ULTRSONIC AND THERMOELASTIC PROPERTIES OF SOME ORGANIC LIQUID : ISOCHORIC ACOUSTICAL PARAMETER
DR. NRAPAL SINGH YADAV DR. R.P.SINGH
Department of Physics , Madhya Pradesh , India 1. BASIC CONCEPT :
liquids can be classified into two categories, polar and non-polar, according to whether their molecules posses a permanent electric dipole moment in their lowest energy level (ground state). Thus a diatomic molecule is polar unless its two atoms are same (e.g. the polar molecules are HCl, CO and the non-polar molecules are H2O2 etc.). triatomic molecules are polar unless their nuclei on a straight line with half way between the two atoms, e.g.
H2O is a triangular polar and C2O is straight non-polar molecule. Debye was the first to recognize the importance of the investigation of dipole moment for a study of the constitution of molecules.According to Andrade’s theory of viscosity, the viscosity of a liquid arises from collisions between twomolecules, during which they are momentarily so closely associated that no relative linear motion is possible between them. This is the form of association recognized as freezing when it involves a large number of molecules instead of a pair. Most polar substances, in freezing , show a sharp drop in their electrical permittivity which falls to nearly the high-frequency value of the gasphase, showing that in the solid state the permanent dipoles no longer contribute to the polarization. Thus freezing may also prevent relative rotation of the molecules.
Hill has identified such substances which lose freedom of dipole rotation upon freezing. They show dramatic change in permittivity at freezing point.
Eyring first described the phenomenon of dielectric relaxation as a chemical rate process. He deduced the relaxation times in ice and water using the rate equations.
Since then, the rate theory has been successfully applied to the data on dielectric behavior of several substances including ice and solid D2O nitrobenzene ortho and meta-nitrotoluene and organic compounds
2. EQUATIONS OF STATE FOR LIQUIDS:
The equations of state for liquids gives us the valuable information about the relation between the change in
thermodynamical variables viz, pressure, volume and temperature Every thermodynamic system has its own equation of state, independent of others.
An equation of state (EOS) expresses the peculiar behaviour of one individual system which distinguishes it from the others. In order to determine the equations of state of a system, the thermodynamic variables of the system are accurately measured and a relation is expressed between them. A variety of mathematical expression (both empirical as well as semi empirical) have been proposed the description of the equation of state of condensed matter. The empirical equation of state involve more than two or three adjustable parameters .
If V denotes the volume of a material at some constant temperature T and pressure P. Then the isothermal bulk modulus BT is defined as
T
T
V
V P
B
at P=P0, then the bulk modulus defined as
0
0 0
P
V P
V P B
The first and second order pressure derivatives of the bulk modulus evaluated at P=P0 are denoted by B0 and B0
respectively. For convenience, one introduce P=P-P0 as the pressure variable for the equation of state and if
B0
0 0B B
B
0Z P
V x V
0then the various equations of state can be written in terms of the dimensionless quantities ,, z and x. these equations are two and three parameter equation of state. The two parameter equation contains the bulk modulus B0 and B0 the first order pressure derivative of bulk modulus, whereas the three parameter
equations of state have one additional parameter B0 the second order pressure derivative of bulk modulus.The semiempirical Tait equation of state seams to have been the most popular equation. Although the Tait equation was originally formulated for water, it has been applied to many other liquids.
