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. Equation (16) further shows that ½(B/A) is greater than both K or K' for a liquid since y is a positive quantity, greater than unity for liquids. The result confirms Madigosky's analysis and conclusion (Madigosky et al 1981) that ½(B/A) is larger than K for various fluorocarbon fluids and low density solids. This shows that volume changes caused by pressure have a greater effect on sound velocity than an equal volume change caused by temperature. From (16) and (19), for K = K' or K" = O, as a special case, it follows that:

(B/A)"/(B/A)'= [(l/y)- 1], (20) B/A = (Ca - 1) = 2K', (21) B/A = 2K, (22) in which the Moelwyn-Hughes parameter C 1 is given by (Sharma 1984b)

CI = [d(1/fl)/dp]r = (2K' + 1). (23)

Equation (20) is identical to that obtained by Nomoto (1966).for liquids. Equations Molecular constant, isothermal volume derivative .... 493

(21) and (22) give the relationship between (B/A) with C1 or K' and K as obtained by Rao and Joarder (1979) and Hartmann (1979) respectively for liquids. This shows that introduction of K" in (16) constitutes an improvement over the work of Hartmann (1979) and Rao and Joarder (1979) for liquids. Thus (16) represents the rigorous form of the generalized equation for the relationship between Beyer's nonlinearity parameter and Rao's acoustical parameter for liquids. Using (1), (5), (7), (11) and (15), the parameters K", Fo, K1, F and 2 have been evaluated for 28 liquids at 293 K. The necessary experimental data on n, 0t and 7 are taken from literature (Allen et al 1960; Gmyrek 1972). The results are presented in table 1. Using (8), (10), (12), (13) and (16), the parameters K, K', n*, f and (B/A) have been evaluated for 17 liquids at 293 K. The necessary experimental data on C, (dC/d T)p are taken from literature (Beyer 1960; Soczkiewicz 1977; Rudnick 1958; Swamy 1973) and data on ~, y and K" from table 1. The results are presented in table 2.

Table 1. Calculated values of the Griineisen parameter and other related thermo-acoustic parameters of liquids at 293 K.

~t x 103- Liquid (K - ~) n" 7b K" F o K l F 2 n-pentane 1"56 1.09 1-504 0"819 1ff563 ff304 1"103 0-547 Isopentane 1-64 1-19 1"351 0-668 10-243 0-356 0-730 0-219 n-Hexane 1'34 1"06 1"286 0-856 11-641 0"305 0'728 0"670 n-Heptane 1"22 1.09 1"271 0-769 12"392 0"281 0-758 0.475 n-Octane 1-13 1"10 1"254 0-725 134)61 0'264 0"767 0'381 n-Nonane t.07 1-11 1-230 0-684 13-569 0-255 0.734 0-295 n-Decane 14)2 1.11 1"224 0.668 14.048 0.244 0.749 0.263 n-Dodeeane 0.94 1.12 1"214 0-611 14.892 0-227 0'777 0.139 n-Tetradeeane 0-89 1-15 1"214 0"500 15'504 0"215 0-821 0'107 1-Octene 1.14 1.09 1-231 0"753 12,982 0-271 0-692 0,448 1-Decene 1.03 1.09 1-230 0.726 13,941 0-245 0.762 0-390 Cyclohexane 1"18 1.14 1-413 0.645 12"677 0"245 1"195 0-143 Benzene 1"22 1.06 1.422 0-842 12'392 0'251 1'180 0-616 Toluene 1.08 1"05 1.353 0-849 13.480 0.234 1"116 0.646 Ethyl benzene 1-01 1.07 1"306 0"779 14.138 0.227 1"034 0.490 m-xylene 0.98 1"04 1.319 0-866 14-448 0-218 1.111 0.689 p-xylene 14)1 1-05 1.310 0-839 14.138 0.226 1,049 0,628 Diethyl ether 1.67 1-05 1.321 0"903 10.131 0'370 0-656 0.774 Acetone 1.42 0-85 1.370 1-424 11.210 0.304 0'889 2.005 Methyl ethyl ketone 1,29 0.89 1"309 1'327 11.937 0"289 0'818 1.755 Cyclohexanone 0"92 1'13 1'274 0'573 15"129 0-212 1"016 0-030 Acetophenone 0.84 14)0 1'225 14)00 16'189 0'201 0'914 14)00 Methyl acetate 1.38 0-95 1-399 1.130 11.420 0.289 0-987 1.312 Ethyl acetate 1.34 1-02 1-393 0.933 11.641 0-210 1,343 0-840 Chloroform 1-26 1-02 1.489 0,947 t2,126 0'333 1-324 0.868 Carbon tetrachloride 1"22 1-10 1.457 0-746 12.392 0.329 1"278 0.375 1,2-Dichloroethane 1.14 1-03 1.193 0.913 12.982 0.275 0-578 0.809 Carbon disulphide 1.18 0"92 1.556 1.252 12.677 0'355 1-608 1-643

"Allen et al (1960); bGrymek (1972) 494 B K Sharma

Table 2. Calculatedvalues of thermo-acousticand nonlinearparameters ofliquids at 293 K.

