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Ultrasonic velocity and isentropic compressibility of binary fluid mixtures at 298.15 K.

Introduction

Physicochemical behavior and molecular interactions occurring in a variety of liquid mixtures and solutions can be studied with the help of ultrasonic velocity. There has been an increasing interest in the study of molecular interactions and a number of experimental techniques have been used to investigate the same in binary liquid mixtures, since data on sound velocity offers a convenient method for determining certain thermodynamical properties of liquids and liquid mixtures, which are not obtained by other methods. Ultrasonic investigations are important in elucidating internal structure of fluid mixtures involving heat transfer, mass transfer etc. which are applicable in many ndustrial applications.

Extensive work has been carried out by Pereira et al. [1], Dzida and Ernest [2], Al-Kandary et al. [3], Pandey et al. [4] and Pan et al. [5] to investigate liquid state through analysis of ultrasonic propagation parameters and to correlate ultrasonic velocity with other physical and thermodynamic parameters. Substantial amount of work have also been done by Shukla et al. [6-12] successfully for the theoretical evaluation of ultrasonic velocity in binary and multicomponent non polar liquid mixtures by various empirical, semi-empirical and statistical mechanical concepts. Shukla et al. [11-12] have further applied these concepts to polar and ionic liquids. As a part of research concerning the thermochemical studies on new working fluid pair, we present here some useful data on speed of sound and isentropic compressibility for the mixture of trimethylbenzene with tetrahydrofuran, tetrachloromethane and dimethyl sulfoxide. These properties have been determined over the whole composition range. The major objective of the present work is to find out the applicability of Prigogine-Flory-Patterson (PFP) theory in polarnonpolar cyclic liquid binary mixtures. The necessary parameters for computation have been taken from the work of Pan et al. [5] and CRC hand book of chemistry & physics [13]. To the best of our knowledge, this is the first report in which PFP model has been applied in polar-nonpolar cyclic liquid mixtures.

Theoretical

In dealing with liquid state, Flory et al. [14-16] defined an element (or segment) as an arbitrary chosen portion of the molecules. Considering such segments in a molecule and following Prigogine's treatment of (y-mer) chain molecules and representing the number of external degree of freedom per segment by 3C, he was able to derive a partition function of the form:

Z = [Z.sub.comb][[[gamma][([v.sup.1/3] - [v.sup.*1/3]).sup.3]].sup.[gamma]nc]exp(-[E.sub.0]/kT) (1)

where k, N, T and [E.sub.0] are, respectively, the Boltzmann constant, number of particles, absolute temperature and excess intermolecular energy. [Z.sub.comb] is combinatorial factor which takes into account the number of ways in which [gamma]N elements intersperse among one another. Flory obtained the following expressions for the intermolecular free energy.

[E.sub.0] = Nrs [eta]/2v (2)

Here [eta] is a constant characterizing the energy of interaction for a pair of neighboring sites, s is the number of intermolecular contact sites per segment and v is the volume per segment. The reduced partition function thus takes the form as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

the reduced equation of state obtained from the resulting partition function is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Changing from the molecular to molar units per segment for v, v* and jj, one gets,

[??] = v/[v.sup.*] = V/[V.sup.*] (5)

[??] = T/[T.sup.*] = 2[v.sup.*]ckT/[S.sub.[eta]] (6)

and

[??] = p/[p.sup.*] = 2[pv.sup.*]/[S.sub.[eta]] (7)

The reduced equation of state at zero pressure becomes,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[??] = [[[alpha]T/3 + 3[alpha]T) + 1].sup.3] (9)

Where [alpha] is the coefficient of thermal expansion at P = 0. Thus the reduced volume [v.sup.~] and reduced temperature [??] can be obtained from the experimental values of [alpha]. The values of [v.sup.~], [??], [v.sup.*] and [T.sup.*] can be computed using eqs (5) and (6). From the reduced equation of state it follows that:

