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Der Pharma Chemica, 2014, 6(5):158-165
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ISSN 0975-413X
CODEN (USA): PCHHAX
Theories of ultrasonic velocities and their application in binary
liquid mixtures of o-chlorophenol with some aliphatic esters at
different temperatures
G. R. Satyanarayana1, K. Balamurali Krishna1, K. Sujatha2 and C. Rambabu1*
1
Department of Chemistry, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India
2
Department of Chemistry, St. Vincent Depaul College, Eluru, Andhra Pradesh, India
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ABSTRACT
Ultrasonic velocities and densities of the binary liquid mixtures of o-chlorophenol with different aliphatic esters like
methylacetate, ethylacetate and butylacetate are measured at temperatures 303.15, 308.15, 313.15 and 308.15K and
atmospheric pressure. Theoretical velocities have been evaluated by using Nomoto (NOM), impedance (IMP), Van
Dael and Vangeel (VDV), Junjie (JUN) and Rao’s specific velocity (RAO) models. A good agreement is found
between experimental and theoretical values. U2exp / U2 imx has also been evaluated for non-ideality in the mixtures.
Chi-square test for the goodness of the fit is applied to understand the relative applicability of these theories to the
present systems. The results are discussed in terms of intermolecular interactions between the component molecules
in these binary liquid mixtures.
Keywords: o-chlorophenol, theoretical velocities, ultrasonics, Chi-square test, interaction parameter.
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INTRODUCTION
The measurement of ultrasonic velocities finds extensive applications in understanding Physico-chemical behaviour
of liquid mixtures [1-3]. The molecular interactions in pure and binary liquid mixtures have been studied by
measuring ultrasonic velocities by several researchers [4-12]. The theoretical values of ultrasonivc velocity have
been evaluated by using Nomoto [13] , impedance relation [14], Van Deal and Vangeel ideal mix relations [15],
Junjie [16] and Rao's Specific velocity relation [17] and the results are interpreted in terms of molecular
interactions.The present work is a continuation of our research programme on a comparison of experimental
ultrasonic velocity with various theoretical models for the binary mixtures of several systems at various
temperatures [18-22].
In this paper, we report the experimental and theoretical ultrasonic velocities of the binary liquid mixtures of ochlorophenol (OCP) with methylacetate (MA), ethylacetate (EA) and n-butylacetate (BA) evaluated by using
various theories such as Nomoto, impedance relation, Van Dael and Vangeel, , Junjie's and Rao's specific velocity
relation at 303.15-318.15K over entair composition range. Further, a comparative study of theoretical results with
experimental values using Chi-square test and the study of molecular interactions from the deviation (α) in the value
of U2exp / U2imx (from unity) have also been studied. Of these models, Nomoto relation and Impdance relation were
reported to be in good agreement with the experimental results for the binary mixture at all temperatures under study
and the results are interpreted in terms of intermolecular interactions between the binary component liquid mixtures.
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MATERIALS AND METHODS
The commercially available pure solvents were used in the present investigation. OCP ( Merk > 99% ) and
MA,EA,BA of AR grade procured from S.D fine chemicals (India) were purified by the standard methods described
by A.Weissberger [23] and the purity of the chemicals was assessed by comparing their measured densities (ρ) and
ultrasonic velocities (U) which were in good agreement with literature values. The mixtures were prepared
gravimetrically using an electronic balance (Shimadzu AY120) with an uncertainty of ± 1× 10-7 Kg and were stored
in air-tight glass bottles. The uncertainty in the mole fraction was estimated to be less than ± 1×10-4. It was ensured
that the components were adequately mixed before being transferred in to the apparatus. The required properties
were measured within one day of the mixture preparation.
The densities, ρ, of pure liquids and their mixtures determined using a 10-5m3 Double - arm pycnometer, and the
values from triplicate replication at each temperature are reproducible within 2 x 10-1kg m3 and the uncertainty in the
measurement of density is found to be 2 parts in 104 parts. The reproducibility in mole fractions was within ±0.0002
Temperature control for the measurement of viscosity and density is achieved by using a microprocessor assisted
circulating water bath, (supplied by Mac, New Delhi) regulated to ±0.01 K, using a proportional temperature
controller. Adequate precautions were taken to minimize evaporation losses during the actual measurements. The
ultrasonic velocity of sound (U) is measured using an ultrasonic interferometer (Mittal Enterprises, New Delhi
model F05) operating at 2 MHz. The measured speeds of sound have a precision of 0.8 m. sec-1 and an uncertainty
less than ± 0.1 m. sec-1. The temperature stability was maintained within ± 0.01K.by circulating water bath around
the measuring cell through a pump.
