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Relationship Between the Molecular Constant, Isothermal Volume

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Relationship Between the Molecular Constant, Isothermal Volume Pramana- J. Phys., Vol. 37, No. 6, December 1991, pp. 489-496. :(_'~ Printed in India. Relationship between the molecular constant, isothermal volume derivative of thermod~tnamic Griineisen parameter, nonlinearity parameter and intermolecular forces in liquids B K SHARMA Physics Group, DPSEE (MLLs Core Group), National Council of Educational Research and Training, New Delhi 110016, India MS received 22 August 1991 Abstract. The isochoric temperature derivative of sound velocity, Beyer's nonlinearity parameter, the isothermal volume derivatives of thermodynamicGriineisen parameter and isochoric heat capacity and the repulsive exponent of intermolecular potential are shown to be related to the molecular constant representing the ratio of internal pressure to cohesive pressure of liquids. The calculated values are reasonably satisfactory and explain the experimental results on sound propagation data of liquids. The results have been used to developfurther understanding of the significanceof molecularconstant, fractionalfree volume and repulsiveexponent of intermolecularpotential in describing various thermoacousticand nonlinear properties and the anharmonic behaviour with regard to molecular order and intermolecular interactions in liquids. Keywords. Molecular constant; Beyer's nonlinearity parameter; fractional free volume; Grfineisen parameter; volume expansivity;sound velocity;liquids. PACS Nos 61-25; 62"60; 62"90; 65"70; 65"90 1. Introduction When sound waves of finite amplitude pass through fluids the non-linear effects like acoustic streaming and distortion of waveform occur due to greater attenuation of high frequency components (Beyer 1974). These non-linear terms in the equation of wave motion can be studied by the use of non-linearity parameters (Beyer 1974; Coppens et al 1965, 1967) which provide certain information about the physical characteristics of the fluid, such as internal pressure, intermolecular spacing and acoustic scattering. Beyer's non-linearity parameter B/A of liquids can be obtained from the distortion of finite amplitude waves (Beyer 1974) and from the variation of sound velocity with pressure and temperature (Rudnick 1958; Beyer 1960; Coppens et a11965, 1967). Endo (1982) has shown on the basis of thermodynamic considerations, that the nonlinearity parameter B/A for liquids may be expressed as a polynomial in terms of the heat capacity ratio and can be calculated from thermodynamic coefficients of sound velocity. Hartmann and Balizer (1987) have determined the values of the non-linearity parameter B/A for n-alkane liquids using a new equation of state. An expression for B/A has been obtained (Sharma 1983a) in terms of the isobaric acoustical parameter K of Rao (1941) and the isothermal acoustical parameter K' of Carnevale-Litovitz (1955) for liquids. This result presents an improvement over the work of Hartmann (1979) who assumed that sound velocity depended on volume 489 490 B K Sharma only and introduced an additional effect on temperature. There is a significant contribution of the isochoric acoustical parameter K" to the thermoacoustic and non-linear properties of liquids and fluorocarbon fluids (Sharma 1983a, b, 1984a, b, 1985, 1987). This result confirms the conclusion on sound propagation data of fluorocarbon fluids and low density solids by Modigosky et al (1981) that ½(B/A) is greater than K through the introduction of K" for the fluids. In this paper, an attempt has been made to relate the thermodynamic Griineisen parameter 1= (used for a structural study of liquids) to the proportionality factor K~ for the relation between sound velocity and internal molar latent heat of vaporisation of liquids. The isochoric acoustical parameter K" for a liquid has been expressed in terms of the molecular constant n, as a measure of the ratio of the internal pressure to cohesive pressure, for the liquid. It is also of present interest to relate the thermodynamic Griineisen parameter 1=, the parameter 2 expressing the isothermal volume derivative of the thermodynamic Griineisen parameter (Sharma 1983b), the Griineisen-like- parameter F0 (Sharma 1983b), the nonlinearity parameter B/A, repulsive exponent of intermolecular potential n*, isothermal acoustical parameter K' and fractional free volume (as a measure of disorder due to increased mobility of molecules) in a liquid with the molecular constant n (as a measure of the ratio of the internal pressure to cohesive pressure) through K" of the liquid. The treatment has the distinct advantage that the nonlinearity parameter B/A can be evaluated from the thermoacoustic data on isobaric temperature coefficient of sound velocity, molecular constant, volume expansivity and heat capacity ratio of liquids. 