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Calculation of some thermodynamic properties and the phase diagram of the PbTe system
Journal of the Less-Common
Metals, 119 (1986)
217
277 - 289
CALCULATION OF SOME THERMODYNAMIC PROPERTIES THE PHASE DIAGRAM OF THE Pb-Te SYSTEM
AND
M. T. CLAVAGUERA-MORA
Termologia, Facultad Barcelona (Spain)
de
Ciencias,
Universidad
Autdnoma
de Barcelona,
Bellaterra,
N. CLAVAGUERA Fisica de1 Estado S6lido, 08028-Barcelona (Spain)
Facultad
de Fisica,
Universidad de Barcelona,
Diagonal 647,
J. ONRUBIA
Termologia, (Spain)
Facultad
de Ciencias Fisicas, Universidad de Valencia, Burjassot,
Valencia
R. COHEN-ADAD
Laboratoire de Physico-Chimie Minerale II, Universite’ Claude Bernard (Lyon I), 43 Boulevard du 11 Novembre 1918, 69621 Villeurbanne (France) (Received
September
16, 1985)
Summary
The strongly associated regular solution model has been used to calculate the phase diagram of the Pb-Te system, and most of the thermodynamic quantities of the Pb-Te liquid alloys at 1200 K, for which experimental values were available in the literature. These are the molar partial and mixing enthalpies, entropies and Gibbs free energies. The dissociation equilibrium constant and the pair interaction parameters used in the calculation were derived from experimental values for both the molar partial enthalpies at infinite dilution and the enthalpy of mixing at 50 at.% Te. The Pb-Te phase diagram has been calculated using the aforementioned parameters, and compared with the data reported in the literature. Agreement is generally quite good between calculated and experimental results, although we assume a negligible solubility range for PbTe in the solid state.
1. Introduction The aim of the present work was to integrate in a unique model for liquid mixtures, both the experimental results on the phase diagram and the thermodynamic mixing functions of liquid solutions of the Pb-Te system. We utilize the strongly associated regular (SAR) solution model, described in ref. 2, to treat the liquid mixtures. The term regular solution is used here in a wide sense, following the nomenclature of Jordan’s work 0022-5088/86/$3.50
0 Elsevier Sequoia/Printed
in The Netherlands
278
[l]. It differs nevertheless from a strictly regular solution in two important ways: (i) there exists an excess entropy of mixing and (ii) the interaction energies vary linearly with temperature. We assume a negligible solubility range for the crystalline phases. The parameters of the model are estimated on the basis of the available partial enthalpies of solution at infinite dilution and the enthalpies of formation of both solid and liquid PbTe. Several studies on the thermodynamic properties of the Pb-Te system have been published. We begin by reviewing the values for the elements and the stoichiometric compound. For lead the values required for the present calculation have been collected in ref. 3 and are as follows: entropy of formation S&x = (15.49 + 0.1) X 4.184 J K-i mol-I; melting ~mperature T, = 600.6 K; enthalpy of fusion A.E& = 1140 X 4.184 kJ mol-‘“; specific heat of solid and liquid state C,’ = (5.629 + 2.328 X lO”3 7’) X 4.184 J K-i mol-f and Cp’= (7.765 0.738 X low3 2’) X 4.184 J K-l mol-* respectively. For tellurium the quantities needed for the present calculation have been collected in ref. 4 and are as follows: S&,x = (11.83 + 0.05) X 4.184 J K-l mol-‘; T, = 722.7 K; AH, = 4.18 X 4.184 J K-’ mol-‘; C,’ = (4.57 + 5.28 X 10m3 2’) X 4.184 J K-l mol-’ ; C,’ = 9.0 X 4.184 J K-l mol-“. For PbTe the solid solubility range is 0.499 - 0.501 at.% Te [43. Other values collected in ref. 4 are as follows: enth~py of formation AH: 298K = --8.2 X 4.184 kJ (g atom)-*; S&sx = (13.15 f 0.3) X 4.184 J K-i (g atom)-i; zn = 1197 K; C,’ = (5.64 + 1.345 X low3 2’) X 4.184 J K-l (g atom)-‘. Some discrepancies appear in the published values for the enthalpy of melting. Steininger [S] proposes a value of 9.4 X 4.184 kJ mol-’ whereas Blachnik and Kluge [63 take 13.7 X 4.184 kJ mol-’ and Shamsuddin et al. [7] obtain 9.9 X 4.184 kJ mol- I. As will be explained later we propose a value of 11.2 X 4.184 kJ mol-‘. The enthalpy of formation has been measured at several temperatures. McAteer and Seltz [ 8] obtained AH; 655K = (-8.44 f 0.20) X 4.184 kJ (g atom)-‘. Castanet et al. [9] found AH: ,s7x = -47.20 f 0.20 kJ (g atom)-“. Furthermore these authors gave a tentative relation for the temperature dependence of that quantity with reference to both pure solid elements, namely AH; T = -32.9 - 4.3 X 10U3 7’ kJ (g atom)-‘. Some the~odynami~ properties of liquid mixtures have also been determined at several ~mperat~es. The activity of tellurium at 1277 K in tellurium~rich mixtures has been measured by Predel et al. [lo] using a comparative method of partial vapour pressure measurements. The integral enthalpy of mixing was measured calorimetrically at 737 K by Castanet et al. [9] and at 1210 K by Blachnik and Gather [ll]. The activity of lead in liquid mixtures was determined by e.m.f. measurements over the whole composition range and between 653 and 1273 K by Moniri and Petot 1121. The results of calorimetric measurements of the partial molar enthalpy values at infinite dilution obtained at several temperatures by different authors and the mixing enthalpy at the stoichiometric composition are reported in Table 1.
