Sound Velocity and Isentropic Compressibility of binary Liquid

Canadian Chemical Transactions

Year 2016 | Volume 4 | Issue 2 | Page 157-167

Research Article

DOI:10.13179/canchemtrans.2016.04.02.0279

Sound Velocity and Isentropic Compressibility of binary Liquid Systems from various Theoretical Models at Temperature range 293.15 to 313.15K Atul Kumar1,2, Urvashi Srivastava1,2, Arun K. Singh3 Kirti Srivastava4 and Rajeev K. Shukla5* 1 2

Department of Chemistry, PSIT, Kanpur- India Department of Physics, PSIT, Kanpur- India

3

Department of Chemistry, BMC, Bharwai, Kaushambi-India Department of Chemistry, JSS Acad. of Tech. Edu., Noida, India 5 Department of Chemistry, V.S.S.D. College, Kanpur-208002- India. 4

*Corresponding Author, Email: [email protected] Received: February 6, 2016

Revised: February 29, 2016

Accepted: March 7, 2016

Published: March 9, 2016

Abstract: Density and speed of sound were measured for the binary liquid mixtures formed by formamide, N-methylacetamide, di-methylformamide and di-methylacetamide with acetonitrile at 293.15, 298.15, 303.15, 308.15 and 313.15 K and atmospheric pressure over the whole concentration range. Prigogine-Flory-Patterson model (PFP), Ramaswamy and Anbananthan (RS) model and model suggested by Glinski, were utilized to generate the data set for the prediction of associational behavior of weakly interacting liquids. The measured properties were fitted to Redlich-Kister polynomial relation to estimate the binary coefficients and standard errors for the validation of our experimental findings. An attempt has also been made to study the molecular interactions involved in the liquid mixture from observed data. Furthermore, McAllister multi body interaction model was also used to correlate the binary properties. Conclusively, these non-associated and associated models were compared and tested for different systems at various temperatures showing that the associated models yield better agreement between theory and experiment as compared to non-associated model. . Keywords: Sound Velocity, Theoretical Models, Binary Mixture, Associated, McAllister, Isentropic Compressibility

1. INTRODUCTION Physicochemical behavior and molecular interactions occurring in a variety of liquid mixtures and solutions can be studied with the help of ultrasonic velocity [1]. Data on sound velocity offers a convenient method for determining certain thermodynamic properties of liquids and liquid mixtures, which are not obtained by other methods. Recently, extensive work has been carried out by many workers Borderless Science Publishing

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[2-6] to investigate liquid state through analysis of ultrasonic propagation parameters and to correlate ultrasonic velocity with other physical and thermodynamic parameters. General applicability and comparison merits for the various models in binary solvent mixtures of aprotic protophobic solvent with aprotic protophilic solvents having distinctly different Gutmann donor numbers yield typical interaction gradually varying with nitrogen base size is the major object of the present work. As a part of research concerning the thermo chemical studies on new working fluid pair, we present here some useful data on ultrasonic velocity and isentropic compressibility for the mixture of formamide, N-methylacetamide (NMA), di-methylformamide (DMF) and di-methylacetamide (DMA) with acetonitrile at 293.15, 298.15, 303.15, 308.15 and 313.15 K and atmospheric pressure over the whole concentration range. These data were analyzed in terms of Ramaswamy and Anbananthan (RS) model [7] model suggested by Glinski [8], Prigogine-Flory-Patterson (PFP) model [9-11], The first two models, RS and model devised by Glinski (associated) are based on the association constant as an adjustable parameters where as PFP and others (non-associated) are based on the additivity of liquids. For that purposes, we selected the liquids having significant importance in many industries, containing poor associating properties. From these results, deviations in ultrasonic velocity, u were calculated and fitted to the Redlich-Kister polynomial equation [12] to derive the binary coefficients and the standard errors. An attempt has also been made to correlate the experimental data with the McAllister multi body interaction model [13] which is based on Eyring٫s theory of absolute reaction rates and for liquids the free energy of activation are additive on a number fraction and that interactions of like and unlike molecules. The mixing behavior of such liquid mixtures is interesting due to presence of cyano group coupled with amide linkage resulting interactions in liquid mixtures. The associational behavior of liquids and their correlation with molecular interactions has also been made using different liquid state models. 2. METHODS 2.1 Experimental 2.1.1 Materials High purity and AR grade samples used in this experiment were obtained from Merck Co. Inc., Germany and purified by distillation in which the middle fraction was collected. The liquids were stored in dark bottles over 0.4 nm molecular sieves to reduce water content and were partially degassed with a vacuum pump. The purity of each compound was checked by gas chromatography and the results indicated that the mole fraction purity was higher than 0.99. 2.1.2 Instruments and Procedure Before each series of experiments, we calibrated the instrument at atmospheric pressure with doubly distilled water. The uncertainty in the density measurement was within ± 6.4 kg.m-3. The densities of the pure components and their mixtures were measured with the bi-capillary pyknometer. The liquid mixtures were prepared by mass in an air tight stopped bottle using an electronic balance model SHIMADZUAX-200.The average uncertainty in the composition of the mixtures was estimated to be less than ±0.0001. Crystal controlled variable path ultrasonic interferometer supplied by M/s Mittal enterprises (model-05F), New Delhi (India), operating at a frequency of 2 MHz was used in the ultrasonic measurements. The reported uncertainty which is better than  1% is the highest uncertainty found from Borderless Science Publishing

