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The solvus entropy for metal-hydrogen systems
Scripta
METALLURGICA
Vol. 15, pp. 145-150, 1981 P r i n t e d in the U.S.A.
P e r g a m o n Press Ltd. All r i g h t s r e s e r v e d
THE SOLVUS ENTROPY FOR METAL-HYDROGEN SYSTEMS
Ted B. Flanagan Department of Chemistry University of Vermont Burlington, Vermont 05405 ( R e c e i v e d A u g u s t 13, ( R e v i s e d D e c e m b e r i,
1980) 1980)
Introduction Solvus determinations are a co...only employed technique for the characterization of metal-H systems (i-17). Usually plots of -in a against reciprocal temperature are made and from the slope of such plots solvus enthalpies are derived; ~ is the terminal hydrogen solubility in the metal phase expressed as the H-to-metal, atom ratio. The solvus enthalpy corresponds, in the limit as a ~ O, to the enthalpy change for the transfer of one mol of H from the hydride phase to the terminal hydrogen solution (1,18). Since the corresponding solvus entropies are generally not explicitly given, there have been no attempts to interpret them. The purpose of this coum~nication is two-fold. Firstly, it is important in view of the widespread use of solvus measurements in metal-H systems (1-17) to derive exact and approximate thermodynamic expressions for the solvus entropy. This has been done for the substitutional solid solution by Freedman and Nowick (19). There are, however, differences between the two types of solutions, e.g., the compositional variable is ~ (mol fraction of solute B) for the substitutional solution and r/8 for the interstltial solutlon, where r is the solute-to-metal, atom ratio and B is the number of interstices per metal atom. Secondly, it will be demonstrated that the (excess) solvus entropy can be readily evaluated from existing solvus data and that it is a useful parameter for the characterization of metal-H systems. Using solvus data for wellcharacterized matal-H systems~ it will be shown that the values of the excess solvus entropies are consistent with the thermodynamic development. Hopefully this parameter will be employed in future studies as an aid in the characterization of metal-H systems. Thermodynamics of the Solvus Entropy Generally the solvus enthalpy, AHsol, is derived from plots of -in ~ against T -I but we will instead use -ln Z where Z = a/(8-~) and 8 is the number of interstices per metal atom; as a ÷ 0, Z ÷ a/8. Following Flanagan and Oates (18), AHso I can be obtained from
where ~ H
= ~H - ½~H2 and the standard designation on V ° refers to unit fugacity. H2
From equation
1 we obtain
where b i s t h e H - t o - m e t a l , atom r a t i o
f o r t h e h y d r i d e phase c o e x i s t i n g w i t h ~ and ~ 2
+ ~ ÷ b
r e f e r s t o t h e f o r m a t i o n o f t h e h y d r i d e phase from H2 (1 atm) and t h e H - s a t u r a t e d m e t a l . the definition of A~H, it follows that
145 0036/9748/81/020145-06502.00/0 C o p y r i g h t (c) 1981 P e r g a m o n Press
Ltd.
From
146
SOLVUS
ENTROPY
Vol.
15,
(~(A~HIT)I~In Z) T = (~Rin P~2 l~la Z) T
No.
2
(3)
The denominator of equation 2 can be evaluated from solubility data if these are available. For very dilute solutions, a ÷ 0, the denominator + R, and the solvus enthalpy corresponds to the transfer of one mol of H from the hydride to the terminal hydrogen solution. The simplest model for interstitial solutions which leads to hydride formation is the regular interstitial solution; in this approx~m-tion the partial entropy is ideal but the partial enthalpy is not, i.e., VH = 4 +
RT in rlCB-rl
+~mrls
(41
where r is the H-to-metal, atom ratio at any H-content, ~Ho is the standard chemical potential, i.e., the value of ~H as r ÷ 0 without the configurational term, and ~ H enthalpy.
is the H-H interaction
It is useful to express AHso I in terms of the regular interstitial solution, RIS,
model because often pressure-composltlon data are not available for the evaluation of nonideallty via equation 3 at the low temperatures where solvus data are obtained. The RIS model leads to the following expression for AHso I from equation 2.
