Comparative Evaluation of Viscosity, Density and Ultrasonic Velocity Using Deviation Modelling for Ethyl-Alcohol Based Binary Mixtures

: This study reports the comparative deviations in experimental viscosity, density and ultrasonic velocity of two


Introduction
Over the last three decades, significant progress has been witnessed in the theoretical understanding of liquid-liquid binary mixtures [1,2], keeping in view that the prediction of the few important properties like density, viscosity, and the excess properties, is related to their engineering performance.Studies on the theoretical modelling of thermodynamic and engineering properties have been carried out on alcohol-based binary/ternary mixtures like water and ethanol [3], 1,3,5-trimethyladamantane + 1,2,3,4-tetrahydronaphthalene + n-octanol and corresponding binary systems [4], n-hexane, ethanol, and cyclopentyl methyl ether [5], pentanol + ethyl cyclohexane + methyl myristate and corresponding binary systems [6] and ethylene glycol-based ternary mixture [7].The knowledge of the thermodynamic and transport properties of alcohol-based binary mixtures can be used to develop theoretical models and for the design of new technologies.Importantly, information on the dynamic viscosity of liquid mixtures can be used in chemical engineering calculations involving fluid, heat, and mass transfer [8].The experimental viscosity data of ethanol-based liquid mixtures (ethanol + 1-hexanol and ethanol + 1-heptanol) have been reported [9].The viscosity for binary mixtures containing n-hexanol and ethyl valerate or hexyl acetate, as well the details on negative viscosity deviations over the entire composition have been presented [10].The viscosity of binary systems containing n-hexanol and other related details has been reported [11][12][13][14].The deviation in the viscosity data was correlated with its composition using polynomial models, and details on the interpretation of molecular interactions, as well as on the transport and ultrasound properties of binary liquid solutions are presented [15][16][17][18][19][20][21].Studies on viscosity, a transport property, have been used in terms of modelling and simulation, and the activation energy values as determined by Arrhenius plots have been detailed [22,23].It has been notified that liquid viscosity is highly affected by the heat and decreases with an increase in temperature [22].Most liquids have an exponential relationship between temperature and viscosity rather than linear dependence.
Ultrasonic properties and their variation within the composition of the binary mixture are useful to design engineering processes and in chemical and biological industries.Ultrasonic velocity is useful to investigate the molecular interactions between the components of the mixture.The measurement of ultrasonic velocity is the only direct method to determine isentropic compressibility, which then provides further access to the related thermodynamic properties of the liquids and liquid mixtures [24][25][26].Ultrasound measurements are one of the most widely used techniques in the investigation of liquids and are essential in the construction and validation of fundamental equations of its state.
The present investigation is focused on the first-time theoretical evaluation of the viscosity, density and ultrasonic properties of two binary liquid mixtures: ethanol + 1-hexanol and ethanol + 1-octanol at 303.15 K, to decide on the deviation of the values of the properties as well to know the average percentage of deviation (APD).In this work, the experimental value of the properties was taken as reference from the literature [27].The experimental viscosities were compared with various theoretical methods like the Hind relation (ηH) [28], Kendall and Monroe relation (ηK-M) [29], Bingham relation (ηB), Arrhenius-Eyring relation (ηAE), Croenauer-Rothfus Kermore relation (ηCRK) and Gambrill relation (ηG) [30].Turning to the other study in this work, the experimental densities were compared by deriving the values through theoretical methods like the Mchaweh-Nasrifar-Mashfeghian model (ρMNM) [31], Hankinson and Thomson model (ρHT) [32], Yamada and Gunn model (ρYG) [33] and Reid et al. model (ρR) [34].Also, this work reports on the comparison of experimental ultrasonic velocities using various theoretical methods like the Nomoto relation (UN) [35], Van Dael and Vangeel relation (Uimr) [36], Impedance relation (UIR) [37], Rao's specific velocity relation (UR) [38] and the Junjie relation (UJ) [39].The APD was determined to identify the most suited method that agrees with the experimental values.From the experimental values of density, viscosity and ultrasonic velocity, various molecular interaction properties like adiabatic compressibility, intermolecular free length, free volume, internal pressure, viscous relaxation time and their excess properties were analysed.