[P.sup.*] = [[alpha]/[[beta].sub.T]]T[[??].sup.2] = [gamma]T[[??].sup.2] (10)

where [[gamma].sub.p] = [([delta]P/[delta]T).sub.V] is the thermal pressure coefficient at P = 0. Characteristic pressure [P.sup.*] is evaluated from this equation for binary liquid mixtures with the component subscript 1 and 2, Flory obtained the following expressions for binary liquid mixtures as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [[PSI].sub.1] and [[PSI].sub.2] are the segment fractions and [[theta].sub.1] and [[theta].sub.2] are the site fractions given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [x.sub.1] and [x.sub.2] are the mole fractions, [X.sub.12] is the interaction parameter and [v.sub.1] and [v.sub.2] are the molar volumes of the components 1 and 2, respectively. The ration ([s.sub.1]/[s.sub.2]) = [([v.sup.*.sub.1]/ [v.sup.*.sub.2]).sup.-1/3] = [([r.sub.1]/[r.sub.2]).sup.-1/3] for spherical molecules. The interaction parameter ([X.sub.12]) can be expressed as:

[X.sub.12] = [P.sup.*.sub.1][[1 - [([P.sup.*.sub.2]/[P.sup.*.sub.1]).sup.1/2][([v.sup.*.sub.2]/[v.sup.*.sub.1]).sup.1/6]].sup.2] (14)

12 is the energy parameter and it is measured of the difference of interaction energy between the unlike pairs and the mean of the like pairs.

The surface tension of binary liquid mixture in terms of Flory's statistical theory can be described by the expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the characteristic and reduced parameters involved in the eqn.

Patterson and Rastogi [17] in their extension of the corresponding state theory dealt with the surface tension by using as a reduction parameter:

[[sigma].sup.*] = [k.sup.1/3][P.sup.*2/3][T.sup.*1/3] (16)

called the characteristic surface tension of the liquid. Here k is Boltzmann constant and [P.sup.*] and [T.sup.*] are the characteristic pressure and temperature, respectively. Starting from the work of Prigogine and Saraga [18] 1952 and Prigogine and Bellmas [19] they derived a reduced surface equation for Van der Waals liquids:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Where M is the fraction of the nearest neighbor that a molecule loses on moving from the bulk of the liquid to the surface. Its most suitable value ranges from 0.25 to 0.29.

Although Flory's statistical theory is not directly related with ultrasonic velocity but its evaluation needed the use of well-known and well-tested Auerbach relation which can be successfully used for the evaluation of ultrasonic velocity in binary liquid mixtures represented as:

u = [([sigma]/6.3 x [10.sup.-4][rho]).sup.2/3] (18)

where [sigma] and [rho] are the surface tension and density of the liquid mixture respectively.

Isentropic compressibility is related to speed of sound and density by the relation: [[beta].sub.s] = [u.sup.-2][[rho].sup.-1] (19)

Material and Methods

Experimental surface tension and experimental density data have been utilized to evaluate experimental sound velocity with the help of well known and well tested relation of Auerbach for six cyclic binary liquid mixtures over the whole composition range at 298.15K. Parameters of pure components are listed in Table 1 along with the literature values. Values of density ([rho]), molar volume ([V.sub.m]), theoretical and experimental speed of sound ([u.sub.cal] and [u.sub.exp]), percent deviation in speed of sound (%[DELTA]u) and isentropic compressibility ([[beta].sub.s]) for six binary mixtures; tetrahydrofuran (THF) + 1,2,4-trimethyl benzene (TMB), tetrahydrofuran (THF) + 1,2,3-trimethyl benzene (TMB), tetrachloromethane (TCM) + 1,2,4-trimethyl benzene (TMB), tetrachloromethane (TCM) + 1,3,5-trimethylbenzene, dimethyl sulfoxide (DMSO) + 1,2,4-trimethyl benzene + dimethyl sulfoxide (DMSO) + 1,3,5-trimethyl benzene at 298.5 K are presented in tables 2-7.

All the necessary data for computation were taken from the work of Pan et al. [5]. A close observation of all the tables reveal that experimental and calculated values of speed of sound are very close to each other indicating the success of PFP model in TMB with THF, TCM and DMSO liquid mixtures.