279 TABLE
1
Heats of mixing liquid elements
of PbTe
and partial
AHM (50 at.% Te) (kJ (g atom)-‘)
T W) 737 1210 1200 879
-26.0 -26.1
f 0.8 + 0.6
enthalpies
Pm &Pb (kJ (g atom)-‘)
at infinite
dilution
with reference
-00
to the
Reference
AH,,
(kJ (g atom)-‘)
-69.0 -73.8
2 0.5 * 3.8
-34.3 -36.9
f 0.3 * 1.3
-59.9
+ 1.1
--36.2
+ 0.6
9 11 12 12
Given the enthalpy, entropy and Gibbs free energy of formation of PbTe at 298 K and the specific heats of lead, tellurium and PbTe below 1197 K, one can calculate the respective formation quantities of solid PbTe at 1197 K with reference to the pure liquid elements. The results are kJ (g atom))‘, AS; 1197K = -23.00 J K-’ (g atom))’ AH; 1197K = -48.92 -21.38 kJ (g atom))‘. Furthermore we know that at 1200 and AG;, 1197K = K the enthalpy and entropy of mixing at the stoichiometric composition are &?&,c x = -26.0 kJ (g atom))’ and AS&sK = -2.6 J K-l (g atom))’ (from ref. 12). We therefore expect an enthalpy of fusion, of roughly 23.5 kJ (g atom))’ (or 11.2 kcal mol-‘) and we will use that value, when necessary, for the calculation. A relatively large number of studies on the phase diagram of the Pb-Te system have been reported. Hansen [13] collected the measurements of Fay and Gillson [ 141, Pelabon [ 151 and Kimura [16]. Miller and Komarek [17] have studied the lead-rich part of the system, including the solid solubility limit of lead in the PbTe compound. The liquidus temperatures for small additions of tellurium to pure lead have been reported by Harris et al. [ 181. Moniri and Petot [ 191 have studied the complete phase diagram. Further studies on the tellurium-rich part of the system have been carried out by Kharif et al. [ 201.
2. The basic formulae for a strongly associated regular solution In this work the binary Pb, -xTe, is treated using the SAR solution model developed in ref. 2. The liquid solutions are assumed to be composed of lead, tellurium and PbTe species, whose mole fractions can be written in terms of the atomic fraction x of tellurium as xp,, = 1 -x
--XXpbTe
(1) XTe
=
x
-
t1
-
x)xPbTe
The species activity coefficients yi (j = Pb, Te, PbTe) are given by
280
(a) for 0 < x < 0.5
2a(l
RT In Tre =
- 2x) +P
l-x a(1 - 2x)2
RTln
YPbTe
=
t1 _xJ2
(b) for 0.5 < x < 1
@(23c-1)
RTlnypb=
RT In Yre =
P(1 -x)2
In %‘bTe
(2b)
X2 PeX
RT
+(y
x
x2
=
II2
where a! and p are the interchange energies for the interactions of the specific species Pb-PbTe and Te-PbTe respectively (both independent of composition), and where we take TBre = 1 at the stoichiometric composition at each temperature, i.e. the reference state of the species PbTe in the hypothetical state of complete association at the stoichiometric composition. The mole fractions of the different species are related through the dissociation equilibrium constant in the liquid by K=-
a+P
XPbxTe XPbTe
-
exp
RT
=
K’exp
a+P -
RT
(3)
where K’=
XPbXTe
-
XPbTe
is the dissociation equilibrium constant for the ideal associated solution (o = p = 0). The partial molar Gibbs free energy of the components lead and tellurium with reference to the corresponding pure liquid components are respectively (a) for 0 < x < 0.5 =Pb
=Te
=RTInxpb+
= RT ln xre +
O!X2
(1 -x)2
(54
24 1 - 2x) l-x
+P
281
(b) for 0.5 < x < 1
dc,b = RT
h xp,, +
X
(5b)
@(I .-x)2
AGTr= RTlnx,,+
x2 The Gibbs free energy of mixing referred mixture can be split into two parts
to 1 mol of the starting
AGM = AG’ + AG=
(6)
where AGi is the Gibbs free energy tion, namely A@ = RT{(i x (XPb
-x)
of mixing for an ideal associated
In xpb + x In xTe} =
In XPb
liquid
+ XTe
In XTe
+ XPbTe
solu-
RT 1 + XPbTe In XPbTe
+ XPbTe
In K’)
(7)
and the regular part AG’ is given by (a) for 0 < x < 0.5 AG’ = (cu + /3)x +
cux(1 - 2x) (8a)
l--X
(b) for 0.5 < x < 1 AC’ = ((u + /3)( 1 -x)
+
p(1 -x)(2x
- 1) (Sb)
X
The calculation of the other thermodynamic quantities is straightforward. The equations for the partial and mixing molar enthalpies and entropies given by the model are presented in Table 2. Furthermore, the partial molar enthaipies at infinite dilution are L’ = l+K’
lim AHp,, = m; X .* 1
L’
lim MT, x -*0
+ (a-AT)+
(9)
= dHFe = ~ + 2((u -AT) 1+ K’
where L’ = Ra(ln h?)/a(llT),
SAR solution,
the following
2(/3-M’)
+ (j3 - BT)
A = &xfaT and B = ap/aT. As K’ < 1 for a approximate formulae are also correct
+@-BT m,“,
=
R
@lnK, U/T)
(10) +CY-AT
because from eqn. (3), In K = In K’ + (CY+ (3)/RT.
2
quantities
vpb
ASM
= axpblaT
and
-(A
-R
=re
pb -RTypbIxpb
-Ax(l
YTe = axT,/aT.
