Sn system versusabsolute for different temperature of lead.

The application of the Prigogine corresponding state principleto this problem for evaluating the surface tension of binary liquidmixture at elevated temperature has been carried out in a prag-matic spirit as was that of the original derivation of Patterson andRastogi [18]. The corresponding state treatment to Pb–Sn system

Fig. 2. Theoretical values of excess surface tension σtheoE of molten Pb–Sn

system versus absolute for different temperature of lead.

4 R.K. Shukla et al. / Journal of Molecular Liquids 135 (2007) 1–4

is somewhat unorthodox, since application of Eq. (3) to the binaryliquid mixture is based on the assumption that they are equivalentto single component liquids and effectively ignores differences inconcentration occurring at the surface of the mixture. Gibbs'enrichment of a mixture surface by the component of lowersurface tension is well known. The normal results show alowering of mixture surface tension, which results in negativedeviation from a linear function of bulkmole fraction. That is whythere is a tendency of our theoretical values to be higher than theobserved values. However, it would not be proper to say that thisis the only reason for higher discrepancies. A part of thediscrepancy may be attributed to approximations made in thecomputation of the interaction parameters X12.

Excess surface tension which is a measure of interactionsinvolved in the liquid mixture is plotted against temperature atvarious compositions and is shown in Fig. 2. Excess surfacetension is evaluated as

rE ¼ ½rtheo−ðx1r1 þ x2r2Þ� ð10ÞOn the basis of the above discussion, it may be concluded

that Flory's statistical theory predicts surface tension of Pb–Snbinary liquid mixtures at elevated temperature up to asubstantial extent. On the basis of our calculation, it can beinferred that the Flory theory affords a useful estimate of thesurface tension without considering such concentration effects.Furthermore, the basic advantage of the Flory theory inestimating the surface tension is that all the necessaryparameters e.g. density, isothermal compressibility and coeffi-cients of thermal expansion for the pure components are used.Our main aim is to show the general applicability of the Florytheory in a qualitative way. It appears that Flory's theory cannotexplain the reverse trend of surface tension with temperaturealthough it is clear from Table 2 that at different mole fractionsvariation of surface tension predicted theoretically is similar tothat obtained from experiment. We can also arrive at theconclusive juncture that Flory's theory has universal applica-bility and it can also be applied in the case of liquid metal alloysover a wide range of temperatures at different mole fractions.

Acknowledgement

Authors are thankful to the Department of Chemistry V.S.S.D.College and Verstehen, a centre for understanding and researchfor kind cooperation and help.

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