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Molecular Interactions from the Experimental and Validation with Estimated Theoretical Sound Velocity

  • L. Palaniappan
  • S. NithiyananthamEmail author
Original Article
  • 85 Downloads

Abstract

The ultrasonic velocity, density and viscosity measurements carried out for the binary mixtures of toluene, m-xylene with some butanols at 303 K. Various theoretical models have been applied to these binary systems; evaluate the sound velocity values and compared with the experimental values. The validity of Nomoto theory (NT), Van Deal-Vangeal (IMR) and Free length theory (FLT) is checked and a comparative study of the above models is made. The non-ideal behavior of the system is explained in terms of molecular interactions of the constituents of the mixture. The interactions further, evident with percentage deviation, molecular interaction parameter and goodness fit test.

Keywords

Molecular interaction Theoretical sound velocity Models Organic fluids 

1 Introduction

The assessment of the nature of molecular interaction present in liquid mixtures can also be done by sound velocity predictions. The departure from linearity in the values of ultrasound velocity when studied as a function of concentration is found to exhibit interesting variations in the case of liquid mixtures [1]. The acoustical and thermo dynamical properties evaluated in the binary and ternary liquid mixtures in the previous chapters signify the physico-chemical and molecular interactions that exist between the components of system.

As an additional confirmation for the presence of specific interaction, an attempt has been made in this present work to correlate the experimental findings of sound velocity with those predicated theoretically on the basis of molecular models. Such comparisons are found to be useful in knowing the thermodynamics of the mixtures and provide a better means to test the validity of the various empirical and semi empirical theories [2, 3, 4].

In recent years, various theories [5, 6] have been in use for computing ultrasonic velocity in liquid mixtures and the deviation in theoretical sound velocity has been attributed mainly to the molecular interactions in the mixtures. In this chapter, the theoretical evaluation of ultrasonic velocity in the binary liquid mixtures have been done using Nomoto’s relation [7], Van Deal Vangeal’s ideal mixture relation [8] and Free length theory [9]. Also, percentage deviation, molecular interaction parameter and Chi square test values are calculated. These statistical parameters support the predictions made in the previous chapters and to check the validity of the theory to the experimental systems.

2 Experimental Techniques

The mixtures of various concentrations in mole fraction were prepared by taking purified AR grade samples at 303 K. In all the mixtures, the mole fraction of the second component, toluene (x2 = 0.3), was kept fixed, while the mole fraction of the remaining two were varied from 0.0 to 0.7 so as to have the mixtures of different composition. Further this fixed 0.3 mol fraction of toluene is to discuss critically the system efficiency in absorption of toluene. The ultrasonic velocity in liquid mixtures have been measured using an ultrasonic interferometer (Mittal type) working at 2 MHz frequency with an accuracy of ± 0.1 ms−1. The density and viscosity are measured using a 5 ml specific gravity bottle and an Ostwald’s viscometer of accuracy of ± 0.1 kg m−3 and ± 0.001 mNsm−2, respectively. All the precautions to minimize the possible experimental error, as explained in our earlier work are perfectly made. The set-up is checked for some standard liquids. The values obtained are compared with literature and found that they match very well with each other.

3 Theory and Calculations

3.1 Nomoto’s Relation [NR]

Nomoto [7] established an empirical formula for ultrasonic velocity in binary liquid mixtures on the assumption of linear dependence of the molar sound velocity on concentration in mole fraction and the additive for molar volume as,
$$ U = \left( {\frac{{x_{1} R_{1} + x_{2} R_{2} }}{{X_{1} V_{1} + x_{2} V_{2} }}} \right)^{3} $$
(1)
where, x1, x2 are the mole fraction of the components. R1, R2 the respective molar sound velocities and V1, V2 the molar volumes, respectively.
Molar sound velocity is related to molecular weight [m] and density [ρ] as,
$$ R = \frac{m}{\rho }U^{1/3} = VU^{1/3} $$
(2)
where, the molar volume obeys the additive property
$$ {\text{V }} = {\text{ x}}_{ 1} {\text{V}}_{ 1} + {\text{ x}}_{ 2} {\text{V}}_{ 2} + {\text{ x}}_{ 3} {\text{V}}_{ 3} $$
(3)