-R(din Z/dT-1)a
=
AHso1
=
-AH(~I 2 + a ÷ b) + AHH(at a) 1 + RHHa(B-a)/RTS 2
(51
From equation 4 for the RIS it is easy to show
AH(~I 2 + a ÷ bl = ~
+ HHH/2 and AHH = AI~ + HHH~/6
where A ~ i s defined by the r e l a t i o n s h i p A.~ = ~
(61
- TASk. Therefore for s ~ l l
~ values,
equation"5 reduces to AHsol = - ~
(7)
a simple result which has not been explicitly noted before. The solvus free energy vanishes at all temperatures because the process of transferring one mol of H from the hydride phase to the terminal interstitial solution is an equilibrium one. Since the solvus enthalpy is obtained from - ~ l n Z/dT -1, this suggests that -RT in Z is also a free energy associated with the solvus. This is an excess free energy change for the solvus. The solvus reaction is
MH6/(b'al ÷ MHa/Cb-a) + [H]at a
(8)
where [H]at a refers to one mol of H at the terminal hydrogen concentration and as a ÷ 0, reaction 8 reduces to the usual definition of the solvus reaction (1,18). is given by AGso I = -A~(~H 2 + a ÷ b) + A~H(at ~1 = 0
AGso I for reaction 8 (91
In order to obtain an expression for the excess solvus free energy change, AG~ol, ideal contributions must be removed from the two terms on the right-hand side of equation 9. interstitial solution has been defined as (201 id ( r ) = v ~ C p ° , V ° , r ~ 0) + RT i n r / ( B - r ) VH
The ideal
(10)
where pO = i arm and V ° is the H-free volume of metal at i arm. Therefore the ideal quantities corresponding to the terms on the rlght-hand-slde of equation 9 are: bib +
id .... d
r,0, -
id and
-
+
Removing these ideal terms from the appropriate terms of equation 9 we obtain
ln
Z
Vol.
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2
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147
E
id = -RT in Z AGsol = AGso I - AG sol
(11)
AS~o I can be obtained from AG~ol, giving -(~AG~ol/~T)p = AS~o I = Rln Z + RT(dln Z/dT)
(12)
(dln Z/dT) can be evaluated from an expression similar to equation i, i.e., (dAUH/dT)at a = (SAVH/~T)at a + (3AVH/~In Z)T(dln Z/dT)
(13)
and therefore using equations 12 and 13 we obtain ASEo Is
=
-AS(~H 2 + a ÷ b) + ASH(at a)
+ R1n
Z
(14)
R-I[3(APH/T)/~In Z] T Experimentally AS~o I can be evaluated from slopes of plots of in Z against T using equation 12 or alternatively from
(15)
AS~o I = AHsol/T + Rln Z
Now AHso I does not contain entroplc conflguratlonal contributions and consequently AHso I = AH~o I. Because of non-ideallty, represented by the denominators of equations 2 and 14, the excess quantities do not represent the transfer of one mol of H from the hydride to s hypothetical interstitial solid solution with el8 = 1/2 except in the limit of a + O. For this reason equations 2 and 14 could be rewritten as AH ° sol (16) AHso I = AH~o I = R-I[~(A~H/T)/~In Z] T
ASEo l+
AS;oI
=
+ Rln
Z
(17)
R-I[~(AUH/T)/~In Z] T and
AC~o I = AH~o I - TAS~o I ~ -RT in Z(as Z + 0)
(18)
where the standard solvus designation refers to the transfer of one mol of H to the hypothetical interstitial standard solution with a/8 = 1/2 in the limit as a ÷ 0. Because solubility data are often not available for the low temperatures where solvus data are determined, an approximate evaluation of equation 14 may be helpful and the RIS model gives AS E = sol i
-Rln Z
+ Rln Z
(19)
+KHHa(8-a)/RTB 2
and as a + O, AS~o I + 0. Another approximate approach is to evaluate the numerator of equation 14 by employing the o relation AS H = AS H - Rln Z and equation 20,
AS(~t,I2
+
+
+)
:
o _ A+ H
-
+
+ln
fs
-
b
(20)
b/B Equation 20 can be derived from AS(~H2+ a + b) = ~
a B ASHd(r/8) where AS H = AS H° -
Rln[r/(8-r)]
However, it is not assumed in equation 20 that (a + b)/8 = 1 as for a RIS model. Equation 20 works quite well for the Pd-H system for which the experimental value of AS(~2H2 + a + b) is -45.6 J(K mol H) -1 (21).