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all the data points. Data were taken from 293.15 K up to 313.15 K in temperature intervals of 5 K. Isentropic compressibility, βs, were calculated from the relation,

 s  u 2  1

(1)

where ρ is the density and u is the ultrasonic velocity. The estimated error in the calculation of isentropic compressibility was found to be ± 2.5 T Pa−1. All results are listed in Table 1 together with literature values [14] for comparison. Table 1. Comparison of Density and Speed of Sound with literature data for pure components at 293.15, 298.15, 303.15, 308.15 and 313.15 K

Compound

T

 x 10-3 K

Acetonitrile

293.15 298.15 303.15 308.15 313.15 293.15 298.15 303.15 308.15 313.15 293.15 298.15 303.15 308.15 313.15 293.15 298.15 303.15 308.15 313.15 293.15 298.15 303.15 308.15 313.15

1.2762 1.2943 1.3151 1.3300 1.3420 0.9431 0.9515 0.9570 0.9594 0.9656 1.1095 1.1224 1.1349 1.1440 1.1364 1.0844 1.0899 1.0970 1.1108 1.1187 1.0839 1.0968 1.1097 1.1190 1.1255

Formamide

NMA

N,N-DMF

N,N-DMA

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βT/TPa-1 108.84 113.54 119.10 123.18 126.54 43.92 45.11 45.89 46.24 47.15 71.51 74.03 76.53 78.38 76.85 66.76 67.79 69.12 71.76 73.31 66.67 69.09 71.56 73.36 74.64

V/cm3 mole-1 51.53 52.55 53.08 53.55 53.97 39.78 39.89 40.03 40.17 40.51 76.44 76.85 76.93 77.20 77.72 76.52 76.92 77.59 78.11 78.38 90.53 91.68 93.07 94.00 94.47

exp/ g.cm-3 0.7865 0.7811 0.7733 0.7665 0.7605 1.1320 1.1290 1.1250 1.1210 1.1117 0.9563 0.9512 0.9502 0.9468 0.9405 0.9551 0.9501 0.9419 0.9357 0.9325 0.9623 0.9502 0.9360 0.9268 0.9221

lit/ g.cm-3 0.7822 0.77649 0.77125 1.1339 1.12915 1.12068 0.9520 0.9460 0.94873 0.94387 0.9412 0.9310 0.9615 0.97633 0.93169 0.9232

uexp/ ms-1 1305.5 1290.1 1263.4 1245.1 1230.5 1625.8 1601.0 1585.2 1577.2 1565.1 1425.8 1401.0 1373.8 1355.8 1370.5 1476.8 1465.2 1454.1 1428.2 1411.2 1470.5 1451.3 1435.1 1421.5 1409.0

ulit/ ms-1 1347.1 1326.3 1293.4 1242.7 1647.9 1626.3 1607.2 1571.3 1455.2 1431.6 1411.8 1375.2 1517.2 1496.4 1481.3 1432.2 1420.6 1501.3 1468.7 1422.6

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2.2 COMPUTATIONAL 2.2.1 Ramswamy and Anbananthan Model (RS Model) Ramswamy and Anbananthan [7] assumed the linearity of acoustic impedance with the mole fraction of components. Further it is assumed, that when solute is added to solvent the molecules interact according to the equilibrium as: A+B ↔ AB and the association constant Kas can be defined as; K as 