Experimental Details
The binary mixtures were prepared from Analar-grade ethanol (E-Merck chemicals, Darmstadt, Germany), 1-hexanol, and 1-octanol (S.D Fine Chemicals Ltd., India) (27).Ethanol, 1-hexanol, and 1-octanol were purified by the methods described in the literature [40,41].The binary mixtures of ethanol with 1-hexanol and 1-octanol were prepared by weighing an appropriate volume of each liquid component and were kept in special airtight bottles.All solutions were prepared in a dry bog.Viscosities were determined using a Cannon-Ubbelhode viscometer [27,42] calibrated with triple-distilled water.The viscometer containing test liquids was kept for about 20 min in a thermostatic water bath and the temperature fluctuation in the viscometer measurement was minimized.The overall experimental uncertainty was estimated to be ±1.5 × 10 −3 .The densities of pure liquids and binary mixtures were measured using a single-stem pycnometer (made of Pyrex glass) with a bulb capacity of 8 × 10 −3 dm 3 and with a graduated stem of 5.0 × 10 −7 dm −3 divisions.The ultrasonic velocities of pure liquids and their mixtures were measured using a single-frequency ultrasonic interferometer operating at 3 MHz with an uncertainty of +0.05% and a temperature (±0.02K) maintained in a thermostatic water bath [27].The values of densities and viscosities at 303.15 K were determined precisely up to ±0.01 kgm −3 and ±3 × 10 −6 Nsm −2 , respectively.The Cannon-Ubbelhode viscometer is conceptually simple: the time it takes a volume of solute to flow through a thin capillary is compared to the time for a solvent flow.It turns out that the flow time for either is proportional to the viscosity, and inversely to the density.

Viscosity Studies
The viscosity of a liquid is affected by many factors such as temperature, size, molecular weight, inter-molecular forces and the presence of impurities.Viscosity determination helps in understanding the molecular interactions and properties of binary and ternary liquid systems.It is to be noted that attractive interactions can cause an increase in the viscosity of these systems.Regardless of the fact that viscosities can be used as the base data in a simulation, equipment design, solution theory or molecular dynamics, it is also essential in designing calculations involving heat transfer, mass transfer and fluid flow.The accurate prediction of the viscosities of binary mixtures is considered very important considering the above facts.A large number of viscosity models have been applied, but few reviews have described the application prospects of the models [43,44].Models that describe the viscosity of liquid mixtures can be classified into two categories: (i) empirical equations using only one adjusting parameter and simple algebraic formulations [28], and (ii) semi-empirical models which are deduced based on different theories, like Eyring's absolute reaction-rate theory, the theory of corresponding states, and molecular dynamic models [45].
Table 1 infers the deviation between the experimental and the literature values for pure components.The following relations were proposed for the evaluation of the viscosity of binary liquid systems by Hind et al. [28]: where x is the mole fraction, η is the viscosity, the subscripts 1 and 2 refer to the components 1 and 2, respectively, and H 12 refers to cross-pair interactions, which can be obtained from Equation (3).
Kendall and Monroe relation (η KM ) [29] Kendall and Monroe derived an equation for the analysis of viscosity of binary liquid systems based on a zero-adjustable parameter: where x 1 , x 2 and η 1 , η 2 are the mole fraction and viscosity of the pure component, respectively.
Bingham relation (η B ) [30] Bingham derived an equation for the analysis of viscosity of binary liquid systems based on a zero-adjustable parameter: Arrhenius-Eyring relation (η AE ) [30] Arrhenius derived an equation for the analysis of viscosity of binary liquid systems based on a zero-adjustable parameter.
Croenauer-Rothfus Kermore relation (η CRK ) [30]: Gambrill relation (η G ) [30]: where v m is the kinematic viscosity of mixture, whereas x i and v i are the mole fraction and kinematic viscosity of individual pure liquids.

Average Percentage of Deviation
Tables 2 and 3 present the experimental and theoretical viscosities predicted by various models for binary liquid systems ethanol + 1-hexanol and ethanol + 1-octanol at 303.15 K. Irrespective of the theoretical models, the common prediction is that the relative η increases with the increase in the concentration of 1-hexanol/1-octanol.The η is the minimum at the lower 1-hexanol/1-octanol concentration, and with the concentration range (1.00) η becomes maximum.The increasing frictional forces are expected to arise from the presence of more carbon atoms in the linear chain of alcohol and the molecular layers formed between the lower-carbon and higher-carbon containing alcohol and to be the reason for the increase in values with concentration of the higher-carbon number containing alcohol [22].From Tables 2 and 3, it can be observed that there are differences in the theoretical values of the viscosities of the ethanol + 1-hexanol and ethanol + 1-octanol binary mixtures at 303.15 K compared to the experimental values.The reason for the difference is ascribed to the limitations and approximation incorporated in these theories.In the ethanol + 1-hexanol binary liquid system, it is observed that the APD, as derived from the theoretical values of η calculated by using various theoretical models (   Regarding the results derived from the ethanol + 1-octanol binary liquid system, the comparison of the experimental η with the value predicted from various theoretical models is presented in Table 3.Additionally, the details of the APD are given in Table 3 which follows the following trend: From the results, it can be concluded that the Gambrill model is the most suitable theoretical method for estimating the viscosity for the Ethanol + 1-octanol binary liquid system.

