Speed of sound, viscosity, and refractive index of multicomponent systems: analysis of experimental data from the Bertrand-Acree-Burchfield equation

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1 Speed of sound, viscosity, and refractive index of multicomponent systems: analysis of experimental data from the Bertrand-Acree-Burchfield equation J.D. PANDEY,' P. JAIN, AND V. VYAS Departtnerzt of Chemistry, Uiliversity of Allc~habad, Allalzabad , U.P., lndia Received February 22, 1994 J.D. PANDEY,' P. JAIN, and V. VYAS. Can. J. Chem. 72, 2486 (1994). The speed of sound in and the viscosity and refractive index of two ternary mixtures (I) toluene + 11-heptane + n- hexane and (11) cyclohexane + il-heptane + tz-hexane, and six binary combinations of these constituents have been measured experimentally. The excess values of these properties are calculated and the results are analysed using the Bertrand-Acree- Burchfield (BAB) model. All the experimental and theoretical values are in fairly reasonable agreement. J.D. PANDEY,' P. JAIN, et V. VYAS. Can. J. Chem. 72, 2486 (1994). On a determine experimentalement la vitesse du son dans ainsi que la viscositt et l'indice de rtfraction de deux mtlanges tertiaires ((I) tolubne + n-heptane + n-hexane et (11) cyclohexane + 11-heptane + n-hexane) ainsi que de six des mtlanges binaires qui y contribuent. On a calcult les valeurs en excbs et on a analyst les risultats ii la lumibre du modble thkorique de Bertrand-Acree-Burchfield (BAB). On a trouvt que I'accord entre les valeurs experimentales et thtoriques de toutes les proprittis en excbs est raisonnable. [Traduit par la redaction] Introduction A thorough knowledge of thermodynamic and transport properties of multicomponent liquid systems is essential in many industrial applications such as design calculation, heat transfer, mass transfer, and fluid flow etc. Various empirical, semi-empirical, and group contribution methods for computing the thermodynamic properties of multicomponent systems have been thoroughly reviewed by Acree (I), Singh et al. (24) used the graphical-theoretical approach to analyse the experimental data on molar excess volume and molar excess enthalpies of ternary non-electrolyte solutions. Flory's statistical mechanical theory (5) in combination with the method used by Patterson and Rastogi (6), has been successfully used to evaluate activity coefficient (7), heat of mixing (7), surface tension (8-lo), sound velocity (11-13), excess volume (14, 15), and viscosity (16) of various multicomponent liquid systems. In all these methods, properties of multicomponent systems have been calculated using properties of the pure components. However, a more promising approach to predict the thermodynamic and physical properties of multicomponent liquid systems using binary combinations of their constituents has been developed by Bertrand, Acree, and Burchfield (1, 17, 18). There are a number of equations (1) to compute excess enthalpy, excess free energy, and other excess thermodynamic properties of multicomponent systems from the excess properties of these binary combinations, provided the composition of the binary combinations is similar to the ternary combinations. Bertrand et al. modified this approach by considering binary systems with components at the same molar ratio as in the ternary systems and evaluated excess enthalpy, excess Gibbs free energy, and excess volume for the ternary systems. The Bertrand- Acree-Burchfield method has not been applied to transport properties. Many important properties such as sound velocity, isothermal compressibility, internal pressure, surface tension etc. have not been estimated using this approach. Recently Acree and Teng (19) tested the applicability and limitations of a newly derived semi-empirical relationship (20) that includes contributions from both two-body and three-body interactions. 