Research Journal of Chemical Sciences ______________________________________________ ISSN 2231-606X Vol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 33 Theoretical Models of Ultrasonic Velocities in binary liquid mixtures Vaidya Rohit, Karthiyayini S. and Millerjothi N.K.3 Department of Chemical Engineering, BITS Pilani Dubai Campus, P.O. Box 345055, Academic City, Dubai, UAE Department of General Sciences, BITS Pilani Dubai Campus, P.O. Box 345055, Academic City, Dubai, UAE Department of Mechanical Engineering, BITS Pilani Dubai Campus, P.O. Box 345055, Academic City, Dubai, UAEAvailable online at: www.isca.in, www.isca.me Received 27th September 2015, revised 3rd October 2015, accepted 15th October 2015 AbstractUltrasonic velocity is one of the important tools for understanding molecular interaction of binary liquid mixtures. Hence, various theoretical models were devised for calculating ultrasonic velocity. In this work, Impedance relation, Nomoto’s Relation, Rao’s specific velocity relation, Van Dael-Vangeel Ideal mixture relation, and Junjie’s relation are considered and the calculated velocities are compared with the experimental values. The theory which gives the closest agreement is analysed and justified. The binary mixtures of octan-1-ol and methyl at 4 different temperaturesover a range of concentrations are considered. A secondary study is also considered to confirm the inferences. Keywords: Nomoto’s, Junjie’s, octan-1-ol, MMA, dipole. Introduction Ultrasonic velocity (USV) is one of the ways to derive information about physical behaviour of liquid mixtures and there exist numerous models of calculating it theoretically from given or observed factors. Studies in USVcan help us in determining interactions between the studied substances on a molecular level. Based on this, theories have been put forth to explain the results obtained. The mathematical models used in the following study include but are not limited to: the Impedance relation, Nomoto’s Relation, Rao’s specific velocity relation, Van Dael-Vangeel Ideal mixture relation, and Junjie’s relation. The experimental values from the paper published by Sridevi Gutta involving octan-1-ol and methyl benzoate will be used for analysis. The theoretical values of USV are compared with the experimental values to calculate the error percentage and conclude which model used is the most accurate for calculating USV in binary mixtures of liquids. Molecular interactions in binary mixtures are studied based on the deviation in the values of Uexp/UIMR. The analysis and justification for the model in the closest agreement with experimental values is offered.The chemical reagents used are 1-octanol and methyl benzoate. The USV and densities for the individual substances are given in table-1 for 4 different temperatures. These have been gathered from various literature sources. The USV for 10 mole fractions of binary mixtures are recorded in table-2 as reported in the lituratures1,2. Table-1 Density and ultrasonic sound speeds values of pure liquids across the temperature range Component 303.15 K 308.15 K Density, (kg/m3) Velocity, U (m/s) Density, (kg/m3) Velocity, U (m/s) Methyl Benzoate 1087.5 1404 1085.9 1376.84 1-Octanol 803.3 1365 801.6 1326.31 Component 313.15 K 318.15 K Density, (kg/m3) Velocity, U (m/s) Density, (kg/m3) Velocity, U (m/s) Methyl Benzoate 1081.4 1367.36 1083.8 1348.42 1-Octanol 800 1303.33 798.1 1291.57 Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 34 Table-2 Experimental values of ultrasonic sounds across the range of temperatures and concentrations 303.15 K 308.15 K 313.15 K 318.15 K Uexp X Uexp X Uexp X Uexp 0 1365 0 1326.32 0 1303.34 0 1291.58 0.1259 1370.53 0.1259 1329.48 0.1259 1306.67 0.1259 1301.06 0.2447 1371.82 0.2447 1333.34 0.2447 1306.67 0.2447 1302.36 0.358 1376.83 0.358 1343.34 0.358 1320.1 0.358 1310.53 0.4635 1380.1 0.4635 1348.44 0.4635 1332.64 0.4635 