Density Studies
Density is an important concept because it allows one to determine whether a substance with a specified density will float/sink in a liquid.Specifically, substances with a lesser density than the density of the liquid will float in that liquid.Also, it is an important physical property used in calculating the acoustic and physical properties of a substance such as the molar refraction, dipole moment, boiling temperature and superficial tension.
The density of the binary liquid mixtures, taken for the present study, was calculated using the Mchaweh-Nasrifar-Mashfeghian model (ρ MNM ) [31], Hankinson and Thomson model (ρ HT ) [32], Yamada and Gunn model (ρ YG ) [33] and Reid et al. model (ρ R ) [34] as given by the following details: Mchaweh-Nasrifar-Mashfeghian model (ρ MNM ) [31] Mchaweh, Nasrifar and Mashfeghian reported the following correlation: where ρ mix is the density of the mixed solution, and ρ cmix is the critical density of the mixture.The critical density of the mixture is calculated with the following equation: where x i is the mole fraction, and ρ ci is the critical density of the ith component.
where the temperature-dependent variable τ mix is calculated by the following expression: In the above equation, T rmix is the reduced temperature of the mixed solution and α SRK is the term from the original Soave Redlich-Kwong equation of the state.The reduced temperature of the mixture is defined as where T ci is the critical temperature of the ith component.The parameter α SRK is defined in terms of the reduced temperature (T r ): The acentric factor of the solution ω mix is calculated with the following expression: where ω i is the acentric factor of the ith component.The acentric factor is a measure of the complexity of the molecule as formed in relation to a molecule with spherical symmetry of a simple fluid for which ω = 0.
Hankinson and Thomson model (ρ HT ) [32] The Hankinson-Thomson model (H-T) [21] is based on the corresponding state principle and is valid for 0.25 < T r < 0.95.The density of the pure compound is defined by ρ = ρ c / V (0) 1 − ωV (1)  (18) The reduced temperature of the component is defined as

Yamada and Gunn model (ρ YG ) [33]
The Yamada-Gunn model extended the Racket equation and requires the molecular weight M, the critical density ρ c , the reduced temperature T r and the acentric factor ω: Reid et al. model (ρ R ) [34] The Reid et al. model proposed an equation also based on the molecular weight, critical density, reduced temperature and acentric factor: The  4 and 5.In both (ethanol + 1-hexanol and ethanol + 1-octanol) binary liquid systems, it was observed that the deviation of density takes the following order: ∆ρ MNM > ∆ρ HT > ∆ρ YG > ∆ρ R .This study demonstrated that the APD of the density is more in the Mchaweh-Nasrifar-Mashfeghian model and least in the Reid et al. model, amongst the models used in this work.Also, the theoretical calculations made on the density of binary mixtures using various models gave the conclusion that the density values estimated from Reid et al. [34] are the most suitable ones for predicting the density of both binary mixtures, Ethanol + h-Hexanol and Ethanol + 1-octanol.Thus, it is worth mentioning that the basic assumptions used in the model of Reid et al. [34] and the assumptions applied in that model are well-suited for estimating the closer value of the density for the studied binary mixtures.