+ B)x
-
-RT2
-R{(l -x) 2x)/(1 -x)
In xpb +x In xTe)-
+ (0 -BT)(l
+ (a -AT)
- A -
2B( 2x -
-(A+B)(1-x)-B(1-x)(2x-l)/x
+ XYT~/XT~)
h XTe - RTYT@/xT, - B(1 - x)~/.x’
In Xpb - RTypb/Xpb
-x)Ypb/xpb
-R
-R
BT)(2X
- x)‘/x’
+ 2@-
+(a-AT+~-BT)(1-x)+(~-BT)(l-xxf(2x-1)/X
-RTZyTe/xTe
-RT2ypb/xpb
0.5GxGl
+ xYdX&
RT{(l
{(I - x)YPt,kPb
2x)/( 1 - x)
%Te,
2x)/(1 -X)-B
- Ax2 I(1 - xl2
In XT~ - RTYT,./xT~ -2A(l
-RInx
=Pb
+(a--AT+P--BT)x+(ff--ATb(l--)/(l--x)
+ (0 - BT) + 2(a - AT)( 1 -
-RTZyTe/xT,
LWTe
AHM
+ (a -AT)xl/(l
’
of Pbl
-RTZypblxpb
-X,2
for SAR solutions
Iw,b
OGxGO.5
Some thermodynamic
TABLE
1 )/x
1 )/X
-
% N
283
3. Evaluation of the parameters of the model The values for the partial molar enthalpies at infinite dilution, and for the molar enthalpy of mixing at the stoichiometric composition provide relative limits for the interchange energies of interaction extrapolated to 0 K and for the temperature dependence of the enthalpy of formation of the associates, namely (x -AT, 0 - BT and Ra(ln K)/a(l/T). The restrictions on the values are as follows: -69
kJ (g atom)-l < AH& < -73.8
-34.3
kJ (g atom)-’
kJ (g atom) -’ < &%?,- < -36.9
AHM = f(K:‘)R
a(ln K) I=-26 W/V
kJ (g atom)-’
kJ (g atom)-’
where f(K’) is a slowly varying function of K and less than 0.5, Furthermore, the best fit to the experimental partial molar Gibbs free energy and entropy of lead at 1200 K within the limitations of the preceding requirements is obtained by using the values shown in Table 3. TABLE
3
Best computational results for the parameters used in calculation of the SAR solutions of liquid Pb-Te alloys at 1200 K. Calculated values of Ra(ln K)/8(1/2’), (Y and fl in kJ (g atom)--l, and A and B values in J K-’ (g atom)---’
0.005
-56.3
11.3
-2.76
-?.5
10.7
4. Results and discussion By use of the values in Table 3 the calculated partial molar enthafpies of each component at infinite dilution are zH,“,=
-71.6
kJ (g atom)-’
A”&L = -36.4
kJ (g atom)-’
and the mixing enthalpy at the sto~~hiometri~ composition is AHM = -26.0
kJ (g atom)-’
Figure 1 shows the general behaviour of the partial molar enthalpies of lead and tellurium calculated using the parameters given in Table 3, together with the experimental data for m,, given by Moniri and Pet& [12]. The agreement between calculated and experimental data is quite good except for x > 0.9 where the discrepancies are of the order of 10%. The mixing enthalpies, entropies and Gibbs free energies calculated from the model at 3200 K are plotted versus the composition in Fig. 2. The experimental values for AHM at 1210 K given by Blachnik and Gather [ll]
284
-75
0
.2
.4
Fltomic
-6 fraction
.8
~ 1
Te
Fig. 1. Partial molar enthalpies of lead and tellurium at 1200 K with reference to the pure components: -, calculated values from the equations of Table 2; x, experimental data from ref. 12. 3
5
t
5 f
a
0
-5
-‘25 0
.2 htomic
.4
.6 fraction
.8
1
Te
Fig. 2. Concentration dependence of the mixing enthalpy AH”, entropy ASM and Gibbs free energy AGM of liquid Pb-Te alloys: -, values calculated at 1200 K from the equations of Table 2 and eqns. (7) and (8); X, experimental data for A?IM at 1210 K from ref. 11.
are also shown in that figure. The concentration dependence of calculated AHM values exhibits a minimum at x = 0.521, with a value of -26.36 kJ (g atom)-‘, and the ASM curve has an indentation with a minimum at the
285
same concentration. Both facts are related to the strong association predicted by the model in the liquid alloy and are in agreement with the results reported in ref. 11, i.e. an experimental minimum for AHM at 52 at.% Te with a value of -26.25 rt 0.95 kJ (g atom))l. Figure 3 shows the calculated partial molar Gibbs free energies and the experimental ii?&, obtained by Moniri and Petot [CC]. The calculated curves are in good agreement with the experimental data to within the estimated uncertainty associated with the measurement, except for values of x around 0.5, where a significant discrepancy occurs. Otherwise further improvement, even if possible, would not be significant in a statistics sense. The calculated activities of lead and ~llurium in the Pb-Te liquid alloys at 1200 K are presented in Fig. 4 and compared with experiments data for apb given by Moniri and Petot [12] at the same temperature, and with those for &re given by Predel et al. [lo] at 1277 K. As observed for the partial molar free energy mph, the agreement is quite good for apb, the discrepancies being less than the experimental uncertainty except for x around 0.5 where the deviations are about 5%. Figure 5 shows the molar partial entropies of Pb-Te liquid alloys at 1200 K and the experimental ds, obtained by Moniri and Petot [ 12). The S-shaped form of both quantities is typical of strong association for x = 0.5. The calculated curves shown in Figs. 1 - 5 have been obtained using a unique set of six parameters specified in Table 3. As a consequence the Gibbs free energy of formation of 1 mol of the complex PbTe is given by AG;_ rr = RT in K = -56.3 + 10.0 X 1Cm3 T kJ mol-‘. To find out if the same set of parameters is able to describe the liquid mixtures in equilibrium with the solid phases we calculated the phase
80
20
0
0
.2
.4 F(tomic
-6 fraction
.8 Te
Fig. 3. Partial molar Gibbs free energies of lead and tellurium at 1200 K with reference to the pure components: -, calculated values from eqns. (5); X, experimental points from ref. 12.