(2)

[A B ] [ A ][ B ]

(3) where [A] is amount of solvent and [B] is amount of solute in the liquid mixture. By applying the condition of linearity in speed of sound with composition, we get, uobs = xA uA + xAB uAB

(4)

where xA, xAB, uA and uAB and uobs are the mole fraction of A, mole fraction of associate AB, ultrasonic velocity of A, ultrasonic velocity of associate AB and observed ultrasonic velocity respectively. The equilibrium reaction in the eq (4) is not complete by definition as there are also molecules of nonassociated component present in the liquid mixture even prevailing in the high solute content. Considering the non-associated component present in the liquid mixture eq takes the form, uRA = [xA uA + xB uB + xAB uAB]

(5)

where xB and uB are the mole fraction of B and ultrasonic velocity of B (non-associated component). The general idea of this model for predicting the values of pure associate AB can be, however, exploited as; K as 

[ AB ] (C A  [ AB ])(C B  [ AB ])

(6)

where CA and CB are initial molar concentrations of the components. One can take any value of K as and calculate the equilibrium value of [AB] for every composition of the mixture as well as [A] =CA-[AB] and [B] =CB-[AB]. Replacing molar concentration by activities for concentrated solution, eq (6) becomes, K as 

(a A

a AB  a AB )( aB  a AB )

(7)

where aA, aB and aAB are the activity of component A, Component B and associate, AB respectively. Taking equimolar activities which are equal to; a´A= aA-aAB and a´B = aB-aAB Borderless Science Publishing

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where a´A and a´B are the activities of [A] and [B] in equi molar quantities respectively. From eq (7) one can obtain the value of Kas as; K as 

a AB 2 a A aB  a A a AB  aB a AB  a A B 

(8)

a AB a ' A .a ' B

Now, assuming any value of ultrasonic velocity in the hypothetical pure component AB, u AB, it is possible to compare the ultrasonic velocity calculated using eq (5) with the experimental values. On changing both the adjustable parameters Kas and uAB gradually, one can get different values of the sum of squares of deviations, S = (uobs - ucal) 2

(9)

where uobs and ucal are the observed and calculated equilibrium properties respectively. The minimum value of S can be obtained theoretically by a pair of the fitted parameters. But we found that for some Kas and uAB, the value of S is high and changes rapidly, and for others, it is low and changes slowly when changing the fitted parameters. In such cases, the value of u AB should not be much lower than the lowest observed ultrasonic velocity of the system or much higher than the highest one. Quantitatively, it should be reasonable to accept the pair of adjustable parameters K as and uAB which has the physical sense and which reproduces the experimental physical property satisfactorily. 2.2.2 Model Devised by Glinski On inspecting the results obtained from Ramaswamy and Anbananthan model, Glisnki [8] suggested the equation assuming additivity with the volume fraction,  of the components, the refined version of Natta and Baccaredda model [15] as, u Glinski

 Au B u AB

u Au B u AB   B u Au AB   ABu Au B

(10)

where ucal is the theoretical ultrasonic velocity of binary liquid mixture, A, B are the volume fractions of component A and B and uA, uB and uAB are the ultrasonic velocity of components A, B and AB. The numerical procedure and determination of association constant, Kas, were similar to that described before and the advantage of this method as compared with the earlier one was that the data on densities of liquid mixture are not necessary except those of pure components needed to calculate the volume fractions. In this context the importance of models assuming associated liquids already mentioned and was further developed and elaborated by Reis et al [16] and others [17]. 2.2.3 Prigogine-Flory-Patterson Model (PFP) Flory and collaborators [9] used the cell partition function of Hirschfelder and Eyring and a ~

~ 1

simple Van der Waals energy- volume relation, U   V Borderless Science Publishing

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equations for the mixing functions and partial molar quantities may be obtained from the general corresponding states equations given by making this particular choice of (m,n). Patterson et al [11] have drawn attention to the close connection between the Flory theory and corresponding state theory of Prigogine employing a simple cell model of the liquid state. The equation of state for the materials conforming to the principle of corresponding states can be expressed in a universal form through the use of suitable characteristic values i.e. (reduction parameters) P*, V*, T* for the pressure, volume and temperature respectively. In order to extend corresponding state theory to deal with the surface tension, Patterson and Rastogi [11] used the reduction parameters as,

