- Email: [email protected]
Isochoric temperature dependence of the thermodynamic Grüneisen parameter and molecular compressibility of liquid argon
Volume 101A, number 8
PHYSICS LETTERS
16 April 1984
ISOCHORIC TEMPERATURE DEPENDENCE OF THE THERMODYNAMIC GRUNEISEN PARAMETER AND MOLECULAR COMPRESSIBILITY OF LIQUID ARGON B.K. SHARMA Department of Physics, Regional College of Education, Bhubaneswar-751007, India Received 11 November 1982 Revised manuscript received 5 November 1983
An attempt has been made to find the isochoriccontribution to the temperature coefficient of internal pressure, sound speed, thermodynamic G~neisen parameter and molecuak compressibility in terms of volume expansivity for liquid argon. The various parameters which describe the thermo-acoustic properties of the liquid are all expressed in terms of a few measurable parametera It is shown that the isochoric temperature derivative of the sound speed is an important factor which contributes significantly to the thermo-acoustic properties. A new dimensionless parameter, Ao, expressed in terms of reduced volume, compressibility and volume expansivity, is introduced which has, on the average, a constant value of 3.77 for liquid argon, while the calculated values of the reduced quantities, molecular constant and molecular compressibility for liquid argon are found to be strong functions of temperature and volume.
The isochoric acoustical parameter K", which measures the temperature dependence o f the sound speed or o f the internal pressure, is an important factor which contributes significantly to various thermoacoustic properties of liquids [1 ]. Harward and Parker [2] have proposed a method to calculate the molecular compressibility/~0 o f the closed packed molecules in a liquid from the isochoric temperature derivative o f the internal pressure. Another basic parameter, C* [3,4], related to the effective translational (external) degrees o f freedom of a molecule in a simple liquid or a liqueried gas, allows a better representation of the equation o f state and accounts satisfactorily for the increased mobility of molecules in the liquid state. Following the earlier suggestion o f introducing this parameter C* into the expressions for the pressure and temperature coefficients o f compressibility when expressing the Moelwyn-Hughes and Anderson-Griineisen parameters for the liquid state [4], this paper presents a direct calculation o f K " and/~0 in terms of C* and o f the volume expansivity o f the liquid. From this, using thermodynamic identities, we extract the explicit temperature dependence of various thermo-acoustical parameters. In particular, three new dimensionless
thermo-acoustical parameters are introduced which turn out to be remarkably constant for liquid argon and thus may have some specific significance for describing thermo-acoustic properties o f other liquids also. There are a number of useful thermo-acoustical parameters which are interrelated. Introducing C* in isochoric temperature, isothermal pressure and isobaric temperature coefficients o f sound speed using relations obtained by Kuczers andKittel as given in ref. [5], the isochoric and isothermal acoustical parameters K " and K ' for a liquid [1 ] may be expressed as K " = K ' - K = (C*Ia)(O In C / a T ) v = ~ C * ,
(1)
K ' = (C*/[3)(0 In C/Op) T = C* [(VIVa) - 1] ,
(2)
in which the sound speed C is given by [6] C = (~/V/Mf3) 112 = (TPiV/mctT) 112 .
(3)
K = -(C*/o0(0 In C/aT)o is the isobaric acoustical parameter, C* = P* V * / R T * and p*, V*, T* are respectively the characteristic pressure, volume and temperature for the liquid [7], 3' = Cp/Cv is the heat capacity ratio, Cp, C v are respectively the isobaric and isochoric 405
Volume 101A, number 8
PHYSICS LETTERS
heat capacity, Va is the free volume, R is the universal gas constant, Pi = otT//3 is the internal pressure, M is the molecular weight and or,/3, V are respectively the volume expansivity, compressibility and molar volume of the liquid at absolute temperature T and pressure p. Assuming 7 does not vary with volume or temperature [3,6], eqs. (1) and (3) yield [1] (a lnPi/aT)v = (a In ot/aT)v + 1/T + 2K"ot/C* = 2 (K'TC* - 1)a,
(4)
in which (a In a/aT)v = - ( 1 + 2T)/T is the isochoric temperature coefficient of c~ for a liquid [ 1]. Using eqs. (1) and (4) the molecular constant n [7] may be expressed as n = [1 - ~T(a In Pi/aT)v]-I = [1 - (K"/C* - 1 ) a T ] - I = (1 - ~-cxT)-1 .