286
1
.8
cn .6 .> .42 0 a .4
.2
Kl
0
0
.2
.4 Atomic
.6
.6
fraction
1
Te
Fig. 4. Concentration dependence of the calculated from eqns. (2); X, experimental data at 1277 K from ref. 10.
activities of lead and tellurium: data at 1200 K from ref. 12;A,
values experimental
20
-20
0
.2
.6
.4 atomic
.6
fraction
1
Te
of lead and tellurium at 1200 K with reference to the pure Fig. 5. Partial molar entropies liquid components: -, values calculated from the equations of Table 2 ; x , experimental data from ref. 12.
diagram of the Pb-Te system. For that purpose we used the following equations for the liquidus curve for equilibrium between liquid solution and solid PbTe compound (a) for 0 < x < 0.5 T=
AHPbTe + cx,(l m AS,~T” _
R
ln(%%Te/x
-
&T,)
2x)2/(1 -A(1
- x)2 -2x)*/(1-x)*
(lla)
287
1
0
.2
.4
fitantic
.6 fraction
.8
1
Te
Fig. 6. Phase diagram of the Pb-Te system. Calculated liquidus temperatures for the following values of h?: - - -, ti = 5 x 10--J; ---, I@ = 1 x 10m3. Experimental points: 0, from ref. 14; a, from ref. 16; X, from ref. 17; +, from ref. 18;Y, from ref. 19; *, from ref. 20.
(b) for 0.5 G x G 1 T=
AHPbTe m + &)(2x - 1)2/X2 _ AS”Te - R h(xpbTe/x&&) - B(2x m
1J2/x2
Wb)
For the liquidus curve for the equilibrium between the liquid solution and solid tellurium T=
Ali: AS?
+ &(l - X)2/X2
- R in &re --.&I
-x)~/x’
with czs= Q!- AT and PO= #3- BT the ~terch~ge energies extrapolated to 0 K, and where X&r,? is the value of Xr,,?reat the stoichiometric composition. Figure 6 compares the theory (for different values of K’) of binary liquidus with experimental results. The eutectic temperature and composition for the PbTe-Te subsystem are respectively 694 K and 0.911 at.% Te when K’ = 1 X 10s3. The experimental eutectic temperature obtained by Moniri and Petot [ 191 and shown in Fig. 6, is 687 K. The most outstanding features of the liquidus curves are the asymmetry with respect to the PbTe compound, and the sharp peak at the 50% composition. The theory of SAR solutions can actually describe the asymmetry because cyf 0. Furthermore, as I(’ is small, the association is large and the peak becomes sharp.
288
5. Conclusions This work is the first published application of the SAR solution model to binary systems to interpret both the thermodynamic mixing functions and the phase diagram (at constant pressure) using a unique set of six parameters basic to the model. In as much as there is no direct measurement of the chemical short-range order in Pb-Te liquid alloys, we have assumed the existence of PbTe complexes in equilibrium with lead and tellurium atoms. Nevertheless, the precise way in which these complexes are formed electronically as well as their lifetimes do not enter explicitly in our treatment so we may expect this treatment to be applicable also to metallic, covalent or even ionic liquid alloys. The results obtained justify the validity of the model, because general expressions for the molar partial Gibbs free energies for the Pb-Te liquid alloys have been obtained which are consistent with both the phase diagram and the thermochemical data. Two substantially different interaction parameters have been introduced in order to reproduce the asymmetries in both liquidus temperatures and thermochemical data. Furthermore these interaction parameters, although dependent on temperature, have values Ia/RTI, I/3/RTI < 1.2 at T = 1200 K, i.e. they are small enough for the regular approximation to be valid. The equilibrium constant K’ for ideal solutions has values lower than the upper limit needed to undertake strong associated treatment, namely K’ < 10-l [ 211. In general, equilibrium constants for ideal solutions depend on the liquid composition [22, 231, but this is not the case for SAR solutions [ 21. However, a small temperature dependence can be allowed because increasing temperature diminishes chemical short-range order.
References 1 A. S. Jordan, Metall. Trans., 1 (1970) 239. 2 M. T. Clavaguera-Mora and N. Clavaguera, J. Phys. Chem. Solids, 43 (1982) 963. 3 R. Hultgren, R. L. Orr, P. D. Anderson and K. K. Kelley, Selected Values of Thermodynamic Properties of Metals and Alloys, Wiley, New York, 1963. 4 K. C. Mills, Thermodynamic Data for Inorganic Sulphides, Selenides and Tellurides, Butterworths, London, 1974. 5 J. Steininger, J. Appl. Phys., 41 (1970) 2713. 6 R. Blachnik and W. Kluge, Thermochim. Acta, 3 (1972) 317. 7 Shamsuddin, P. Ramachandraro, S. Misra and T. R. Anantharaman, J. Mater. Sci., 10 (1975) 1849. 8 J. H. McAteer and H. Seltz, J. Am. Chem. Sot., 58 (1936) 2081. 9 R. Castanet, Y. Claire and M. Laffitte, High Temp.-High Pressures, 4 (1972) 343. 10 B. Predel, J. Piehl and M. J. Pool, 2. Metal&d., 66 (1975) 347. 11 R. Blachnik and B. Gather, J. Less-Common Met., 92 (1983) 207. 12 N. Moniri and C. Petot, Thermochim. Acta, 77 (1984) 151. 13 M. Hansen, Constitution of Binary Alloys, McGraw-Hill, New York, 1958. 14 H. Fay and C. B. Gillson,Am. Chem. J., 27 (1902) 81. 15 H. Pelabon, Ann. Chim. Phys., 17 (1909) 557. 16 M. Kimura, Mem. Coll. Sci. Kyoto Univ., 1 (1915) 149.