The RIS approximation predicts that AS(~H 2 + a * b) = As~ = -53.6
J(K mol H) -1 (21) and using equation 20, we find AS(~H + a + b) = -44.2 J(K mol H) -1 where values of a and b (273 K) have been taken from ref. 21.
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2
Excess Solvus Entropies for Representative Metal-H Systems In contrast to most metal-H systems, pressure-composltion-temperature data can be determined for Pd-H at quite low temperatures and, therefore, experimental values can be compared with values predicted from equation 14 using experimental data for evaluation of the non-ldeal term. The equilibrium hydrogen pressures are extremely small for the group VB-H systems even at 300 K and, therefore, the non-ldeality terms (the denominator of equation 14) are not available directly from low temperature pressure-composltion data via equation 3. For these cases the non-ldeallty most be estimated, e.g., as in equation 19. Experimental data for Pd-H(D) were determined by pressure-composltlon measurements at low temperatures where values of a are relatively small, 0.005 to 0.0005. The samples were annealed prior to use and the solvus data were determined during cooling (absorption of hydrogen). The solvus was located by the change in slopes of the pressure-temperature relationships at constant hydrogen contents. Values of A ~ , AS~, ~H(~H 2 + a ÷ b) and ~S(~H 2 + a ÷ b) and the corresponding quantities for deuterium are available for the same sample from previous work (22). For Pd-H(D) it is known that B = i, i.e., the octahedral interstices are occupied (23). A summary of the results are shown in Table I. The experimental solvus values were determined from 190 to 250 K and the reported values are the average of about five determinations in this temperature interval. TABLE I Solvus Data for Pd-H and Pd-D (190 to 250 K) (Enthalples are in units of kJ(mol H) -1 and entropies in units of 3 ( K m o l H) -1) Pd-H
Pd-D
Reference
AHso I
- 9.3
- 8.7
present work
AH(~H 2 + a ÷ b)
-18.6
-17.1
22
AS(½H 2 + a ~ b)
-45.9
-46.9
22
~ ° i ( 2 0 0 K)
-10.6
- 9.0
22
~(250
K)
-I0.0
- 8.7
22
AS~(200 K)
-55.0
-56.9
22
AS~(250 K)
-54.4
-56.8
22
AHItH
-46.0
-46.0
21
AS~oI(190-250 K), exp.
- 5.2
- 6.4
present work
AS~oI(190-250 K), calc. 1
- 4.3
-8.2 ± 2.0
present work
AS~oI(190-250 K), calc. 2
- 4.7
-5.2
present work
1Calculated from equation 14 using experimental values for ASC~H 9 + a ÷ b) and ASH(at a). The former values were taken from Table I and the l~tter from experimental data near the low temperature phase boundary a. The non-ldeal correction for the denominator of equation 14 was determined from experimental data using equation 3. 2Calculated from equations 14 and 20 using the hydride phase boundaries determined by Wicke and Nernst (21) at ~200 K and the terminal hydrogen solubilities determined here.
The v a l u e s f o r d e u t e r i u m a r e more a c c u r a t e t h a n t h o s e f o r hydrogen b e c a u s e t h e e q u i l i b r i u m pressures of deuterium are greater at a given content and temperature. It is clear that the values of ASEols are negative.
This arises in part from the dependence of 0H, the Einstein
Vol.
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SOLVUS
ENTROPY
149
temperature, upon the H-content; 8w changes from 795 to 661 K in passing from the dilute solution to the hydride phase (24)~ This leads t O a n entropy contribution of -1.7 J(K mol H) -I which has bean included in all of the calculated values. The calculated values shown in the last row of Table I are probably the best and the agreement with experiment is good. The major contribution to ASEo Io
arises from the fact that ~ + b ~ 8 = 1 for the real system, i.e., the
phase diagram is unsymmetrical for Pd-H. zero as predicted by the RIS model.