(5)
Eq. (5) can be utilized to evaluate n for a simple liquid like argon from the knowledge of ot alone. Eq. (5) also shows that n should increase with temperature, a fact which is in qualitative agreement with recent work [8]. Assuming C V does not vary with either volume or temperature, the first order isobaric and isochoric temperature derivatives of V, using eqs. (1) and (4), are related as u
In D a r ) p = 0
+ gxC , ,
(6)
0 =--.(c*/otxa In F/aT)v =C* [(aT) -1 - 2 (K'TC* - 1)] = C* [(c~T) -1 - ~1 ,
(7)
16 April 1984
where F = otV//3Cv =Pi V/CvT is the thermodynamic Griineisen parameter [3] and (a In Pi/aT) = -2ct is the isobaric temperature coefficient of intPernal pressure [8] for a liquid. Eq. (2) may be rewritten in terms of K ' and C* as
Va/V = C*(K' + C*) -1 .
(8)
The expression for K ' for liquid argon on the basis of the hard sphere model with attractive interactions, using eqs. (1) and (7) may be expressed as [3]
K' = ~C* [7(0/C* + 1) + ! ~ + otT(4 + 1~S'C*)] , (9) where S* is given by [3] S* = a / 3 V = otT/3~= ~(3 + 4 a T ) .
(10)
The molecular compressibility/30 [2], using eqs. (4), (5) and (10) may be expressed in terms of or as
~'= 2@ In lar)v/(a lnPilar) v =n(l+2°tT)-(l+2czT) . -
6(l+2aT)
1
(11) '
N
in which ~ = (a In V/aT)p = aT* is the reduced volume expansivity [3] ,.~=/3//30 is the reduced compressibility and V = V/V*, T = T/T* are respectively the reduced volume and temperature for a simple liquid. Eq. (3), using eq. (11), may be expressed as
V*otT/(3omC2 =A 0 = C'S"/n = (4/7 + 2 ~ ' ) / V ,
(12)
in which ~' = (1 + 6otTff7 is a factor introduced by Nomoto [9], S ' = ~/'yV = 3S*/'y and S" = ~ ' ~ are the reduced parameters introduced which may be
Table 1 Calculated values of molecular compressibility and related thermo-acoustical parameters of liquid argon at various temperatures. T (K) 83.8 85 87.3
90 95 100 105 100
406
~
~
'~
n
:Jo s -I, (I0- bar
k
u
0
Va/V
#7
S'
S"
Ao
4.37 4.42 4.49
1.281 1.289 1.303
5.576 5.613 5.702
1.065 1.067 1.070
0.714 0.749 0.813
2.29 2.25 2.19
2.13 2.09 2.03
1.37 1.33 1.27
0.25 0.26 0.26
1.88 1.87 1.83
2.03 2.02 2.01
2.00 1.99 1.96
3.80 3.77 3.96
4.58 4.80 5.08 5.41 5.90
1.319 1.349 1.382 1.416 1.457
5.817 6.109 6.452 6.871 7.494
1.074 1.082 1.092 1.105 1.121
0.895 1.086 1.320 1.650 2.098
2.11 1.98 1.80 1.74 1.62
1.96 1.20 0.26 1.83 1.07 0.27 1.70 0.94 0.29 1.58 0.82 0.29 1.46 0.69 0.30
1.88 1.89 1.89 1.86 1.85
2.01 2.02 3.78 2.01 2.04 3.80 2.01 2.06 3.79 2.05 2.05 3.81 2.01 2.07 3.71
average value
1.87 2.02 2.02 3.77
(10-3 K -1)
Volume 101A, number 8
PHYSICS LETTERS
used to account for the variation o f ~ and ~'with temperature and volume but these will turn out to be remarkably constant for the liquid argon and thus have special significance in the study of thermo-acoustic properties of simple liquids. Table 1 gives the calculated values of the parameters n, ~, V, [30,K, Va/V, O, la, S', S" andA 0 for liquid argon using eqs. (5)-(9), (11) and (12). The calculated values of K ' and V* = 21.99 cm 3 mo1-1, T* = 1270 K and C* ---0.573 molecule -1 have been taken from earlier work [3,7] and experimental data on V, ol, # and 3' from the literature [10]. The calculated values of V, ~, n, Va/Vand {30 are found sensitive to changes in both volume and temperature. However, the values of K,/a and 0 decrease with increasing temperature - a trend which is in agreement with earlier work for liquefied gases and liquids [3,4], and low density fluorocarbon fluids [11]. The estimated values o f n are found quite reasonable and lie in the range 1.1 + 0.1 in close agreement with data and analysis by Gee et al. [12] and Sharma [7,8] for simple liquids. Also the calculated values of the factors S' and S" are very close to 2 and remain fairly constant within experimental error. The average values of S' and S" are identical, which suggests that these factors are not independent but are related through reduced parameters. However, the average values of the factors A 0 and ~ a r e about 3.77 and 1.87, respectively, which differ by a factor of 2.02. This further implies that all the parameters S', S", A 0 and ~ a r e related to each other. The present results show that the relationships,K ~ ta + C*(Va/V) and A 0 ~ S'(~'T'), hold true in the case of liquid argon.