 *  k 1 / 3 P *2 / 3 T *1 / 3

(11)

called the characteristic surface tension of the liquid. Here k is the Boltzmann constant. A segment experiences an increase in the configurational energy equal to  M U (V ) due to the loss of a molar fraction, M, of its nearest neighbors at the surface while moving from the bulk phase to the surface phase. With the particular (3, ∞) choice of m, n potential, we get; ~

~

~ 5 / 3

~

 (V )  [ M V

~ 1/ 3

(

V

 1 .0 ~

V

~ 1/ 3

) ln(

2

V ~ 1/ 3

V

 0 .5

~

)]

 1 .0

(12) Thus on the basis of Flory theory, surface tension of liquid mixture is given by the eq.  

*

~

~

 (V )

(13) and surface tension is related to the ultrasonic velocity by well known and well tested [2-6] relation of Auerbach [18]

   u PFP   4   6.3x10  

2/3

(14) All the notations used in the above equations have their usual significance as detailed out by Flory. 2.2.4 McAllister – three body model McAllister considered a number of different three bodied planar encounters in the study of the viscosity of a mixture of molecules type (1) and (2). He proposed that the total free energy of activation will be dependent on the free energy of activation (Gi, Gij or sijk) of individual interactions and their fraction

of

G 

x12 G1*

*



total 

* x12 x2 G121

* 2 x1 x22 G123



G  *

or

occurrences 

(

x i3 , x i2 x j , x 2j , x i x 2j

* 2 x12 x2 G112



or

*  3x12 x2 G12

).

Hence,

* x1 x22 G212

x23G2*

x13G1*

xi x j xk (15)

*  3x1 x22 G21



x23G2*

(16)

Now applying eq (15) for each set of interactions (i.e.111,121,211, 112; 212, 122, 221; and 222) and then taking logarithms of equations so obtained to eliminate free energy terms, following equation is obtained Borderless Science Publishing

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ln u  x13 ln u1  3x12 x2 ln u12  3x1 x2 ln u21  x23 ln u2  ln[ x1  x2 M 2 / M1 ]  3x12 x2 ln[( 2  M 2 / M1 ) / 3]

 3x1 x22 ln[(1  2M 2 / M1 ) / 3]  x23 ln[ M 2 / M1 ]

(17)

where 2M1  M 2 3 M 1  2M 2  3

M 12  M 21

and

(18)

2.2.5 McAllister –four body model If there is much difference in size of two molecules then four body model approaches more nearly a 3-dimensional treatment. Again in considering different interactions and their fraction of total occurrences, energy of activation may be written as sum of energy of activations of various interactions * * * G*  x14 G1*  4 x13 x2 G1112  6 x12 x22 G112  4 x1 x23G2221

 x24 G2*

(19)

by techniques entirely analogous to method given above, the following equation is derived; ln umix  x14 ln u1  4 x13 x2 ln u1112  6 x12 x2 ln u1122  4 x1 x22 ln u222  x24 ln u2  ln( x1  x2 M 2 / M1 )  4 x13 x2 ln[( 3  M 2 / M 1 ) / 4]  6 x12 x22 ln[(1  M 2 / M 1 ) / 2]

 4 x1 x23 ln[(1  3M 2 / M 1 ) / 4]  x24 ln( M 2 / M 1 )

(20)

where u,x1, u1,M1, x2, u2 and M2 are the ultrasonic velocity of mixture, mole fraction, ultrasonic velocity and molecular weight of pure component 1 and 2 respectively. McAllister coefficients are adjustable parameters that are characteristic of the system. 3. RESULT AND DISCUSSION Associations in liquids were analyzed earlier [19] by considering van der Waals equation of state which was based only on simple averaged geometrical deviations without analyzing the system in terms of equilibrium. It is related usually to the deviation of different quantities from additivity. The Ramaswami and Anbananthan model was further corrected by Glinski [8] and tested [20] to predict the associational behavior of liquid mixtures. The quantities analyzed were refractive index, molar volume, viscosity, intermolecular free length and many others [20-22, 23-24]. The results of fittings obtained from the model were utilized properly. The calculations were carried out by performing a computer program which allows the fittings easily for both the adjustable parameters simultaneously. Initially, we considered the association constant, Kas and speed of sound of hypothetical associate, uA,B as the fitted parameters. On changing these parameters, the equilibrium concentrations of species [A], [B] and [AB] will change and the ultrasonic velocity can be computed. The difference between experimental and theoretical values for ultrasonic velocity is used to obtain the sum of squares of deviation. It is assumed that in solution Borderless Science Publishing