3.2 Ideal Mixture Relation [IMR]

Van deal and Vangeal [8] assumed the adiabatic compressibility [β] of the mixture as,
$$ \beta = \phi_{1} \beta_{1} + \phi_{2} \cdot \beta_{2} $$
(4)
where, ϕ refers to volume fraction. Using the ratio of specific heats,γ, Van Deal suggested the following relation for ultrasonic velocity in homogeneous liquid mixtures
$$ \beta_{IM} = \phi \frac{\gamma }{{\gamma_{IM} }}\beta_{1} + \phi_{2} \frac{\gamma }{{\gamma_{IM} }}\beta_{2} $$
(5)
This equation is true if the mixture is an ideal one and also,
$$ \gamma_{1} = \gamma_{2} = \gamma_{IM} $$
(6)
Equation (6) reveals that the volume fractions in Eq. (5) can be transformed into a linear combination of the mole fraction and hence we have,
$$ \beta_{IM} = X_{1} \beta_{1} + X_{2} \beta_{2} $$
(7)
On the basis of this equation, the sound velocity is given as,
$$ \frac{1}{{(x_{1} m_{1} + x_{2} m_{2} )U_{IMR}^{2} }} = \frac{{X_{1} }}{{m_{1} U_{1}^{2} }} + \frac{{X_{2} }}{{m_{1} U_{1}^{2} }} $$
(8)
The degree of molecular interaction α is given as,
$$ \alpha = \left[ {\frac{{U^{2}_{\exp t} }}{{U^{2}_{IMR} }}} \right] - 1 $$
(9)

3.3 Free Length Theory [FLT]

Jacobson [9, 10] introduced the concept of intermolecular free length to determine the ultrasonic velocity in pure liquids and liquid mixtures. Further, he related the velocity of the pure liquids to the free length Lf by the equation,
$$ {\text{UL}}_{\text{f}} \rho^{1/2} = {\text{ K}}_{\text{T}} $$
(10)
where, KT is the temperature dependent Jacobson’s constant which takes a value of 199.976 × 10−8 in M.K.S. units at 303 K. For liquid mixtures, the above equation can be written as,
$$ U_{FLT} = \frac{{K_{T} }}{{L_{fmix} \rho^{1/2} }} $$
(11)
where, Lf mix and ρ represents the free length and density of the mixture.
The free length Lf mix of the liquid mixtures is given by,
$$ L_{fmix} = 2\left[ {\frac{{V_{m} - (X_{1} V_{0.1} + X_{2} V_{0.2} + X_{3} V_{0.3} }}{{(X_{1} Y_{1} + X_{2} Y_{2} + X_{3} Y_{3)} }}} \right] $$
(12)
where, V0.1, V0.2 and V0.3 represent the volume [V0 = Vm Uexpt/Uα] of the pure components at absolute zero and,
$$ Y = \left[ {\frac{{2V_{m} }}{{L_{fmix} }}} \right] \cdot \left[ {1 - \frac{{U_{\exp t} }}{{U_{a} }}} \right] $$
(13)
where, Y is the surface area per mole.

4 Percentage Deviation

The percentage deviation in sound velocity between the experimental and computed values are calculated as [11],
$$ \left[ {\frac{\Delta U}{U}} \right]\% = \left[ {\frac{{U_{\exp t} - U_{\text{the}} }}{{U_{\exp t} }}} \right] \cdot 100\% $$
(14)

5 Chi Square Test for Goodness of Fit

The Karl Pearson [12] test of goodness fit is a very powerful tool to determine the deviations of the theoretical values from the experimental ones. If Oi [i = 1,2,3…n] is a set of observed values and Ei [i = 1,2,3…n] is the corresponding set of theoretical values, then the Chi square is given by,
$$ (\chi )^{2} = \mathop \varSigma \limits_{i = 1}^{n} \left[ {(O_{i} - E_{i} )^{2} /E_{i} } \right] $$
(15)

Chi Square follows Chi Square distribution with (n–1) degrees of freedom. It is calculated through Eq. (15) is less than the tabulated value for the same degree 2.167 then there is a good correlation between the theory and the experiment as the calculated values are acceptable as having only 5% error. For the best correlation, the calculated 2 should be less than 1.239 for 1% error.