Murnaghan has proposed an equation of state, known as Murnaghan equation and this equation can represent quite accurately on the comparison of liquids over a wide pressure range. A well known Birch equation, which has been used successfully as an empirical equation in geophysics to represent pressure-volume behaviour for elements and compound under high pressure. Another important equations of this class is Keane equation of state. The Birch and Keane equations as well as the Murnaghan equations were originally derived for use with solids, but they are applicable to liquids as well. The value of the bulk modulus becomes physically unreasonable at extremely high pressure. On the other hand, the Keane equation does not imply any such physically and thermodynamically unreasonable condition at very high pressures. Excellent reviews of the semiempirical and theoretical equations of state have been given by Barsch and Chang and Mac-Donald. Hayward has pointed out that the well known Tait equation frequently used in the literature is not Tait’s original equation.Murnaghan has developed a theory of finite strain and has proposed an equation of state for isotropic solids, known as Murnaghan’s equation. This equation has been where neglected in literature and seems to be unknown to there who have used Tait equation. Cook and Rogers independently derived the Murnaghan equation. Mac- Donald has a well known equation among the class in the “Birch” equation, which has been used successfully as an empirical equation in geophysics to represent pressure-volume behaviour for elements and compounds under high pressure. Another important equation of this class in the equation of “Keane”
which has become prominent in recently year There are many types of equations of state as follows-
2.1 NONLINEAR EQUATIONS:- There are many different ways for writing the equations of state. For consistency,
they are all given here with (V/V0) on the left and involving the parameters ,0,
and q. alternatively, they may all be solved for P as a function of (V/V0). The present form of the equations assumes that V0 is known exactly and other parameters are follows:
A simpler symbol for B0 the pressure derivative of bulk modulus at P=P0
(dimensionless) Compressibility
q :
2B0B(B0)2
12 (2
2)12 BT :V
TV P
(isothermal bulk modulus)B0 : (B)P=0
B0 :
B0 P0B :
P
TB
B
:P
TB
2 2
V : Specific volume or volume V0 : Volume of V appropriate at P=P0
P : Pressure
n : number of disposable parameters in
an equation of state
(K=0,1,2,……n-1)
a similar symbol for K0K
(dimensionless) T : Temperature
2.2. POLYNOMIAL EQUATION: The four polynomial equation considered are of the form
n 1o k
k k
X A y
When we take x
V0 V V
they y=P and the resulting equations is slater’s equation and for x=P, y
V0 V V0
, the resulting equation is the Bridgman equation. For X
V0 V V
, y=P becomes the Davis-Gardan equation.2.3 MODIFIED EQUATION OF STATE: In connection with other work, it was found that a modified form of semi empirical Tait equation holds remarkably well at high densities for a large number of organic liquids. It appears, in fact, that this modified equation is the most generally applicable wide pressure range two parameter equation for liquid. The negative volume catastrophe of the TE makes it theoretically unappealing even
though PL is usually of the order of millions of atmospheres and so beyond the range of most measurements. It is therefore describle to modify the TE slightly in a way which will avoid this difficulty of the equation. One may modify the equation.
r P
V r V
0 1
0
1 ln
1
as to avoid V<0 behaiour, write P r
r
xln(1
0 )1 and replace xV
V 0 1 by V V0 1x, the same to first order in x. This result, itself a possibly useful equation of state may be further modification by replacing
V0 V 1
by ln
V0 V
x
, again the same to first order in. on making the transformation r
in x, one finally obtains 0 1
0
1
P
V V
0 1 0 01
V P V
Considerably simpler that equation
0
10
0
1
r P
V
V
Yet again equal to 0 , ehen P=0. The transformation r
has been made in order to distinguish between these parameters and expand V0/V in a power series in 0P using first then. on comparing the results, term by term, one fields equalities for zero and first orders value of r and . The second order terms may also be made equal if we take =r-1.When this value is substituted in the third order term of the modified equation, the coefficient of (0P)3 becomes (r2/3)- (7/6)r+1. That of the Tait equation is virtually the same except for a difference of r/6. Thus the relation=r-1makes the modified equation and Tait equation identical to second order and almost identical to third.
Although was derived independently by Mac-Donald and Barlow to obtain an improvement to Tait equation and although the modified equation was stated by Mac-Donald to fit almost all of Bridgeman’s data on compression of liquids very well it was later discovered to be by no means a new equations. As Gilvarry has mentioned, it is identical to Murnaghan equation and Gilvarry feels
that the equations obtained from quantum-mechanics or a lattice model are simplified version of a very general equation of state given by Gilvarry
Gilvarry has shown how may be modified to take temperature dependence into account and assumes to be temperature independent. However, the only temperature dependence then remaining on the right hand side is that of
0. Its dependence is frequently known or can be readily measured independently.
However, sometimes h does apparently vary with temperature as well. From it follows that
T
P
T
1
,10
Where the isothermal (instantaneous) compressibility is defined by
TT
V V T
B
1
.is identical with the linear Tangent Modulus equation. Although there seems to be some disagreement as to how for this linearity persists, it is generally agreed upon that at pressure up to several hundred bars
T1for any liquids essentially a linear function of pressure.The pressure dependence of
T in the low pressure region may now be calculated from yielding
2 T T PT
Experimental values of center around ≈ 10 with no or very small temperature dependence. The remarkably simple relation, seemed intriguing, inviting investigations as to whether it could be deduced from a simple model of the liquid state, such as a Van der Waals type model.
