Thermoacoustical approach to the intermolecular free-length of liquid mixtures

Abstract

The intermolecular free-length is evaluated by making use of thermoacoustical parameters followed by a comparative study. To achieve this objective, thermoacoustical parameters of four ternary and two quaternary liquid mixtures have been computed. These parameters have been utilized to calculate the available volume (V_a), which in turn has been used to compute intermolecular free-length (L_f) for the systems under investigation. The computed values of L_f have then been compared with the values estimated from well-established thermodynamic and ultrasonic methods. To the best of our knowledge, this investigation is a pioneering attempt in evaluation of intermolecular free-length involving multicomponent liquid mixtures by making use of thermoacoustical parameters.

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Introduction

New interest has been revived in the study of thermoacoustical parameters in the view of temperature and pressure coefficient of the bulk modulus. Several approaches are found in literature¹ to evaluate thermoacoustical parameters for a wide variety of liquids. Pandey et al.^2,3 made an effort to estimate these parameters for liquids over a wide range of temperature and pressure, and to analyse the interrelationship among various thermoacoustical properties. Tabhane et al.⁴ used expansivity data to compute thermoacoustical parameters in binary liquid mixtures. Very recently, Pandey et al.¹ determined the available volume of few liquids and binary mixture of acetone + methyl iodide at varying temperatures by computing some of the thermoacoustical parameters.

Available volume is related to another very useful quantity, intermolecular free-length (L_f), which has been extensively used in literature^5–7 to study the intermolecular interactions in mixtures. The present paper is a continuation of our previous work¹ involving computation of available volume of few pure organic liquids and a binary liquid mixture, and aims to compute thermoacoustical parameters for four ternary and two quaternary liquid mixtures. Available volume, computed from thermoacoustical parameters, has been used to obtain intermolecular free-length of systems under consideration. The calculated values of L_f are compared with the values estimated from well-established thermodynamic and ultrasonic methods.⁸ To the best of our knowledge, this is the first time when intermolecular free-length has been evaluated using thermoacoustical parameters, and compared with the values obtained by other methods.

Theoretical

The Moelwyn-Hughes parameter (C₁) and reduced molar volume (Ṽ) may be expressed as,Here, α represents thermal expansion coefficient.

The isochoric temperature coefficient of internal pressure, X, is given by

Expression of isothermal (K′), isochoric (K″) and isobaric (K) acoustical parameters¹ are

andwhere

The isothermal acoustical parameter (K′) is related to the available molar volume (V_a) of the substance as,¹

According to Schaaffs,⁸V_a can be expressed as,

where u is the ultrasonic velocity at temperature T and u_∞ is 1600 m s⁻¹.

Thermodynamically,⁸ the value of V_a can be calculated from the following equation:

where T_c is the critical temperature. In the present work, the critical temperature of systems under consideration is supposed to be mole fraction additive of the values of its pure components.

The intermolecular free-length (L_f) is given by the relation,⁸

where V_a represents the available volume per mole and Y is the surface area per mole given by

V_a = V − V₀ (11)

and

Here, N is the Avogadro number and V₀ and V are the molar volumes at zero temperature and at temperature T, respectively.

Eqns. (1) to (7), (8) and (9) have been employed to obtain the available volume in the case of the thermoacoustical, ultrasonic and thermodynamic methods, respectively. Using the computed values of available volume, intermolecular free-lengths of liquid mixtures have been estimated, videeqns. (10) to (12).

Results and discussion

Four ternary systems, viz. benzene + chloroform + cyclohexane (I), toluene + chloroform + cyclohexane (II), chlorobenzene + chloroform + cyclohexane (III) and dioxane + chloroform + cyclohexane (IV) at 303.15 K, and two quaternary systems, viz. n-pentane + n-hexane + cyclohexane + benzene (V) and n-pentane + n-hexane + benzene + toluene (VI) at 298.15 K, have been considered for the present investigation. The experimental data of systems under consideration have been taken from the literature.^9,10 The data of contributory pure components required for computation are given in Table 1. The coefficients of thermal expansivity, α, of the systems under investigation have been calculated by a method given elsewhere.¹¹ The intermolecular free-lengths (L_f), evaluated through three different methods, are recorded in Tables 2 and 3 for ternary and quaternary mixtures, respectively. A comparative study of the three methods has also been presented graphically in Figs. 1 and 2 for ternary and quaternary systems, respectively. The average percentage values (APD) of thermoacoustical and ultrasonic methods with respect to the thermodynamic method for the four ternary and two quaternary systems are found to be −2.9, −47.2 (System (I)); −5.6, −48.6 (System (II)); −9.7, −53.9 (System (III)); −5.2, −45.9 (System (IV)); −3.0, −13.0 (System (V)) and −5.8, −14.4 (System (VI)).