'Author to whom correspondence may be addressed. This work is to determine density, sound velocity, viscosity, and refractive index of two ternary non-electrolyte solutions (I) toluene + n-heptane + n-hexane and (11) cyclohexane + n-heptane + n-hexane. To use the Bertrand-Acree-Burchfield method to estimate sound velocity, refractive index, and viscosity of these ternary systems, we measured the same properties in the respective binary mixtures. The theoretical sound velocity, refractive index, and viscosity values obtained from the Bertrand-Acree-Burchfield method were compared to the experimental values. A slight modification in the Bertrand- Acree-Burchfield method was adopted. Experimental AnalaR grade toluene, 11-heptane, n-hexane, and cyclohexane were obtained from MIS BDH Chemicals Ltd., Poole, England. These were purified by double distillation and dried in accordance with the usual procedure. All measurements were carried out while using a thermostatically-controlled water-bath where temperature had been set and maintained by a Beckmann thermometer, which was calibrated against a platinum resistance thermometer. The variation in the temperature was not allowed to exceed f 0.0l0C during the measurements. The speed of sound (c) in all liquid systems was measured using an ultrasonic interferometer. To determine the velocity of sound a single crystal-controlled variable path ultrasonic interferometer having a frequency of 2.0 MHz has been used. This method has been described in earlier papers (21, 22), and the estimated error is found to be f 0.07%. Viscosities (q) of the various liquid systems were determined using a pre-calibrated Ubbelhode-type viscometer. This method is based on the measurement of the time in which a constant volume of liquid flows through a capillary. Uncertainty in the viscosity measurement has been found to be 10.01%. The refractive indices of various liquid systems have been measured using Abbe's refractometer. The temperature was kept constant by maintaining a constant flow of water from a thermostat through the outer jacket of the cell. Measured refractive indices were found to be accurate to f units. Other experimental details have been discussed in an earlier paper (17). Theoretical A number of equations have been proposed (1-19) to predict the thermodynamic excess properties of multicomponent systems. These methods have been critically reviewed by Acree

2 PANDEY ET AL (1). However, the most accurate method to estimate the phys- In a reasonable modified approach volume fraction can be ical and thermodynamic excess properties was developed by used in place of weighted mole fractions and the equation takes Bertrand, Acree, and Burchfield (17-19). They have developed the following form: an equation based on a model ternary system to study the ther- modynamic properties of solute in simple binary solvents. This equation obeys the general mixing equation, [91 m:...~ = C C (xi + xj)($i + $j)(mf) i=l j>i [ L1 m;x3 = (nlyl + my2 + n3~3~3)-~(nlyln2~2~12 where x, and x, are the mole fractions of multicomponent systems; and $, and $, are their respective volume fractions and + n1yin3y3a13 + n2y2n3y3a23) AZf can be more appropriately taken as the experimental exwhere Zll represents any extensive thermodynamic property cess property of the constituent at the described in terms of interaction parameters All and weighting Same of the components as in a multicom~onent factor y,. By suitable manipulations, eq. [l] can be rearranged 'ystem. to In subsequent studies eq. [9] has been used to determine the excess ultrasonic velocity, excess viscosity, and excess refrac- [21 my;, {(nl = + n2)(f1 +f2)~;~2*~1~2a12) tive indices of two ternary systems for which eq. [9] takes the (X;YI + ~3'2) form, where and the subscript * represents the composition of a binary system which would be formed if all the third components were removed, so that for an i-j system, Review of eq. [2] shows that for the systems obeying this equation, the properties of contributing binary systems would obey (per mole of ternary solution) the following equation: For one mole of ternary solution eq. [2] can be rearranged to or in general, for multicomponent systems, N A number of methods have been suggested for the evaluation of weighting factors which are a rough measure of the skew of the binary excess mixing property from a symmetric curve with an extremum at the equimolar