Theoretical Estimation of Surface Tension of

Asian J. Research Chem. 6 (10): October 2013

www.ajrconline.org

ISSN 0974-4169

RESEARCH ARTICLE

Theoretical Estimation of Surface Tension of Cryogenic Liquid MixturesNitrogen+Argon and Argon+Methane B.R. Chaturvedi1, Pavan Kumar Gautam1*, Rama Shankar Saroj1 and Sunil2 1

Department of Chemistry, University of Allahabad, Allahabad-211002 2 Department of Chemistry, M.G.C.G. University, Chitrakoot, M.P. *Corresponding Author E-mail: [email protected]

ABSTRACT: Surface tension of two binary (N2+Ar and Ar+CH4) cryogenic liquid mixtures has been estimated using two theoretical models. A comparative study of calculated surface tension values, obtained from Flory theory and Brock & Bird empirical relation, has been made. The results are discussed on the basis of intermolecular interactions between the components of the mixtures. Flory’s statistical theory has been found to be the most superior.

KEYWORDS: Surface tension, Cryogenic liquid mixtures, Brock-Bird relation, Flory theory. INTRODUCTION: Surface tension is a diagnostic parameter for describing various properties of liquids and liquid mixtures. The surface tension can be exactly related to the intermolecular forces through statistical thermodynamics. Such an equation is derived by Kirkwood and Buff equation [1] but it is difficult to use it because one must know how molecules are distributed through the surface. Kirkwood and Buff equation is valid only for simple spherical molecules. However, much more general equations have recently [2-5] been derived which are applicable to more complex molecules. These equations are again difficult to use for numerical work. Because of these difficulties a variety of approximate procedures have been developed. These methods differ as to the amount of numerical efforts involved and the degree of approximations required. Summarised discussion about the theories of surface tension has been given by Defay and Prigogine[6], Hildbrand and Scott[7], Onno and Kondo[8], Guggenheim[9,10], Buff et al.[11,12], Croxton[13], Toxvaerd[14], Gibbs[15], and Gubbins and Haile[16], in their review articles. It appears to us that nobody has attempted to evaluate surface tension of cryogenic liquid mixtures on the basis of any theoretical model due to scarcity of experimental data. Perhaps, it is our first attempt to apply empirical and statistical mechanical models to estimate the values of surface tension of cryogenic liquid mixtures for which the experimental data are available. Received on 03.08.2013 Accepted on 22.08.2013

Modified on 18.08.2013 © AJRC All right reserved

Asian J. Research Chem. 6(10): October 2013; Page 952-955

It is possible to write down exact relationship for the thermodynamic properties and molecular distributions in the interface. The theories of surface tension reviewed by different investigators are all based to some extent on the statistical thermodynamics, but also involve some degree of empirical fitting in most cases. Moreover, the reliable methods for predicting the surface tension of liquids of more complex molecules, and especially for cryogenic liquid mixtures are scare in literature. Among, all the methods known, so far, for the theoretical evaluation of surface tensions of mixtures, the most sophisticated are Flory-Patterson statistical theory [17-19]. However, to avoid the different difficulties of mathematical interpretation, empirical and semi-empirical relations can also been used to predict different thermodynamic properties of liquids and liquid mixtures. Empirical methods for estimating the surface tension have been discussed by Reid et al [21]. Simple two-parameter corresponding state theory [11, 22] applies only for molecules having nearly spherical symmetry. A critical review of the various empirical and other theoretical methods employed for the estimation of surface tension of liquids and mixtures has been presented by various workers during recent past [4,23-26]. The method of extending the principle to polyatomic and polar liquids is made through inclusion of a third parameter by Brock and Bird[27], designed for the estimation of surface tension of pure liquids, can also be extended for predicting the surface tension of binary liquid mixtures. The surface tension data of Sprow and Prausnitz[28] for two binary cryogenic liquid mixtures have been used. In the present investigation, the applicability of Brock and Bird relation and Flory theory

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has been examined for two cryogenic liquid mixtures of Surface tension of pure liquid is calculated by Florynitrogen-argon (I) at 83.82 K and argon-methane (II) at Patterson [21-23] theory with the help of following expression: 90.67 K.

THEORETICAL: Brock and Bird found the following correlations between the critical constant and surface tension

σ  P T  

 0.432  =  −0.951 +  (1 − Tr )119 1 Z C     3

1   3

 P T  

= ( −0.281 + 0.133α k )(1 − Tr )11

2 C C 

T TC

ZC =

PCVC RTC

 d ln 

αk =    d ln T C

...2

9

are characteristic and reduced surface

1

2

σ ∗ = k 3 P 3T

1 3

.10

Here ‘k’ is the Boltzmann constant.

{

}{

is given by

}

σ%(v%) =Mv%−5/3 ( v%1/3 −1) ln ( v%1/3 −0.5) /( v%1/3 −1)

...3

.11

where ‘M’ is the fraction of nearest neighbours that a molecule loses on moving from the bulk of the liquid to the surface. It is most suitable values is 0.29, which is used in our calculations.