Liquid K "-d K' (K/K') n* f (B/A)

n-Pentane 2.671 3.490 0-765 11.940 0-223 7.806 n-Hexane 2"942 3"798 0.775 13.788 0.208 8-086 n-Heptane 2.841 3.610 0.787 12.660 0.217 7.637 n-Octane 2-895 3.620 0.800 12.720 0"216 7"608 n-Nonane 2"878 3-562 0"808 12.372 0-219 7"439 n-Deeane 2"890 3"558 0-812 12.348 0-219 7"415 n-Dodecane 3-028 3-639 0.832 12-834 0-216 7-540 n-Tetradecane 3"039 3-539 0-859 12.234 0.220 7"292 Cyciohexane 3.053 3.698 0"826 13.188 0.213 7"929 Benzene 3'000 3"842 ff781 14"052 0.206 8"395 Toluene 3.100 3-950 0.785 14-700 0.202 8.500 p-xylene 3.760 4.599 0-818 18-594 0.179 9.718 Chloroform 2.696 3.643 0-740 12.858 0.215 8"212 Carbon tetrachloride 2.750 3.496 0.787 11.976 0.222 7.674 Ether 2-724 3-627 0.751 12.762 0.216 7.834 Acetone 2.970 4.394 0-676 17.364 0-185 9-842 Carbon disulphide 2"453 3.705 0.662 13.230 ff212 8-802

"Beyer (1960); bSoczkiewicz (1977); c Rudnick (1958); dSwamy (1973).

3. Results and discussion

The calculated values of the isochoric acoustical parameter K" for the liquids given in table 1 range from about 0"50 to 1.33 and those of K1 range from about 0.20 to 0.37 which are of the same order as observed for most of the fluorocarbon fluids, n-alkanes and other quasi-spherical molecular liquids (Sharma 1985, 1987). The calculated values of the parameter 2 vary from about 0.12 to 2-00, similar to that observed for most of the alkali halides, fluorocarbon fluids and quasi-spherical molecular liquids (Sharma 1983b, 1985; Roberts and Ruppin 1971). However, the quantity ;t representing the first order isothermal volume derivative of 1~ is found to be positive for liquids which shows that 1~ decreases with the volume in liquids. It is interesting to note from table 1 that the calculated values of Fo in the case of homologous series of saturated hydrocarbons increase with increasing length of the chain that forms a molecule of a given chemical compound. This indicates that the degree of space filling increases with the increasing number of the homologue and confirms the studies of Bondi (1954) and Soczkiewicz (1977) on space filling in a homologous series of saturated hydrocarbons. The calculated values of the Beyer's nonlinearity parameter B/A for the liquids given in table 2, are in reasonably satisfactory agreement with those experimentally observed (Beyer 1960; Coppens et al 1965; Endo 1982; Rudnick 1958; Sharma 1983a) for these liquids. The calculated values of the ratio (K/K') vary from about 0.66 to 0.86 as compared to unity as suggested by Nomoto (1966) and Hartmann (1979) for liquids. For most of the liquids under present investigation, the values of f are around 0"21, which are of the same order as observed for saturated hydrocarbons (Soczkiewicz 1977) and other liquids (Sharma 1985). The quasi-constancy of the quantity f at 293 K for a wide variety of liquids has echoes in the universal value of the fractional free volume at the glass temperature in the WLF theory (Williams et a11955). However, Molecular constant, isothermal volume derivative .... 495 the calculated values of the repulsive exponent n* vary from about 11 to 19 as compared to the range from about 13 to 19 for other liquids (Soczkiewicz 1977) and from about 16 to 56 for polymers (Sharma 1984b, 1986). Higher values of n and K' for p-xylene as compared to carbon tetrachloride and other liquids show its anharmonic bchaviour with strong molecular interactions which is stronger than other liquids. This implies that p-xylene having higher values of K', n* and smaller value off is strongly anharmonic as compared to carbon tetrachloride having smaller values of K' and n*, similar to that observed for other liquids and polymers (Soczkiewicz 1977; Sharma 1984b, 1986). An inspection of tables 1 and 2 reveals the importance of n and K" which contribute significantly to the thermo-acoustic and nonlinear properties of these liquids. The present treatment offers a convenient means for establishing relationship between the thermo-acoustic and nonlinear properties of liquids and correlating with fractional free volume and repulsive exponent of intermolecular potential through the molecular constant as a useful parameter for investigating several thermo-acoustic properties of liquids. The present theoretical approach gives a formalism which satisfactorily accounts for the empirical results on sound propagation data of liquids without using any adjustable parameters.

Acknowledgements

The author expresses his sincere thanks to Dr B Hartmann, Naval Surface Weapon Centre, USA, Dr J Gmyrek and Dr E Soczkiewicz, Institute of Physics, Silesian Technical University, Poland for helpful correspondence.

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