3: DAVIS-GARDON AND BRIDGEMAN EQUATIONS :Davis and Gordon have given some equations of state. They find the following equation
K w
P 1
2 1 1
0
Thus may be denoted as second order- Davis Gordon equation and truncated Taylor series expansion around, (V0/V)=1 and was used by Bridgman many years ago. A similar equation involving (V0-V)/
V0 instead of w has been discussed by slater . ritten in inverse form, the second order Davis-Gordon equation becomes
1 12 0
0
1 (
22 )
L L L P
V
V
(2.66) Where L (-1)-1. Its series expansion is
1 ...
2 1 1
2
1 1
2 2 30
Z Z Z
V
V
(2.67)
Davis-Gordon have also considered Bridgman’s second degree Taylors expansion of V in powers of P around P=0.
4.ISOTHERMAL, ISOBARIC AND ISOCHORIC ACOUSTICAL PARAMETER (KISTH KISBA& KISCH)
Sharma developed a relation involving sound speed which may be used to compute the parameters of nonlinearity expressed in terms of both the acoustical parameter Kisch of Rao and Kisoth of Carnavale and T.A. Litovitz. For this purpose, the temperature and pressure dependence of thermal coefficient of expansion may be conveniently utilized to obtain these relationships. Such a study is expected to give quantative and qualitative accurate results for explaining satisfactorily the physical basis for investigating the internal structure and intermolecular interactions in terms of molecular cohesion and order in liquids.
Two well known and interesting emprical relations involving the velocity of sound in liquids, were denoted by Rao and the other by Carnavale and Litovitz and these two has have been the subjects of some interest in recent times.Rao observed that for a wide variety of organic liquids near room temperature, there is a relationship between coefficient of sound velocity and volume expansivity. Yamato and Wada have pointed out K is about "3" for unassociated organic liquids near room temperature and "2" for liquified gases and takes a rather scattered value for molton metals. The value of K depends on intermolecular potential. The value of K varies from polymer to polymer and K remains constant below and above transition point for each polymer. Warfield and Hartmann that the bulk modulus, Grüneisen constant and Rao's constant have a strong temperature dependence.
There are three important thermoacoustic parameters (Kisoch., Kisoth and Kisob.) which are directly related to the derivatives of the ultrasonic velocity with respect to
volume, temperature and pressure respectively as follows:
P P
isobar
T C ln 1 V ln
C
K ln
T T
isoth
P C ln 1 V ln
C
K ln
V isoch
T C ln
K 1
where Kisob, Kisoth and Kisoch are designated as Rao's isobaric, carnavale and litovitz isothermal and Sharma's isochoric acoustical parameters respectively. These equations yield a thermodynamic relationship as
Kisoth = Kisob + Kisoch
or
Kisoch = Kisoth + Kisob
5.BEYER'S NONLINEAR PARAMETERS The parameter (B/A)0 is used as a parameter to described the non-linearity of the medium. It can be obtained from the finite amplitude waves and the variation of sound velocity with temperature and pressure. The nonlinear parameter plays a significant role in nonlinear acoustics and its determination is of increasing interest in a number of area ranging from under water acoustics to medical science. From the knowledge of nonlinear parameter one can gain information about some physical propertiesof the liquids such as internal pressure, intermolecular spacing, acoustic scattering and structural behaviour etc. in liquids and liquid mixtures.
In the recent past various workers have propsed a number of theoretical methods for estimating the nonlinear parameter (B/A)0 for liquids and liquid mixture. Tong et al. proposed a simplified method for calculating the (B/A)0 values of pure organic liquids making use of Schaaff's equation for sound velocity.
Recently using this method, Jugan et al.
and Pandey et al calculated (B/A)0 values of some liquids.