Values of AS~o 1o
are relatively small (Table I) but not
Pressure-composltlon data are not available for group VB metal-H systems at the temperatures where most solvus data have been determined.
Nonetheless experlmentaI~values of AS~o I can be
determined from equation 15 using+ when available, equations given by the authors to express
their solvus data and compared to values calculated from equation 14 using AS(~H2+ a ÷ b) values from an equation similar to 20 but for V-H and Ta-H we have assumed that the hydride phases are fully ordered and their conflguratlonal entropies are, therefore, zero. Thus for these systems only the contribution from the termlnal hydrogen solution is included in equation 20 for the evaluation of AS(~H 2 + a + b). Non-zero values of
AS2oI
for real systems can arise from the following factors:
(i) the
differing vibrational characteristics of H between the dilute and hydride phases, (il) nonideallty, (ill) unsymmetrical phase diagrams, (iv) non-ideal partial conflguratlonal entropies, (v) ordered hydride phases. Items (1), (li), and (ill) have been allowed for in the calculated values of AS~o I~
for Pd-H and the agreement with experiment is good; (ill) is the most important
factor giving rise to the negative values of
~S2oI.
For the group VB-H systems the situation
is more complicated due partly to the experimental scatter of the solvus values reported in the literature (Table II). After allowance for non-ideallty in an approximate way, the most important factor determining the sign of AS~o I appears to be the fact that ordered hydride phases are formed for V and Ta. This makes the term -AS(~H~ + ~ ÷ ~) + AS H in equation 14 more positive than for when a disordered hydride phase fo~ms. It seems safe to say that the positive values of AS~o I_
for V and Ta are associated with formation of ordered hydride phases.
Nb-H is
an intermediate case since the hydride phase is ordered but does not correspond to the stolchlometry of the hydride phase which forms so that some disorder must obtain. For this reason we I~ have not attempted to calculate values of dS~o more exact values of dS~o I~
for N-b-H.
It should be possible to calculate
for these systems when more precise thermodynamic data are available.
TABLE II Solvus Values for Group VB-H Systems AS~ol/J(K mol H) -I, exp. Nb-H(3) Nb-H(II) V-H(8)
V-H(16) Ta-H(12) Ta-H(17)
-3.9 -0.5 4.7
3.2 5.1 16.8
AS~ol/J(K mol H) -I, calc. -
-
5.7 3.7 -
In conclusion we have attempted to show that dS~o I can be as useful an experimental thermodynamic parameter as AHso I although it has not been generally evaluated for metal-H systems. using best characterized systems available we have shown that the experimental values of AS~o 1 are in general agreement with values expected from our thermodynamic analysis and the known properties of these systems. It should now be possible to employ this parameter as an aid in the characterization of other metal-H and alloy-H systems.
By
150
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15, No.
2
Acknowledgements The N.S.F. is thanked for financial support of this research. the referee for a series of useful suggestions.
The author wishes to thank
References 1. 2. 3. 4. 5. 6. 7. 8. 9. i0. Ii. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
D. T. Peterson and D. G. Westlake, Trans. AIME 215, 444 (1959). D. G. Westlake, Trans. AIME 239, 1341 (1967). D. G. Westlake, Trans. AIME 245, 287 (1969). O. Buck, D. O. Thompson, and C. A. Weft, J. Phys. Chem. Solids 32, 2331 (1971). N. E. Paton, B. S. Hickman, and D. H. Leslie, Met. Trans. 2, 2791 (1971). K. Faber and H. Schultz, Scripta Met. 6, 1065 (1972). D. G. Westlake and S. T. Ockers, Met. Trans. 4, 1355 (1973). H. Y. Chan 8 and C. A. Weft, Acta Met. 21, 1233 (1973). D. G. Westlake and S. T. Ockers, Met. Trans. 