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The entropy change on melting of simple substances
Volume 76A, number 2
PHYSICS LEUERS
17 March 1980
THE ENTROPY CHANGE ON MELTING OF SIMPLE SUBSTANCES Jeffery L. TALLON’ Department of Structural Properties ofMaterials, The Technical University ofDenmark, DK-2800 Lyngby, Denmark Received 29 November 1979
5m’ for argon persists, unIt is shown that the volume independent component, R In 2, of the entropy of melting, ~ changed, some distance into the fluid phase as a configurational entropy. For argon, sodium, and caesium in its normal melting regime ~Sm = R In 2+ nBT~Vm, where ~ Vm is the volume change on melting and aBT is the temperature independent product of the thermal expansion coefficient and the isothermal bulk modulus.
Stishov et a!. [1] have suggested that the volume independent change in entropy on melting of simple substances is R In 2, where R is the molar gas constant. This proposition was based on the fact that, for sodium and argon, extrapolation of the entropy change, 1.~Sm, as a function of volume change on melting, ~ V~,gives the valueR In 2 when ~Vm = 0. This has been partially confirmed by Lasocka [2] for a number of metals. He plotted the zero pressure values of ~Sm against ~Vm for the metals investigated and found that the points lie scattered about a line which passes through Z~Sm =Rln 2 at L~Vm= 0. There is as yet, no sound theoretical basis for the origin of this fundamental entropy change. It is, evidently, the minimum entropy change associated with the minimum disorder still compatible with a complete loss of long-range order. It may be the change m entropy associated with the transition from a state with a uniform distribution of cell size per atom to a state with a random distribution of cell size, where the two states have the same average cell size. Tsang and Tang [3] have shown that such a random distribution is essential to the first order character of melting. A more likely proposition has been advanced which identifies this fundamental entropy change with the existence of bifurcations in the on-site potential seen by each atom in the liquid phase [4]. In the pres1
Permanent address: Physics and Engineering Laboratory, Department of Scientific and Industrial Research, Private Bag, Lower Hutt, New Zealand.
ent note, we provide further evidence for the existence of a configurational entropy of R In 2 and we also show that the volume dependent, vibrational component of the entropy for the liquid may be characterized using a simple property of the equation of state for cubic solids. Swenson [5]has shown that the equation of state of the alkali metals and inert gases can be approximated above the Debye temperature by an isothermal bulk modulus, Br, which is temperature independent at constant volume. One of the consequences of this behavior is that the product, aB~of the volume thermal expansion coefficient, a and the bulk modulus is constant independent of temperature, T, volume, V, and of course pressure, P. Thus,
aBT
=
(aP/aT)~= constant.
(1)
In order to demonstrate the generality of this result, we have plotted in fig. 1 the temperature dependence of aB~at atmospheric pressure for a number of other simple cubic materials, including a range of ionic crystals, copper, silver, gold and lead. For these, aBr does vary with temperature, but from above the Debye temperatures, only very weakly so. The near constancy of aBT has an important consequence in the present context, in that the crystal entropy, S, has a linear dependence on isothermal volume. This arises immediately from the Maxwell relation [12]
~asia V)r
=
aBr,
(2) 139
Volume 76A, number 2
PHYSICS LETTERS
6-
60-
5
Zn
-
-
~~Na1
1
-
••~~
-
-.. -~
30
T/ K
20 20
in ref. [8). For MgO the isothermal bulk modulus is from ref. [9] and the thermal expansion from ref. [101. The high temperature data for lead are from ref. [11].
so that
aB [V(T,P)
—
V(T, 0)].
