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Liquid curves of eutectic system Na-K under pressure
Volume 47A, number 3
PHYSICS LETTERS
25 March 1974
L I Q U I D C U R V E S O F E U T E C T I C S Y S T E M Na-K U N D E R P R E S S U R E A.B. BHATIA Theoretical Physics Institute, Umversity o/Alberta, Edmonton, Alberta, Canada
N.H. MARCH and L. RIVAUD Department o/Physics, Imperial College o/Science and Technology, South Kensington, London S. W.7, UK Received 28 January 1974 Data on the phase diagram of Na-K under pressure is used to demonstrate that the entropy change on melting of pure Na and pure K are almost independent of pressure over a substantial range. Chemical association, involving NaK2, is a likely explanation of the peak in the Uquidus curves at high pressures at a K concentration of about 2[3.
The model of conformal ~olutions [1] has been applied to study the concentration fluctuations in liquid Na-K alloys [2]. The interchange energy w may be extracted from a knowledge of the partial structure factors or from thermodynamic data. At normal pressure, and in units of RT~l a, T 1 being the freezing temperature, w has a value of approximately 1.1. Estimates were also made of the first and second pressure derivatives of w from the measured molar volume and compressibility of the liquid alloy. This theory was subsequently used [3] to calculate the liquidus curves of Na-K, and it was found that the above value o f w corrected the ideal curves (see fig. 1) to yield satisfactory agreement with experiment. Since the completion of this work, it has come to our attention that the phase diagram of Na-K has been determined under pressure [4, 5]. In the present note this pressure data is analyzed using the above model and this allows some assessment of the range of validity of the conformal solution model to be made. In this analysis, it proves helpful to replot the data [4, 5] in terms of reduced temperature. At the Narich end, this is simply t = T/TI(P) and similarly at the K-rich end. Out to a concentration of K, c K = 0.2, the curves lie almost on top of one another as can be seen from fig. 1, for pressures of 1,5 000 and 10 000 atm. Two curves from conformal solution theory [3] are also shown. The first takes I = L 1/RTNa as 0.86 in [3]. While this is consistent with the observed latent heat L 1, it is worth pointing out that the curve with l = 0.95 fits the data somewhat better, though we attach no major significance to this. Plots for the
L.0
z.0 I~
\,~
I'1 i ATIR.
~
THEORY ~ m 0,86
X 5,000
---
THEORY~,~" 0,95
1'~
f
o lo.ooo
/I
• so.ooo
if/ 0,9
0.9
0,8
0,S
~ 0-4
0.7
ClD L-0,86
0,7
\
\ \
o.~
o12
\ o'.,
o16
I
0,8
1,o
[:},6
C POTASS[ UH
Fig. 1. Equilibrium phase diagram for Na-K at sewral pressures plotted in terms of the reduced temperature t = T/T1 (p); TI(P) is the melting temperature of pure Na for the curves and data at sodium rich end, and similarly of pure K at potassium rich end.