composition. A method for the calculation of the ratio of raw weighting factors has been given by Bertrand et al. (15). N Values of the sound velocity, viscosity, and refractive index are then calculated using the following equations: where The results so obtained were compared with the experimental findings and with the results of those theories which utilise the properties of single components to predict the thermodynamic properties of multicomponent liquid mixtures. Finally, the theoretically predicted results were examined in the light of average percentage deviations (0) and standard error (6) calculated from the expressions: and

3 2488 CAN. J. CHEM. VOL. 72, 1991 where Z is any measured and theoretically computed property and N is the total number of sets. Results and discussion Sound velocity, viscosity, and refractive index of two ternary liquid mixtures - toluene + n-heptane + n-hexane (I) and cyclohexane + n-heptane + tz-hexane (11) have been predicted using the respective experimental excess properties of their constituent binary liquid mixtures. In this paper a modified Bertrand-Acree-Burchfield approach has been utilized, where weighted mole fractions have been replaced by volume fractions and values of the experimental excess properties for constituent binary liquid systems have been used at the same molar ratio as these are in the ternary liquid mixtures. The values of sound velocity (c), viscosity (q), and refractive index (n) are presented in Table 1. Mole fractions of the constituent binary mixtures were calculated to ensure that they are in the same molar ratio as in the ternary liquid mixture. The values of c, q, and n at the new mole fractions were determined graphically from the experimental values. Using these values the excess properties of TABLE 1. Parameters of the pure components at K Ultrasonic velocity Viscosity x 10" Refractive Component (m S-I) (N s m-') index Toluene Cyclohexane n-heptane n-hexane binary liquid mixtures were evaluated. These values, compiled in Tables 2b, 2c, 2d and 3b, 3c, 3d were then used to predict ncfz3, and for systems I and I1 using eqs. [lo] to Respective values of sound velocity, viscosity, and refractive index of mixtures were then calculated using eqs to These values are compiled in Tables 2a and 3a for systems I and I1 respectively, and their percentage deviations were recorded in Tables 4 and 5. A close look at Tables 4 and 5 reveals that the percentage deviations in the computation of sound velocity is 1.92 and 0.70 respectively for TABLE 2. (a) Computed ultrasonic velocity, viscosity, and refractive index of the ternary system - toluene (x,) + n-heptane (x') + tz-hexane (x3) at K AcE lo3 rlcalcd x lo3 XI x2 (m s-i) (N s M-~) (N s m-?) AIP "calcd (b) Excess sound velocity, excess viscosity, and excess refractive index of the binary system - n-heptane (x,) + toluene (x') at different molar ratios

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5 CAN. J. CHEM. VOL TABLE 3. (Concluded) (b) Excess sound velocity, excess viscosity, and excess refractive index of the binary system - cyclohexane (x,) + n-heptane (x2) at different molar ratios (c) Excess sound velocity, excess viscosity, and excess refractive index of the binary system - n-heptane (x,) + n-hexane (x2) at different molar ratios (d) Excess sound velocity, excess viscosity, and excess refractive index of the binary system - cyclohexane (x,) + n-hexane (x2) at different molar ratios

6 PANDEY ET AL. TABLE 4. Mean percentage deviation and standard percentage error in the computation of ultrasonic velocity, viscosity, and refractive index of the ternary system - toluene (xl) + n-heptane (x2) + n-hexane (xj) at K Mean percentage deviation (o) Standard percentage error (6) TABLE 5. Mean percentage deviation and standard percentage error in the computation of ultrasonic velocity, viscosity, and refractive index of the ternary system - cyclohexane (x,) + n-heptane (x,) + n-hexane (xj) at K Mean percentage deviation (o) Standard percentage error (6) systems I and 11. This shows that the results are in excellent agreement with the experimental findings, and are much better than those