Ultrasonic Velocity Studies
Studies on ultrasonic velocity are useful for extensive applications towards the evaluation of the thermodynamic and physicochemical properties of simple, binary and ternary mixtures [49,50].Considering the extensive reports in the literature on binary mixtures, it is understood that relatively less attention has been focused on the mixtures based on ethanol [51][52][53].In the present work, the experimental ultrasonic velocities are compared with values derived through various theoretical methods: the Nomoto relation (U N ) [35], Van Dael and Vangeel relation (U IMR ) [36], Impedance relation (U IR ) [37], Rao's specific velocity relation (U R ) [38] and the Junjie relation (U J ) [39].The experimental values, along with the theoretical values calculated using various models, are presented in Tables 6 and 7 along with standard relations [35][36][37][38][39].For the Ethanol + 1-hexanol binary liquid system, the following trend was noticed: ∆U R > ∆U N > ∆U IMR > ∆U IR > ∆U J .The calculations reveal that the APD of ultrasonic velocity is the most when using Rao's specific velocity relation [38] and is the least in Jungie's relation [39].This leads to the conclusion that Jungie's relation is the most suitable theoretical method for estimating the ultrasonic velocity for the Ethanol + 1-hexanol binary liquid system.In the case of the Ethanol + 1-octanol binary liquid system, it was observed that deviation takes the order ∆U IMR > ∆U R > ∆U N > ∆U IR > ∆U J .The APD of ultrasonic velocity values as calculated from the Van Dael and Vangeel [36] relation were relatively higher than those predicted by Jungie's relation [39].Thus, it can be concluded that Jungie's relation [39] is the most suitable theoretical method for estimating the ultrasonic velocity for Ethanol + 1-octanol binary liquid system.

Molecular Interaction Properties
From the experimental values of density, viscosity and ultrasonic velocity [27], various molecular interaction parameters like adiabatic compressibility, intermolecular free length, free volume, internal pressure, and viscous relaxation time were determined and are presented in Table 8.
Normally, a decrease in adiabatic compressibility indicates closed packing and decreased ionic repulsion.In the present study, the adiabatic compressibility for both systems (Ethanol + 1-Hexanol and Ethanol + 1-Octanol) increases with an increase with the concentration of ethanol.This indicates that the molecules are loosely packed in the solution.The adiabatic compressibility shows an inverse behaviour when compared to ultrasonic velocity.This indicates that there is a significant interaction between the binary liquids.This increasing trend suggests a moderate strong electrolytic nature in which the solutes tend to attract the solvent molecules.The intermolecular free length shows a similar behaviour to adiabatic compressibility.From Table 8, the free volume shows an increasing trend with the increase in the concentration of ethanol.This may be compactness due to association at a higher concentration [54].This increasing trend is due to stronger intramolecular interaction than intermolecular interaction which can be attributed to the loose packing of molecules inside the shield, which suggests a weak molecular interaction in the components of mixtures [55].The internal pressure is a measure of cohesive forces between the constituent molecules in liquids.It is also defined as the energy required to vaporize a unit volume of a substance.The values of internal pressure increase with an increase in the mole fractions of ethanol.The value of internal pressure was found to be greater for the Ethanol+1-hexanol than Ethanol+1-octanol liquid system.This suggests that there is a strong interaction between the solute and solvent molecules or that there is an increase in the extent of complexation with the increase in concentration [55].The internal pressure of a liquid reflects the molecular interaction.The dispersion of the ultrasonic speed of sound in the binary system gives information about the characteristics of relaxation time (τ), which explains the cause of dispersion.The decreasing trend of relaxation time was observed in the present case.It may be due to the structural changes occurring in the mixtures resulting in the weakening of intermolecular forces [56].

Studies on Excess Parameters
In order to elucidate the nature of molecular interactions between the components of the liquid mixtures, it is of considerable interest to study the excess parameters rather than the actual values [57].Non-ideal liquid mixtures show a significant deviation from linearity in their physical behaviour with respect to the concentration, and temperature interoperates with the presence of strong or weak interactions.
The excess values of β E , L f E , τ E and π E are recorded in Table 9.The positive excess values represent the dispersion forces, while the negative values indicate the dipole-dipole interaction, charge transfer interaction and hydrogen bonding between the unlike molecules [58].Excess parameters have been calculated using the following relation: A E = A exp − A id A id = Σ A i X i where A i represents any acoustical parameter and xi is the corresponding mole fraction.
The excess values of adiabatic compressibility, free length, internal pressure and relaxation time were found to be negative in both systems (Ethanol + 1-Hexanol and Ethanol + 1-Octanol).This indicates the presence of a strong interaction between the components of the mixtures [59].