1312.95 0.5642 1383.5 0.5642 1351.74 0.5642 1335.79 0.5642 1314.87 0.6601 1387.06 0.6601 1354.64 0.6601 1335.79 0.6601 1320.1 0.7513 1390.2 0.7513 1357.88 0.7513 1338.95 0.7513 1323.1 0.8381 1396.67 0.8381 1369.44 0.8381 1346.67 0.8381 1326.1 0.920 1400.1 0.920 1373.69 0.920 1354.74 0.920 1330.1 1 1404 1 1376.85 1 1367.37 1 1348.43 In the first study scenario referred to here, methyl benzoate is “1” and octan-1-ol is “2”.” In the secondary study considered to confirm our findings, the substances used are methyl methacrylate and 3 aryl esters/alcohols - 2-methoxy ethanol, 2-ethoxy ethanol, 2-butoxy ethanol. This study is conducted at a fixed temperature with gradually increasing mole fractions. Methodology Nomoto’s Relation: This relation was set up by O. Nomoto in 1958. The relation is as follows:  =� (1) \r\r  , Where, M = molar mass; = density; V = molar volume; x = molar fraction Impedance relation: The impedance relation is: !"!" =� (2)Where %&Z = acoustic impedance Van Dael Vangeel Relation: The Van dael Vangeel Ideal Mixing relation (IMR)6,7 is given by: ') + ',** -*,*** +. =� (3)Rao’s Specific Sound Velocity: !/0 (3) \r\r1  r = Rao’s specific sound velocityJunjie’s Relation: Junjie’s relation9,10 is given by: )) ',*$ -**,**$* +. =� (5)Calculations: The calculations were performed using MATLAB R2011a and the results were imported and plotted in graphs using Microsoft Excel. Experimental observations were taken from literature sources and compared with the calculated values in the form of percentage errors and chi square values11,12. Observations along with the experimental values were calculated and tabulated. In the following section we will see the graphical representation of percentage error and chi square relation for each equation. Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 35 Results and Discussion Percentage error: This kind of test has been carried out in binary mixtures before however it has been carried out in a mixture of fatty primary alcohol and ester only in some cases. The structure of Methyl Benzoate, an ester, is as follows: When the two liquid substances are mixed and forma binary mixture, various intermolecular forces such as hydrogen bonding, charge transfer, dispersive forces, and dipole-induced dipole interactions, come into play. This causes a change in volume in the mixture and thus the observed values deviate from the calculated values13. This can be explained by observing the structures of the molecules involved and their interactions. The structure allows oxygen atoms to cause dipole moments towards themselves and this gives the molecule a net dipole of approximately 6.214This induces a dipole in the non-polar octan-1-ol and thus causes a closer bond than simple Vander Vaal’s forces. This changes the volume from the ideal condition and thus causes the changes in calculated results from all the theoretical models. However since the deviation observed was smallest in this model for the concentration of 0.3 to 0.4 (approximately at 0.357), it indicates that the denser liquid 1 (methyl benzoate) is in the optimum concentration with minimum molecular interaction to affect the molar volume of that species in such a way that it reduces the ultrasonic velocity and gives a value closer to the observed one. Similarly in the impedance relation, the change in volume and the non-additivity of the acoustic impedance factor cause the deviations from observed values. In the case of Rao’s specific velocity the error occurs due to non-additivity of the specific velocity factor and fact that mole ratios cannot be the only deciding factors in the final equation. Figure-1 Nomoto's Relation Figure-2 3-D structure of Methyl Benzoate Molecule (Red=oxygen) 0.20.40.60.81.200.20.40.60.811.2Percentage ErrorMole Fraction 303.15K 308.15K 313.15K 318.15K Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 36 Figure-3 Acoustic Impedance