- Email: [email protected]
Temperature dependence of density and thermal expansion in some liquid metals
Journal of Non-Crystalhne Sohds 156-158 (1993) 417-420 North-Holland
~
]
~OURNAL
OF
~
U
Temperature dependence of density and thermal expansion in some liquid metals Mithlesh Kumari and Narsingh Dass
PhystcsDepartment, Unwersltyof Roorkee,Roorkee247667,Indm
In the present paper, a relahon suggested by the authors has been applied m some hqmd metals to study the density and coefficmnt of thermal expansion as a function of temperature The llqmd metals studied in the temperature range between the melting and the boiling points are sodmm, potassium, cesmm, magnesmm, lndmm, copper, barmm, arsemc and uranmm. The overall conclusion drawn ~s that there ~s an excellent agreement in the enure range of temperature m each case. Further, a theoretmal justfficaUon of the relahon used ~s discussed
1. Introduction
2. Theory
Recent technological as well as scientific advances have yielded much increased interest in the liquid metals. An accurate knowledge of the physical properties of these liquids has been vital and important in many of the applications. One such property is the density or volume. Hence, a knowledge of the temperature dependence of the density in liquid metals is required in many branches of scientific research. Further, the temperature dependence of density will give the coefficient of thermal expansion which is an important parameter from the theoretical as well as from the practical point of view. With this aim in mind, we have studied the temperature dependence of the density and the coefficient of thermal expansion in liquid metals such as sodium, potassium, cesium, magnesium, indium, copper, barium, arsenic and uranium in the temperature range between their melting to boiling points.
The present theory of the liquid metals is based on the assumption that the product of the measured isothermal bulk modulus, B x (P, T), and the thermal expansion coefficient, a (P, T), is approximately independent of temperature:
BT( P, T)a( P, T) = B T ( P , To)c~( P, T(,) =~(P).
(1)
Experiments show that eq. (1) is nearly independent of temperature for many solids for T > 0o, where 0 o is the Debye temperature. For example, Anderson [1] has shown that eq. (1) is valid in case of MgO, A1203 and porous MgzSiO 4 solids. Similarly, Yagi [2] has reported that the product aB T varies only slightly with volume in case of NaF and CsCI. All these results support the assumption taken at present in case of liquid metals. Differentiation of eq. (1) with respect to temperature keeping the pressure constant gives
Correspondence to
Dr M. Kumarl, Department of Physics, Umverslty of Roorkee, Roorkee 247667 (U.P), India Tel +91-1332 72130 Telefax" +91-1332 73560 Telex 0597-201 IN
l
0022-3093/93/$06.00 © 1993 - Elsevier Scmnce Pubhshers B V All rights reserved
[a°l(e'T)]
=($T(p, T)
(2)
M. Kumart, N Dass / Temperature dependence of denstty and thermal expanston
418
where 6T(P, T) is the Anderson-Griineisen parameter and is given as CST(P , T) 1
[ 0BT(P , T)]
= - a(P, T)BT(P, T) [
0T
p.
(3)
Integration of eq. (2) requires the temperature dependence of ~T (P, T) which is taken in the form 6T(P, T) = ~T(P, To) ec(r-T°),
(4)
where TO is an ambient temperature and C is a constant of temperature. Putting eq. (4) into eq. (2) and integrating in the limit T = T and T = To, we obtain
a( e, To) a(P, T) = 1-~7 [e c(r-r°) - 1].'