Results and Discussions

The percent deviations in speed of sound at 298.15 K from figures 1 and 2 shows that they are positive for THF and TCM with 1,2,4-TMB and are negative for THF, TCM and DMSO with 1,3,5-TMB while DMSO with 1,2,4-TMB has both positive and negative values. The maximum positive value of percent deviation in speed of sound follow the order DMSO + 1,2,4-TMB > TCM + 1,2,4-TMB > THF + 1,2,4-TMB and maximum negative values of percent deviation in speed of sound follow the order DMSO + 1,3,5TMB > TCM + 1,3,5-TMB > THF + 1,3,5-TMB. In both the cases, the trend is similar except for the system DMSO + 1,2,4-TMB where both positive and negative deviations are observed. The possible reason may be that there are no strong intermolecular forces between DMSO and TMB because 1,3,5-TMB is a symmetrical nonpolar molecule. Conversely, stronger forces exists for the systems with 1,2,4-TMB, so the values are positive. Negative values of the systems with 1,3,5-TMB are possibly owing to the greater repulsion due to bulky methyl group as compared to 1,2,4-TMB. These short range repulsive forces are responsible for less compact structure of 1,3,5 TMB. With the increase of concentration, density increases and molar volume decreases due to shrinkage in the volume of DMSO and TCM. Hence, an increase in the speed of sound may be correlated to the structure former of the 1,3,5-TMB. Positive deviations are due to dipolar-dipolar interactions because 1,2,4-TMB is a polar molecule.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The lack of smoothness in deviations is due to the interaction between the component molecules. Isentropic compressibility increases regularly with the increase of mole fraction. Increment becomes larger with 1,2,4-TMB. Strong discrepancy is observed in Table 6 where the value of isentropic compressibility decreases. Possibly, it is due to dipolar repulsion which lowers the energy hence regular trend is not observed, while density and speed of sound show regular behavior.

Conclusion

Thus it can be concluded from the preceding discussion that PFP model can be applied successfully to polar-nonpolar cyclic liquid binary mixtures.

Acknowledgements

One of author (R. K. Shukla) is thankful to U.G.C. for financial assistance and Department of Chemistry of V. S. S. D. College for cooperation and help.

References and Notes

[1] Pereira, S. M.; Rivas, M. A.; Iglesias, T. P. J. Chem. Eng. Data 2005, 47, 1363.

[2] Dzida, M.; Ernest, S. J. Chem. Eng. Data 2003, 48, 1453.

[3] Al-Kandary, J. A.; Al-Jimaz, A. S.; Latiq, H. M. A. J. Chem Eng. Data 2006, 51, 2074.

[4] Pandey, J. D.; Singh, A. K.; Day, R. Indian J. Chem. Tech. 2005, 12, 588.

[5] Pan, C.; Ke, Q.; Ouyang, G.; Zhan, X.; Yang, Y.; Huang, Z. J. Chem. Eng. Data 2004, 49, 1839.

[6] Shukla R. K.; Kumar, A.; Srivastava, K.; Yadava, S. J. Mol. Liq. 2008, 140, 25.

[7] Shukla, R. K.; Kumar, A.; Singh, N.; Shukla, A. J. Mol. Liq. 2008, 140, 117.

[8] Shukla, R. K.; Kumar, A.; Srivastava, K.; Singh, N. J. Soln. Chem. 2007, 36, 1103.

[9] Shukla, R. K.; Rai, R. D.; Shukla, A. K.; Mishra, N.; Pandey, J. D. Indian J. Pure Appl. Phys. 1993, 31, 54.

[10] Shukla, R. K.; Rai, R. D.; Shukla, A. K.; Pandey, J. D. J. Chem. Thermodyn. 1989, 21, 125.

[11] Shukla, R. K.; Rai, R. D.; Pandey, J. D. Can. J. Chem. 1989, 67, 437.

[12] Shukla, R. K.; Shukla, S.; Pandey, V.; Awasthi, P. J. Phys. Chem. Liq. 2007, 45, 169.

[13] Chemical Rubber Publishing Company Handbook of Chemistry and Physics, Florida: CRC, Boca Raton, 1979.