6 Results and Discussions

6.1 Results

The experimentally measured ultrasonic velocity values and the estimated ultrasonic velocity obtained from the various theoretical models for the binary systems taken up in this study are given in Tables 1, 2 and 3, respectively. The percentage deviation of ultrasonic velocity, the molecular interaction parameter (α) and the Chi square deviation (\( \chi^{2} \)) of the theoretical velocities from the experimental values for the respective systems are also presented in the same Tables.
Table 1

Estimated sound velocities and their validation parameters for toluene binaries

Mole fraction

Ultrasonic velocity in ms−1

Percentage deviation

α

Ultrasonic velocity in ms−1

Percentage deviation

α

x1

x3

Uexpt

UNR

UIMR

UFLT

UNR

UIMR

UFLT

Uexpt

UNR

UIMR

UFLT

UNR

UIMR

UFLT

1-Butanol system

2-Butanol system

0.0

1.0

1226.2

1226.2

1226.2

1154.8

0.000

0.000

5.823

0.000

1185.0

1185.0

1185.0

1138.5

0.000

0.000

3.924

0.000

0.1

0.9

1232.1

1233.1

1228.3

1158.0

− 0.081

0.308

6.014

0.006

1196.2

1196.3

1190.0

1140.4

− 0.008

0.518

4.665

0.010

0.2

0.8

1238.0

1239.8

1231.1

1169.0

− 0.145

0.557

5.574

0.011

1205.4

1207.3

1195.9

1157.4

− 0.158

0.788

3.982

0.016

0.3

0.7

1243.5

1246.2

1234.8

1179.1

− 0.217

0.700

5.179

0.014

1216.3

1218.2

1202.9

1167.9

− 0.156

1.102

3.979

0.022

0.4

0.6

1250.2

1252.6

1239.4

1198.8

− 0.192

0.864

4.111

0.018

1226.8

1228.9

1211.1

1185.6

− 0.171

1.280

3.358

0.026

0.5

0.5

1257.1

1258.7

1244.9

1205.0

− 0.127

0.970

4.144

0.020

1237.1

1239.0

1220.2

1203.7

− 0.154

1.366

2.700

0.028

0.6

0.4

1263.2

1264.7

1251.4

1231.6

− 0.119

0.934

2.502

0.019

1246.5

1249.0

1230.7

1224.6

− 0.201

1.268

1.757

0.026

0.7

0.3

1268.6

1270.6

1258.8

1253.9

− 0.158

0.773

1.159

0.016

1255.6

1258.9

1242.6

1248.3

− 0.263

1.035

0.581

0.021

0.8

0.2

1274.2

1276.3

1267.1

1274.4

− 0.165

0.557

− 0.016

0.011

1265.7

1268.5

1255.8

1273.1

− 0.221

0.782

− 0.585

0.016

0.9

0.1

1280.8

1281.8

1276.6

1307.1

− 0.078

0.328

− 2.053

0.007

1277.8

1278.0

1270.7

1293.2

− 0.016

0.556

− 1.205

0.011

1.0

0.0

1287.2

1287.2

1287.2

1329.2

0.000

0.000

− 3.263

0.000

1287.2

1287.2

1287.2

1329.2

0.000

0.000

− 3.263

0.000

  

