K. ALLARD, A. MAELAND, J. W, SIMONS,AND T. B. FLANAGAN
136
r*
T U V
Vf
W
a Y
Length parameter in the Mie potential Absolute tempekature Mie potential function Volume per molecule Free volume per molecule Molecular mass Communal entropy Numerical constant that depends on the geometry of the lattice
€
P Q
4 (r) 4(O)N/2 4*(O)N/2 W
Energy parameter in the Mie potential Number density Rigid-sphere diameter Interaction energy of a molecule with its neighbors Lattice energy (one shell) Lattice energy (all shells) Experimentally determined constant equal to 1/C
Application of the Electron-Donation Model for Hydrogen Absorption to Palladium-Rich Alloys.
Hydrogen-Gold-Palladium
by K. Allard, A. Maeland, J. W. Simons, and Ted B. Flanagan Chemistry Department, University of Vermont, Burlington, Vermont 06401 (Received May $1, 1067)
Detailed data for absorption of hydrogen by a series of gold-palladium alloys have been obtained in the low-content a phase. Heats of absorption at infinite dilution determined from the extrapolation of the isosteric heats are: 5980 (cal/mole of H2) (5.7), 7000 (15.3), 7540 (lS.S), 9040 (26.5), and 9340 (44.7), where the number in parentheses refers to the atom per cent of gold in the alloy. Entropies of absorption have been obtained and are compared to values calculated from a model of localized protons treated as three-dimensional oscillators. Results are interpreted in terms of a model where both gold and hydrogen are assumed to donate electrons to the empty combined s and d bands of the palladium.
Introduction The absorption of hydrogen by palladium-rich alloys is of interest from several points of view. Such investigations serve as useful tests of the electrondonation model for proton absorption proposed by Mott’ for pure palladium-hydrogen and extended to a statistical mechanical model by Lacher.2 It may also prove to be feasible to probe the band shape of palladium and its alloys by utilizing the data on hydrogen absorption. Finally, it may be pointed out that palladium alloys have frequently been utilized to show the influence of the electronic band of metals upon heterogeneous reactions.8 It appears to the present authors that the role of the electronic band structure of metals in influencing reactions of chemical interest should be probed with absorption systems such as those described here before the more complex problem of the correlation of electronic structure with surface catalysis is attempted. The main reasons for this suggested sequence The Journal of PhyaiCal Chemistry
is that absorption data can be treated using bulk metallic properties, whereas heterogeneous catalytic data must be interpreted using more poorly characterized surface properties. In addition, these absorption systems are not as subject to the experimental problems associated with surfaces, e.g., irreproducibility arising from contamination. Although there have been data available on hydrogen absorption by palladium alloys, aside from the early work of Sieverts at elevated temperature^,^ the majority of the data at temperatures below approximately 120” (1) N. F. Mott and H. Jones, “Theory of Metals and Alloys,” Clarendon Press, Oxford, 1936; N. F. Mott, Advan. Phys., 13, 326 (1964). (2) J. R. Lacher, Proc. Roy. Soc. (London), A161, 525 (1937). (3) E.Q.,G.Bond, “Catalysis by Metals,” Academic Press, London, 1962. (4) Sieverts’ extensive contributions are reviewed by F. A. Lewis, “The Palladium Hydrogen System,” Academic Press, London, 1967.