(5)
where , / = a ( e , TO)ST(P, To)/C. Now, writing a(P, T) = 1/V(P, T)[OV (P, T)/OT]p and integrating eq. (5), we obtain the temperature dependence of the volume as
V(P, T) V(P, To)
=
[(i + r/) e-C(r-To)-r/] -8,
changes in the results of density and a(P, T) are found by taking that C tends to zero. Hence, we may take C---0 without introducing a significant error in the results. Now, under the condition that C --- 0, eqs. (5) and (6) become or(P, T) =
To)
(7)
1-t~T(P, To)a( P, To)( r-ro)
and
V( P, T ) / V ( P, To) [1-~ST(P , ro)a(P, To)(r-To)] '/*'6e'w°)
=
(8) In the present paper, we are interested in studying the temperature dependence of the density and the coefficient of thermal expansion at one atmospheric pressure which is taken to be zero pressure. Hence, eqs. (7) and (8) at P = 0 become
To) a(O, T) = [l_~ST(O ' To)a(O ' To)(T_To)] (9)
(6) and
where fl = a(P, To)/C(I + ~7). However, it has been experienced that (~T(P, T) is a slow temperature-dependent parameter. In simple words, the fitted value of C has a small magnitude. Therefore, no significant
p(o, T)/p(O, To) = [1--~T(O , To)a(O,
T o ) ( T - T o ) ] 1/8T(0" To)
(10)
Table 1 Input data Lnquid metal
Temperature range (K)
TO (K)
a(O, To) (10-5/K)
~T(O, T0)
RMSD (g/cm 3) X 10 -4
Na K Cs Mg Cu Ba In As U(natural) U 238
373 00-1273 00 865.78-1525.61 1131.33-1535 11 923.00-1390.00 1356.00-2855.00 1013.00-1201.00 429.30-2286 00 1090.00-1320.00 1406.00-4200.00 1406.00-4200.00 1406.00-4200.00 1406.00-4200.00
373 00 865 78 1131.33 923.00 1356.00 1013.00 429.30 1090.00 1406.00 1406 O0 1406.00 1406.00
25 303 35.125 47 500 16.806 10 024 6.323 9.704 10.042 5 768 5.753 5 779 5.774
1.1746 1.2041 1.6083 0.7994 0.9918 12.8780 0.9607 7.8138 1.0140 1.0324 0.9723 0.9989
1.5 7.8 12.6 2.5 3.5 9.5 4.7 12 0 3.8 18.1 4.1 6.4
U 235
U 233
M. Kuman, N. Dass / Temperature dependence of denstty and thermal expanston
419
Equations (7) and (8) are the same as those used earlier by the authors [3] in case of liquid mercury in the temperature range 0-150°C and pressure range 0-8 kbar. It may be mentioned here that an interesting result is obtained when the present assumption is combined with that of earlier one given by Dass and Kumari [4]. The result is that the product of the measured isothermal bulk modulus, BT(P, T), and the thermal expansion coefficient, a(P, T), is approximately independent of both the temperature and the pressure: BT(P, T)a(P, T) = B T ( O , To)a(O, To). (11) Equation (11) will have a wide range of applications.
3. Calculations
In the present paper, eq. (10) is applied to the experimental density data of sodium [5], potassium [6], cesium [6], magnesium [7], indium [8], copper [9], barium [10], arsenic [11] and uranium [12] liquids.
.=
3.1. Density The best fitted values of a(O, To) and 6T(O , T0) obtained for these liquid metals are reported in table 1 along with the temperature range studied and the root-mean square deviation obtained in each case. The calculated values of density are compared in table 2. The overall conclusion drawn is that there is an excellent agreement in the entire range of temperature for all the liquid metals. Due to lack of space, the results for density and thermal expansion are reported only for magnesium, sodium and indium.
3.2. Thermal expansion Once the values of a(O, To) and 6T(O, To) become known, eq. (9) can be easily applied to compute the coefficient of thermal expansion, a(O, T), as a function of temperature. The computed values are compared with the experimental
=. ,..<
x t"-O
"O
420
M Kumart, 19. Dass / Temperature dependence of density and thermal expanston
data in table 2 for those liquids for which the experimental results are available. It m a y be concluded from table 2 that the overall a g r e e m e n t is very good.
New Delhi, for the award of a R e s e a r c h Associateship.