Table 1. Variation of compressibility (𝑁−1𝑚2) or (106𝑀𝑃𝑎−1) with pressure for chloroform at different temperatures T1 = 303.65 K, T2 = 323.65 K and T3 = 343.75 K
P (MPa) T1 T2 T3
50 751.9 800.5 900.1
200 400.0 409.6 425.4
400 254.9 257.4 260.4
600 190.2 190.9 191.2
800 153.0 153.1 152.5
1000 128.7 128.5 127.5
1200 111.5 111.2 110.1
1400 98.6 98.2 97.1
1600 88.6 88.2 87.0
1800 80.5 80.1 79.0
2000 73.9 73.5 72.4
Table 2: Variation of compressibility (𝑁−1𝑚2) or (106𝑀𝑃𝑎−1) with pressure for n- hexane at T = 423.15 K
P (MPa) A B C D
50 2061.2 1575.3 1718.5 1705.9 200 1073.4 584.3 566.8 574.6 400 764.5 348.8 318.3 324.0 600 625.7 257.2 226.4 230.8 800 542.5 207.1 177.7 181.2 1000 485.6 175.0 147.2 150.1 1200 443.5 152.5 126.1 128.7 1400 410.7 135.7 110.7 113.0 1600 384.3 122.6 98.9 100.9 1800 362.4 112.2 89.5 91.4 2000 343.9 103.6 81.8 83.6
6. SUMMARY AND CONCLUSIONS:- The present thesis embodies the results of an investigation of ultrasonic and thermoelastic properties of some organic liquids. These properties of liquieds play a vital role in understanding their molecular structure and other physical properties [1-4]. We have investigated in detail the properties such as refractive index polarizability parameter, bulk modulus, coefficient of the mal expansion, compressibility and equation of state for different kinds of organic liquids in order mto understand the significance of microscopic factors related to internal stucutre, molecular order, anharmonicity, molecular cohesion and intermolecular interactions. The liquids studies in the present work have been selected according to the availability of tge experimental data over wide ranges of pressure and temperature from melting point to critical point. They are composed of molecules with different geometries and interactions. The organic liquids are methanol, ethanol, 1-propanol, 1-butanol, hexane, cyclohexane, chloroform, carbon tetrachloride, toluene, benzene, ethylbenzene, propylbezene dichnoloromethane, cyclopentane, cyclohepatne, cyclooctane, cyclodecane and tetrachnoloromethane.
An equation of state for liquids gives us valuable information about the
relationship between the chan ges in thermodynamical variables viz, pressure, volume and temperature [5]. Every thermodynamic system has its own equation of state, independent of one individual system which distinguishes it from the others. In order to determine the equations of a system , the thermodynamic variables of the system are accurately measured and a relation is formulated between them. A variety of mathematical expressions both empirical as well as scmiempirical have been examined for the description of the equation of state of condensed matter.
The empirical equation of state involve two or three adjustable parameters viz the bulk modulus Bg and Bg the first order pressure derivative of bulk modulus, whereas the three parameter equations of state have one additional parameter B`0
the second order pressure derivative of the bulk modulus, all at zero pressure.
These are determined using the potential energy functions.
The variation of density or volume with pressurc can be determined with the help of an equation of state. Several forms of the equations of state have been discussed by Barsch and Chang [6] and Mac Donald[7] , Munaghan and Brich [8]
described the elastic and elasto-optic behavior of liquids in its entire stability field. Since then various equations of state have been used for the study isotropic liquids. Most recent and advanced developments in the field of equations of state have been made by Stacey [9] using the Keane approach [10]
based on the pressure derivative of bulk modulus. Pandey et at. [11] have presented an expression for the pressure- density relationship retaining the terms up to quadratic in density. We find that the results obtained from the Pandey et al. expression deviate substantially from the corresponding values derived from the Keane EOS. To make a critical test we have also calculated the bulk modulus which represents a higher order property to confirm the discrepancy, It is emphasized that higher order term such as the cubic term in density must be added in the pressure-density relationship. When this is done in the present study, the results not only for pressure-density but also for the bulk modulus-density come remarkably closer to the results based on the Keane EOS. It
is always useful to have a polynomial expression for the pressure-density relationship. Such an expression in the the form of Taylors series expansion provides a clear understanding of various order terms, say harmonic (up to quandratic) as well as anharmonic for cubic and higher order. The Keane EOS yields accurate results for liquids up to very high pressures [12]. This analysis is useful for describing thermal, elastic and harmonic behavior of liquids. It has been found that the values of P/B0, B/B0 and P/B increase with increasing value of density but the value of B’ decreases with increasing compression or density but the value of B’ decreases with increasing compression or density. We have developed a common equation of state for different liquids takin B’0=8.0 and B’=4.8. These values are found to be most suitable for the liquids under study.