-
-
22
24
26
28
30
3 mol1 Fig. 2. Various isotherms of molar V/ entropy 1O~m versus molar volume
1000
Fig. 1. The temperature dependence of aBT for various ionic crystals and metals. The data used for these materials are those used and referenced in previous work [6,7]. For the alkali halides the curves at lower temperatures are based on the data
(3)
T
For crystalline argon, the entropy is indeed very nearly linear in isothermal volume [131. Our purpose here is to show that this relationship is satisfied in the liquid phase and through the melting expansion, using crystal data for the aBT product. This may be rationalized on the basis that if aBr is constant, then it is also independent of entropy change and can be expected to remain nearly constant through the melting expansion. We recall, also, that the bulk modulus is a simple continuous function of volume through the melting expansion [6,7]. Fig. 2 shows a number of isotherms of entropy as a function of volume for solid argon at the melting line and for liquid argon at, and above, the freezing line. 140
-
~
KBr
500
v
soii~/ KCI
—
+
-
o0
NoCI
0
0
50
NaP
~
2
S=S
LIOUi~
70
-
I:
17 March 1980
for liquid and solid argon (open data points). ~: 83.8 K; 0: 100 K; o: 120 K;v: 140 K, and 0: 160 K. The curved solid line ~5 the melting lIne and the curved dotted line is the freezing line. The full data points are obtained by subtracting R In 2 from the liquid data. The dashed line has the slope csBT = 2.45 x 106 Pa/K taken from ref. [15].
The experimental data, which were reported by Crawford and Daniels [14], are shown by the open symbols, while the full symbols are obtained by subtractingRln2 from each of the liquid phase data. Evidently, the data points for the solid lie precisely on the extrapolated modified liquid isotherms. (There is only one datum point for the liquid isotherm at 83.8 K, so a line parallel to the 100 K isotherm is drawn through this point. The intersection of the melting curve is not quite as good as in the other cases.) Moreover, the gradients of these isotherms agree very well with the value aBT = 2.45 X 106 Pa/K taken from data for the high temperature solid reported by Peterson et al. [15]. For the purpose of comparison, a dashed line of this slope is included in the figure. We conclude, then, that the configurational entropy of R In 2 appears on melting and persists, unchanged, in the liquid phase. On this basis, in the absence of the communal en-
Volume 76A, number 2
PHYSICS LETTERS
tropy, we may expect the entropy of melting for a simple substance to be = R In 2 + aB ~ V (4~ m
T
m
‘
‘
/
provided that the aBr product is fairly constant in the solid phase. Such appears to be the case for the metals sodium and caesium. In fig. 3 we show the entropy of melting as a function of volume change on melting for these metals along their melting curves. For sodium, the data are from Ivanov et al. [16] and for caesium from Makarenko et a!. [17]. The two solid lines have been constructed with the gradients aBr = 1 .47 X 106 Pa/K for sodium and aBT = 6.02 X l0~Pa/K for caesium. These products were evaluated from the room temperature values of a and Br given in the data compilation by Gsclmeidner [18]. The intercepts of the lines were adjusted for best fit of the data points. Evidently the volume dependent part of the entropy change in both cases is extremely well characterized by eq. (4). At higher compressions the data for caesium swings away from the line, and this is due tot the electronic transition which is presumed to occur in the liquid phase and which accounts for the occurrence of a melting maximum [19]. The intercepts are near, but by no means exactly equal to In 2. There may be some curvature at very high pressures (i.e. small ~ Vm) due to break-down in the assumption of a constant aBr product, or the ____________________________________
0.85
-
0.80
-
7 /
-
U
• •
• 77
/. /
0.75
~‘~“
-
I
0.70
0
0.4
0.8
I 1 ~2
I
1
1.6
/~V/ 10~m~mo1 Fig. 3. The entropy change on melting versus volume change on melting for sodium: • and for caesium: .. The solid lines have gradients equal to the room temperature product ceBT for each metal [18].