203
Volume 47A, number 3
PHYSICS LETTERS
K-rich end, with the same l values, are shown; there is rather more spread in the pressure curves here but ~t is again not very much for small Na concentrations. Thus our first conclusion is that l = L/R T1 is almost independent of pressure. Though we are not able to give a convincing theoretical argument why this should be so, we note first that L/T 1 is the entropy change AS of the appropriate pure metal on melting. If one takes a formula hke that due to Mott [6], namely AS ~ 3R
In (VS/VL)
(1)
where vL and vS are Einstein frequencies for the hqmd and solid at the melting point, then although v L and us would each change with pressure, ~t does not seem unreasonable that their ratio is a weak function ofp. Anderson et al. [5] give results over a wade pressure range at potassmm concentrations in the range 0.6 < e K < 0.8. As fig. 1 shows, there are new features at the higher pressures and, in particular, at 50 000 arm, a maximum appears for c K near to 0.7. This ~s a rather clear indication that the conformal solution model is breaking down at such high pressures. Now if there is a strong tendency to form NaK 2 then this would indeed lead to a peak m the liquidus curves at c K = 2/3 [e.g., 7]. This as a likely explanation for the data at 50 000 atm. If this tendency to assocmte is really strong, the electrical resistivity should also show anomalous behaviour near c K = 2/3 at this pressure, but we have found no measurements to test this. In this connection, it should be noted that the energy o f formation Hf of the chemical complexes Na 2 K and NaK 2 has been measured at 1 atm [8]. For a gmole of (Na+K), Le., for example ~Na+~K -~ ~NaK 2, Hf/RT~ 1.2 at T = 300°K and ~ 0.3 for T = 380°K for both compounds. A value Hf < RT implies that there is only a weak tendency to form chemical associations so that the conformal solution model can be expected to be a good approximation at 1 atm for temperatures higher than about 300°K [9]. The final point concerns w/RT 1. First, scaling with reduced temperature shown in fig. 1 is only approximate. In the concentration range 0.2 < c K < 0.4 and 0.8 < ci¢ < 1, the data between 1 and 10 000 atm indicate that W/Tl(P) (TI(P) = T1Na at Na rich end etc.) decreases with increasing pressure, since the data at higher p is closer to the ideal curves, as seen from 204
25 March 1974
fig. 1. However, it is difficult to make a quantitatwe statement for several reasons. First, in the conformal solution model, w is a function of b o t h p and T. l f w e take the heat o f mixing A H data [10] and assume w = AT+BT 2, then it is found that
w ~ I.9 RT(1
2s T ) s7 30-0
(2)
since A H = c(1 - c ) ( w - Tdw/dT). This means that one should strictly take account o f the temperature dependence of w even at 1 atm, in interpreting the hquidus curve at different concentrations, and this would lead to a small lowering of the value of w/RTNa say, from 1.1 to 0.9. Secondly the work on the pressure dependence of w [2] yields
w(p)
w(p=l-~----~ "~ 1 - 0.7 X 1 0 - 4 p - 0.34 X 1 0 - 8 p 2
(3)
with p in atmospheres. This is a strong variation but this form is determined near 1 atm and it is not valid to use such an extrapolation to pressures as large as 104 atm. In fact, one knows in the alkali metals that changes in other properties, for example, TI(P), tend to saturate at high pressures. Thus it seems possible to say only that, up to about 104 atm, the data indicates for 0.2 < c K < 0.4 and 0.7 < e K < 1, that w/RT 1 decreases with increasing pressure and that this is not unexpected in view of the results (2) and (3) above. It would seem of interest to test the conclusions here by &rect measurement of the latent heats of pure Na and K as a function of pressure and also to investigate the electrical resistivity of the Na-K liquid alloy a t p ~ 5 × 104 atm near t o e K = 2/3. [l] H.C. Longuet-Hlggins, Proc. Roy. Soc. A205 (1951) 247. [2] A.B. Bhatia, W.H. Hargrove and N.H. March, J. Phys.
C6 (1973) 621. [3] A.B. Bhatia and N.H. March, Phys. Lett. 41A (1972)
397. [4] C.H. Kean, Phys. Rev. 55 (1939) 750. [51 D.R. Anderson, J.B. Ott, J.R. Goates and H.T. Hall, J. Chem. Phys. 54 (1971) 234. [6] N.F. Mott, Proc. Roy. Soc. A146 (1934) 465. [7] A. Reisman, Phase equilibria (Academic Press, New York, 1970) p. 230. [8] R.L. McKisson and L.A. Bromley, J. Am. Chem. Soc. 73(1951) 314. [9] A.B. Bhatia and W.H. Hargrove, Lett. Nuovo Cim., to be pubfished. [10] R. Hultgren, R.L. Orr, P.D. Anderson and K.K. Kelly, Selected values of thermodynamic properties of metals and alloys (Wiley, New York, 1963).