Chi–square values (χ)2

0.025

0.567

24.042

Chi–square values (χ)2

0.032

1.151

12.972

3-Butanol system

ISO-Butanol system

0.0

1.0

1094.2

1094.2

1094.2

1019.4

0.000

0.000

6.836

0.000

1157.5

1157.5

1157.5

1823.3

0.000

0.000

− 57.521

0.000

0.1

0.9

1112.6

1114.6

1104.5

1033.0

− 0.180

0.728

7.154

0.015

1170.2

1171.7

1164.2

1765.9

− 0.128

0.513

− 50.906

0.010

0.2

0.8

1130.2

1134.6

1116.3

1052.2

− 0.389

1.230

6.901

0.025

1183.6

1185.7

1172.1

1715.8

− 0.177

0.972

− 44.965

0.020

0.3

0.7

1147.5

1154.6

1129.7

1074.8

− 0.619

1.551

6.336

0.032

1196.1

1199.2

1181.1

1661.6

− 0.259

1.254

− 38.918

0.026

0.4

0.6

1165.3

1174.6

1145.2

1103.5

− 0.798

1.725

5.303

0.035

1209.5

1212.5

1191.4

1610.0

− 0.248

1.496

− 33.113

0.031

0.5

0.5

1183.4

1194.0

1162.3

1129.9

− 0.896

1.783

4.521

0.037

1222.2

1225.7

1203.2

1558.3

− 0.286

1.555

− 27.500

0.032

0.6

0.4

1204.6

1212.7

1181.2

1156.9

− 0.672

1.943

3.960

0.040

1235.6

1238.5

1216.4

1506.2

− 0.235

1.554

− 21.900

0.032

0.7

0.3

1225.2

1231.9

1203.3

1197.7

− 0.547

1.787

2.245

0.037

1248.4

1251.1

1231.3

1460.5

− 0.216

1.370

− 16.990

0.028

0.8

0.2

1246.5

1250.0

1227.1

1232.3

− 0.281

1.556

1.139

0.032

1261.5

1263.2

1247.7

1413.9

− 0.135

1.094

− 12.081

0.022

0.9

0.1

1268.5

1269.0

1255.7

1267.3

− 0.039

1.009

0.095

0.020

1274.8

1275.4

1266.5

1366.8

− 0.047

0.651

− 7.217

0.013

1.0

0.0

1287.2

1287.2

1287.2

1329.2

0.000

0.000

− 3.263

0.000

1287.2

1287.2

1287.2

1329.2

0.000

0.000

− 3.263

0.000

  

Chi–square values (χ)2

0.333

2.548

32.406

Chi–square values (χ)2

0.046

1.657

1015.04

 
Table 2

Estimated sound velocities and their validation parameters for m-Xylene binaries

Mole fraction

Ultrasonic velocity in ms−1

Percentage deviation

α

Ultrasonic velocity in ms−1

Percentage deviation

α

x1

x3

Uexpt

UNR

UIMR

UFLT

UNR

UIMR

UFLT

Uexpt

UNR

UIMR

UFLT

UNR

UIMR

UFLT

1-Butanol system

2-Butanol system

0.0

1.0

1226.2

1226.2

1226.2

1154.8

0.000

0.000

5.823

0.000

1185.0

1185.0

1185.0

1138.5

0.000

0.000

3.924

0.000

0.1

0.9

1236.7

1239.0

1225.6

1141.4

− 0.186

0.898

7.706

0.018

1199.6

1202.7

1186.8

1128.4

− 0.258

1.067

5.935

0.022

0.2

0.8

1245.4

1251.1

1227.1

1134.4

− 0.458

1.469

8.913

0.030

1213.2

1219.4

1190.9

1124.5

− 0.511

1.838

7.311

0.038

0.3

0.7

1257.6

1262.6

1230.7

1134.0

− 0.398

2.139

9.828

0.044

1227.6

1235.5

1197.5

1119.1

− 0.644

2.452

8.838

0.051

0.4

0.6

1266.5

1273.6

1236.8

1133.2

− 0.561

2.345

10.525

0.049

1241.5

1250.2

1206.3

1120.6

− 0.701

2.835

9.738

0.059

0.5

0.5

1277.3

1283.8

1245.0

1132.0

− 0.509

2.529

11.376

0.053

1255.6

1264.9

1218.1

1128.1

− 0.741

2.987

10.155

0.063

0.6

0.4

1286.5

1293.4

1255.6

1134.8

− 0.536

2.402

11.792

0.050

1268.3

1278.7

1232.8

1125.5

− 0.820

2.799

11.259

0.058

0.7

0.3

1295.3

1302.8

1269.1

1139.3

− 0.579

2.023

12.044

0.042

1282.4

1291.9

1250.7

1139.3

− 0.741

2.472

11.159

0.051

0.8

0.2

1305.1

1311.5

1285.3

1143.4

− 0.490

1.517

12.390

0.031

1296.8

1304.5

1272.2

1147.7

− 0.594

1.897

11.498

0.039

0.9

0.1

1315.6

1319.9

1304.8

1154.3

− 0.327

0.821

12.261

0.017

1311.2

1316.4

1297.4

1164.3

− 0.397

1.052

11.203

0.021

1.0

0.0

1328.0

1328.0

1328.0

1169.2

0.000

0.000

11.958

0.000

1328.0

1328.0

1328.0

1169.2

0.000

0.000

11.958

0.000

  