4. Conclusions
References
T h e important conclusions drawn are as follows (i) A physical justification is given for eqs. (7) and (8) which are earlier applied successfully in case o f liquid mercury in the t e m p e r a t u r e range 0 - 1 5 0 ° C and the pressure range 0 - 8 Kbar. (ii) T h e very g o o d a g r e e m e n t between the c o m p u t e d and the experimental data of density and coefficient of thermal expansion also tends to support the idea that the p r o d u c t of the measured isothermal bulk m o d u l u s and the thermal expansion coefficient is i n d e p e n d e n t of both the t e m p e r a t u r e and the pressure. (iii) A t present, we are unable to explain the high value of 6-r (O, T 0) in case of barium and arsenic. It seems that these liquids are m o r e compressible than the other liquid metals. O n e of the authors (M.K.) is grateful to the Council of Scientific and Industrial Research,
[1] O L Anderson, J Geophys. Res. 84 (1979) 3537. [2] T. Yagl, J. Phys. Chem. Solids 39 (1978) 563 [3] M Kumarl and N Dass, J Non-Cryst. Sohds 117 & 118 (1990) 563 [4] N Dass and M. Kumari, Phys Status Sohdl (b)124 (1984) 531. [5] L Lelbowitz, M.G Chasanov and R Blomqulst, J. Appl. Phys. 42 (1971) 2135 [6] J.P Stone, C.T Ewlng, J R. Spann, E.W Stelnkuller, D D. Wllhams and R R Miller, J Chem Eng Data 11 (1966) 320. [7] P J. McGonigal, A.D Kirshenbaum and A.V. Gross, J. Phys Chem 66 (1962) 737 [8] P.J. McGonlgal, J.A. Cahill and A.D Karshenbaum, J Inorg. Nucl. Chem. 24 (1962) 1012. [9] J A Cahlll and AD. Klrshenbaum, J Phys Chem 68 (1962) 1080. [10] C.C Addison and R.J Pulhan, J Chem Soc (1962) 3873 [11] P J McGonlgal and A V Gross, J. Phys Chem. 67 (1963) 924 [12] A.V Gross, J A. Cahlll and A.D Kirshenbaum, J. Am. Chem Soc 83 (1961) 4665.
~
]
~OURNAL
OF
~
U
Temperature dependence of density and thermal expansion in some liquid metals Mithlesh Kumari and Narsingh Dass
PhystcsDepartment, Unwersltyof Roorkee,Roorkee247667,Indm
In the present paper, a relahon suggested by the authors has been applied m some hqmd metals to study the density and coefficmnt of thermal expansion as a function of temperature The llqmd metals studied in the temperature range between the melting and the boiling points are sodmm, potassium, cesmm, magnesmm, lndmm, copper, barmm, arsemc and uranmm. The overall conclusion drawn ~s that there ~s an excellent agreement in the enure range of temperature m each case. Further, a theoretmal justfficaUon of the relahon used ~s discussed
1. Introduction
2. Theory
Recent technological as well as scientific advances have yielded much increased interest in the liquid metals. An accurate knowledge of the physical properties of these liquids has been vital and important in many of the applications. One such property is the density or volume. Hence, a knowledge of the temperature dependence of the density in liquid metals is required in many branches of scientific research. Further, the temperature dependence of density will give the coefficient of thermal expansion which is an important parameter from the theoretical as well as from the practical point of view. With this aim in mind, we have studied the temperature dependence of the density and the coefficient of thermal expansion in liquid metals such as sodium, potassium, cesium, magnesium, indium, copper, barium, arsenic and uranium in the temperature range between their melting to boiling points.
The present theory of the liquid metals is based on the assumption that the product of the measured isothermal bulk modulus, B x (P, T), and the thermal expansion coefficient, a (P, T), is approximately independent of temperature:
BT( P, T)a( P, T) = B T ( P , To)c~( P, T(,) =~(P).