17 March 1980
value of the intercept in excess of In 2 may be due to the appearance of a small part of communal entropy of melting. The question of communal entropy is by no means resolved for, on the other hand, computer simulations of the melting of hard spheres discount the communal entropy [20] and on the other, simulations of the melting of alkali halides suggest the entire cornmunal entropy appears on melting [21]. In fig. 3 we have considered separate extrapolations back to zero volume change for each material. Lasocka [2], by plotting Z~VmIV5 against ~Sm, found that he could fit a single line through the scattered atmospheric pressure data points for a number of metals and this line intercepted the entropy axis at RIn 2. That this is possible, by allowing some scatter, may be seen by rewriting eq. (4) as = R In 2 + B ~ ~ ~ ‘V m a ~ 5’ m’ RIn2 + 7Cv(~Vm/Vs) (5) R In 2 + 3 7R (~ ~ IV) m
Here V5 is the molar volume of the solid at the melting point, ‘y is the GrUneisen constant, and C~,is the constant volume heat capacity. Thus, since ~ydoes not vary much from metal to metal, the crude correlation demonstrated by Lasocka is possible. We must emphasize that eq. (4) is not universal, as an attempt to apply it to the alkali halides wifi reveal. There are a number of possible reasons for this. A portion of the communal entropy may appear on melting, the material in question may be in a regime near a melting maximum in which the melting line is curved due to continuous change in coordination in the liquid at the freezing point [22,23], or the material may simply not be characterized by a relatively constant aBr product at high temperatures. The author wishes to thank Professor R.M.J. Cotterril for many stimulating discussions on this subject, the New Zealand Department of Scientific and Industrial Research for financial assistance, and the Technical University of Denmark for financial assistance and hospitality during the course of this work. References [1] S.M. Stishov, I.N. Makarenko, V.A. Ivanov and A.M. Nikolaenko, Ploys. Lett. 45A (1973) 18. 141
Volume 76A, number 2
PHYSICS LETTERS
[2] M. Lasocka, Phys. Lett. 5iA (1975) 137. [31 T. Tsang and H.T. Tang, Phys. Rev. A18 (1978) 2315. [4] R.M.J. Cotterill and J.L. Talon, Nature, to be published. [5] C.A. Swenson, J. Phys. Chem. Solids 29 (1968) 1337. [6] J.L. Tallon, W.H. Robinson and S.I. Smedley, Phil. Mag. 36 (1977) 741. [71 J.L. Tallon, Phil. Mag. 39 (1979) 151. [8] B. Yates and C.H. Panter, Proc. Phys. Soc. 80 (1962) 373. [9] D. Chung and W.G. Lawrence, J. Am. Ceram. Soc. 47 (1964) 448. [10] Y.S. Touloukian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, Thermophysical properties of matter, Vol. 13 (Plenum, New York, 1977). [111 C.L. Void, M.E. Glicksman, E.W. Kammer and L.C. Cardinal, J. Phys. Chem. Solids 38 (1977) 157. [121 E. Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1959) p. 87. [13] R.K. Crawford, W.F. Lewis and W.B. Daniels, J. Phys. C9 (1976) 1381.
142
17 March 1980
[14] R.K. Crawford and W.B. Daniels, J. Chem. Phys. 50 (1969) 3171. [15] O.G. Peterson, D.N. Batchelder and R.O. Simmons, Phys. Rev. 150 (1966) 703. [16] V.A. Ivanov, I.N. Makarenko, A.M. Nikolaenko and S.M. Stishov, JETP Lett. 12 (1970) 7. [17] I.N. Makarenko, V.A. Ivanov and S.M. Stishov, JETP Lett. 18 (1973) 187. [18] K.A. Gschneidner, Solid State Phys. 16 (1964) 275. [19] A. Jayaraman, R.C. Newton and J.M. McDonough, Phys. Rev. 159 (1967) 527. [20] W.G. Hoover and F.H. Ree, J. Chem. Phys. 49 (1968) 3609. [21] L.V. Woodcock and K. Singer, Trans. Faraday Soc. 67 (1971) 12. [22] E. Rapoport, in: Liquid metals, ed. S.Z. Beer (Dekker, New York, 1972) p. 373. [23] J.L. Talon, Phys. Lett. 72A (1979) 150.