THEORETICAL CONSIDERATIONS:
1.1 Nomoto theory : Nomoto’s empirical formula is based on the assumption of the linear dependence of the
molecular sound velocity on concentration and the additivity of the molar volume in the liquid mixture. The sound
velocity U is given by
n
U=
[
∑x R
i =1
n
i
i
∑xV
i =1
i
]
3
i
where the molar sound velocity R = x1R1+x2R2.
Hence, ultrasonic velocity (U) is given by
U=
[
]
x1 R1 + x2 R2 3
x1V1 + x2V2
.............. (1)
In the above equation Ri = (Mi/ρi) Ui1/3 = Vi (Ui)1/3
1.2 Impedance relation: The specific acoustic impedance of the pure liquids are used for evaluating the ultrasonic
velocity in the liquid mixtures by the following relation:
U= ∑xiZi/∑xii
............. (2)
where Zi is acoustic impedance and ρi is the density of the mixture.
1.3 Van Dael and Vangeel relation: Van Dael and Vangeel obtained the formula for ultrasonic velocity in the liquid
mixtures adopting the adiabatic compressibilities of the pure liquids based on ideal mixing of the liquids. Van Dael
and Vangeel assumed that the adiabatic compressibility (βad) of the mixture is given by
βad = φA (βad)A + φB (βad)B
and suggested the following relation for sound velocity in homogeneous liquid mixtures.
im
γB
γA
= φA im
(β ad ) A + φB im
(β ad ) B
β
ad
γ
γ
Where φ and γ refer the volume function and principal specific ratio.
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It holds true if the mixture is an ideal one and also γA = γB = γim. It can be transformed into a linear combination of
the mole fractions if the additional assumption vA = vB is made
βad im = xA (βad)A + xB (βad)B
The sound velocities appropriate to the above equations are given by
x A v A + xB vB
1
vA
vB
= ϕA
+ ϕB
2
2
xA M A + xB M B (U im )
M AU A
M BU B2
1
1
xA
xB
=
+
2
2
im
x A M A + xB M B (U )
M AU A M BU B2
and
.............. (3)
1.4 Junjie relation: This relation derived by Junjie for the ultrasonic velocity of the mixture in terms of the mole
fraction, molecular weight and density of the mixture.
n
∑x V
i i
i =1
U=
n
(
∑
............... (4)
n
xi M i )1/ 2 (
i =1
∑
xiVi / ρiui 2 )1 / 2
i =1
where the symbols have their usual meanings.
1.5 Rao’s relation: Using the ratio of the temperature coefficient of velocity and expansion coefficient, Rao derived
a formula for ultrasonic velocity (U)
R
U=  
V 
3
............... (5)
where V is the molar volume and R is called Rao’s constant or molar sound velocity, which is constant for a liquid
at a temperature.
CHI-SQUARE TEST FOR GOODNESS OF FIT:
According to Karl Pearson, Chi-square value is evaluated for the binary liquid mixtures under study using the
formula
n
χ2 = ∑(( U(obs) - U(cal))2 / U(cal))
................ (6)
i=1
where n is the number of data used,
and ‘U (obs) = experimental values of ultrasonic velocities
U(cal) = computed values of ultrasonic velocities
AVERAGE PRECENTAGE OF ERROR (SdU):
The Average percentage error is calculated by using the relation
SdU = 1/n∑((U (obs) - U (cal)) / U (obs)) X 100%
................ (7)
where n is the number of data used.
U (obs) = experimental values of ultrasonic velocities
MOLECULAR ASSOCIATIONS:
The degree of intermolecular interaction or molecular association is given by
α = [U2exp/U2imx]-1
.................(8)
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RESULTS AND DISCUSSION
The experimental ultrasonic velocities and the theoretical values evaluated by Nomoto’s Relation (NOM),
Impedance Relation (IMP), Vandeal Vangael Ideal Mixing Relation (VDV), Junjie’s relation (JUN), Rao’s specific
velocity method (RAO) are compared for all the three binaries OCP+MA, OCP+EA, OCP+BA along with the
percentage of deviations are presented in TABLES 1-3 at all the four temperatures 303.15, 308.15, 313.15, 318.15 K
and atmospheric pressure. The validity of different theoretical formulae is checked by the chi-square test for all the
mixtures at all the temperatures and the values are given in TABLE-4.