[14] Flory, P. J. J. Am. Chem. Soc. 1965, 82, 1833.

[15] Abe, A.; Flory, P. J. J. Am. Chem. Soc. 1965, 82, 1838.

[16] Flory, P. J.; Orwoll, R. A.; Vrij, A. J. Am. Chem. Soc. 1964, 86, 3515.

[17] Patterson, D.; Rastogi, A. K. J. Phys. Chem. 1970, 74, 1067.

[18] Prigogine, I.; Saraga, L. J. Chem. Phys. 1952, 49, 399.

[19] Prigogine, I.; Bellmas, A.; Method, V. Molecular Theory of Solutions. Amsterdam: North-Holland, 1957.

[20] Riddick, J.; Bunger, W. B.; Sakano, T. K. Technique of Chemical Organic Solvent; Physical Properties and Methods of Purification, 4th ed. John Willy and Sons. 1986

[21] Nair, A. K. J. Phy. Chem. Liq. 2007, 45, 371.

[22] Moattar, M. T. Z.; Cegineara, R. M. J. Chem. Eng. Data 2007, 52, 2359.

Rajeev Kumar Shukla (a) *, S. N. Dixit (b), Pratima Jain (b), Preeti Mishra (b) and Sweta Sharma (b)

(a) Department of Chemistry, V. S. S. D. College, Kanpur--208002, India

(b) Department of Chemistry, Govt. Science College, Gwalior, M. P. India

Received: 28 August 2010; accepted: 31 November 2010.

* Corresponding author. E-mail: rajeevshukla47@rediffmail.com. Phone: +91 9838516217
Table 1. Parameters of pure components at 298.15 K

Liquid       p (a)     p (a)     [10.sup.4]   [[beta].sub.T]/
             (exp)     (lit)       a/K-1       [TPa.sup.-1]

1,2,4-TMB   0.87164    087174      11.168         81.445
1,3,5-TMB   0.86103   0.86109      11.320         85.961
THF         0.88206   0.88197      11.464         90.440
TCM         1.58380   1.58429      11.504         91.704
DMSO        1.09554   1.095560     9.8922         50.134

Liquid       [V.sub.m]/    u/[ms.sup.-1]
             [cm.sup.3]        (exp)
            [mol.sup.-1]

1,2,4-TMB     137.893         1415.7
1,3,5-TMB     139.592         1389.3
THF            81.752         1291.2
TCM            97.121          921.6
DMSO           71.316         1480.9

Liquid      u/[ms.sup.-1]   u/[ms.sup.-1]
                (cal)           (lit)

1,2,4-TMB      1412.3         1421 (b)
1,3,5-TMB      1390.8         1397 (c)
THF            1283.6         1278 (d)
TCM             916.6          926 (e)
DMSO           1498.4         1487 (d)

(a)--ref [5], (b)--ref [20], (c)--ref [21], (d)--ref [22],
(e)--ref [7]

Table 2. Values of mole fraction ([X.sub.1]), density
([rho]), molar volume ([V.sub.m]), theoretical and
experimental speed of sound (u-cal, u-exp), %[DELTA]u
and isentropic compressibility of tetrahydrofuran and
1,2,4-trimethyl benzene

[X.sub.1]   [[rho].sub.m]/      [V.sub.m]/        u(cal)/
            g.[cm.sup.-3]    c.c.[mol.sup.-1]   [ms.sup.-1]

0.0988          0.8727            132.3           1404.5
0.2018          0.8737            126.6           1396.5
0.3003          0.8748            121.0           1388.8
0.3995          0.8758            115.5           1381.1
0.5002          0.8769            109.8           1373.3
0.6005          0.8779            104.2           1365.7
0.6986          0.8789            98.67           1358.5
0.7996          0.8800            93.00           1351.4
0.9006          0.8810            87.33           1344.6

[X.sub.1]     u(exp)/     (%[DELTA]u)     [beta]s/
            [ms.sup.-1]                 [Mpa.sup.-1]