2. Molecular constant and thermoacoustic parameters The thermodynamic Griineisen parameter 1= used for structural study of liquids (Knopoff and Shapiro 1970; Sharma 1985, 1987) can be expressed in terms of the parameter K1 and the heat capacity ratio 7 as = aV/flCv = Li/TCv = MC2a/Cp = (~ - l)/aT = (~ - l)/~,K~ (1) in which the sound velocity C, heat capacity r~itio ?, the proportionality constant K~ and the internal molar latent heat of vaporisation Li are related as Li = F TCv = (aT/fl) V = K~ MC 2, (2) C = (TVIMfl) 1/2 = (TLilMotT) 1/2 = (LilMK1) 1/2, (3) = (Cp/Cv) -- 1 + (Tct2MC2/Cp), (4) K 1 = FTCv/MC 2 = ~tT/7. (5) Where M is the molecular weight, Cp, Cv, or, fl, V are respectively the isobaric and isochoric heat capacity, volume expansivity, isothermal compressibility and molar volume of the liquid at absolute temperature T and pressure p. Using (1), (4) and (5), the expression (Collins et al 1956) relating the ratio (V*/V) of the hard core (or incompressible) volume V* at absolute zero temperature to molar volume V with the sound velocity C and the Griineisen-like-parameter Fo may be Molecular constant, isothermal volume derivative .... 491 expressed as (V*/V) 1/3 = 1 - R(TVg2 MC 2 + VCp)/VgCpMC 2 = 1 - (R/Cv)[KI + (rCv)- 1] = 1 - (R/FCv) = 1 - (ro)- 1 (6) in which the Griineisen-like-parameter Fo for a liquid (Sharma 1983b) may be expressed as F o = (ctV/flR) = ~,x/a/(~,t/a _ 1) = (3 + 4ctT)/~tT = 4 + (3/yKl). (7) Assuming the sound velocity C as a function of both volume V and temperature T in a liquid, the isobaric (Rao 1941), isothermal (Carnevale and Litovitz 1955) and isochoric (Sharma 1983a) acoustical parameters are related as K' = K + K" = - (d In C/d In V)r = (1/fl)(d In C/dp) r (8) in which the isochoric (K") and isobaric (K) acoustical parameters are given by the relations K" = r' - r = (1/0t)(d In C/dT)v = (1/fl)(d In C/dp)v, (9) K = - (d In C/d In V)p = - (1/~t)(d In C/dT)r (10) Using the expression obtained (Sharma 1983a, b) for the molecular constant n as a measure of the isochoric temperature derivative of internal pressure for a simple, non-polar and unassociated liquid, the isochoric acoustical parameter K" may be expressed in terms of n and ~t as K" = 1 + (1 - n)/n~T. (11) Equation (11) shows that K" can be evaluated from the experimental data on n and ~t available in literature (Allen et al 1960). For a simple, non-polar and unassociated liquid, for which n is more than unity, (11) imparts a value of K" less than unity in agreement with experimental value (Sharma 1983a; Hartmann 1979). Using these calculated values of K" and those of K from experimental data on C and (dC/dT)p as reported in literature (Swamy 1973; Beyer 1960; Soczkiewicz 1977; Rudnick 1958) K' can then be evaluated, using (8), for the liquids. The fractional (available) volume f as a measure of disorder due to increased mobility of molecules in a liquid (Sharma 1983b, 1985) and the repulsive exponent n* of the intermolecular potential (Sharma 1984b) can be expressed, using (8), in terms of K' as (Sharma 1986) f = (Va/V) = (K' + 1)- 1 = [K + 2 + (1 - n)/notT] - 1, (12) n* = 3(2K' - 3)= 3 [(2/f)- 5]. (13) where Va = (V- V*) is the free (available) volume of the molecules of the liquid. Equations (12) and (13) show that the repulsive exponent n* of the intermolecular potential is related to f, K' and molecular constant n of a liquid and can be determined from thermo-acoustic data of the liquid. 492 B K Sharma Using (1), (3), (8)-(11), the expressions obtained (Sharma 1985) for the isothermal volume derivatives of isochoric heat capacity Cv and thermodynamic Griineisen parameter 1= for a liquid may be expressed in terms of y, n and ~ as Y' = (d In Cv/dln V) T = F(K" - 1)~T = (V - 1)(g" - 1) = (7 - 1)(1 - n)/n~T, (14) 2 = - (d In l=/d In V)T = [(d In Cv/d In V)r + 2K" - 1] = K"(? + 1) - ? = ! + (7 + 1)(1 - n)/naT. (15) Beyer's nonlinearity parameter (B/A) of liquids (Coppens et al 1965; Sharma 1983a, 1985, 1987), which is expressed as a particular combination of the temperature and pressure derivatives of the sound velocity using (1), (4), (5), (8)-(11) and (I 5), may be given as B/A = (B/A)'+ (B/A)" = (B/A)' [1 + (B/A)"/(B/A)'] = 2(K + 7K") = 2[K + y + (1 - n)/nKt] = 2[K' + (7 - 1)K"] = 2[K' + (Y - 1) + I~(1 - n)/n] = 2K'[7 - (y - 1)(K/K')] = 2[K'+ (~ - 1)(2 + y)/(y + 1)], (16) (B/A)' = (2MC2/V)(dlnC/dp)T = 2~K' = 27[K + 1 + (1 - n)/nc~T], (17) (B/A)" = (2M CZ otT/Cp)(d In C/d T)v = - 2Kff otT = - 2K(y - 1) = -2yKfK1, (18) [ (B/ A)"/(B/A)'] = - FK t (K/K' ) = [(I/y)- 1](K/K'). (19) Equation (16) shows that the nonlinearity parameter (B/A) is related to K, K', 1=, K1 and 2 of a liquid and can be evaluated from the thermo-acoustic data on 7, K, n and of the liquid.
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