PHYSICS LEUERS
17 March 1980
THE ENTROPY CHANGE ON MELTING OF SIMPLE SUBSTANCES Jeffery L. TALLON’ Department of Structural Properties ofMaterials, The Technical University ofDenmark, DK-2800 Lyngby, Denmark Received 29 November 1979
5m’ for argon persists, unIt is shown that the volume independent component, R In 2, of the entropy of melting, ~ changed, some distance into the fluid phase as a configurational entropy. For argon, sodium, and caesium in its normal melting regime ~Sm = R In 2+ nBT~Vm, where ~ Vm is the volume change on melting and aBT is the temperature independent product of the thermal expansion coefficient and the isothermal bulk modulus.
Stishov et a!. [1] have suggested that the volume independent change in entropy on melting of simple substances is R In 2, where R is the molar gas constant. This proposition was based on the fact that, for sodium and argon, extrapolation of the entropy change, 1.~Sm, as a function of volume change on melting, ~ V~,gives the valueR In 2 when ~Vm = 0. This has been partially confirmed by Lasocka [2] for a number of metals. He plotted the zero pressure values of ~Sm against ~Vm for the metals investigated and found that the points lie scattered about a line which passes through Z~Sm =Rln 2 at L~Vm= 0. There is as yet, no sound theoretical basis for the origin of this fundamental entropy change. It is, evidently, the minimum entropy change associated with the minimum disorder still compatible with a complete loss of long-range order. It may be the change m entropy associated with the transition from a state with a uniform distribution of cell size per atom to a state with a random distribution of cell size, where the two states have the same average cell size. Tsang and Tang [3] have shown that such a random distribution is essential to the first order character of melting. A more likely proposition has been advanced which identifies this fundamental entropy change with the existence of bifurcations in the on-site potential seen by each atom in the liquid phase [4]. In the pres1
Permanent address: Physics and Engineering Laboratory, Department of Scientific and Industrial Research, Private Bag, Lower Hutt, New Zealand.
ent note, we provide further evidence for the existence of a configurational entropy of R In 2 and we also show that the volume dependent, vibrational component of the entropy for the liquid may be characterized using a simple property of the equation of state for cubic solids. Swenson [5]has shown that the equation of state of the alkali metals and inert gases can be approximated above the Debye temperature by an isothermal bulk modulus, Br, which is temperature independent at constant volume. One of the consequences of this behavior is that the product, aB~of the volume thermal expansion coefficient, a and the bulk modulus is constant independent of temperature, T, volume, V, and of course pressure, P. Thus,
aBT
=
(aP/aT)~= constant.
(1)
In order to demonstrate the generality of this result, we have plotted in fig. 1 the temperature dependence of aB~at atmospheric pressure for a number of other simple cubic materials, including a range of ionic crystals, copper, silver, gold and lead. For these, aBr does vary with temperature, but from above the Debye temperatures, only very weakly so. The near constancy of aBT has an important consequence in the present context, in that the crystal entropy, S, has a linear dependence on isothermal volume. This arises immediately from the Maxwell relation [12]
~asia V)r
=
aBr,
(2) 139
Volume 76A, number 2
PHYSICS LETTERS
6-
60-
5
Zn
-
-
~~Na1
1
-
••~~
-
-.. -~
30
T/ K
20 20
in ref. [8). For MgO the isothermal bulk modulus is from ref. [9] and the thermal expansion from ref. [101. The high temperature data for lead are from ref. [11].
so that
aB [V(T,P)
—
V(T, 0)].