...4

...5

In equation (4) R is the gas constant. If the values of PC, VC and TC of pure liquids are known, one may easily evaluate the surface tension of pure liquids in the light of equations (1), (3) and (4). In the case of binary liquid mixtures the pseudo critical constant values can be taken. The simplest procedure is to assume that Pcm, Tcm and Vcm are all mole fraction average i.e.

Pcm = X 1 PC (1) + X 2 Pc (2)

and

Patterson and Rastogi [23] in their extension of the corresponding state theory defined characteristic surface tension of the pure liquid as

In the above equation , , and are the reduced temperature, the critical compressibility factor and redical factor respectively. These are expressed as

Tr =

where tension.

..9

... 1

2 C C 

σ

σ = σ ∗σ% (v% )

In the prediction of surface tension of aforesaid liquid mixtures by Flory-Patterson theory, only thermal expansion coefficient and isothermal compressibility of pure components are required. The procedure for calculating the reduced and characteristic parameters is as follows starting with the reduced equation of state:

(

...7

Vcm = X 1VC (1) + X 2Vc (2)

.. 8

3

v%1/3 − 1 % T= 4 V% 3

...6

Tcm = X 1TC (1) + X 2Tc ( 2)

)

 V% = 1 + α T 3(1 + α T )  

P∗ =

α TV% 2 βT

...12

..13

...14

RESULTS AND DISCUSSIONS:

The values of experimental surface tension of first three Here X1 and X2 are the mole fractions of first and second cryogenic liquid mixtures have been taken from the work of component respectively. Thus, the values of surface tension Sprow and Prausnitz[28]. The values of molar volumes and of binary liquid mixtures can be obtained from equation (1) critical constants have been taken from the literature [21]. in confirmation with equations (3), (4) and (6).

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Table.1 Parameters of Pure Cryogenic Liquids along with experimental and predicted surface tensions Cryogenic Liquid

T

Pc

Vc

Tc

Nitrogen Argon

83.82 83.82 90.67 90.67

33.52 48.34

0.0901 0.0771

126.2 150.8

45.66

0.099

190.8

Methane

1.3872 1.2927 1.3252 1.2269

0.0745 0.06345 0.67611 0.0537

The values of critical constants, reduced and characteristic parameters along with the experimental and predicted surface tension values by Brock and Bird relation and Flory statistical theory for the mixtures (Ar+ CH4 & N2+ Ar) are shown in Table 2 and 3 respectively. Tables 2 and 3 also list the values of characteristic pressures, temperatures, surface tensions and reduced surface tensions for this respective mixture at different compositions. A thorough examination of last column of Tables 2 and 3 reveal that the percentage deviation between experimental and predicted surface tension values lies in between 0.1 to 9.1 and 0.1 to 4.7 for the systems nitrogen-argon and argon-methane respectively. Equation (1) has been found to be satisfactory for pure liquids and simple homogenous organic liquid mixtures (systems I and II), yet it is essentially semi-empirical. The values of surface tension, vide equation (1), are more sensitive to the value of critical constants, especially to the critical volume. Thus, the accurate values of critical constants of the mixtures at different compositions are required for obtaining better results. Table 2 and 3 also

T*

P*×10-9 Dynecm-2

1125.20 1321.10 1341.20 1688.20

2.5045 3.0355 2.9203 2.8322

Dynecm-1 99.13 118.88 116.45 123.19

0.08514 0.10771 0.09893 0.12964

dynecm-1 8.44 12.81 11.52 15.97

dynecm-1 7.42 13.39 11.65 17.78

show that the characteristic and reduced values of surface tension for both the system decrease linearly with increase in mole percentage of first named component hence showing the same trend as observed experimentally. Moreover, the theoretical values are more sensitive to M. Auxiliary calculations shows that for the set of liquids under present investigations, the overall agreement is better by using a value of 0.32 for this parameter. The discrepancy may be partly attributed to the approximation made in the computation of adjustable parameters X12 and partly to the non-orthodoxy of the theory, since application of equation (8) to the equivalent single component liquid effectively ignores differences in concentrations occurring at the surface of the mixture. On the basis of our calculations it can be concluded that the theory affords a useful estimate of the surface tension of simple cryogenic liquid mixtures without considering such concentration effects and without adjusting any parameter to fit the surface properties of the mixtures.