6A, 399 (1975). E. Heller and H. Wipf, Phys. Stat. Sol. (a) 33, 525 (1976). J. L. Whltton, J. B. M/tchell, T. Schober, and H. Wenzl, Acta Met. 24, 483 (1976). K. Rosan and H. Wipf, Phys. Star. Sol. (a) 38, 611 (1976). G. Pfeiffer and H. Wipf, J. Phys. F: Metal Phys. 6, 167 (1976). H. K. Birnbaum, M. L. Grossbeck, and M. Amano, J. Less Common Metals 49, 357 (1976). D. G. Westlake, Scripta Met. 10, 75 (1976). T. Schober and A. Carl, Phys. Stat. Sol. (a) 43, 443 (1977). T. Schober and A. Carl, Scripta Met. 11, 397 (1977). T. B. Flanagan and W. A. Oates, Scrlpta Met. 12, 873 (1978). J. E. Freedman and A. S. Nowlck, Acta Met. 5, 176 (1958). W. A. Oates and T. B. Flanagan, Scrlpta Met. 12, 759 (1978). E. Wicke and G. Nernst, Bet. Bunsenges Physlk Chem. 68, 224 (1964). J. D. Clewley, T. Curran, T. B. Flanagan, and W. A. Oates, J.C.S. Faraday Trans. I 69, 449 (1973). J. E. Worsham, M. G. Wilkinson, and C. G. Shu11, J. Phys. Chem. Solids 3, 303 (1957). W. Drexel, A. Murianai, D. Tocchettl, W. Kley, I. Sosnowska, and D. K. Ross, 3. Phys. Chem. Solids 37, 1135 (1976). T. Schober and H. Wenzl, in "Hydrogen in Metals," Vol. II, G. Alefeld and J. VSlkl, Sprlnger-Verlag, Berlin, 1978, Chap. 2.
METALLURGICA
Vol. 15, pp. 145-150, 1981 P r i n t e d in the U.S.A.
P e r g a m o n Press Ltd. All r i g h t s r e s e r v e d
THE SOLVUS ENTROPY FOR METAL-HYDROGEN SYSTEMS
Ted B. Flanagan Department of Chemistry University of Vermont Burlington, Vermont 05405 ( R e c e i v e d A u g u s t 13, ( R e v i s e d D e c e m b e r i,
1980) 1980)
Introduction Solvus determinations are a co...only employed technique for the characterization of metal-H systems (i-17). Usually plots of -in a against reciprocal temperature are made and from the slope of such plots solvus enthalpies are derived; ~ is the terminal hydrogen solubility in the metal phase expressed as the H-to-metal, atom ratio. The solvus enthalpy corresponds, in the limit as a ~ O, to the enthalpy change for the transfer of one mol of H from the hydride phase to the terminal hydrogen solution (1,18). Since the corresponding solvus entropies are generally not explicitly given, there have been no attempts to interpret them. The purpose of this coum~nication is two-fold. Firstly, it is important in view of the widespread use of solvus measurements in metal-H systems (1-17) to derive exact and approximate thermodynamic expressions for the solvus entropy. This has been done for the substitutional solid solution by Freedman and Nowick (19). There are, however, differences between the two types of solutions, e.g., the compositional variable is ~ (mol fraction of solute B) for the substitutional solution and r/8 for the interstltial solutlon, where r is the solute-to-metal, atom ratio and B is the number of interstices per metal atom. Secondly, it will be demonstrated that the (excess) solvus entropy can be readily evaluated from existing solvus data and that it is a useful parameter for the characterization of metal-H systems. Using solvus data for wellcharacterized matal-H systems~ it will be shown that the values of the excess solvus entropies are consistent with the thermodynamic development. Hopefully this parameter will be employed in future studies as an aid in the characterization of metal-H systems. Thermodynamics of the Solvus Entropy Generally the solvus enthalpy, AHsol, is derived from plots of -in ~ against T -I but we will instead use -ln Z where Z = a/(8-~) and 8 is the number of interstices per metal atom; as a ÷ 0, Z ÷ a/8. Following Flanagan and Oates (18), AHso I can be obtained from
where ~ H
= ~H - ½~H2 and the standard designation on V ° refers to unit fugacity. H2
From equation
1 we obtain
where b i s t h e H - t o - m e t a l , atom r a t i o
f o r t h e h y d r i d e phase c o e x i s t i n g w i t h ~ and ~ 2
+ ~ ÷ b
r e f e r s t o t h e f o r m a t i o n o f t h e h y d r i d e phase from H2 (1 atm) and t h e H - s a t u r a t e d m e t a l . the definition of A~H, it follows that
145 0036/9748/81/020145-06502.00/0 C o p y r i g h t (c) 1981 P e r g a m o n Press
Ltd.