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three associates instead of two are formed (A, B and AB). The associate can be treated as a fitted one with the value of Kas.

values of ultrasonic velocity in pure

Values of thermal expansion coefficient () and isothermal compressibility needed in the PFP model were obtained from the equation which have already been tested in many cases by us [19, 22] and others [24-26]. The mixing function  can be represented mathematically by Redlich-Kister polynomial equation [12] for correlating the experimental data as; p

y  xi (1  x1 ) Ai ( 2 x1  1) i

(21) where y refers to deviation in ultrasonic velocity (u), x1 is the mole fraction and Ai is the coefficient. i 0

The values of coefficients were determined by a multiple regression analysis based on the least square method and are summarized along with the standard deviations between the experimental and fitted values of the respective function in Table S1 (supporting information). The values of standard deviations lie between 2.26-30.24 with the largest value corresponds to acetonitrile + NMA mixture at 308.15 K for ultrasonic velocity and 2.7-49.0 with the largest value corresponds to acetonitrile + DMA mixture at 308.15 K for isentropic compressibility respectively. McAllister coefficients a, b, and c were calculated using the least square procedure and the results of estimated parameters and standard deviation between the calculated and experimental values are presented in Table S2 (supporting information). It is observed that the four body model is correlated the mixture ultrasonic velocity to a significantly higher degree of accuracy for all the systems than the three body model. Generally McAllister model is adequate in correlating the systems having small deviations. Mixture data are presented in Table S3-S4 (supporting information). With the increase of mole fraction, the values of speed of sound obtained from all the models decrease at all temperatures except at a few places. The absolute average percent deviations (AAPD) in ultrasonic velocity and isentropic compressibility obtained from different models are provided in Table 4. It is observed that associated processes provide fairly good results as compared to non-associated. Higher deviation values in PFP model can be explained as the model was developed for non-electrolyte meric spherical chain molecules and the system under investigation have interacting and associating properties. Moreover, the expression used for the computation of  and T are also empirical in nature. Positive deviations in speed of sound are a result of molecular association and complex formation whereas negative deviations are due to molecular dissociation. The actual sign and magnitude of deviations depend upon relative strength of two opposite effect. The lacks of smoothness in deviations are due to the interaction between the component molecules. Isentropic compressibility increases regularly with the increase of mole fraction while density and ultrasonic velocity show regular behavior. Results of ultrasonic velocity obtained from different models along with percent deviation are reported in Table 5. A careful perusal of the results clearly indicates the close proximity of our results with the experimental findings. Borderless Science Publishing

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4. CONCLUSION Models assuming associated processes give more reliable results as compared to non-associated processes and helpful in deducing the internal structure of associates through the fitted values of ultrasonic velocity and isentropic compressibility in a hypothetical pure associate and observed dependence of concentration on composition of a mixture. Conclusively, it is further stated that all the models used in the present work successfully agree well with the experimental findings provided the liquids should have poor associating properties. In future, a general equation is to be simulated and developed by taking suitable fittings which could explain the liquid state properties more reliable. 5. ACKNOWLEDGMENT Authors are very thankful to Department of Chemistry, V.S.S.D. College, PSIT and JSS Academy of technical education. REFERENCES [1]

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the PFP and BAB Models in Predicting the Surface and Transport Properties of Liquid Ternary Systems. J. Solu. Chem. 2007, 36, 1103-1116. [26] Pandey, J. D. Chandra, P. Srivastava, Tanu. Soni, N. K.; Singh, A. K. Estimation of Surface Tension of Ternary Liquid Systems by Corresponding State Group Contributions. Fluid. Phase. Equilibria. 2008,273 44-51. [27] Dey, R. Soni, N. K. Mishra, R. K.; Dwivedi, D. K. Modified Flory Theory and Pseudo Spinodal Equation of State for the evaluation of Isothermal Compressibility, Internal Pressure and Pseudogruneisen Parameter J.Pure & Appl. Ultrason. 2006, 28, 20-28. [28] Pandey, J. D.; Verma, Richa. Inversion of the Kirkwood-Buff theory of Solutions: Application to Binary Systems. Chem. Phys. 2001, 270, 429-438.

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