289
17 E. Miller and K. L. Komarek, Trans. Metall. Sec. AIME, 236 (1966) 832. 18 J. S. Harris, J. T. Longo, E. R. Gertner and J. E. Clark, J. Ctyst. Growth, 27 (1975) 334. 19 N. Moniri and C. Petot, J. Calor. Anal. Thermique, 9 195. N. Moniri, T&se docteur-ingknieur, Paris VI, 1978. 20 Ya. L. Kharif, P. V. Kovtunenko, A. A. Maier and I. Kh. Avetisov, Zh. Fiz. Khim., 56 21 22 23
(1982) 2180. J. Onrubia, Ph.D. Thesis, Universidad de Valencia, 1983. F. Sommer, 2. Metallkd., 72 (1982) 73, 77. S. Lele, G. V. S. Sastry and P. Ramachandrarao, 2. Metallkd.,
75 (1984)
446.
Metals, 119 (1986)
217
277 - 289
CALCULATION OF SOME THERMODYNAMIC PROPERTIES THE PHASE DIAGRAM OF THE Pb-Te SYSTEM
AND
M. T. CLAVAGUERA-MORA
Termologia, Facultad Barcelona (Spain)
de
Ciencias,
Universidad
Autdnoma
de Barcelona,
Bellaterra,
N. CLAVAGUERA Fisica de1 Estado S6lido, 08028-Barcelona (Spain)
Facultad
de Fisica,
Universidad de Barcelona,
Diagonal 647,
J. ONRUBIA
Termologia, (Spain)
Facultad
de Ciencias Fisicas, Universidad de Valencia, Burjassot,
Valencia
R. COHEN-ADAD
Laboratoire de Physico-Chimie Minerale II, Universite’ Claude Bernard (Lyon I), 43 Boulevard du 11 Novembre 1918, 69621 Villeurbanne (France) (Received
September
16, 1985)
Summary
The strongly associated regular solution model has been used to calculate the phase diagram of the Pb-Te system, and most of the thermodynamic quantities of the Pb-Te liquid alloys at 1200 K, for which experimental values were available in the literature. These are the molar partial and mixing enthalpies, entropies and Gibbs free energies. The dissociation equilibrium constant and the pair interaction parameters used in the calculation were derived from experimental values for both the molar partial enthalpies at infinite dilution and the enthalpy of mixing at 50 at.% Te. The Pb-Te phase diagram has been calculated using the aforementioned parameters, and compared with the data reported in the literature. Agreement is generally quite good between calculated and experimental results, although we assume a negligible solubility range for PbTe in the solid state.
1. Introduction The aim of the present work was to integrate in a unique model for liquid mixtures, both the experimental results on the phase diagram and the thermodynamic mixing functions of liquid solutions of the Pb-Te system. We utilize the strongly associated regular (SAR) solution model, described in ref. 2, to treat the liquid mixtures. The term regular solution is used here in a wide sense, following the nomenclature of Jordan’s work 0022-5088/86/$3.50
0 Elsevier Sequoia/Printed
in The Netherlands
278
[l]. It differs nevertheless from a strictly regular solution in two important ways: (i) there exists an excess entropy of mixing and (ii) the interaction energies vary linearly with temperature. We assume a negligible solubility range for the crystalline phases. The parameters of the model are estimated on the basis of the available partial enthalpies of solution at infinite dilution and the enthalpies of formation of both solid and liquid PbTe. Several studies on the thermodynamic properties of the Pb-Te system have been published. We begin by reviewing the values for the elements and the stoichiometric compound. For lead the values required for the present calculation have been collected in ref. 3 and are as follows: entropy of formation S&x = (15.49 + 0.1) X 4.184 J K-i mol-I; melting ~mperature T, = 600.6 K; enthalpy of fusion A.E& = 1140 X 4.184 kJ mol-‘“; specific heat of solid and liquid state C,’ = (5.629 + 2.328 X lO”3 7’) X 4.184 J K-i mol-f and Cp’= (7.765 0.738 X low3 2’) X 4.184 J K-l mol-* respectively. For tellurium the quantities needed for the present calculation have been collected in ref. 4 and are as follows: S&,x = (11.83 + 0.05) X 4.184 J K-l mol-‘; T, = 722.7 K; AH, = 4.18 X 4.184 J K-’ mol-‘; C,’ = (4.57 + 5.28 X 10m3 2’) X 4.184 J K-l mol-’ ; C,’ = 9.0 X 4.184 J K-l mol-“. For PbTe the solid solubility range is 0.499 - 0.501 at.% Te [43. Other values collected in ref. 4 are as follows: enth~py of formation AH: 298K = --8.2 X 4.184 kJ (g atom)-*; S&sx = (13.15 f 0.3) X 4.184 J K-i (g atom)-i; zn = 1197 K; C,’ = (5.64 + 1.345 X low3 2’) X 4.184 J K-l (g atom)-‘. Some discrepancies appear in the published values for the enthalpy of melting. Steininger [S] proposes a value of 9.4 X 4.184 kJ mol-’ whereas Blachnik and Kluge [63 take 13.7 X 4.184 kJ mol-’ and Shamsuddin et al. [7] obtain 9.9 X 4.184 kJ mol- I. As will be explained later we propose a value of 11.2 X 4.184 kJ mol-‘. The enthalpy of formation has been measured at several temperatures. McAteer and Seltz [ 8] obtained AH; 655K = (-8.44 f 0.20) X 4.184 kJ (g atom)-‘. Castanet et al. [9] found AH: ,s7x = -47.20 f 0.20 kJ (g atom)-“. Furthermore these authors gave a tentative relation for the temperature dependence of that quantity with reference to both pure solid elements, namely AH; T = -32.9 - 4.3 X 10U3 7’ kJ (g atom)-‘. Some the~odynami~ properties of liquid mixtures have also been determined at several ~mperat~es. The activity of tellurium at 1277 K in tellurium~rich mixtures has been measured by Predel et al. [lo] using a comparative method of partial vapour pressure measurements. The integral enthalpy of mixing was measured calorimetrically at 737 K by Castanet et al. [9] and at 1210 K by Blachnik and Gather [ll]. The activity of lead in liquid mixtures was determined by e.m.f. measurements over the whole composition range and between 653 and 1273 K by Moniri and Petot 1121. The results of calorimetric measurements of the partial molar enthalpy values at infinite dilution obtained at several temperatures by different authors and the mixing enthalpy at the stoichiometric composition are reported in Table 1.