Chi–square values (χ)2

0.248

4.208

179.65

− 

Chi–square values (χ)2

0.439

5.942

146.92

3-Butanol system

ISO-Butanol system

0.0

1.0

1094.2

1094.2

1094.2

1019.4

0.000

0.000

6.836

0.000

1157.5

1157.5

1157.5

1823.3

0.000

0.000

− 57.521

0.000

0.1

0.9

1092.0

1121.7

1100.6

1017.9

− 2.720

− 0.788

6.786

− 0.016

1173.4

1178.4

1160.8

1746.3

− 0.426

1.074

− 48.824

0.022

0.2

0.8

1138.1

1148.2

1109.7

1020.2

− 0.887

2.495

10.359

0.052

1189.6

1197.7

1166.4

1673.9

− 0.681

1.950

− 40.711

0.040

0.3

0.7

1161.2

1173.9

1121.8

1024.6

− 1.094

3.393

11.764

0.071

1205.2

1217.1

1174.8

1602.7

− 0.987

2.522

− 32.982

0.052

0.4

0.6

1183.3

1198.2

1136.9

1035.4

− 1.259

3.921

12.499

0.083

1221.4

1235.2

1185.9

1535.8

− 1.130

2.907

− 25.741

0.061

0.5

0.5

1208.2

1221.9

1155.7

1050.2

− 1.134

4.345

13.077

0.093

1237.6

1252.4

1199.8

1466.2

− 1.196

3.054

− 18.471

0.064

0.6

0.4

1232.6

1244.7

1178.5

1069.5

− 0.982

4.389

13.232

0.094

1253.0

1269.2

1217.3

1402.9

− 1.293

2.849

− 11.963

0.060

0.7

0.3

1258.5

1266.5

1205.9

1088.9

− 0.636

4.180

13.476

0.089

1269.2

1284.5

1237.5

1340.2

− 1.205

2.498

− 5.594

0.052

0.8

0.2

1280.2

1287.8

1239.2

1114.0

− 0.594

3.203

12.982

0.067

1287.5

1300.0

1263.2

1276.0

− 0.971

1.887

0.893

0.039

0.9

0.1

1303.5

1308.3

1279.5

1141.3

− 0.368

1.841

12.443

0.038

1306.1

1314.1

1292.5

1222.6

− 0.613

1.041

6.393

0.021

1.0

0.0

1328.0

1328.0

1328.0

1169.2

0.000

0.000

11.958

0.000

1328.0

1328.0

1328.0

1169.2

0.000

0.000

11.958

0.000

  