0.0988        1410.1         0.390         0.5809
0.2018        1405.3         0.632         0.5869
0.3003        1400.4         0.828         0.5927
0.3995        1394.4         0.958         0.5986
0.5002        1386.8         0.974         0.6047
0.6005        1377.9         0.885         0.6107
0.6986        1367.8         0.677         0.6165
0.7996        1355.9         0.336         0.6223
0.9006        1343.7        -0.067         0.6278

Table 3. Values of mole fraction ([X.sub.1]), density
([rho]), molar volume ([V.sub.m]), theoretical
experimental speed of sound (u-cal, u-exp), %[DELTA]u
and isentropic compressibility of tetrahydrofu and
1,3,5-trimethylbenzene

[X.sub.1]   [[rho].sub.m]/      [V.sub.m]/        u(cal)/
            g.[cm.sup.-3]    c.c.[mol.sup.-1]   [ms.sup.-1]

0.0481          0.8620            136.8           1388.0
0.2021          0.8653            127.9           1379.0
0.3002          0.8673            122.2           1373.4
0.4000          0.8694            116.5           1367.8
0.5004          0.8716            110.6           1362.3
0.6010          0.8737            104.8           1356.9
0.7003          0.8758            99.09           1351.8
0.7989          0.8778            93.38           1347.0
0.9000          0.8800            87.54           1342.5

[X.sub.1]     u(exp)/     (%[DELTA]u)     [beta]s/
            [ms.sup.-1]                 [Mpa.sup.-1]

0.0481        1381.3        -0.484        0.60217
0.2021        1372.6        -0.473        0.60769
0.3002        1365.4        -0.587        0.61121
0.4000        1358.9        -0.653        0.61475
0.5004        1352.8        -0.702        0.61826
0.6010        1347.6        -0.687        0.62167
0.7003        1342.9        -0.665        0.62488
0.7989        1338.8        -0.615        0.62784
0.9000        1335.6        -0.515        0.63054

Table 4. Values of mole fraction ([X.sub.1]), density
([rho]), molar volume ([V.sub.m]), theoretical and
experimental speed of sound (u-cal, u-exp), %[DELTA]u,
isentrtopic compressibility of tetrachloromethane and
1,2,4-trimethyl benzene

[X.sub.1]   [[rho].sub.m]/      [V.sub.m]/        u(cal)/
            g.[cm.sup.-3]    c.c.[mol.sup.-1]   [ms.sup.-1]

0.0496          0.9070            135.9           1371.8
0.0996          0.9426            133.8           1333.6
0.1986          1.0131            129.8           1264.4
0.3001          1.0854            125.7           1201.2
0.4006          1.1569            121.6           1145.1
0.5987          1.2980            113.5           1049.6
0.6986          1.3692            109.4           1007.6
0.7985          1.4403            105.3            969.2
0.9011          1.5134            101.2            932.9

[X.sub.1]     u(exp)/     (%[DELTA]u)     [beta]s/
            [ms.sup.-1]                 [Mpa.sup.-1]

0.0496        1374.9         0.222        0.58587
0.0996        1337.3         0.277        0.59654
0.1986        1270.1         0.449        0.61742
0.3001        1208.3         0.588        0.63854
0.4006        1152.0         0.604        0.65921
0.5987        1060.5         1.029        0.69938
0.6986        1014.9         0.721        0.71937
0.7985         973.0         0.394        0.73918
0.9011         932.1        -0.088        0.75929

Table 5. Values of mole fraction ([X.sub.1]), density
([rho]), molar volume ([V.sub.m]), theoretical and
experimental speed of sound (u-cal, u-exp), %[DELTA]u,
isentropic compressibility of tetrachloromethane and
1,3,5-trimethyl benzene

[X.sub.1]   [[rho].sub.m]/      [V.sub.m]/        u(cal)/
            g.[cm.sup.-3]    c.c.[mol.sup.-1]   [ms.sup.-1]

0.1009          0.9340            137.5           1311.8
0.2014          1.0066            133.3           1242.8
0.2994          1.0774            129.0           1183.2
0.4007          1.1506            124.7           1128.2
0.4998          1.2223            120.5           1080.1
0.5995          1.2943            116.2           1036.5
0.6988          1.3661            112.0            997.3
0.7970          1.4371            107.8            962.4
0.9012          1.5124            103.5            929.0