Fig. 1 Intermolecular free-length (L_f) vs. mole fraction of first component (x₁) for ternary systems.

Fig. 2 Intermolecular free-length (L_f) vs. mole fraction of first component (x₁) for quaternary systems.

Table 1 Parameters of pure components

Components	T/K	T c/K	ρ/g cm^−3	u/m s^−1	α 10^3/K^−1
Cyclohexane	303.15	553.50	0.7662	1230.3	1.2390
Chloroform	303.15	536.40	1.4665	966.6	1.3000
Benzene	303.15	562.05	0.8677	1278.3	1.3200
Toluene	303.15	591.75	0.8539	1285.3	1.0650
Chlorobenzene	303.15	632.40	1.0957	1250.0	0.9900
Dioxane	303.15	587.00	1.0185	1324.7	1.1150
n-Pentane	298.15	469.60	0.6215	991.2	1.6226
n-Hexane	298.15	507.40	0.6552	1076.2	1.3897
Cyclohexane	298.15	553.40	0.7736	1253.0	1.2150
Benzene	298.15	562.09	0.8783	1295.0	1.2265
Toluene	298.15	591.70	0.8626	1304.3	1.0740

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Table 2 Intermolecular free-length of ternary liquid mixtures at 303.15 K

x 1	x 2	ρ/g cm^−3	u/m s^−1	L f/Å
				Thermoacoustical	Ultrasonic	Thermodynamic
Benzene(1) + chloroform (2) + cyclohexane (3)
0.1015	0.3019	0.9453	1124.6	0.5831	0.8534	0.5707
0.2019	0.2990	0.9574	1121.9	0.5793	0.8539	0.5654
0.2998	0.2993	0.9656	1128.6	0.5766	0.8344	0.5612
0.4001	0.3002	0.9813	1134.0	0.5725	0.8162	0.5557
0.5045	0.2983	0.9963	1141.8	0.5683	0.7928	0.5500
0.6031	0.2976	1.0119	1146.3	0.5643	0.7771	0.5446

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Toluene (1) + chloroform (2) + cyclohexane (3)
0.0911	0.3685	0.9815	1106.4	0.5828	0.8960	0.5684
0.2198	0.2939	0.9478	1138.8	0.5858	0.8377	0.5618
0.3004	0.3036	0.9627	1142.9	0.5843	0.8155	0.5562
0.4013	0.3026	0.9691	1144.8	0.5841	0.8116	0.5504
0.5016	0.2999	0.9813	1151.9	0.5825	0.7942	0.5435
0.5976	0.3035	0.9948	1158.7	0.5811	0.7779	0.5373

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Chlorobenzene (1) + chloroform (2)+ cyclohexane (3)
0.1001	0.3034	0.9659	1126.4	0.5853	0.8535	0.5621
0.1988	0.3034	0.9998	1131.5	0.5835	0.8401	0.5487
0.3150	0.2875	1.0326	1141.0	0.5816	0.8170	0.5337
0.4005	0.2994	1.0768	1141.7	0.5782	0.8114	0.5219
0.5023	0.2982	1.1116	1151.2	0.5763	0.7888	0.5101
0.5997	0.3007	1.1488	1156.4	0.5741	0.7755	0.4990

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Dioxane (1) + chloroform (2)+ cyclohexane (3)
0.1023	0.2998	0.9589	1132.0	0.5819	0.8353	0.5656
0.2024	0.3000	0.9818	1129.9	0.5779	0.8346	0.5566
0.2981	0.3011	1.0132	1134.0	0.5724	0.8179	0.5466
0.3991	0.2912	1.0328	1144.5	0.5686	0.7897	0.5378
0.5001	0.3007	1.0784	1152.4	0.5616	0.7633	0.5267
0.6046	0.2966	1.1108	1177.3	0.5562	0.7043	0.5169

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Table 3 Intermolecular free-length of quaternary liquid mixtures at 298.15 K