PHYSICS LETTERS
25 March 1974
L I Q U I D C U R V E S O F E U T E C T I C S Y S T E M Na-K U N D E R P R E S S U R E A.B. BHATIA Theoretical Physics Institute, Umversity o/Alberta, Edmonton, Alberta, Canada
N.H. MARCH and L. RIVAUD Department o/Physics, Imperial College o/Science and Technology, South Kensington, London S. W.7, UK Received 28 January 1974 Data on the phase diagram of Na-K under pressure is used to demonstrate that the entropy change on melting of pure Na and pure K are almost independent of pressure over a substantial range. Chemical association, involving NaK2, is a likely explanation of the peak in the Uquidus curves at high pressures at a K concentration of about 2[3.
The model of conformal ~olutions [1] has been applied to study the concentration fluctuations in liquid Na-K alloys [2]. The interchange energy w may be extracted from a knowledge of the partial structure factors or from thermodynamic data. At normal pressure, and in units of RT~l a, T 1 being the freezing temperature, w has a value of approximately 1.1. Estimates were also made of the first and second pressure derivatives of w from the measured molar volume and compressibility of the liquid alloy. This theory was subsequently used [3] to calculate the liquidus curves of Na-K, and it was found that the above value o f w corrected the ideal curves (see fig. 1) to yield satisfactory agreement with experiment. Since the completion of this work, it has come to our attention that the phase diagram of Na-K has been determined under pressure [4, 5]. In the present note this pressure data is analyzed using the above model and this allows some assessment of the range of validity of the conformal solution model to be made. In this analysis, it proves helpful to replot the data [4, 5] in terms of reduced temperature. At the Narich end, this is simply t = T/TI(P) and similarly at the K-rich end. Out to a concentration of K, c K = 0.2, the curves lie almost on top of one another as can be seen from fig. 1, for pressures of 1,5 000 and 10 000 atm. Two curves from conformal solution theory [3] are also shown. The first takes I = L 1/RTNa as 0.86 in [3]. While this is consistent with the observed latent heat L 1, it is worth pointing out that the curve with l = 0.95 fits the data somewhat better, though we attach no major significance to this. Plots for the
L.0
z.0 I~
\,~
I'1 i ATIR.
~
THEORY ~ m 0,86
X 5,000
---
THEORY~,~" 0,95
1'~
f
o lo.ooo
/I
• so.ooo
if/ 0,9
0.9
0,8
0,S
~ 0-4
0.7
ClD L-0,86
0,7
\
\ \
o.~
o12
\ o'.,
o16
I
0,8
1,o
[:},6
C POTASS[ UH
Fig. 1. Equilibrium phase diagram for Na-K at sewral pressures plotted in terms of the reduced temperature t = T/T1 (p); TI(P) is the melting temperature of pure Na for the curves and data at sodium rich end, and similarly of pure K at potassium rich end.