Relation Figure-4 Rao's Specific Sound VelocityFigure-5 Ideal Mixture Relation -0.20.20.40.60.81.21.400.20.40.60.811.2Percentage ErrorMole Fraction 303.15K 308.15 313.15K 318.15K 0.20.40.60.81.200.20.40.60.811.2Percentage ErrorMole Fraction 303.15K 308.15K 313.15K 318.15K 0.20.40.60.81.200.20.40.60.811.2Percentage ErrorMole Fraction 303.15K 308.15K 313.15K 318.15K Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 37 In the Van Dael-Vangeel IMR method, deviations from experimental values of the theoretical values may be because of the compressibility of the component liquids in the present mixture. This means that isothermal compressibility at each of the temperatures is changed when minor changes in pressure cause minor changes in volume which affect the density and hence the molar volume. This can be avoided by using components where the molecular sizes are comparable, and the polar interactions are kept to a minimum. Chi square relations: When we compare the chi square values for all mathematical models considered, we observe that Nomoto’s relation shows the least amount of deviation, followed by Vangeel IMR and then impedance and Rao’s relation, and lastly Junjie’s relation. This gives the accuracy of each theoretical model over all the range of data points combined. Table-3 Chi square calculations for Nomoto’s Relation Nomoto's Relation X1 303.15K 308.15K 313.15K 318.15K 0 9.11E-15 6.92E-15 6.78E-17 3.94E-14 0.1258 0.001995 0.002441 0.006829 0.011624 0.2446 0.000648 0.006375 0.066147 0.000144 0.357 3.30E-05 0.003259 0.004177 0.003363 0.4634 0.000192 0.003531 0.011477 0.000984 0.5643 0.000572 0.000191 0.00033 0.018116 0.6602 0.001152 0.001934 0.024642 0.022372 0.7514 0.003559 0.008641 0.060369 0.050026 0.8382 0.000192 0.006763 0.044313 0.088744 0.921 3.33E-06 0.00367 0.027751 0.11913 1 2.12E-12 7.72E-05 3.44E-17 9.82E-14 0.008346 0.036882 0.246036 0.314503 Table-4 Chi square calculations for Van Dael-Vangeel Ideal Mixing Relation Vangeel Ideal Mixing Relation X1 303.15K 308.15K 313.15K 318.15K 0 3.79E-29 3.90E-29 0 0 0.1258 0.00092 0.004747 0.011016 0.007514 0.2446 0.002568 0.013075 0.087835 0.002416 0.357 0.000732 0.00014 0.013377 9.40E-05 0.4634 0.0026 5.94E-05 0.00243 0.007383 0.5643 0.003868 0.001616 0.001711 0.036389 0.6602 0.004942 0.009283 0.045535 0.041175 0.7514 0.008304 0.019557 0.086717 0.072796 0.8382 9.80E-05 0.001972 0.061155 0.110659 0.921 0.000226 0.001263 0.034996 0.132849 1 0 3.75E-29 0 3.83E-29 0.024258 0.051714 0.344771 0.411274 To further confirm the methods of analysis used, another similar experimental set up was considered and the theoretical models were applied for it in the same way as presented above. 15The liquids used were similar but differed in the fact that there was no aromatic compound used. The mixtures were of methylmethacrylate with 3 aryl esters/alcohols - 2-methoxy ethanol, 2-ethoxy ethanol, 2-butoxy ethanol. The temperature was kept constant at 303.15K and the molar concentrations of all 3 mixtures were varied from 0.1 to 0.9 in increments of 0.1. Instead of Rao’s specific velocity and the impedance relation, free length theory and collision theory were considered16, 17The following is a series of tables of the experimental values observed for varying concentration of methyl methacrylate (X1), and its comparison to the calculated value using the various theories. The percentage error is also specified. These are followed by the plots of percentage error vs. mole fraction. This graphical representation gives us a good idea about the accuracy of each theory. Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 38 Table-5 Chi square calculations for Impedance relation Impedance Relation X1 303.15K 308.15K 313.15K 318.15K 0 0 0 0 0 0.1258 0.000514 0.019437 0.038546 2.68E-05 0.2446 0.018855 0.052752 0.199946 0.033275 0.357 0.017404 0.01624 0.088069 0.022979 0.4634 0.026264 0.019539 0.02019 0.066084 0.5643 0.028829 0.033561 0.051604 0.127606 0.6602 0.027777 0.050457 0.145121 0.124026 0.7514 0.028924 0.060069 0.18715 0.154194 0.8382 0.004379 0.000929 0.119751 0.177571 0.921 0.002007 1.66E-05 0.057284 0.170131 1 0 0 0 0 0.154953 0.253001 0.907661 0.875893 Table-6 Chi square calculations for Junjie’s relation X1 303.15K 308.15K 313.15K 318.15K 0 4.63E-09 1.39E-10 5.10E-09 2.22E-09 0.1258 0.050925 0.019215 0.013101 0.091973 0.2446 0.088281 0.065483 0.009019 0.113812 0.357 0.185888 0.25114 0.162236 0.271231 0.4634 0.225362 0.32518 0.419517 0.251811 0.5643 0.23833 0.301962 0.344136 0.18133 0.6602 0.212789 0.2246 0.155215 0.155747 0.7514 0.142294 0.132058 0.058645 0.066679 0.8382 0.122714 0.1873 0.027835 0.005502 0.921 0.035882 0.065763 0.002416 0.017617 1 4.38E-11 1.18E-10 1.71E-10 1.66E-10 1.302465 1.5727 1.192119 1.155703 Table-7 Chi square calculations for Rao’s Specific velocity relationRao's Specific Velocity Relation X1 303.15K 308.15K 313.15K 318.15K 0 1.51E-28 3.51E-28 3.57E-28 0 0.1258 0.000312 0.00734 0.016226 0.004502 0.2446 0.005194 0.020383 0.111805 0.00678 0.357 0.002898 0.000569 0.026735 0.001023 0.4634 0.006428 0.000978 1.03E-05 0.017277 0.5643 0.008323 0.006248 0.008787 0.055751 0.6602 0.0094 0.017434 0.068399 0.059861 0.7514 0.012871 0.028823 0.112158 0.092926 0.8382 0.000687 0.000505 0.07669 0.128363 0.921 0.000568 0.000561 0.04126 0.143138 1 1.47E-28 1.84E-27 2.42E-27 1.38E-27 0.046682 0.082842 0.462069 0.50962 In our secondary set up, observing the percentage error plots tells us that the free length theory with a maximum of over 3.5 % deviation is the least accurate method for binary mixtures where dipole-dipole interactions are present18. In the case of these mixtures, the methyl methacrylate and other alcohols all have strong dipoles which give rise to dipole-dipole interactions. This causes changes in volumes which are not a result of pressure changes. This causes the deviations in adiabatic compressibility and therefore a change in the value of intermolecular free length. If we consider the collision factor theory the “actual volume” factor in the equation again does not consider the very strong hydrogen bonding and dipole-dipole interactions in the compounds and thus it gives a deviation. This would not have been the case in case of a mixture with slightly weaker interactions such as dipole-induced dipole and weak Van Der Waal’s forces. Thus from the graph we see that our previous conclusion of Nomoto’s theory being the most accurate one is supported by our secondary observations as well. Thus in both cases, despite there being a strong presence of dipole-dipole interactions and even hydrogen bonding in the second case, Nomoto’s theory gives the least amount of error. Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 39 Figure-6 Junjie's Relation Table-8 For 2-methoxy ethanol X1 exp(m/s) FLT UNomoto’s UCFT UIMR UJunjie’s PENomoto’s PEFLT PECFT PEIMR PEJunjie’s 0.1 1421 1416 1418.01 1418 1413 1418 0.21 0.34 0.23 0.62 0.23 0.2 1408 1388 1402.23 1400 1402 1399 0.41 1.43 0.56 0.43 0.64 0.3 1246 1210 1240.64 1234 1246 1236 0.43 2.89 0.94 0.79 0.78 0.4 1245 1205 1238.27 1229 1234 1229 0.54 3.21 1.32 0.91 1.31 0.5 1235 1193 1228.94 1216 1220 1214 0.49 3.39 1.56 1.24 1.67 0.6 1069 1024 1065.58 1055 1055 1083 0.32 4.23 1.32 1.31 -1.32 0.7 1008 975 1005.89 999 998 1018 0.21 3.24 0.89 0.98 -1.03 0.8 930 911 928.23 926 923 938 0.19 2.04 0.46 0.76 -0.86 0.9 920 916 919.12 917 917 923.8 0.11 0.42 0.38 0.38 -0.41 AVG PE 0.32 2.35 0.85 0.82 0.92 0.20.40.60.81.21.41.61.800.20.40.60.811.2Percentage ErrorMole Faction 303.15 K 308.15 K 313.115 K 318.15 K Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 40 Figure-7 For 2-methoxy ethanol Table-9 For 2-ethoxy ethanol X1 Uexp(m/s) UFLT UNomoto’s UCFT UIMR UJunjie’s PENomoto’s PEFLT PECFT PEIMR PEJunjie’s 0.1 1638 1631 1632.26 1634 1631 1634.7 0.35 0.4 0.3 0.4 0.2 0.2 1500 1458 1495.37 1491 1488 1492.5 0.31 