obtained when computed using the Flory-Patterson approach by Pandey et al. (11-16). This method yields better results because the predictions are made using the thermodynamic properties of pure components and excess velocities of constituent binary liquid mixtures. Here, all possible binary interactions are considered and are easily accountable by using the experimental excess velocities. More encouraging results were obtained while predicting the viscosity and refractive in- dex of ternary systems. The standard error in computing viscosity was found to be 3.58 and 1.51 for systems I and 11. Viscosity of the same ternary liquid mixtures was predicted using empirical, semi-empirical, and statistical approaches and the standard error in most cases was found to be more than 5%. Thus, this approach has an advantage over other methods to predict the viscosity of the multicomponent systems using data of the pure components. The deviations may be due to the polarity and size effects of the components. In systems I and 11, an increase in q with increasing concentration of the

7 2492 CAN. J. CHEM. VOL first and second component contributes to the filling up of the third component into the empty sites afforded by the first and second components. For refractive index, the standard errors were found to be 0.12 and 0.93 for systems I and 11, respectively, indicating an excellent agreement with the experimental results. Hence, this.approach can be used to predict refractive indices more accurately. The most important feature of this type of equation is that it is applicable to both integral and differential mixing properties and is independent of the manner in which the binary mixing data are represented. It provides reasonably accurate predictions. The excess properties of the multicomponent systems containing more than three liquids can also be successfully expressed in terms of the contributing ternary solutions as However, this equation is not expected to be as useful as eq. [9] due to the scarcity of ternary data and tediousness of obtaining the same molar ratios in the ternary liquid mixture as is present in multicomponent liquid systems. We conclude, therefore, that the modified Bertrand-Acree-Burchfield approach can be successfully applied to predict various excess thermodynamical parameters in the case of multicomponent liquid systems. 1. W.E. Acree. It1 Thermodynamic properties of non-electrolyte solutions. Academic Press, New York pp P.P. Singh and V.K. Sharma. Can. J. Chern. 61, 2321 (1983). 3. P.P. Singh, H.P. Dahiya, and S. Dagar. Fluid Phase Equil. 43, 341 (1988). 4. P.P. Singh, S. Maken, and M. Bhatia. Indian J. Chern. 29 A, 108 (1990). 5. P.J. Flory. J. Am. Chem. Soc. 87, 1833 (1965). 6. D. Patterson and A.K. Rastogi. J. Phys. Chern. 74, 1067 (1970). 7. W. Brostow and J.S. Sochanski. J. Mater. Sci. 10, 2134 (1975). 8. J.D. Pandey and N. Pant. J. Am. Chern. Soc. 104, 3299 (1982). 9. J.D. Pandey, R.K. Shukla, and R.D. Rai. J. Chern. Soc. Faraday Trans. I, 85, 331 (1989). 10. M.S. Khanwalkar. Indian J. Pure Appl. Phys. 29, 616 (1991). 11. J.D. Pandey, V.N.P. Srivastava, (Mrs.) V. Vyas, and N. Pant. Indian J. Pure Appl. Phys. 25, 467 (1987). 12. J.D. Pandey, N. Pant, (Mrs.) N. Agrawal, and (Mrs.) Shikha. Acustica, 68, 225 (1989). 13. J.D. Pandey, A.K. Shukla, R.K. Shukla, and R.D. Rai. J. Chern. Thermodyn. 21, 125 (1989). 14. J.D. Pandey, R.K. Shukla, A.K. Shukla, and R.D. Rai. J. Chern. Soc. Faraday Trans. I, 84, 1853 (1988). 15. J.D. Pandey, R.D. Rai, A.K. Shukla, and R.K. Shukla. J. Phys. Chern. 91, 4627 (1989). 16. J.D. Pandey, R.D. Rai, A.K. Shukla, R.K. Shukla, and (Mrs.) N. Mishra. Indian J. Pure Appl. Phys. 31, 54 (1993). 17. G.L. Bertrand, W.E. Acree (Jr.), and T.E. Burchfield. J. Soln. Chern. 12, 327 (1987). 18. W.E. Acree and G.L. Bertrand. J. Soln. Chern. 12, 755 (1983). 19. I.L. Ten and W.E. Acree (Jr.). Phys. Chern. Liq. 25, 101 (1993). 20. G.A. Mansoori, K.R. Hall, J.C. Holste, and C.A. Hwang. Fluid Phase Equil. 62, 173 (1991). 21. J.D. Pandey, A.K. Puri, and K. Misra. Thermochirn. Acta, 74, 313 (1984). 22. J.D. Pandey, K. Misra, A.K. Shukla, and R.D. Rai. Can. J. Chern. 65, 303 (1987).