279 TABLE
1
Heats of mixing liquid elements
of PbTe
and partial
AHM (50 at.% Te) (kJ (g atom)-‘)
T W) 737 1210 1200 879
-26.0 -26.1
f 0.8 + 0.6
enthalpies
Pm &Pb (kJ (g atom)-‘)
at infinite
dilution
with reference
-00
to the
Reference
AH,,
(kJ (g atom)-‘)
-69.0 -73.8
2 0.5 * 3.8
-34.3 -36.9
f 0.3 * 1.3
-59.9
+ 1.1
--36.2
+ 0.6
9 11 12 12
Given the enthalpy, entropy and Gibbs free energy of formation of PbTe at 298 K and the specific heats of lead, tellurium and PbTe below 1197 K, one can calculate the respective formation quantities of solid PbTe at 1197 K with reference to the pure liquid elements. The results are kJ (g atom))‘, AS; 1197K = -23.00 J K-’ (g atom))’ AH; 1197K = -48.92 -21.38 kJ (g atom))‘. Furthermore we know that at 1200 and AG;, 1197K = K the enthalpy and entropy of mixing at the stoichiometric composition are &?&,c x = -26.0 kJ (g atom))’ and AS&sK = -2.6 J K-l (g atom))’ (from ref. 12). We therefore expect an enthalpy of fusion, of roughly 23.5 kJ (g atom))’ (or 11.2 kcal mol-‘) and we will use that value, when necessary, for the calculation. A relatively large number of studies on the phase diagram of the Pb-Te system have been reported. Hansen [13] collected the measurements of Fay and Gillson [ 141, Pelabon [ 151 and Kimura [16]. Miller and Komarek [17] have studied the lead-rich part of the system, including the solid solubility limit of lead in the PbTe compound. The liquidus temperatures for small additions of tellurium to pure lead have been reported by Harris et al. [ 181. Moniri and Petot [ 191 have studied the complete phase diagram. Further studies on the tellurium-rich part of the system have been carried out by Kharif et al. [ 201.
2. The basic formulae for a strongly associated regular solution In this work the binary Pb, -xTe, is treated using the SAR solution model developed in ref. 2. The liquid solutions are assumed to be composed of lead, tellurium and PbTe species, whose mole fractions can be written in terms of the atomic fraction x of tellurium as xp,, = 1 -x
--XXpbTe
(1) XTe
=
x
-
t1
-
x)xPbTe
The species activity coefficients yi (j = Pb, Te, PbTe) are given by
280
(a) for 0 < x < 0.5
2a(l
RT In Tre =
- 2x) +P
l-x a(1 - 2x)2
RTln
YPbTe
=
t1 _xJ2
(b) for 0.5 < x < 1
@(23c-1)
RTlnypb=
RT In Yre =
P(1 -x)2
In %‘bTe
(2b)
X2 PeX
RT
+(y
x
x2
=
II2
where a! and p are the interchange energies for the interactions of the specific species Pb-PbTe and Te-PbTe respectively (both independent of composition), and where we take TBre = 1 at the stoichiometric composition at each temperature, i.e. the reference state of the species PbTe in the hypothetical state of complete association at the stoichiometric composition. The mole fractions of the different species are related through the dissociation equilibrium constant in the liquid by K=-
a+P
XPbxTe XPbTe
-
exp
RT
=
K’exp
a+P -
RT
(3)
where K’=
XPbXTe
-
XPbTe
is the dissociation equilibrium constant for the ideal associated solution (o = p = 0). The partial molar Gibbs free energy of the components lead and tellurium with reference to the corresponding pure liquid components are respectively (a) for 0 < x < 0.5 =Pb
=Te
=RTInxpb+
= RT ln xre +
O!X2
(1 -x)2
(54
24 1 - 2x) l-x
+P
281
(b) for 0.5 < x < 1
dc,b = RT
h xp,, +
X
(5b)
@(I .-x)2
AGTr= RTlnx,,+
x2 The Gibbs free energy of mixing referred mixture can be split into two parts
to 1 mol of the starting
AGM = AG’ + AG=
(6)
where AGi is the Gibbs free energy tion, namely A@ = RT{(i x (XPb
-x)
of mixing for an ideal associated
In xpb + x In xTe} =
In XPb
liquid
+ XTe
In XTe
+ XPbTe
solu-
RT 1 + XPbTe In XPbTe
+ XPbTe
In K’)
(7)
and the regular part AG’ is given by (a) for 0 < x < 0.5 AG’ = (cu + /3)x +
cux(1 - 2x) (8a)
l--X
(b) for 0.5 < x < 1 AC’ = ((u + /3)( 1 -x)
+
p(1 -x)(2x
- 1) (Sb)
X
The calculation of the other thermodynamic quantities is straightforward. The equations for the partial and mixing molar enthalpies and entropies given by the model are presented in Table 2. Furthermore, the partial molar enthaipies at infinite dilution are L’ = l+K’
lim AHp,, = m; X .* 1
L’
lim MT, x -*0
+ (a-AT)+
(9)
= dHFe = ~ + 2((u -AT) 1+ K’
where L’ = Ra(ln h?)/a(llT),
SAR solution,
the following
2(/3-M’)
+ (j3 - BT)
A = &xfaT and B = ap/aT. As K’ < 1 for a approximate formulae are also correct
+@-BT m,“,
=
R
@lnK, U/T)
(10) +CY-AT
because from eqn. (3), In K = In K’ + (CY+ (3)/RT.