Chi–square values (χ)2

1.582

13.04

208.32

Chi–square values (χ)2

1.079

6.108

816.16

Table 3

Estimated sound velocities and their validation parameters for aniline binaries

Mole fraction

Ultrasonic velocity in ms−1

Percentage deviation

α

Ultrasonic velocity in ms−1

Percentage deviation

α

x1

x3

Uexpt

UNR

UIMR

UFLT

UNR

UIMR

UFLT

Uexpt

UNR

UIMR

UFLT

UNR

UIMR

UFLT

1-Butanol system

2-Butanol system

0.0

1.0

1226.4

1226.2

1226.2

1154.8

0.016

0.016

5.838

0.000

1185.0

1185.0

1185.0

1138.5

0.000

0.000

3.924

0.000

0.1

0.9

1251.5

1261.8

1244.9

1135.9

− 0.823

0.527

9.237

0.011

1211.2

1223.4

1204.9

1131.1

− 1.007

0.520

6.613

0.010

0.2

0.8

1294.4

1297.9

1266.3

1162.5

− 0.270

2.171

10.190

0.045

1264.5

1264.9

1227.9

1119.3

− 0.032

2.894

11.483

0.061

0.3

0.7

1326.4

1334.9

1290.9

1183.3

− 0.641

2.676

10.789

0.056

1298.6

1302.8

1253.9

1115.2

− 0.323

3.442

14.123

0.073

0.4

0.6

1360.8

1372.6

1319.1

1202.8

− 0.867

3.064

11.611

0.064

1349.3

1344.7

1284.7

1164.8

0.341

4.788

13.674

0.103

0.5

0.5

1397.3

1410.6

1351.1

1190.2

− 0.952

3.306

14.821

0.070

1388.5

1387.7

1320.4

1126.3

0.058

4.905

18.884

0.106

0.6

0.4

1436.5

1450.3

1389.0

1264.2

− 0.961

3.307

11.994

0.070

1435.8

1430.2

1360.3

1100.2

0.390

5.258

23.374

0.114

0.7

0.3

1480.2

1490.0

1432.1

1047.1

− 0.662

3.250

29.260

0.068

1479.2

1475.2

1408.7

1240.0

0.270

4.766

16.171

0.103

0.8

0.2

1522.5

1530.6

1482.8

931.3

− 0.532

2.608

38.831

0.054

1519.3

1520.3

1464.8

1315.3

− 0.066

3.587

13.427

0.076

0.9

0.1

1565.5

1572.0

1542.7

1445.1

− 0.415

1.456

7.691

0.030

1554.3

1566.6

1532.0

1595.0

− 0.791

1.435

− 2.619

0.029

1.0

0.0

1614.0

1614.0

1614.0

1491.7

0.000

0.000

7.577

0.000

1614.0

1614.0

1614.0

1491.7

0.000

0.000

7.577

0.000

  

Chi–square values (χ)2

0.640

9.173

703.21

Chi–square values (χ)2

0.282

19.549

338.45

3-Butanol system

ISO-Butanol system

0.0

1.0

1094.2

1094.2

1094.2

1019.4

0.000

0.000

6.836

0.000

1157.5

1157.5

1157.5

1823.3

0.000

0.000

− 57.521

0.000

0.1

0.9

1146.2

1138.6

1116.4

988.7

0.663

2.600

13.741

0.054

1196.4

1197.9

1178.2

1797.7

− 0.125

1.521

− 50.259

0.031

0.2

0.8

1200.5

1182.9

1141.1

1034.2

1.466

4.948

13.853

0.107

1248.2

1239.7

1202.2

1882.3

0.681

3.685

− 50.801

0.078

0.3

0.7

1235.2

1232.6

1172.0

1016.6

0.210

5.117

17.698

0.111

1283.5

1282.0

1229.3

1838.9

0.117

4.223

− 43.272

0.090

0.4

0.6

1275.5

1316.9

1233.8

959.4

− 3.246

3.269

24.782

0.069

1330.2

1326.4

1261.4

1917.0

0.286

5.172

− 44.114

0.112

0.5

0.5

1316.7

1332.3

1246.6

1064.0

− 1.185

5.324

19.192

0.116

1364.6

1371.4

1298.2

1902.8

− 0.498

4.866

− 39.440

0.105

0.6

0.4

1387.8

1383.6

1292.9

1110.8

0.303

6.838

19.960

0.152

1416.6

1417.3

1341.0

2047.4

− 0.049

5.337

− 44.529

0.116

0.7

0.3

1429.7

1439.2

1351.0

1152.5

− 0.664

5.505

19.389

0.120

1451.5

1465.1

1392.0

2231.1

− 0.937

4.099

− 53.710

0.087

0.8

0.2

1485.5

1498.8

1424.4

1295.9

− 0.895

4.113

12.763

0.088

1498.8

1512.6

1450.8

2561.4

− 0.921

3.203

− 70.897

0.067

0.9

0.1

1551.6

1554.0

1505.7

1657.9

− 0.155

2.958

− 6.851

0.062

1560.3

1562.8

1524.1

3028.3

− 0.160

2.320

− 94.084

0.048

1.0

0.0

1614.0

1614.0

1614.0

1491.7

0.000

0.000

7.577

0.000

1614.0

1614.0

1614.0

1491.7

0.000

0.000

7.577

0.000

  