(1)
Experiments show that eq. (1) is nearly independent of temperature for many solids for T > 0o, where 0 o is the Debye temperature. For example, Anderson [1] has shown that eq. (1) is valid in case of MgO, A1203 and porous MgzSiO 4 solids. Similarly, Yagi [2] has reported that the product aB T varies only slightly with volume in case of NaF and CsCI. All these results support the assumption taken at present in case of liquid metals. Differentiation of eq. (1) with respect to temperature keeping the pressure constant gives
Correspondence to
Dr M. Kumarl, Department of Physics, Umverslty of Roorkee, Roorkee 247667 (U.P), India Tel +91-1332 72130 Telefax" +91-1332 73560 Telex 0597-201 IN
l
0022-3093/93/$06.00 © 1993 - Elsevier Scmnce Pubhshers B V All rights reserved
[a°l(e'T)]
=($T(p, T)
(2)
M. Kumart, N Dass / Temperature dependence of denstty and thermal expanston
418
where 6T(P, T) is the Anderson-Griineisen parameter and is given as CST(P , T) 1
[ 0BT(P , T)]
= - a(P, T)BT(P, T) [
0T
p.
(3)
Integration of eq. (2) requires the temperature dependence of ~T (P, T) which is taken in the form 6T(P, T) = ~T(P, To) ec(r-T°),
(4)
where TO is an ambient temperature and C is a constant of temperature. Putting eq. (4) into eq. (2) and integrating in the limit T = T and T = To, we obtain
a( e, To) a(P, T) = 1-~7 [e c(r-r°) - 1].'
(5)
where , / = a ( e , TO)ST(P, To)/C. Now, writing a(P, T) = 1/V(P, T)[OV (P, T)/OT]p and integrating eq. (5), we obtain the temperature dependence of the volume as
V(P, T) V(P, To)
=
[(i + r/) e-C(r-To)-r/] -8,
changes in the results of density and a(P, T) are found by taking that C tends to zero. Hence, we may take C---0 without introducing a significant error in the results. Now, under the condition that C --- 0, eqs. (5) and (6) become or(P, T) =
To)
(7)
1-t~T(P, To)a( P, To)( r-ro)
and
V( P, T ) / V ( P, To) [1-~ST(P , ro)a(P, To)(r-To)] '/*'6e'w°)
=
(8) In the present paper, we are interested in studying the temperature dependence of the density and the coefficient of thermal expansion at one atmospheric pressure which is taken to be zero pressure. Hence, eqs. (7) and (8) at P = 0 become
To) a(O, T) = [l_~ST(O ' To)a(O ' To)(T_To)] (9)
(6) and
where fl = a(P, To)/C(I + ~7). However, it has been experienced that (~T(P, T) is a slow temperature-dependent parameter. In simple words, the fitted value of C has a small magnitude. Therefore, no significant
p(o, T)/p(O, To) = [1--~T(O , To)a(O,
T o ) ( T - T o ) ] 1/8T(0" To)
(10)
Table 1 Input data Lnquid metal
Temperature range (K)
TO (K)
a(O, To) (10-5/K)
~T(O, T0)
RMSD (g/cm 3) X 10 -4
Na K Cs Mg Cu Ba In As U(natural) U 238
373 00-1273 00 865.78-1525.61 1131.33-1535 11 923.00-1390.00 1356.00-2855.00 1013.00-1201.00 429.30-2286 00 1090.00-1320.00 1406.00-4200.00 1406.00-4200.00 1406.00-4200.00 1406.00-4200.00
373 00 865 78 1131.33 923.00 1356.00 1013.00 429.30 1090.00 1406.00 1406 O0 1406.00 1406.00
25 303 35.125 47 500 16.806 10 024 6.323 9.704 10.042 5 768 5.753 5 779 5.774
1.1746 1.2041 1.6083 0.7994 0.9918 12.8780 0.9607 7.8138 1.0140 1.0324 0.9723 0.9989
1.5 7.8 12.6 2.5 3.5 9.5 4.7 12 0 3.8 18.1 4.1 6.4
U 235
U 233
M. Kuman, N. Dass / Temperature dependence of denstty and thermal expanston
419
Equations (7) and (8) are the same as those used earlier by the authors [3] in case of liquid mercury in the temperature range 0-150°C and pressure range 0-8 kbar. It may be mentioned here that an interesting result is obtained when the present assumption is combined with that of earlier one given by Dass and Kumari [4]. The result is that the product of the measured isothermal bulk modulus, BT(P, T), and the thermal expansion coefficient, a(P, T), is approximately independent of both the temperature and the pressure: BT(P, T)a(P, T) = B T ( O , To)a(O, To). (11) Equation (11) will have a wide range of applications.