From
146
SOLVUS
ENTROPY
Vol.
15,
(~(A~HIT)I~In Z) T = (~Rin P~2 l~la Z) T
No.
2
(3)
The denominator of equation 2 can be evaluated from solubility data if these are available. For very dilute solutions, a ÷ 0, the denominator + R, and the solvus enthalpy corresponds to the transfer of one mol of H from the hydride to the terminal hydrogen solution. The simplest model for interstitial solutions which leads to hydride formation is the regular interstitial solution; in this approx~m-tion the partial entropy is ideal but the partial enthalpy is not, i.e., VH = 4 +
RT in rlCB-rl
+~mrls
(41
where r is the H-to-metal, atom ratio at any H-content, ~Ho is the standard chemical potential, i.e., the value of ~H as r ÷ 0 without the configurational term, and ~ H enthalpy.
is the H-H interaction
It is useful to express AHso I in terms of the regular interstitial solution, RIS,
model because often pressure-composltlon data are not available for the evaluation of nonideallty via equation 3 at the low temperatures where solvus data are obtained. The RIS model leads to the following expression for AHso I from equation 2.
-R(din Z/dT-1)a
=
AHso1
=
-AH(~I 2 + a ÷ b) + AHH(at a) 1 + RHHa(B-a)/RTS 2
(51
From equation 4 for the RIS it is easy to show
AH(~I 2 + a ÷ bl = ~
+ HHH/2 and AHH = AI~ + HHH~/6
where A ~ i s defined by the r e l a t i o n s h i p A.~ = ~
(61
- TASk. Therefore for s ~ l l
~ values,
equation"5 reduces to AHsol = - ~
(7)
a simple result which has not been explicitly noted before. The solvus free energy vanishes at all temperatures because the process of transferring one mol of H from the hydride phase to the terminal interstitial solution is an equilibrium one. Since the solvus enthalpy is obtained from - ~ l n Z/dT -1, this suggests that -RT in Z is also a free energy associated with the solvus. This is an excess free energy change for the solvus. The solvus reaction is
MH6/(b'al ÷ MHa/Cb-a) + [H]at a
(8)
where [H]at a refers to one mol of H at the terminal hydrogen concentration and as a ÷ 0, reaction 8 reduces to the usual definition of the solvus reaction (1,18). is given by AGso I = -A~(~H 2 + a ÷ b) + A~H(at ~1 = 0
AGso I for reaction 8 (91
In order to obtain an expression for the excess solvus free energy change, AG~ol, ideal contributions must be removed from the two terms on the right-hand side of equation 9. interstitial solution has been defined as (201 id ( r ) = v ~ C p ° , V ° , r ~ 0) + RT i n r / ( B - r ) VH
The ideal
(10)
where pO = i arm and V ° is the H-free volume of metal at i arm. Therefore the ideal quantities corresponding to the terms on the rlght-hand-slde of equation 9 are: bib +
id .... d
r,0, -
id and
-
+
Removing these ideal terms from the appropriate terms of equation 9 we obtain
ln
Z
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147
E
id = -RT in Z AGsol = AGso I - AG sol
(11)
AS~o I can be obtained from AG~ol, giving -(~AG~ol/~T)p = AS~o I = Rln Z + RT(dln Z/dT)
(12)
(dln Z/dT) can be evaluated from an expression similar to equation i, i.e., (dAUH/dT)at a = (SAVH/~T)at a + (3AVH/~In Z)T(dln Z/dT)
(13)
and therefore using equations 12 and 13 we obtain ASEo Is
=
-AS(~H 2 + a ÷ b) + ASH(at a)
+ R1n
Z
(14)
R-I[3(APH/T)/~In Z] T Experimentally AS~o I can be evaluated from slopes of plots of in Z against T using equation 12 or alternatively from
(15)
AS~o I = AHsol/T + Rln Z