2
quantities
vpb
ASM
= axpblaT
and
-(A
-R
=re
pb -RTypbIxpb
-Ax(l
YTe = axT,/aT.
+ B)x
-
-RT2
-R{(l -x) 2x)/(1 -x)
In xpb +x In xTe)-
+ (0 -BT)(l
+ (a -AT)
- A -
2B( 2x -
-(A+B)(1-x)-B(1-x)(2x-l)/x
+ XYT~/XT~)
h XTe - RTYT@/xT, - B(1 - x)~/.x’
In Xpb - RTypb/Xpb
-x)Ypb/xpb
-R
-R
BT)(2X
- x)‘/x’
+ 2@-
+(a-AT+~-BT)(1-x)+(~-BT)(l-xxf(2x-1)/X
-RTZyTe/xTe
-RT2ypb/xpb
0.5GxGl
+ xYdX&
RT{(l
{(I - x)YPt,kPb
2x)/( 1 - x)
%Te,
2x)/(1 -X)-B
- Ax2 I(1 - xl2
In XT~ - RTYT,./xT~ -2A(l
-RInx
=Pb
+(a--AT+P--BT)x+(ff--ATb(l--)/(l--x)
+ (0 - BT) + 2(a - AT)( 1 -
-RTZyTe/xT,
LWTe
AHM
+ (a -AT)xl/(l
’
of Pbl
-RTZypblxpb
-X,2
for SAR solutions
Iw,b
OGxGO.5
Some thermodynamic
TABLE
1 )/x
1 )/X
-
% N
283
3. Evaluation of the parameters of the model The values for the partial molar enthalpies at infinite dilution, and for the molar enthalpy of mixing at the stoichiometric composition provide relative limits for the interchange energies of interaction extrapolated to 0 K and for the temperature dependence of the enthalpy of formation of the associates, namely (x -AT, 0 - BT and Ra(ln K)/a(l/T). The restrictions on the values are as follows: -69
kJ (g atom)-l < AH& < -73.8
-34.3
kJ (g atom)-’
kJ (g atom) -’ < &%?,- < -36.9
AHM = f(K:‘)R
a(ln K) I=-26 W/V
kJ (g atom)-’
kJ (g atom)-’
where f(K’) is a slowly varying function of K and less than 0.5, Furthermore, the best fit to the experimental partial molar Gibbs free energy and entropy of lead at 1200 K within the limitations of the preceding requirements is obtained by using the values shown in Table 3. TABLE
3
Best computational results for the parameters used in calculation of the SAR solutions of liquid Pb-Te alloys at 1200 K. Calculated values of Ra(ln K)/8(1/2’), (Y and fl in kJ (g atom)--l, and A and B values in J K-’ (g atom)---’
0.005
-56.3
11.3
-2.76
-?.5
10.7
4. Results and discussion By use of the values in Table 3 the calculated partial molar enthafpies of each component at infinite dilution are zH,“,=
-71.6
kJ (g atom)-’
A”&L = -36.4
kJ (g atom)-’
and the mixing enthalpy at the sto~~hiometri~ composition is AHM = -26.0
kJ (g atom)-’
Figure 1 shows the general behaviour of the partial molar enthalpies of lead and tellurium calculated using the parameters given in Table 3, together with the experimental data for m,, given by Moniri and Pet& [12]. The agreement between calculated and experimental data is quite good except for x > 0.9 where the discrepancies are of the order of 10%. The mixing enthalpies, entropies and Gibbs free energies calculated from the model at 3200 K are plotted versus the composition in Fig. 2. The experimental values for AHM at 1210 K given by Blachnik and Gather [ll]
284
-75
0
.2
.4
Fltomic
-6 fraction
.8
~ 1
Te
Fig. 1. Partial molar enthalpies of lead and tellurium at 1200 K with reference to the pure components: -, calculated values from the equations of Table 2; x, experimental data from ref. 12. 3
5
t
5 f
a
0
-5
-‘25 0
.2 htomic
.4
.6 fraction
.8
1
Te
Fig. 2. Concentration dependence of the mixing enthalpy AH”, entropy ASM and Gibbs free energy AGM of liquid Pb-Te alloys: -, values calculated at 1200 K from the equations of Table 2 and eqns. (7) and (8); X, experimental data for A?IM at 1210 K from ref. 11.
are also shown in that figure. The concentration dependence of calculated AHM values exhibits a minimum at x = 0.521, with a value of -26.36 kJ (g atom)-‘, and the ASM curve has an indentation with a minimum at the
285
same concentration. Both facts are related to the strong association predicted by the model in the liquid alloy and are in agreement with the results reported in ref. 11, i.e. an experimental minimum for AHM at 52 at.% Te with a value of -26.25 rt 0.95 kJ (g atom))l. Figure 3 shows the calculated partial molar Gibbs free energies and the experimental ii?&, obtained by Moniri and Petot [CC]. The calculated curves are in good agreement with the experimental data to within the estimated uncertainty associated with the measurement, except for values of x around 0.5, where a significant discrepancy occurs. Otherwise further improvement, even if possible, would not be significant in a statistics sense. The calculated activities of lead and ~llurium in the Pb-Te liquid alloys at 1200 K are presented in Fig. 4 and compared with experiments data for apb given by Moniri and Petot [12] at the same temperature, and with those for &re given by Predel et al. [lo] at 1277 K. As observed for the partial molar free energy mph, the agreement is quite good for apb, the discrepancies being less than the experimental uncertainty except for x around 0.5 where the deviations are about 5%. Figure 5 shows the molar partial entropies of Pb-Te liquid alloys at 1200 K and the experimental ds, obtained by Moniri and Petot [ 12). The S-shaped form of both quantities is typical of strong association for x = 0.5. The calculated curves shown in Figs. 1 - 5 have been obtained using a unique set of six parameters specified in Table 3. As a consequence the Gibbs free energy of formation of 1 mol of the complex PbTe is given by AG;_ rr = RT in K = -56.3 + 10.0 X 1Cm3 T kJ mol-‘. To find out if the same set of parameters is able to describe the liquid mixtures in equilibrium with the solid phases we calculated the phase
80
20
0
0
.2
.4 F(tomic
-6 fraction
.8 Te
Fig. 3. Partial molar Gibbs free energies of lead and tellurium at 1200 K with reference to the pure components: -, calculated values from eqns. (5); X, experimental points from ref. 12.