16 April 1984
The consequences of the constancy of the parameters
S' and S" and also the validity of the approximate relationship between A 0 and S', and between K and # need examining. The present treatment has the distinct advantage of calculating all the parameters "~,~, K, K', K", n, la, O, Va/V and S" from C* and ot only, this describing simply the thermo-acoustic properties of liquid argon. The author expresses sincere thanks to Professor R.N. Haward, Centre of Materials Sciences, University of Birmingham, UK for interest and helpful correspondence.
References [1] B.K. Sharma, J. Acoust. Soc. Am. 73 (1983) 106.
[2] R.N. Haward and B.M. Parker, J. Phys. Chem. 72 (1968) 1842. [3] B.K. Sharma, Phys. Lett. 89A (1982) 16; 95A (1983) 107; 95A (1983) 107; 96A (1983) 133. [4] B.K. Sharma, Pramana 20 (1983) 91; 14 (1980) 477. [5] E. Soczkiewicz, Archly. Acoust. 2 (1977) 325. [6] B. Hartmann, J. Acoust. Soc. Am. 65 (1979) 1392. [7] B.K. Shanna, Indian J. Pure AppL Phys. 15 (1977) 633; 19 (1981) 668, 670. [8] B.K. Sharma, Acustica 48 (1981) 121; 49 (1981) 164; 50 (1982) 160. [9] O. Nomoto, J. Phys. Soc. Japan 18 (1963) 1526. [ 10] J.S. Rowlinson, Liquid and liquid mixtures (Butterworths, London, 1969) p. 46. [11] W.M. Madigosky, I. Rosenbaum and R. Lucas, J. Acoust.
Soc. Am. 69 (1981) 1639. [12] G. Allen, G. Gee and G.J. Wilson, Polymer 1 (1960) 456.
407
PHYSICS LETTERS
16 April 1984
ISOCHORIC TEMPERATURE DEPENDENCE OF THE THERMODYNAMIC GRUNEISEN PARAMETER AND MOLECULAR COMPRESSIBILITY OF LIQUID ARGON B.K. SHARMA Department of Physics, Regional College of Education, Bhubaneswar-751007, India Received 11 November 1982 Revised manuscript received 5 November 1983
An attempt has been made to find the isochoriccontribution to the temperature coefficient of internal pressure, sound speed, thermodynamic G~neisen parameter and molecuak compressibility in terms of volume expansivity for liquid argon. The various parameters which describe the thermo-acoustic properties of the liquid are all expressed in terms of a few measurable parametera It is shown that the isochoric temperature derivative of the sound speed is an important factor which contributes significantly to the thermo-acoustic properties. A new dimensionless parameter, Ao, expressed in terms of reduced volume, compressibility and volume expansivity, is introduced which has, on the average, a constant value of 3.77 for liquid argon, while the calculated values of the reduced quantities, molecular constant and molecular compressibility for liquid argon are found to be strong functions of temperature and volume.
The isochoric acoustical parameter K", which measures the temperature dependence o f the sound speed or o f the internal pressure, is an important factor which contributes significantly to various thermoacoustic properties of liquids [1 ]. Harward and Parker [2] have proposed a method to calculate the molecular compressibility/~0 o f the closed packed molecules in a liquid from the isochoric temperature derivative o f the internal pressure. Another basic parameter, C* [3,4], related to the effective translational (external) degrees o f freedom of a molecule in a simple liquid or a liqueried gas, allows a better representation of the equation o f state and accounts satisfactorily for the increased mobility of molecules in the liquid state. Following the earlier suggestion o f introducing this parameter C* into the expressions for the pressure and temperature coefficients o f compressibility when expressing the Moelwyn-Hughes and Anderson-Griineisen parameters for the liquid state [4], this paper presents a direct calculation o f K " and/~0 in terms of C* and o f the volume expansivity o f the liquid. From this, using thermodynamic identities, we extract the explicit temperature dependence of various thermo-acoustical parameters. In particular, three new dimensionless
thermo-acoustical parameters are introduced which turn out to be remarkably constant for liquid argon and thus may have some specific significance for describing thermo-acoustic properties o f other liquids also. There are a number of useful thermo-acoustical parameters which are interrelated. Introducing C* in isochoric temperature, isothermal pressure and isobaric temperature coefficients o f sound speed using relations obtained by Kuczers andKittel as given in ref. [5], the isochoric and isothermal acoustical parameters K " and K ' for a liquid [1 ] may be expressed as K " = K ' - K = (C*Ia)(O In C / a T ) v = ~ C * ,
(1)
K ' = (C*/[3)(0 In C/Op) T = C* [(VIVa) - 1] ,
(2)
in which the sound speed C is given by [6] C = (~/V/Mf3) 112 = (TPiV/mctT) 112 .