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The data reveals that the velocities computed from Nomoto’s relation (NOM) and Impdance relation (IMP) exhibit
more satisfactory agreement with the experimental values in the temperature range 303.15K to 318.15K than other
approaches in the binary systems. It is observed that the experimental values show deviation with the theoretical
values of ultrasonic velocities which confirms the existence of molecular interactions [22]. This may be due to
interactions occurring between the hetero molecules of the binaries. Higher deviations are observed in Rao’s specific
and slight variations in Junjie’s theories. There are higher variations in some intermediate concentration range
suggesting the existence of strong tendency of association between component molecules as a result of dipole-dipole
interactions. However, there is reasonably a good agreement between the experimental and theoretical velocities of
Nomoto’s relation and Impedance relation. Nomoto’s theory proposes that the volume does not change upon mixing.
Therefore, no interaction between the components of liquid mixtures has been taken into account. Similarly, the
assumption for the formation of ideal mixing relation is that, the ratios of specific heats of ideal mixtures and the
volumes are also equal. Again no molecular interactions are taken into account. But upon mixing, interactions
between the molecules occur because of the presence of various types of forces such as dispersion forces, charge
transfer, hydrogen bonding dipole-dipole and dipole-induced dipole interactions. Thus, the observed deviation of
theoretical values of velocity from the experimental values shows that the molecular interactions are taking place
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between the unlike molecules in the liquid mixtures. From the Tables it is observed that maximum positive deviation
exhibiting a minimum of approximately 0.5 mole fraction for all the three systems at all the temperatures. The ratio
U2exp/U2imx is an important tool to measure the non ideality in the mixtures especially in such cases where the
properties other than sound velocity are not known.
Figures a, b and c represent the variation of U2exp/U2imx with the mole fraction of OCP for all three binary systems
studied, and the ratio of U2exp/U2imx gives an idea of extent of interaction taking place between molecules of the
mixtures. The positive deviation for three systems infers strong interactions between the components. The
percentage of deviation in velocity is reflecting both negative and positive magnitudes, indicating non ideal
behaviour of liquid mixtures. The evaluated interaction parameters are positive for all the systems, indicating strong
interactions between the mixing molecules. The negative values of interaction parameter indicates the dominance of
dispersion forces arising from the breakage of hydrogen bonds in the associates. But a positive value of (α) in all the
system clearly indicates the existence of strong tendency for the formation of association in mixture through strong
dipole-dipole / hydrogen bonding interactions and higher values of percentage deviation indicates maximum
departure of the particular theory from experiment at that particular concentration and magnitude of the chi-square
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value finally determines the overall validity of the theory. The chi square values along with average percentage error
are given in TABLE- 4.
Fig (a):Plots of U2exp/U2imx vs X1 for the studied system OCP+MA,
at temperatures 303.15K , 308.15K, 313.15K,318.15K
1
U2exp/U2imx
0.996
Series1
0.992
Series2
0.988
Series3
Series4
0.984
0
0.2
0.4
0.6
0.8
1
mole fraction of OCP
Fig (b):Plots of U2exp/U2imx vs X1 for the studied system OCP+EA,
at temperatures 303.15K , 308.15K, 313.15K,318.15K
1
0.992
U2exp/U2imx
0.984
0.976
Series1
0.968
Series2
0.96
Series3
0.952
0
0.2
0.4
0.6
0.8
mole fraction of OCP
1
Series4
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Fig (c):Plots of U2exp/U2imx vs X1 for the studied system OCP+BA,
at temperatures 303.15K , 308.15K, 313.15K,318.15K
1
U2exp/U2imx
0.992
0.984
Series1
0.976
Series2
0.968
Series3
0.96
Series4
0
0.2
0.4
0.6
0.8
mole fraction of OCP
1
CONCLUSION
From the values of experimental and evaluated velocity values, it may be concluded that, the Nomoto’s relation,
Impedance relation have provided good agreement. Thus, the linearity of molar sound velocity and additivity of
molar volumes, as suggested by Nomoto, and Impedance relation in deriving the empirical relations have been truly
observed in the aforementioned binary liquid mixtures. The success of Nomoto’s relation in predicting the
experimental ultrasonic velocities for polar-polar liquid mixtures has also been emphasized by others [24].
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