[X.sub.1]     u(exp)/     (%[DELTA]u)     [beta]s/
            [ms.sup.-1]                 [Mpa.sup.-1]

0.1009        1306.0        -0.445        0.62221
0.2014        1235.2        -0.615        0.64319
0.2994        1173.6        -0.818        0.66299
0.4007        1118.0        -0.912        0.68274
0.4998        1071.0        -0.847        0.70131
0.5995        1028.8        -0.744        0.71914
0.6988         991.3        -0.613        0.73592
0.7970         957.6        -0.494        0.75134
0.9012         925.1        -0.418        0.76616

Table 6. Values of mole fraction ([X.sub.1]), Density
([rho]), molar volume ([V.sub.m]), theoretical and
experimental speed  of sound (u-cal, u-exp), %[DELTA]u,
isentropic compressibility of dimethylsulfoxide and
1,2,4-trimethylbenzene

[X.sub.1]   [[rho].sub.m]/      [V.sub.m]/        u(cal)/
            g.[cm.sup.-3]    c.c.[mol.sup.-1]   [ms.sup.-1]

0.1007          0.8942            131.2           1259.5
0.2007          0.9166            124.5           1264.7
0.3007          0.9390            117.9           1270.4
0.3995          0.9611            111.3           1276.6
0.5011          0.9838            104.5           1283.7
0.6012          1.0062             97.9           1291.4
0.6992          1.0282             91.3           1300.0
0.7992          1.0506             84.7           1309.9
0.9013          1.0734             77.9           1321.7

[X.sub.1]     u(exp)/     (%[DELTA]u)     [beta]s/
            [ms.sup.-1]                 [Mpa.sup.-1]

0.1007        1367.2         7.88         0.70503
0.2007        1354.0         6.59         0.68209
0.3007        1333.6         4.74         0.65984
0.3995        1321.9         3.42         0.63841
0.5011        1312.2         2.17         0.61685
0.6012        1289.1         -0.18        0.59588
0.6992        1271.0         -2.28        0.57552
0.7992        1272.4         -2.95        0.55474
0.9013        1407.5         6.10         0.53331

Table 7. Values of mole fraction ([X.sub.1]), density
([rho]), molar volume ([V.sub.m]), theoretical and
experimental speed of sound (u-cal, u-exp), %[DELTa]u,
isentropic compressibility of dimethylsulfoxide and
1,3,5-trimethylbenzene

[X.sub.1]   [[rho].sub.m]/      [V.sub.m]/        u(cal)/
            g.[cm.sup.-3]    c.c.[mol.sup.-1]   [ms.sup.-1]

0.1023          0.8850            132.6           1399.7
0.1987          0.9076            126.0           1408.4
0.3027          0.9320            118.9           1418.1
0.401           0.9551            112.2           1427.7
0.4999          0.9783            105.5           1437.7
0.6003          1.0018             98.6           1448.5
0.6990          1.0250             91.9           1459.7
0.8003          1.0487             85.0           1471.9
0.8998          1.0720             78.2           1484.6

[X.sub.1]     u(exp)/     (%[DELTA]u)     [beta]s/
            [ms.sup.-1]                 [Mpa.sup.-1]

0.1023        1338.4         -4.58        0.57675
0.1987        1303.8         -8.02        0.55548
0.3027        1274.0        -11.31        0.53355
0.401         1275.5        -11.93        0.51371
0.4999        1285.3        -11.86        0.49452
0.6003        1314.8        -10.17        0.47575
0.6990        1350.6         -8.08        0.45791
0.8003        1404.3         -4.81        0.44016
0.8998        1466.3         -1.25        0.42321
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Author:Shukla, Rajeev Kumar; Dixit, S.N.; Jain, Pratima; Mishra, Preeti; Sharma, Sweta
Publication:Orbital: The Electronic Journal of Chemistry
Date:Oct 1, 2010
Words:3669
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