203
Volume 47A, number 3
PHYSICS LETTERS
K-rich end, with the same l values, are shown; there is rather more spread in the pressure curves here but ~t is again not very much for small Na concentrations. Thus our first conclusion is that l = L/R T1 is almost independent of pressure. Though we are not able to give a convincing theoretical argument why this should be so, we note first that L/T 1 is the entropy change AS of the appropriate pure metal on melting. If one takes a formula hke that due to Mott [6], namely AS ~ 3R
In (VS/VL)
(1)
where vL and vS are Einstein frequencies for the hqmd and solid at the melting point, then although v L and us would each change with pressure, ~t does not seem unreasonable that their ratio is a weak function ofp. Anderson et al. [5] give results over a wade pressure range at potassmm concentrations in the range 0.6 < e K < 0.8. As fig. 1 shows, there are new features at the higher pressures and, in particular, at 50 000 arm, a maximum appears for c K near to 0.7. This ~s a rather clear indication that the conformal solution model is breaking down at such high pressures. Now if there is a strong tendency to form NaK 2 then this would indeed lead to a peak m the liquidus curves at c K = 2/3 [e.g., 7]. This as a likely explanation for the data at 50 000 atm. If this tendency to assocmte is really strong, the electrical resistivity should also show anomalous behaviour near c K = 2/3 at this pressure, but we have found no measurements to test this. In this connection, it should be noted that the energy o f formation Hf of the chemical complexes Na 2 K and NaK 2 has been measured at 1 atm [8]. For a gmole of (Na+K), Le., for example ~Na+~K -~ ~NaK 2, Hf/RT~ 1.2 at T = 300°K and ~ 0.3 for T = 380°K for both compounds. A value Hf < RT implies that there is only a weak tendency to form chemical associations so that the conformal solution model can be expected to be a good approximation at 1 atm for temperatures higher than about 300°K [9]. The final point concerns w/RT 1. First, scaling with reduced temperature shown in fig. 1 is only approximate. In the concentration range 0.2 < c K < 0.4 and 0.8 < ci¢ < 1, the data between 1 and 10 000 atm indicate that W/Tl(P) (TI(P) = T1Na at Na rich end etc.) decreases with increasing pressure, since the data at higher p is closer to the ideal curves, as seen from 204
25 March 1974
fig. 1. However, it is difficult to make a quantitatwe statement for several reasons. First, in the conformal solution model, w is a function of b o t h p and T. l f w e take the heat o f mixing A H data [10] and assume w = AT+BT 2, then it is found that
w ~ I.9 RT(1
2s T ) s7 30-0
(2)
since A H = c(1 - c ) ( w - Tdw/dT). This means that one should strictly take account o f the temperature dependence of w even at 1 atm, in interpreting the hquidus curve at different concentrations, and this would lead to a small lowering of the value of w/RTNa say, from 1.1 to 0.9. Secondly the work on the pressure dependence of w [2] yields
w(p)
w(p=l-~----~ "~ 1 - 0.7 X 1 0 - 4 p - 0.34 X 1 0 - 8 p 2
(3)
with p in atmospheres. This is a strong variation but this form is determined near 1 atm and it is not valid to use such an extrapolation to pressures as large as 104 atm. In fact, one knows in the alkali metals that changes in other properties, for example, TI(P), tend to saturate at high pressures. Thus it seems possible to say only that, up to about 104 atm, the data indicates for 0.2 < c K < 0.4 and 0.7 < e K < 1, that w/RT 1 decreases with increasing pressure and that this is not unexpected in view of the results (2) and (3) above. It would seem of interest to test the conclusions here by &rect measurement of the latent heats of pure Na and K as a function of pressure and also to investigate the electrical resistivity of the Na-K liquid alloy a t p ~ 5 × 104 atm near t o e K = 2/3. [l] H.C. Longuet-Hlggins, Proc. Roy. Soc. A205 (1951) 247. [2] A.B. Bhatia, W.H. Hargrove and N.H. March, J. Phys.
C6 (1973) 621. [3] A.B. Bhatia and N.H. March, Phys. Lett. 41A (1972)
397. [4] C.H. Kean, Phys. Rev. 55 (1939) 750. [51 D.R. Anderson, J.B. Ott, J.R. Goates and H.T. Hall, J. Chem. Phys. 54 (1971) 234. [6] N.F. Mott, Proc. Roy. Soc. A146 (1934) 465. [7] A. Reisman, Phase equilibria (Academic Press, New York, 1970) p. 230. [8] R.L. McKisson and L.A. Bromley, J. Am. Chem. Soc. 73(1951) 314. [9] A.B. Bhatia and W.H. Hargrove, Lett. Nuovo Cim., to be pubfished. [10] R. Hultgren, R.L. Orr, P.D. Anderson and K.K. Kelly, Selected values of thermodynamic properties of metals and alloys (Wiley, New York, 1963).