x 1	x 2	x 3	ρ/g cm^−3	u/m s^−1	L f/Å
					Thermoacoustical	Ultrasonic	Thermodynamic
n-Pentane (1) + n-hexane (2) + cyclohexane (3) + benzene (4)
0.0488	0.1238	0.1831	0.8051	1240.1	0.5822	0.6046	0.5550
0.0658	0.1078	0.2036	0.8004	1239.9	0.5832	0.6059	0.5573
0.0813	0.0934	0.2238	0.7988	1237.2	0.5834	0.6115	0.5588
0.1006	0.0778	0.2430	0.7995	1236.6	0.5830	0.6122	0.5601
0.1180	0.0629	0.2615	0.7941	1230.2	0.5842	0.6263	0.5627
0.1243	0.0466	0.2842	0.7908	1240.1	0.5849	0.6070	0.5632
0.1410	0.1304	0.3129	0.7709	1205.4	0.5921	0.6862	0.5805
0.1560	0.1262	0.1513	0.7828	1213.5	0.5866	0.6627	0.5747
0.1285	0.1192	0.5888	0.7549	1206.2	0.6001	0.6940	0.5897
0.1537	0.0925	0.1685	0.7906	1226.4	0.5842	0.6355	0.5691
0.1649	0.1013	0.5177	0.7547	1194.3	0.5985	0.7175	0.5910
0.1368	0.1258	0.1507	0.7851	1210.5	0.5861	0.6686	0.5712
0.0910	0.1721	0.6137	0.7509	1201.6	0.6030	0.7075	0.5920
0.0649	0.1378	0.1103	0.8048	1263.7	0.5815	0.5570	0.5567
0.1810	0.1656	0.2971	0.7542	1197.4	0.5968	0.7083	0.5944

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n-Pentane (1) + n-hexane (2) + benzene (3) + toluene (4)
0.0943	0.0918	0.4587	0.8216	1260.1	0.5843	0.5689	0.5438
0.1300	0.1373	0.2974	0.8011	1232.2	0.5936	0.6312	0.5565
0.1278	0.1288	0.3589	0.8054	1237.3	0.5908	0.6198	0.5551
0.1450	0.1291	0.3376	0.7991	1225.7	0.5924	0.6452	0.5588
0.1492	0.1384	0.3421	0.7974	1217.9	0.5925	0.6614	0.5611
0.1843	0.1484	0.2711	0.7855	1200.1	0.5963	0.7032	0.5693
0.1823	0.1640	0.3613	0.7838	1197.9	0.5940	0.7047	0.5729
0.1819	0.1606	0.3842	0.7824	1200.1	0.5937	0.6996	0.5732
0.1250	0.1665	0.2455	0.7975	1223.1	0.5960	0.6548	0.5592
0.1691	0.2041	0.2218	0.7769	1201.3	0.6001	0.7051	0.5752
0.1866	0.0826	0.1250	0.7995	1229.3	0.5974	0.6438	0.5559
0.1372	0.1580	0.5548	0.7926	1211.4	0.5875	0.6686	0.5662
0.0660	0.1053	0.7033	0.8273	1268.1	0.5776	0.5453	0.5436
0.0524	0.1434	0.4201	0.8176	1260.5	0.5885	0.5714	0.5441
0.1568	0.0468	0.4582	0.8184	1256.3	0.5843	0.5749	0.5482

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A close observation of Tables 2 and 3 and Figs. 1 and 2 clearly illustrates a fairly good agreement in the values of intermolecular free-length, computed from thermoacoustical and thermodynamic methods. The credibility of the results is further enhanced by the APD (average percentage deviation) values of the six systems under consideration. The values obtained from thermoacoustical method lie in between the values of thermodynamic and ultrasonic methods. The values computed from ultrasonic method seem to be on the higher side, as compared to those determined from thermoacoustical and thermodynamic methods. The higher values obtained with ultrasonic method over thermodynamic method have also been observed for liquid methane and liquid tetrafluoromethane.⁸ The superiority of thermodynamic method over ultrasonic method has been established in literature.⁸ The thermodynamic approach appears to be good for the estimation of L_f, provided T is not very close to T_C. The ultrasonic method fails completely when the ultrasonic velocity in liquids or liquid mixtures exceeds 1600 m s⁻¹, since at T > T_C the right hand side of eqn. (8) disappears, and when u > u_∞, the right hand side of eqn. (9) vanishes. Therefore, the thermoacoustical method is found to be more effective than the other two methods. It is also evident from the tables that L_f decreases with increasing mole fraction of the first component for all systems, while, ultrasonic velocity exhibits the reverse trend. The variation of ultrasonic velocity is related to free-length. The increase in ultrasonic velocity and corresponding decrease in free-length for all the systems under study, resembles the view proposed by Eyring and Kincaid,¹² according to which the ultrasonic velocity increases with decrease in free-length and vice versa. The increases or decreases in the values of ultrasonic velocity and intermolecular free-length with composition, for all the systems studied, indicate interactions between contributing molecules.⁷

In conclusion, the present investigation throws light on the evaluation of thermoacoustical parameters in multicomponent systems, and the determination of intermolecular free-length through three different methods. The present approach is found to be accurate and simple for computation of intermolecular free-length, and thus, provides a novel pathway for meaningful study of intermolecular interactions in liquid mixtures.

Acknowledgements

One of the authors (J. C.) wishes to acknowledge the financial assistance provided by CSIR-New Delhi, India.

References

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