-
-
22
24
26
28
30
3 mol1 Fig. 2. Various isotherms of molar V/ entropy 1O~m versus molar volume
1000
Fig. 1. The temperature dependence of aBT for various ionic crystals and metals. The data used for these materials are those used and referenced in previous work [6,7]. For the alkali halides the curves at lower temperatures are based on the data
(3)
T
For crystalline argon, the entropy is indeed very nearly linear in isothermal volume [131. Our purpose here is to show that this relationship is satisfied in the liquid phase and through the melting expansion, using crystal data for the aBT product. This may be rationalized on the basis that if aBr is constant, then it is also independent of entropy change and can be expected to remain nearly constant through the melting expansion. We recall, also, that the bulk modulus is a simple continuous function of volume through the melting expansion [6,7]. Fig. 2 shows a number of isotherms of entropy as a function of volume for solid argon at the melting line and for liquid argon at, and above, the freezing line. 140
-
~
KBr
500
v
soii~/ KCI
—
+
-
o0
NoCI
0
0
50
NaP
~
2
S=S
LIOUi~
70
-
I:
17 March 1980
for liquid and solid argon (open data points). ~: 83.8 K; 0: 100 K; o: 120 K;v: 140 K, and 0: 160 K. The curved solid line ~5 the melting lIne and the curved dotted line is the freezing line. The full data points are obtained by subtracting R In 2 from the liquid data. The dashed line has the slope csBT = 2.45 x 106 Pa/K taken from ref. [15].
The experimental data, which were reported by Crawford and Daniels [14], are shown by the open symbols, while the full symbols are obtained by subtractingRln2 from each of the liquid phase data. Evidently, the data points for the solid lie precisely on the extrapolated modified liquid isotherms. (There is only one datum point for the liquid isotherm at 83.8 K, so a line parallel to the 100 K isotherm is drawn through this point. The intersection of the melting curve is not quite as good as in the other cases.) Moreover, the gradients of these isotherms agree very well with the value aBT = 2.45 X 106 Pa/K taken from data for the high temperature solid reported by Peterson et al. [15]. For the purpose of comparison, a dashed line of this slope is included in the figure. We conclude, then, that the configurational entropy of R In 2 appears on melting and persists, unchanged, in the liquid phase. On this basis, in the absence of the communal en-
Volume 76A, number 2
PHYSICS LETTERS
tropy, we may expect the entropy of melting for a simple substance to be = R In 2 + aB ~ V (4~ m
T
m
‘
‘
/
provided that the aBr product is fairly constant in the solid phase. Such appears to be the case for the metals sodium and caesium. In fig. 3 we show the entropy of melting as a function of volume change on melting for these metals along their melting curves. For sodium, the data are from Ivanov et al. [16] and for caesium from Makarenko et a!. [17]. The two solid lines have been constructed with the gradients aBr = 1 .47 X 106 Pa/K for sodium and aBT = 6.02 X l0~Pa/K for caesium. These products were evaluated from the room temperature values of a and Br given in the data compilation by Gsclmeidner [18]. The intercepts of the lines were adjusted for best fit of the data points. Evidently the volume dependent part of the entropy change in both cases is extremely well characterized by eq. (4). At higher compressions the data for caesium swings away from the line, and this is due tot the electronic transition which is presumed to occur in the liquid phase and which accounts for the occurrence of a melting maximum [19]. The intercepts are near, but by no means exactly equal to In 2. There may be some curvature at very high pressures (i.e. small ~ Vm) due to break-down in the assumption of a constant aBr product, or the ____________________________________
0.85
-
0.80
-
7 /
-
U
• •
• 77
/. /
0.75
~‘~“
-
I
0.70
0
0.4
0.8
I 1 ~2
I
1
1.6
/~V/ 10~m~mo1 Fig. 3. The entropy change on melting versus volume change on melting for sodium: • and for caesium: .. The solid lines have gradients equal to the room temperature product ceBT for each metal [18].