Chi–square values (χ)2

1.999

28.217

448.21

Chi–square values (χ)2

0.363

20.833

2786.12

Toluene system

m-Xylene system

0.0

1.0

1287.2

1287.2

1287.2

1329.2

0.000

0.000

− 3.263

0.000

1328.0

1328.0

1328.0

1169.2

0.000

0.000

11.958

0.000

0.1

0.9

1324.3

1313.6

1311.0

1325.3

0.808

1.004

− 0.076

0.020

1356.5

1348.6

1351.7

1172.1

0.582

0.354

13.594

0.007

0.2

0.8

1355.4

1341.1

1336.1

1335.8

1.055

1.424

1.446

0.029

1388.2

1370.5

1376.3

1172.6

1.275

0.857

15.531

0.017

0.3

0.7

1388.6

1370.3

1363.2

1354.7

1.318

1.829

2.441

0.038

1412.5

1394.1

1402.0

1191.4

1.303

0.743

15.653

0.015

0.4

0.6

1422.5

1400.3

1391.7

1395.5

1.561

2.165

1.898

0.045

1446.4

1419.1

1428.6

1189.7

1.887

1.231

17.748

0.025

0.5

0.5

1454.6

1432.0

1422.4

1449.0

1.554

2.214

0.385

0.046

1480.2

1446.0

1456.4

1214.7

2.310

1.608

17.937

0.033

0.6

0.4

1480.2

1465.5

1455.5

1497.4

0.993

1.669

− 1.162

0.034

1510.8

1475.1

1485.7

1261.3

2.363

1.661

16.514

0.034

0.7

0.3

1512.4

1499.8

1490.4

1626.8

0.833

1.455

− 7.564

0.030

1535.8

1505.8

1515.5

1343.1

1.953

1.322

12.547

0.027

0.8

0.2

1548.6

1536.0

1528.3

2008.1

0.814

1.311

− 29.672

0.027

1558.3

1538.9

1546.8

1590.3

1.245

0.738

− 2.054

0.015

0.9

0.1

1585.5

1574.1

1569.4

3116.3

0.719

1.015

− 96.550

0.021

1586.4

1575.0

1579.7

2591.5

0.719

0.422

− 63.357

0.009

1.0

0.0

1614.0

1614.0

1614.0

1491.7

0.000

0.000

7.577

0.000

1614.0

1614.0

1614.0

1491.7

0.000

0.000

7.577

0.000

  

Chi–square values (χ)2

1.632

3.476

878.23

Chi–square values (χ)2

3.641

1.619

722.32

 
Figure 1a shows the trend of (α) for all the four binary systems of toluene. In the same way Fig. 1b, c are pertaining to m-xylene binaries and aniline binaries, respectively.
Fig. 1

Trend of molecular interaction parameter (α) in a Toluene, b m-Xylene and c Aniline Binaries

The perusal of the Tables 1, 2 and 3 reveals good and worst agreement between the experimental and calculated sound velocities, owing to the several assumptions and approximations made in the respective theories. In NR, it is supposed that the volume does not change on mixing. The agreement between experimental and theoretical velocities of Nomoto relation in all the binary systems, suggests that R is additive property in all the systems. It can be noticed that in all the binary systems predictions by NR gives excellent closeness with the experimental values and the FLT predictions are fully unacceptable. FLT prediction is found to fail even for pure liquids.

IMR predictions are highly acceptable than NR, only for aniline + m-xylene binary and in all other systems NR seems to be the best one. In binaries, the values predicted by IMR are always lesser than the experimental values but NR predictions, in most cases, are higher than experimental values among the various theories taken into consideration, NR is found to give an excellent prediction of sound velocity. The order of merit for the prediction of sound velocity goes down as NR and IMR and FLT is totally invalid for all the systems considered.

6.2 Discussions on Percentage Deviations

The average percentage deviations between the observed and calculated sound velocities may be attributed to the molecular interactions [11, 13, 14, 15] between the components of the liquid mixture. Such type of interactions has not been taken into account in the formalism of the respective theories. Further, they are inadequate to account comprehensively for the experimental manifestation of molecular interaction in various ultrasonic processes. The extent of deviation in velocities may be attributed to the presumption made in the theories for the polar—polar and polar—weak polar interaction between the molecules [16].

6.3 Discussions on Molecular Interaction Parameter

Molecular interaction parameter (α) which is given as the deviation in the value of \( {\text{U}}_{ \exp }^{2} /{\text{U}}_{\text{IMR}}^{2} \) from unity, is a useful validation parameter that will ensure the extent of interactions between the components of the mixture and it also concludes whether the chosen mixture is ideal or not (Tables 1, 2 and 3). These values are also presented in the same Tables. This is efficient to state the extent of molecular interaction in the liquid mixtures, especially in those cases where the properties other then the sound velocity are not known.

Another validation arises from the Chi square deviations. It is observed that the predictions based on percentage deviation and \( \chi^{2} \) tests are similar, however those of Chi square test are evident. Chi square test of goodness of fit enable to find whether the deviation of the theoretical values from the experimental ones are due to chance or really due to the inadequacy of the theory to fit in the data [17].

6.4 Discussions on Chi Square Deviations

It follows Chi square distribution with (n–1) degrees of freedom. For a set of eleven readings of the present investigations for binary mixtures, it is noted that if the \( \chi^{2} \) value is less than 2.558, the theoretical values are acceptable as having only 1% error. To accept 5% error, \( \chi^{2} \) values may go up to 3.940. Further, if the range of acceptance goes up to 5% then, the values may go up to 2.167 (Tables 1, 2 and 3).