2.8 0.6 0.8 0.5 0.3 1460 1422 1451.53 1450 1440 1448.3 0.58 2.6 0.7 1.4 0.8 0.4 1360 1315 1351.56 1348 1334 1336.9 0.62 3.3 0.9 1.9 1.7 0.5 1232 1188 1226.33 1221 1202 1208.6 0.46 3.6 0.8 2.4 1.9 0.6 1080 1043 1075.47 1070 1051 1060.6 0.42 3.4 0.9 2.7 1.8 0.7 1000 971 995.63 992 977 985.2 0.44 2.9 0.7 2.3 1.4 0.8 920 894 917.056 914 912 912.66 0.32 2.8 0.5 0.9 0.8 0.9 600 598 598.44 597 598 597.63 0.26 0.3 0.4 0.4 0.4 AVG PE 0.41 2.45 0.64 1.46 1.05 Figure-8 For 2-ethoxy ethanol 0.10.20.30.40.50.60.70.80.9Percentage ErrorMole Fraction PEJunjie’s PEIMR PECFT PEFLT PENomoto’s 100.10.20.30.40.50.60.70.80.9Percentage ErrorMole Fraction PENomoto’s PEFLT PECFT PEIMR PEJunjie’s Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 41 Figure-9 For 2-butoxy ethanolTable-10 For 2-butoxy ethanol X1 Uexp(m/s) UFLT UNomoto’s UCFT UIMR UJunjie’s PENomoto’s PEFLT PECFT PEIMR PEJunjie’s 0.1 2240 2236 2231.49 2236 2231 2235.5 0.38 0.2 0.2 0.4 0.2 0.2 1530 1483 1524.8 1522 1516 1520.9 0.34 2.9 0.5 0.9 0.6 0.3 1498 1450 1488.71 1486 1473 1480.1 0.62 3.2 0.8 1.7 1.2 0.4 1474 1422 1464.42 1456 1458 1453.4 0.65 3.5 1.2 1.1 1.4 0.5 1440 1384 1432.22 1419 1434 1412.9 0.54 3.9 1.5 0.43 1.9 0.6 1418 1360 1411.05 1400 1441 1437.9 0.49 4.1 1.3 -1.6 -1.4 0.7 1230 1212 1224.96 1219 1246 1243.6 0.41 3.2 0.9 -1.3 -1.1 0.8 990 969 985.535 982 998 997.01 0.35 2.1 0.8 -0.84 -0.7 0.9 680 677 675.784 678 683 683.52 0.26 0.43 0.46 -0.43 -0.5 AVG PE 0.44 2.61 0.85 0.96 1 Figure-10 2 Ethoxy ethanol Figure-11 Methyl MethacrylateConclusion Thus we can conclude that Nomoto’s relation is the closest to theoretical values, followed by Vangeel IMR and impedance relation, and then Rao’s specific velocity relation and finally Junjies’s relation. Further improvements to accuracy can be made through the establishing of a relationship between dipole-induced dipole interactions and changes in volume, and appending these factors into the existing theories.References 1.Sridevi G and Fakruddin Sk, Experimental and Theoretical Studies of Ultrasonic Velocity in Binary Liquid Mixtures of Methyl Benzoate at Different Temperatures, Journal of Chemical and Pharmaceutical 100.10.20.30.40.50.60.70.80.9Percentage ErrorMole Fraction PENomoto’ PEFLT PECFT PEIMR Research Journal of Chemical Sciences ___________________________________________________________ ISSN 2231-606XVol. 5(10), 33-42, October (2015) Res. J. Chem. Sci. International Science Congress Association 42 Research, 4(8), 3792-3796(2012)2.Gutta Sridevi, Ultrasonic Study of Acoustical Parameters of Binary Liquid Mixtures of Methyl Benzoate with 1Octanol at 303.15K, 308.15K, 313.15K and 318.15K, Research Journal of Chemical Sciences, 3(3), 14-19(2013) 3.Shaik Babu S.V., Kumara Shastry, Ha SieTiong and Sreehari Sastry S., Experimental and Theoretical Studies of Ultrasonic Velocity in Binary Liquid Mixtures of Ethyl benzoate, E-Journal of Chemistry, 9(4), 2309-2314(2012)4.Nomoto O, J Phys Soc Jpn.,13, 1528 (1958)5.Shanmuga Priya C., Nitya S., Velraj G. and Kannappan A.N., Molecular interaction studies in liquid mixture using ultrasonic technique, International Journal of Advanced Science and Technology, 18, 59-73 (2010)6.Zareena Begum P.B., Sandhya Sri and Rambabu C., Theoretical Evaluation of Ultrasonic Velocities in Binary Liquid Mixtures of Anisaldehyde with Some Alcoxyethanols at Different Temperatures, ISRN Physical Chemistry, vol. 2012, Article ID 943429, 12 pages, (2012)7.Van Dael W and Vangeel E, Proc IntConf on calorimetry and thermodynamics, Warasa, 1955, 555 8.Durga Bhavani M., Ratnakar A. and Kavitha Ch., International Letters of Chemistry, Physics and Astronomy, 5, 1-6 (2013)9.GV Rama Rao, PB Sandhya Sri, A Vishwanatha Sarma and C Rambabu, Indian Journal of Pure and Applied Physics, 45,135-1422007)10.Junjie Z, J China UnivSci Tech., 14, 298 (1984)11.Praveen Babu G., Pavan Kumar B. and Nagarjun K. 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