3. Calculations
In the present paper, eq. (10) is applied to the experimental density data of sodium [5], potassium [6], cesium [6], magnesium [7], indium [8], copper [9], barium [10], arsenic [11] and uranium [12] liquids.
.=
3.1. Density The best fitted values of a(O, To) and 6T(O , T0) obtained for these liquid metals are reported in table 1 along with the temperature range studied and the root-mean square deviation obtained in each case. The calculated values of density are compared in table 2. The overall conclusion drawn is that there is an excellent agreement in the entire range of temperature for all the liquid metals. Due to lack of space, the results for density and thermal expansion are reported only for magnesium, sodium and indium.
3.2. Thermal expansion Once the values of a(O, To) and 6T(O, To) become known, eq. (9) can be easily applied to compute the coefficient of thermal expansion, a(O, T), as a function of temperature. The computed values are compared with the experimental
=. ,..<
x t"-O
"O
420
M Kumart, 19. Dass / Temperature dependence of density and thermal expanston
data in table 2 for those liquids for which the experimental results are available. It m a y be concluded from table 2 that the overall a g r e e m e n t is very good.
New Delhi, for the award of a R e s e a r c h Associateship.
4. Conclusions
References
T h e important conclusions drawn are as follows (i) A physical justification is given for eqs. (7) and (8) which are earlier applied successfully in case o f liquid mercury in the t e m p e r a t u r e range 0 - 1 5 0 ° C and the pressure range 0 - 8 Kbar. (ii) T h e very g o o d a g r e e m e n t between the c o m p u t e d and the experimental data of density and coefficient of thermal expansion also tends to support the idea that the p r o d u c t of the measured isothermal bulk m o d u l u s and the thermal expansion coefficient is i n d e p e n d e n t of both the t e m p e r a t u r e and the pressure. (iii) A t present, we are unable to explain the high value of 6-r (O, T 0) in case of barium and arsenic. It seems that these liquids are m o r e compressible than the other liquid metals. O n e of the authors (M.K.) is grateful to the Council of Scientific and Industrial Research,
[1] O L Anderson, J Geophys. Res. 84 (1979) 3537. [2] T. Yagl, J. Phys. Chem. Solids 39 (1978) 563 [3] M Kumarl and N Dass, J Non-Cryst. Sohds 117 & 118 (1990) 563 [4] N Dass and M. Kumari, Phys Status Sohdl (b)124 (1984) 531. [5] L Lelbowitz, M.G Chasanov and R Blomqulst, J. Appl. Phys. 42 (1971) 2135 [6] J.P Stone, C.T Ewlng, J R. Spann, E.W Stelnkuller, D D. Wllhams and R R Miller, J Chem Eng Data 11 (1966) 320. [7] P J. McGonigal, A.D Kirshenbaum and A.V. Gross, J. Phys Chem 66 (1962) 737 [8] P.J. McGonlgal, J.A. Cahill and A.D Karshenbaum, J Inorg. Nucl. Chem. 24 (1962) 1012. [9] J A Cahlll and AD. Klrshenbaum, J Phys Chem 68 (1962) 1080. [10] C.C Addison and R.J Pulhan, J Chem Soc (1962) 3873 [11] P J McGonlgal and A V Gross, J. Phys Chem. 67 (1963) 924 [12] A.V Gross, J A. Cahlll and A.D Kirshenbaum, J. Am. Chem Soc 83 (1961) 4665.