6.5 Validations of others

Tables 1, 2 and 3 are pertaining to the binary mixtures with butanol (polar) as second component, whereas the first component is weak polar in first two systems, (viz. toluene and m-xylene) and strong polar (aniline) in second system. A perusal of these Tables reveals that the percentage deviation for NR is comparatively lower than that for IMR whereas they are totally invalid for FLT [18].

FLT method is based on the assumption that the molecules are behaving as rigid spheres [9]. Thus the rigidness of components cannot be accounted due to the failure of FLT and the component seems to be more elastic and flexible. This elasticity is expected to make the system to be more and more ideal.

For an ideal mixture, IMR predictions should match with experimental values. This also goes bad in the present systems as IMR predictions have only second merit and hence the systems are non-ideal. This loss in elasticity may be attributed to the fact that the components are involving in azeotropic/complex formation.

It can be seen from these Tables 1, 2 and 3, that NR is unanimously fit well for all these systems whereas FLT is not fit for polar–polar type (aniline + alcohol systems) and for near ideal mixtures. As Nomoto’s method basically depends on the linear dependence of sound velocity on mole fraction, the increasing mole fraction increases the structure existing in the system and thus, the increase of first component supports for the regular structure–formation as concluded in the respective binary systems [19].

Further, for NR predictions, % deviations are mostly negative and low for all the toluene and m-xylene binaries. However for few aniline binaries they are positive and still higher than IMR predictions. NR method demands the dependence of molar sound velocity and hence this may be obviously true for the chosen binaries. The very small magnitudes of % deviation reveal closeness to experimental values and hence it is evident that molar sound velocity plays the key role in deciding the molecular interactions [20].

The molecular interaction parameter α is small but positive in all the mixtures. This observed α values clearly indicate the existence of strong interactions existing between the polar—polar and polar—weak-polar molecules. The percentage deviation and Chi square values obtained for IMR is very high, whereas for FLT it is totally obscured which indicates that these theories are invalid for the system considered. This is also reflected in the large percentage deviation of sound velocity.

Since similar interpretations have been made for the present system in theoretical study further confirm the existence of strong interactions between the component molecules in the mixtures that arises due to the disturbance in the molecular symmetry of the polar molecules.

It is unanimously found that FLT method is applicable for the binary systems also. It reveals that the molecules are not rigid spheres, but more elastic. This elastic nature of molecules is found to be suppressed in the systems as IMR predictions deviates largely. Thus the better agreement of NR is an indication that the molecules in the mixtures are behaving much more complex than in pure state. This complexation nature of molecules can lend a good support for tight packing of molecules and at the same time with desired extent of interaction [21].

As the IMR method totally fails, the suggestions of \( \alpha \) using IMR are not treated as final. They have to be further refined. This can be done by the obtained \( \chi^{2} \) values. As stated earlier, a \( \chi^{2} \) equal to or less than 2.558 for binaries indicates an error of 1% in predictions where as for 5% error it may be up to 3.940. Hence all the binary systems, except aniline + m-xylene, fully satisfy NR assumptions. Thus, the dependence of sound velocity on mole fraction is ascertained. The system aniline + m-xylene seem to be more ideal [22].

The intermolecular interaction attraction terms appear to exhibit certain deviations from ideality [23]. The perusal of Fig. 1a–b indicates that the magnitude of interaction increases with chain structure deformations, i.e., maximum interaction is existing for iso-ol and minimum for 1-ol. Thus, the chances for azeotropic/complex destruction of binary complexes by aniline are quite high in toluene system than in m-xylene systems. Further, these variations confirm the dependence of chain length on molecular interaction as discussed above.

7 Conclusions

Among the three theories taken up for the prediction of sound velocity, Namoto Relation is found to yield excellent comparison with the experimental value for the systems investigated. The existences of strong molecular interactions between polar–polar and polar-weak polar molecules are evident. All the binary systems found to be not rigid, non-ideal and depend on molar sound velocity. The structure formation with concentrations is confirming the azeotropic destruction is quite ascertained in the considered mixtures.

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Copyright information

© The Tunisian Chemical Society and Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of PhysicsAnnamalai UniversityChidambaramIndia
  2. 2.Post Graduate and Research Department of PhysicsThiru. Vi. Kalyanasundaram Govt Arts and Science CollegeThiruvaurIndia

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