Table.2. Comparison of Calculated and Experimental Surface Tensions of Cryogenic Liquid Systems

Nitrogen-Argon(1) system, T= 83.82 K Mole % of Nitrogen σ(expl)

5.80 8.48 14.80 19.20 27.92 30.00 43.40 49.60 55.80 62.90 65.10 68.20 75.70 79.60 80.90 87.60 91.20

12.80 12.54 12.03 11.69 11.03 10.94 10.07 9.74 9.45 9.08 8.94 8.79 8.46 8.29 8.23 7.90 7.76

σ(caltd) Brock-Bird T* (K)

12.27 12.08 11.65 11.37 10.81 10.80 9.92 9.59 9.27 8.93 8.81 8.67 8.33 8.15 8.10 7.81 7.66

1307.7 1301.7 1287.6 1279.0 1259.4 1255.0 1227.5 1215.4 1203.4 1190.0 1186.0 1180.2 1166.7 1159.8 1157.5 1145.8 1139.7

P*×10

-9

σ*

2.9984 2.9816 2.9427 2.9162 2.8652 2.8532 2.7784 2.7458 2.7134 2.6774 2.6650 2.6511 2.6150 2.5966 2.5905 2.5596 2.5434

A comparison of the results obtained from Brock & Bird relation with the results of Flory theory reveals that both approaches are giving reasonable agreement with the experimental results. This verifies the utility of the semiempirical relation of Brock & Bird. But Flory’s statistical theory is yet to be preferred as this theory does not involve much approximation as compared to Brock and Bird relation. Moreover, the basic advantage of this theory over Brock and Bird relation is that all the essential parameters

117.521 116.900 115.464 114.481 112.580 112.104 108.365 107.152 106.045 105.002 105.900 104.605 103.070 102.507 102.050 101.090 100.505

σ(caltd) Flory

0.1059 0.1051 0.1033 0.1020 0.0995 0.0990 0.0958 0.0945 0.0928 0.0898 0.0882 0.0878 0.0872 0.0861 0.0850 0.0847 0.0843

12.45 12.29 11.93 11.68 11.20 11.09 10.38 10.12 9.84 9.43 9.26 9.18 8.99 8.82 8.67 8.56 8.47

Δ% (Brock-Bird) Δ% (Flory)

4.1 3.6 3.1 2.7 2.0 0.9 1.5 1.5 1.9 1.6 1.4 1.3 1.5 1.7 1.6 1.1 1.3

2.8 2.0 0.8 0.1 -1.5 -1.3 -3.1 -3.9 -4.1 -3.8 -3.5 -4.4 -6.2 -6.4 -5.3 -8.4 -9.1

viz. densities, thermal expansion coefficient and isothermal compressibility of pure components, can be determined experimentally in very precise way, and these experimental values of pure components are sufficient to obtain all the parameters of the mixture at different mole fractions, whereas in case of Brock and Bird relation the values of pseudo critical constants have been used in the absence of experimental critical constants at different compositions.

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Table.3 Comparison of Calculated and Experimental Surface Tensions of Argon-Methane Argon-Methane(II) system, T= 90.67 K * -9 P ×10 Mole % of Argon σ(expl) σ(caltd) Brock-Bird T* (K) σ*

8.32 11.70 20.40 28.90 41.10 50.80 71.20 85.20 91.30

16.80 16.42 15.66 14.96 14.13 13.57 12.60 12.11 11.90

17.62 17.39 16.70 16.11 15.20 14.61 13.14 12.18 11.76

1658.10 1706.22 1618.00 1588.60 1546.40 1512.70 1441.80 1393.00 1371.70

2.8351 2.8401 2.8462 2.8525 2.8621 2.8703 2.8891 2.9033 2.9102

CONCLUSION: From the above analysis of Brock and Bird relation, it can be concluded that this relation can be used, in satisfactory way, to predict the surface tension of cryogenic liquid mixtures. This relation is also able to predict the variation of surface tension of mixtures with composition and temperature showing same trend as observed experimentally. Moreover, the relation of Brock and Bird is very simple in elaboration and thus avoids the botheration of calculating different thermodynamic parameters. It can also be concluded that the Flory theory affords a useful estimate of the surface tensions of simple cryogenic binary liquid mixtures without considering such concentration effects and without adjusting any parameter to fit the surface properties of the mixtures. So, after analysis the Flory statistical theory is yet to be preferred as this theory does not involve much approximation as compared to Brock and Bird relation.

ACKNOWLEDGEMENT: The authors are thankful to Prof. J.D. Pandey, Former Head and U.G.C. Emeritus Fellow, Dept. of Chemistry, University of Allahabad for his keen interest and guidance in the progress of the work.

122.53 123.85 121.85 121.46 120.48 119.82 118.44 117.47 117.02

0.1343 0.1263 0.1237 0.1218 0.1175 0.1146 0.1082 0.1038 0.1018

σ(caltd) Flory Δ% (Brock-Bird) Δ% (Flory) 16.46 -4.8 2.00 15.64 -5.9 4.70 15.07 -6.6 3.70 14.72 -7.6 1.60 14.15 -7.5 -0.10 13.73 -7.6 -1.20 12.81 -4.2 -1.60 12.19 -0.6 -0.70 11.90 1.2 -0.10

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