Now AHso I does not contain entroplc conflguratlonal contributions and consequently AHso I = AH~o I. Because of non-ideallty, represented by the denominators of equations 2 and 14, the excess quantities do not represent the transfer of one mol of H from the hydride to s hypothetical interstitial solid solution with el8 = 1/2 except in the limit of a + O. For this reason equations 2 and 14 could be rewritten as AH ° sol (16) AHso I = AH~o I = R-I[~(A~H/T)/~In Z] T
ASEo l+
AS;oI
=
+ Rln
Z
(17)
R-I[~(AUH/T)/~In Z] T and
AC~o I = AH~o I - TAS~o I ~ -RT in Z(as Z + 0)
(18)
where the standard solvus designation refers to the transfer of one mol of H to the hypothetical interstitial standard solution with a/8 = 1/2 in the limit as a ÷ 0. Because solubility data are often not available for the low temperatures where solvus data are determined, an approximate evaluation of equation 14 may be helpful and the RIS model gives AS E = sol i
-Rln Z
+ Rln Z
(19)
+KHHa(8-a)/RTB 2
and as a + O, AS~o I + 0. Another approximate approach is to evaluate the numerator of equation 14 by employing the o relation AS H = AS H - Rln Z and equation 20,
AS(~t,I2
+
+
+)
:
o _ A+ H
-
+
+ln
fs
-
b
(20)
b/B Equation 20 can be derived from AS(~H2+ a + b) = ~
a B ASHd(r/8) where AS H = AS H° -
Rln[r/(8-r)]
However, it is not assumed in equation 20 that (a + b)/8 = 1 as for a RIS model. Equation 20 works quite well for the Pd-H system for which the experimental value of AS(~2H2 + a + b) is -45.6 J(K mol H) -1 (21).
The RIS approximation predicts that AS(~H 2 + a * b) = As~ = -53.6
J(K mol H) -1 (21) and using equation 20, we find AS(~H + a + b) = -44.2 J(K mol H) -1 where values of a and b (273 K) have been taken from ref. 21.
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Excess Solvus Entropies for Representative Metal-H Systems In contrast to most metal-H systems, pressure-composltion-temperature data can be determined for Pd-H at quite low temperatures and, therefore, experimental values can be compared with values predicted from equation 14 using experimental data for evaluation of the non-ldeal term. The equilibrium hydrogen pressures are extremely small for the group VB-H systems even at 300 K and, therefore, the non-ldeality terms (the denominator of equation 14) are not available directly from low temperature pressure-composltion data via equation 3. For these cases the non-ldeallty most be estimated, e.g., as in equation 19. Experimental data for Pd-H(D) were determined by pressure-composltlon measurements at low temperatures where values of a are relatively small, 0.005 to 0.0005. The samples were annealed prior to use and the solvus data were determined during cooling (absorption of hydrogen). The solvus was located by the change in slopes of the pressure-temperature relationships at constant hydrogen contents. Values of A ~ , AS~, ~H(~H 2 + a ÷ b) and ~S(~H 2 + a ÷ b) and the corresponding quantities for deuterium are available for the same sample from previous work (22). For Pd-H(D) it is known that B = i, i.e., the octahedral interstices are occupied (23). A summary of the results are shown in Table I. The experimental solvus values were determined from 190 to 250 K and the reported values are the average of about five determinations in this temperature interval. TABLE I Solvus Data for Pd-H and Pd-D (190 to 250 K) (Enthalples are in units of kJ(mol H) -1 and entropies in units of 3 ( K m o l H) -1) Pd-H
Pd-D
Reference
AHso I
- 9.3
- 8.7
present work
AH(~H 2 + a ÷ b)
-18.6
-17.1
22
AS(½H 2 + a ~ b)
-45.9
-46.9
22
~ ° i ( 2 0 0 K)
-10.6
- 9.0
22
~(250
K)
-I0.0
- 8.7
22
AS~(200 K)
-55.0
-56.9
22
AS~(250 K)
-54.4
-56.8
22
AHItH
-46.0
-46.0
21
AS~oI(190-250 K), exp.