(3)
K = -(C*/o0(0 In C/aT)o is the isobaric acoustical parameter, C* = P* V * / R T * and p*, V*, T* are respectively the characteristic pressure, volume and temperature for the liquid [7], 3' = Cp/Cv is the heat capacity ratio, Cp, C v are respectively the isobaric and isochoric 405
Volume 101A, number 8
PHYSICS LETTERS
heat capacity, Va is the free volume, R is the universal gas constant, Pi = otT//3 is the internal pressure, M is the molecular weight and or,/3, V are respectively the volume expansivity, compressibility and molar volume of the liquid at absolute temperature T and pressure p. Assuming 7 does not vary with volume or temperature [3,6], eqs. (1) and (3) yield [1] (a lnPi/aT)v = (a In ot/aT)v + 1/T + 2K"ot/C* = 2 (K'TC* - 1)a,
(4)
in which (a In a/aT)v = - ( 1 + 2T)/T is the isochoric temperature coefficient of c~ for a liquid [ 1]. Using eqs. (1) and (4) the molecular constant n [7] may be expressed as n = [1 - ~T(a In Pi/aT)v]-I = [1 - (K"/C* - 1 ) a T ] - I = (1 - ~-cxT)-1 .
(5)
Eq. (5) can be utilized to evaluate n for a simple liquid like argon from the knowledge of ot alone. Eq. (5) also shows that n should increase with temperature, a fact which is in qualitative agreement with recent work [8]. Assuming C V does not vary with either volume or temperature, the first order isobaric and isochoric temperature derivatives of V, using eqs. (1) and (4), are related as u
In D a r ) p = 0
+ gxC , ,
(6)
0 =--.(c*/otxa In F/aT)v =C* [(aT) -1 - 2 (K'TC* - 1)] = C* [(c~T) -1 - ~1 ,
(7)
16 April 1984
where F = otV//3Cv =Pi V/CvT is the thermodynamic Griineisen parameter [3] and (a In Pi/aT) = -2ct is the isobaric temperature coefficient of intPernal pressure [8] for a liquid. Eq. (2) may be rewritten in terms of K ' and C* as
Va/V = C*(K' + C*) -1 .
(8)
The expression for K ' for liquid argon on the basis of the hard sphere model with attractive interactions, using eqs. (1) and (7) may be expressed as [3]
K' = ~C* [7(0/C* + 1) + ! ~ + otT(4 + 1~S'C*)] , (9) where S* is given by [3] S* = a / 3 V = otT/3~= ~(3 + 4 a T ) .