286
1
.8
cn .6 .> .42 0 a .4
.2
Kl
0
0
.2
.4 Atomic
.6
.6
fraction
1
Te
Fig. 4. Concentration dependence of the calculated from eqns. (2); X, experimental data at 1277 K from ref. 10.
activities of lead and tellurium: data at 1200 K from ref. 12;A,
values experimental
20
-20
0
.2
.6
.4 atomic
.6
fraction
1
Te
of lead and tellurium at 1200 K with reference to the pure Fig. 5. Partial molar entropies liquid components: -, values calculated from the equations of Table 2 ; x , experimental data from ref. 12.
diagram of the Pb-Te system. For that purpose we used the following equations for the liquidus curve for equilibrium between liquid solution and solid PbTe compound (a) for 0 < x < 0.5 T=
AHPbTe + cx,(l m AS,~T” _
R
ln(%%Te/x
-
&T,)
2x)2/(1 -A(1
- x)2 -2x)*/(1-x)*
(lla)
287
1
0
.2
.4
fitantic
.6 fraction
.8
1
Te
Fig. 6. Phase diagram of the Pb-Te system. Calculated liquidus temperatures for the following values of h?: - - -, ti = 5 x 10--J; ---, I@ = 1 x 10m3. Experimental points: 0, from ref. 14; a, from ref. 16; X, from ref. 17; +, from ref. 18;Y, from ref. 19; *, from ref. 20.
(b) for 0.5 G x G 1 T=
AHPbTe m + &)(2x - 1)2/X2 _ AS”Te - R h(xpbTe/x&&) - B(2x m
1J2/x2
Wb)
For the liquidus curve for the equilibrium between the liquid solution and solid tellurium T=
Ali: AS?
+ &(l - X)2/X2
- R in &re --.&I
-x)~/x’
with czs= Q!- AT and PO= #3- BT the ~terch~ge energies extrapolated to 0 K, and where X&r,? is the value of Xr,,?reat the stoichiometric composition. Figure 6 compares the theory (for different values of K’) of binary liquidus with experimental results. The eutectic temperature and composition for the PbTe-Te subsystem are respectively 694 K and 0.911 at.% Te when K’ = 1 X 10s3. The experimental eutectic temperature obtained by Moniri and Petot [ 191 and shown in Fig. 6, is 687 K. The most outstanding features of the liquidus curves are the asymmetry with respect to the PbTe compound, and the sharp peak at the 50% composition. The theory of SAR solutions can actually describe the asymmetry because cyf 0. Furthermore, as I(’ is small, the association is large and the peak becomes sharp.
288
5. Conclusions This work is the first published application of the SAR solution model to binary systems to interpret both the thermodynamic mixing functions and the phase diagram (at constant pressure) using a unique set of six parameters basic to the model. In as much as there is no direct measurement of the chemical short-range order in Pb-Te liquid alloys, we have assumed the existence of PbTe complexes in equilibrium with lead and tellurium atoms. Nevertheless, the precise way in which these complexes are formed electronically as well as their lifetimes do not enter explicitly in our treatment so we may expect this treatment to be applicable also to metallic, covalent or even ionic liquid alloys. The results obtained justify the validity of the model, because general expressions for the molar partial Gibbs free energies for the Pb-Te liquid alloys have been obtained which are consistent with both the phase diagram and the thermochemical data. Two substantially different interaction parameters have been introduced in order to reproduce the asymmetries in both liquidus temperatures and thermochemical data. Furthermore these interaction parameters, although dependent on temperature, have values Ia/RTI, I/3/RTI < 1.2 at T = 1200 K, i.e. they are small enough for the regular approximation to be valid. The equilibrium constant K’ for ideal solutions has values lower than the upper limit needed to undertake strong associated treatment, namely K’ < 10-l [ 211. In general, equilibrium constants for ideal solutions depend on the liquid composition [22, 231, but this is not the case for SAR solutions [ 21. However, a small temperature dependence can be allowed because increasing temperature diminishes chemical short-range order.
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17 E. Miller and K. L. Komarek, Trans. Metall. Sec. AIME, 236 (1966) 832. 18 J. S. Harris, J. T. Longo, E. R. Gertner and J. E. Clark, J. Ctyst. Growth, 27 (1975) 334. 19 N. Moniri and C. Petot, J. Calor. Anal. Thermique, 9 195. N. Moniri, T&se docteur-ingknieur, Paris VI, 1978. 20 Ya. L. Kharif, P. V. Kovtunenko, A. A. Maier and I. Kh. Avetisov, Zh. Fiz. Khim., 56 21 22 23
(1982) 2180. J. Onrubia, Ph.D. Thesis, Universidad de Valencia, 1983. F. Sommer, 2. Metallkd., 72 (1982) 73, 77. S. Lele, G. V. S. Sastry and P. Ramachandrarao, 2. Metallkd.,
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