Conclusions
In this work, the experimental viscosity of ethanol-based binary liquid mixtures (Ethanol + 1-hexanol and Ethanol + 1-octanol at 303.15 K) was compared with a value predicted using various theoretical models.With regard to Ethanol + 1-hexanol binary liquid system, it was found that the average percentage of deviation (APD) of viscosity is more in the Hind and Bingham model and less in the Kendall and Monroe model.It was therefore concluded that the Kendall and Monroe model is the most suitable theoretical method for estimating viscosity for the Ethanol + 1-hexanol binary liquid system.Regarding the results derived for the Ethanol + 1-octanol binary liquid system, it was observed that the APD of η is more in the Hind model and Bingham model and less in the Gambrill model.From the results, it was concluded that the Gambrill model is the most suitable theoretical method for estimating the viscosity of the Ethanol + 1-octanol binary liquid system.Upon comparing the experimental density of the binary liquid mixtures (Ethanol + 1-hexanol and Ethanol + 1-octanol at 303.15 K) by applying various theoretical models, it was inferred that the APD of the density predicted by the Mchaweh-Nasrifar-Mashfeghian model is larger and is the least in the Reid model.It was concluded that the Reid et al. model is the most suited to predict the density closest to the experimental results for both the binary liquid systems.On comparing the experimental ultrasonic velocity through values derived through various theoretical models of binary liquid mixtures (Ethanol + 1-hexanol and Ethanol + 1-octanol) at 303.15 K, it was concluded that the APD of the ultrasonic velocity for the Ethanol + 1-hexanol binary liquid system is the most in Rao's specific velocity relation and the least in Jungie's relation, indicating the best suitability of Jungie's relation for estimating the ultrasonic velocity.On applying various theoretical models for the Ethanol + 1-octanol binary liquid system, it was observed that the average percentage of deviation of the ultrasonic velocity is the most in the Van Dael and Vangeel relation and the least in Jungie's relation.Hence, Jungie's relation is the most suitable theoretical method for estimating the ultrasonic velocity for the Ethanol + 1-octanol binary liquid system.
Variation in molecular interaction parameters with the molar concentration of ethanol suggested the presence of specific solute-solvent interactions at a higher concentration, and the effect of concentration was analysed.The calculated excess values and their signs indicate the possible involvement of specific hydrogen-bonding interactions in the binary mixture components.The results of the present study provide insights and inference on knowing the best or most suited theoretical model that could predict the closest values of thermo-acoustic parameters for ethanol-based binary liquid mixtures.Importantly, the study informs the specific choice of theoretical model that could give the closest thermoacoustic property values for ethanol-based binary mixtures, having aliphatic linear chain alcohols with varying numbers of carbons.The results achieved in this study into the ultrasonic velocity, density and viscosity of ethanol-based binary liquid mixtures are expected to justify the practical application of simple models to estimate the few important properties involved in industrial applications.
density values calculated using the Mchaweh-Nasrifar-Mashfeghian model, Hankinson and Thomson model, Yamada and Gunn model and Reid et al. model and their deviations are presented in Tables

Table 1 . Experimental values of the viscosity, density and ultrasonic velocity of pure liquids. Liquid Density (kg/m 3 ) Viscosity (×10 −3 Nsm −2 ) Ultrasonic Velocity (m/s) Expt [27] Literature Expt [27] Literature Expt [27] Literature
Hind relation, Kendall and Monroe relation, Bingham relation, Arrhenius-Eyring relation, Croenauer-Rothfus Kermore relation and Gambrill relation), follow the trend ∆η H = ∆η B > ∆η AE >∆η CRK > ∆η G > ∆η KM .This study infers that the APD of η is more in the Hind model and Bingham model and less in the Kendall and Monroe model.Furthermore, within Table 2, it can be noticed that the Kendall and Monroe model is the most suitable theoretical model for predicting viscosity for ethanol + 1-hexanol binary liquid systems.

Table 2 .
Experimental and theoretical viscosity of binary liquid system of Ethanol + 1-Hexanol at 303.15 K.

Table 3 .
Experimental and theoretical viscosity of binary liquid system of Ethanol + 1-Octanol at 303.15 K.

Table 4 .
Experimental and theoretical density of binary liquid system of Ethanol + 1-Hexanol at 303.15 K.

Table 5 .
Experimental and theoretical density of binary liquid system of Ethanol + 1-Octanol at 303.15 K.

Table 6 .
Experimental and theoretical ultrasonic velocity of binary liquid system of Ethanol + 1-Hexanol at 303.15 K.

Table 7 .
Experimental and Theoretical Ultrasonic velocity of Binary liquid system of Ethanol + 1-Octanol at 303.15 K.

Table 8 .
Molecular interaction properties of binary liquid systems of Ethanol + 1-Hexanol and Ethanol + 1-Octanol at 303.15 K.