- 5.2
- 6.4
present work
AS~oI(190-250 K), calc. 1
- 4.3
-8.2 ± 2.0
present work
AS~oI(190-250 K), calc. 2
- 4.7
-5.2
present work
1Calculated from equation 14 using experimental values for ASC~H 9 + a ÷ b) and ASH(at a). The former values were taken from Table I and the l~tter from experimental data near the low temperature phase boundary a. The non-ldeal correction for the denominator of equation 14 was determined from experimental data using equation 3. 2Calculated from equations 14 and 20 using the hydride phase boundaries determined by Wicke and Nernst (21) at ~200 K and the terminal hydrogen solubilities determined here.
The v a l u e s f o r d e u t e r i u m a r e more a c c u r a t e t h a n t h o s e f o r hydrogen b e c a u s e t h e e q u i l i b r i u m pressures of deuterium are greater at a given content and temperature. It is clear that the values of ASEols are negative.
This arises in part from the dependence of 0H, the Einstein
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temperature, upon the H-content; 8w changes from 795 to 661 K in passing from the dilute solution to the hydride phase (24)~ This leads t O a n entropy contribution of -1.7 J(K mol H) -I which has bean included in all of the calculated values. The calculated values shown in the last row of Table I are probably the best and the agreement with experiment is good. The major contribution to ASEo Io
arises from the fact that ~ + b ~ 8 = 1 for the real system, i.e., the
phase diagram is unsymmetrical for Pd-H. zero as predicted by the RIS model.
Values of AS~o 1o
are relatively small (Table I) but not
Pressure-composltlon data are not available for group VB metal-H systems at the temperatures where most solvus data have been determined.
Nonetheless experlmentaI~values of AS~o I can be
determined from equation 15 using+ when available, equations given by the authors to express
their solvus data and compared to values calculated from equation 14 using AS(~H2+ a ÷ b) values from an equation similar to 20 but for V-H and Ta-H we have assumed that the hydride phases are fully ordered and their conflguratlonal entropies are, therefore, zero. Thus for these systems only the contribution from the termlnal hydrogen solution is included in equation 20 for the evaluation of AS(~H 2 + a + b). Non-zero values of
AS2oI
for real systems can arise from the following factors:
(i) the
differing vibrational characteristics of H between the dilute and hydride phases, (il) nonideallty, (ill) unsymmetrical phase diagrams, (iv) non-ideal partial conflguratlonal entropies, (v) ordered hydride phases. Items (1), (li), and (ill) have been allowed for in the calculated values of AS~o I~
for Pd-H and the agreement with experiment is good; (ill) is the most important
factor giving rise to the negative values of
~S2oI.
For the group VB-H systems the situation
is more complicated due partly to the experimental scatter of the solvus values reported in the literature (Table II). After allowance for non-ideallty in an approximate way, the most important factor determining the sign of AS~o I appears to be the fact that ordered hydride phases are formed for V and Ta. This makes the term -AS(~H~ + ~ ÷ ~) + AS H in equation 14 more positive than for when a disordered hydride phase fo~ms. It seems safe to say that the positive values of AS~o I_
for V and Ta are associated with formation of ordered hydride phases.
Nb-H is
an intermediate case since the hydride phase is ordered but does not correspond to the stolchlometry of the hydride phase which forms so that some disorder must obtain. For this reason we I~ have not attempted to calculate values of dS~o more exact values of dS~o I~
for N-b-H.
It should be possible to calculate
for these systems when more precise thermodynamic data are available.
TABLE II Solvus Values for Group VB-H Systems AS~ol/J(K mol H) -I, exp. Nb-H(3) Nb-H(II) V-H(8)
V-H(16) Ta-H(12) Ta-H(17)
-3.9 -0.5 4.7
3.2 5.1 16.8
AS~ol/J(K mol H) -I, calc. -
-
5.7 3.7 -
In conclusion we have attempted to show that dS~o I can be as useful an experimental thermodynamic parameter as AHso I although it has not been generally evaluated for metal-H systems. using best characterized systems available we have shown that the experimental values of AS~o 1 are in general agreement with values expected from our thermodynamic analysis and the known properties of these systems. It should now be possible to employ this parameter as an aid in the characterization of other metal-H and alloy-H systems.
By
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Acknowledgements The N.S.F. is thanked for financial support of this research. the referee for a series of useful suggestions.
The author wishes to thank
References 1. 2. 3. 4. 5. 6. 7. 8. 9. i0. Ii. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
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