(10)
The molecular compressibility/30 [2], using eqs. (4), (5) and (10) may be expressed in terms of or as
~'= 2@ In lar)v/(a lnPilar) v =n(l+2°tT)-(l+2czT) . -
6(l+2aT)
1
(11) '
N
in which ~ = (a In V/aT)p = aT* is the reduced volume expansivity [3] ,.~=/3//30 is the reduced compressibility and V = V/V*, T = T/T* are respectively the reduced volume and temperature for a simple liquid. Eq. (3), using eq. (11), may be expressed as
V*otT/(3omC2 =A 0 = C'S"/n = (4/7 + 2 ~ ' ) / V ,
(12)
in which ~' = (1 + 6otTff7 is a factor introduced by Nomoto [9], S ' = ~/'yV = 3S*/'y and S" = ~ ' ~ are the reduced parameters introduced which may be
Table 1 Calculated values of molecular compressibility and related thermo-acoustical parameters of liquid argon at various temperatures. T (K) 83.8 85 87.3
90 95 100 105 100
406
~
~
'~
n
:Jo s -I, (I0- bar
k
u
0
Va/V
#7
S'
S"
Ao
4.37 4.42 4.49
1.281 1.289 1.303
5.576 5.613 5.702
1.065 1.067 1.070
0.714 0.749 0.813
2.29 2.25 2.19
2.13 2.09 2.03
1.37 1.33 1.27
0.25 0.26 0.26
1.88 1.87 1.83
2.03 2.02 2.01
2.00 1.99 1.96
3.80 3.77 3.96
4.58 4.80 5.08 5.41 5.90
1.319 1.349 1.382 1.416 1.457
5.817 6.109 6.452 6.871 7.494
1.074 1.082 1.092 1.105 1.121
0.895 1.086 1.320 1.650 2.098
2.11 1.98 1.80 1.74 1.62
1.96 1.20 0.26 1.83 1.07 0.27 1.70 0.94 0.29 1.58 0.82 0.29 1.46 0.69 0.30
1.88 1.89 1.89 1.86 1.85
2.01 2.02 3.78 2.01 2.04 3.80 2.01 2.06 3.79 2.05 2.05 3.81 2.01 2.07 3.71
average value
1.87 2.02 2.02 3.77
(10-3 K -1)
Volume 101A, number 8
PHYSICS LETTERS
used to account for the variation o f ~ and ~'with temperature and volume but these will turn out to be remarkably constant for the liquid argon and thus have special significance in the study of thermo-acoustic properties of simple liquids. Table 1 gives the calculated values of the parameters n, ~, V, [30,K, Va/V, O, la, S', S" andA 0 for liquid argon using eqs. (5)-(9), (11) and (12). The calculated values of K ' and V* = 21.99 cm 3 mo1-1, T* = 1270 K and C* ---0.573 molecule -1 have been taken from earlier work [3,7] and experimental data on V, ol, # and 3' from the literature [10]. The calculated values of V, ~, n, Va/Vand {30 are found sensitive to changes in both volume and temperature. However, the values of K,/a and 0 decrease with increasing temperature - a trend which is in agreement with earlier work for liquefied gases and liquids [3,4], and low density fluorocarbon fluids [11]. The estimated values o f n are found quite reasonable and lie in the range 1.1 + 0.1 in close agreement with data and analysis by Gee et al. [12] and Sharma [7,8] for simple liquids. Also the calculated values of the factors S' and S" are very close to 2 and remain fairly constant within experimental error. The average values of S' and S" are identical, which suggests that these factors are not independent but are related through reduced parameters. However, the average values of the factors A 0 and ~ a r e about 3.77 and 1.87, respectively, which differ by a factor of 2.02. This further implies that all the parameters S', S", A 0 and ~ a r e related to each other. The present results show that the relationships,K ~ ta + C*(Va/V) and A 0 ~ S'(~'T'), hold true in the case of liquid argon.
16 April 1984
The consequences of the constancy of the parameters
S' and S" and also the validity of the approximate relationship between A 0 and S', and between K and # need examining. The present treatment has the distinct advantage of calculating all the parameters "~,~, K, K', K", n, la, O, Va/V and S" from C* and ot only, this describing simply the thermo-acoustic properties of liquid argon. The author expresses sincere thanks to Professor R.N. Haward, Centre of Materials Sciences, University of Birmingham, UK for interest and helpful correspondence.
References [1] B.K. Sharma, J. Acoust. Soc. Am. 73 (1983) 106.
[2] R.N. Haward and B.M. Parker, J. Phys. Chem. 72 (1968) 1842. [3] B.K. Sharma, Phys. Lett. 89A (1982) 16; 95A (1983) 107; 95A (1983) 107; 96A (1983) 133. [4] B.K. Sharma, Pramana 20 (1983) 91; 14 (1980) 477. [5] E. Soczkiewicz, Archly. Acoust. 2 (1977) 325. [6] B. Hartmann, J. Acoust. Soc. Am. 65 (1979) 1392. [7] B.K. Shanna, Indian J. Pure AppL Phys. 15 (1977) 633; 19 (1981) 668, 670. [8] B.K. Sharma, Acustica 48 (1981) 121; 49 (1981) 164; 50 (1982) 160. [9] O. Nomoto, J. Phys. Soc. Japan 18 (1963) 1526. [ 10] J.S. Rowlinson, Liquid and liquid mixtures (Butterworths, London, 1969) p. 46. [11] W.M. Madigosky, I. Rosenbaum and R. Lucas, J. Acoust.
Soc. Am. 69 (1981) 1639. [12] G. Allen, G. Gee and G.J. Wilson, Polymer 1 (1960) 456.
407