17 March 1980
value of the intercept in excess of In 2 may be due to the appearance of a small part of communal entropy of melting. The question of communal entropy is by no means resolved for, on the other hand, computer simulations of the melting of hard spheres discount the communal entropy [20] and on the other, simulations of the melting of alkali halides suggest the entire cornmunal entropy appears on melting [21]. In fig. 3 we have considered separate extrapolations back to zero volume change for each material. Lasocka [2], by plotting Z~VmIV5 against ~Sm, found that he could fit a single line through the scattered atmospheric pressure data points for a number of metals and this line intercepted the entropy axis at RIn 2. That this is possible, by allowing some scatter, may be seen by rewriting eq. (4) as = R In 2 + B ~ ~ ~ ‘V m a ~ 5’ m’ RIn2 + 7Cv(~Vm/Vs) (5) R In 2 + 3 7R (~ ~ IV) m
Here V5 is the molar volume of the solid at the melting point, ‘y is the GrUneisen constant, and C~,is the constant volume heat capacity. Thus, since ~ydoes not vary much from metal to metal, the crude correlation demonstrated by Lasocka is possible. We must emphasize that eq. (4) is not universal, as an attempt to apply it to the alkali halides wifi reveal. There are a number of possible reasons for this. A portion of the communal entropy may appear on melting, the material in question may be in a regime near a melting maximum in which the melting line is curved due to continuous change in coordination in the liquid at the freezing point [22,23], or the material may simply not be characterized by a relatively constant aBr product at high temperatures. The author wishes to thank Professor R.M.J. Cotterril for many stimulating discussions on this subject, the New Zealand Department of Scientific and Industrial Research for financial assistance, and the Technical University of Denmark for financial assistance and hospitality during the course of this work. References [1] S.M. Stishov, I.N. Makarenko, V.A. Ivanov and A.M. Nikolaenko, Ploys. Lett. 45A (1973) 18. 141
Volume 76A, number 2
PHYSICS LETTERS
[2] M. Lasocka, Phys. Lett. 5iA (1975) 137. [31 T. Tsang and H.T. Tang, Phys. Rev. A18 (1978) 2315. [4] R.M.J. Cotterill and J.L. Talon, Nature, to be published. [5] C.A. Swenson, J. Phys. Chem. Solids 29 (1968) 1337. [6] J.L. Tallon, W.H. Robinson and S.I. Smedley, Phil. Mag. 36 (1977) 741. [71 J.L. Tallon, Phil. Mag. 39 (1979) 151. [8] B. Yates and C.H. Panter, Proc. Phys. Soc. 80 (1962) 373. [9] D. Chung and W.G. Lawrence, J. Am. Ceram. Soc. 47 (1964) 448. [10] Y.S. Touloukian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, Thermophysical properties of matter, Vol. 13 (Plenum, New York, 1977). [111 C.L. Void, M.E. Glicksman, E.W. Kammer and L.C. Cardinal, J. Phys. Chem. Solids 38 (1977) 157. [121 E. Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1959) p. 87. [13] R.K. Crawford, W.F. Lewis and W.B. Daniels, J. Phys. C9 (1976) 1381.
142
17 March 1980
[14] R.K. Crawford and W.B. Daniels, J. Chem. Phys. 50 (1969) 3171. [15] O.G. Peterson, D.N. Batchelder and R.O. Simmons, Phys. Rev. 150 (1966) 703. [16] V.A. Ivanov, I.N. Makarenko, A.M. Nikolaenko and S.M. Stishov, JETP Lett. 12 (1970) 7. [17] I.N. Makarenko, V.A. Ivanov and S.M. Stishov, JETP Lett. 18 (1973) 187. [18] K.A. Gschneidner, Solid State Phys. 16 (1964) 275. [19] A. Jayaraman, R.C. Newton and J.M. McDonough, Phys. Rev. 159 (1967) 527. [20] W.G. Hoover and F.H. Ree, J. Chem. Phys. 49 (1968) 3609. [21] L.V. Woodcock and K. Singer, Trans. Faraday Soc. 67 (1971) 12. [22] E. Rapoport, in: Liquid metals, ed. S.Z. Beer (Dekker, New York, 1972) p. 373. [23] J.L. Talon, Phys. Lett. 72A (1979) 150.