The lineqr B’ versus P/B relationship needs to be replaced with an equation by which the gradient of a graph of B’ Versus P/B decreases with P/B. Stacey [13] first tried a B’ versus P/B equation and subsequently an expmential one [14] but these were seen to be unsatisfactory when the thermodynamic limit on B’. was recognized [15] requiring curvature to the reciprocal form. The validity of the linear relationship between 1/B’ and P/B has been found. We have also. presented a comparison of the result based on the Keane B-primed equation and the Stacey reciprocal B-primed equation for B/B0
and B` with the change in density for liquids with a common value of B`m. The two equations yield the results in good agreement with each other.
Analysis of thermo-acoustic parameters is useful and important for understanding the behavior of organic liquids under the effect of pressure and temperature. There are three important thermo-acoustic parameters which are directly related to the derivatives of the ultrasonic velocity with respect to pressure, volume and temperature [16], The values of ultrasonic velocity ©, density(p) and derivatives of C with respect to p are the basic important data needed for calculating various thermodynamic quantities of liquids [17].
It is therefore, desirable to investigate a relationship for the dependence of ultrasonic velocity on density. The relationship between C and p has been found to hold with the help of
experimental values of ultransonic velocity and density for organic liquids in the present study. The present study proveds a simple method for determining the themoacoustic parameters and B`
which have been found to be related with other thermo-dynamic properties and non-lincarity parameters for the organic liquids [18-21].
The relations between refractive index.
density, polarizability, packing fraction and wavclenghth are useful to study the dielectric and structural behavior of organic liquids [22-26] . A considerable interest in such studies has been creasted due to the structural behavior of complex polar molecules in different non-polar solvents. The density dependence of refractive index has been analyzed using the relations between refractive index and polarizability parameters. The polarizability parameters have bee calculated at two differet emperature (293.15K abd 298.15K) for each liquid using Eykaman equation, Lorentz-lorenz equation and Drude equation at two wavelengths. It has been found that the values of polarizability parameters decrease with the incrcasing temperature [27-29]. We have determined the values of packing fraction by revising the formulation due to Kirkwood and Brown [30] and due to Onsager and Bottcher [31]. Thus an attempt has bee made to remove the discrepancy between the Onsagar- Bottcher and the Kirkwook- Brown equations by modifying the values of polarizability parameters. The values of refractive index have determined for different liquids using the Omini equation and compared with the available experimental data [32,33]. The calculated ad experimental values have been found to present close agreement with each other. We have thus calculated the polrizability as well as refractive index for a number of liquids. The Variation of refractive index with frequency or wavelength has also been studied at two temperatures (9293.15K and 298.15) using the Drude type and Lorentz-Lorenz type models for liquids. 1-propanol and 1-butanol. The variations predicted from both the equations agree closely with each other.
An alalysis has also been presented for the relationship between refractive index versus Eulerian strain parameter, refractive index versus Eulerian strain
parameter, refractive index versus pressure and refractive index versus V/V0. The pressure dependence of the refractive index for twelve liquids has been measured experimentally and reported in the literature [34]. Values of V/V0 as a function of pressure have been calculated from the Keane equation modified in the present study. The Eulcrain strain parameters have been calculated and related with the values of refractive ndex. The results have been perented for differen t liquids viz Bezence, carbontetracholoride, toluene, chlorobenzene, water, n-octance, n- nonane, n-decane, n-pentane, n-hexane, n- n-hepatane and methl alcohol. It has been found that in every case the refractive index increases with pressure with pronounce linearity, and the refractive index increases with the Eulerain strain with pronounced non- lineraity. Values of refractive index and polarizability parameters have been calculated as a function of pressure using the results based on the equation of state.
Values of polarizability parameters are found to increase with the increase in pressure approximately in the same manner as the density changes with pressure. Values of thermooptic coefficients related to temperature derivatives of refractive index and piezo- optic coefficients related to pressure derivatives of refractive index habe been calculated from the density derivatives of refractive index using the input data on isothermal compressibility and volume thermal expansiviites.
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