APPLICATION OF THE ELECTRON-DONATION MODELTO PALLADIUM-RICH ALLOYB have been determined using electrochemical techn i q u e ~ . ~This , ~ research has established the general features of the absorption behavior, such as the free energy and enthalpy changes corresponding to the a --t 6 phase transformation and the final hydrogen content at equilibrium (25O, 1 atm). The present investigation represents the first attempt to determine electrochemically equilibrium data of a series of palladium-rich alloys using the &--He dilution technique developed by Simons and Flanagan6 for the a phase of hydrogen-palladium. An important exception to the above is the hydrogentransfer catalyst method utilized by Wicke, Brodowsky, and their co-workers’ t o determine hydrogen absorption behavior near room temperature. Aside from constituting an important experimental advance in this area of research, their analysis of their data also represents an important contribution. Specifically, these workers have investigated rather thoroughly hydrogen absorption by silver-palladium’b and have done some research of a more preliminary nature on nickelpalladium,8 rYnodium-palladium,8 tin-palladium,8 and lead-palladiurn alloys.6 The gold-palladium alloy system behaves similarly to the silver-palladium system with respect to hydrogen absorption, in that the absorption of hydrogen into the two-phase region becomes more exothermic as gold is addedQ despite the fact that neither gold nor silver absorb hydrogen in their pure states. This present work reports detailed isothermal data of absorption of hydrogen by the gold-palladium alloy system in the low hydrogen content a phase, where problems associated with hysteresis are usually absent, and its interpretation in terms of the extended statistical mechanical model following a treatment of the absorption problem for pure palladium.6~10
137
has been observed experimentally.12 With reference to hydrogen absorption, it has been shown that the addition of tin or lead to palladium is about four times as effective as hydrogen or silver in the filling of d bands.* There are still difficulties present in the model, l 3 but the above observations constitute strong evidence in its favor. A successful interpretation of the data obtained for palladium alloys in terms of this model will offer further evidence in its favor. Wicke and Nernst12* have employed an approach based on an earlier treatment by Wagner based on eq 1
RT In pHal”
= pH =
where n is H/Pd (or H/metal for alloys), and Brodowsky14 has further divided ApH(n) into A p ~ + ( n ) Ape-(%),where A p ~ + ( ncorresponds ) to the H-H interaction term and Ape-(n)corresponds to electronic terms due to band filling. From plots of RT In p”’(1 - n ) / n us. n (at small n), A p o ~and A p ~ ( n )are obtained. Brodowsky argues that the initial dependence of Ape-(n) upon n is small and will be neglected. This allows A p ~ + ( nto) be evaluated from the quasi-chemical approximation; i e . , the initial slope of A p ~ + ( n )iS 12RT[1 - e ( - W / R T ) ] where Wis the apparent pair interaction energy in the lattice, which arises from strain and consequent aggregation of interstitial protons. 14,15 Since A p ~ + ( nis) now known, Ape-(n) can be evaluated experimentally. Essentially the same treatment is employed by these workers for hydrogen absorption by palladium-rich alloys, e.g., ref 7b. An important new finding arises from their study of the silver-palladium alloys, namely, that the heat of absorption at infinite dilution (n + 0) becomes more exothermic as the silver content increases. This trend may be inferred from
+
Electron-Donation Models of Proton Absorption
It is useful t o discuss and develop the theories of hydrogen absorption in order to have the necessary equations for the interpretation of the data of the present study. The authors favor the model in which the hydrogen is absorbed as protons and its electrons are donated to the empty combined s-d bands of the metal.’ (In the metal, the protons must be screened by electrons from the metallic bands; Mott has suggested that d-band electrons pile up t o screen the protons.) It appears to us that the evidence in favor of this model for palladium and its alloys is rather convincing. Mackleit and Schindler” recently have shown directly from low-temperature electronic heat-capacity measurements that the density of states in the d band of palladium decreases with absorption of hydrogen. The present authorslo have shown that if the effect due to the filling of the empty s-d bands of palladium is included in the Lacher model, it is predicted that the heat of absorption decreases at high hydrogen contents, as
(5) E.g., F. A. Lewis, Platinum Metals Rev., 4, 132 (1960); 5, 21 (1961). (6) J. W. Simons and T. B. Flanagan, J . Phys. Chem., 69, 3773 (1965). (7) See, for example: (a) Techn. Bull. EngZehard I n d . Inc., 7, 41 (1966); (b) H. Brodowsky and E. Poeschel, 2. Physik. Chem., 44, 143 (1965) ; (c) H. Brodowsky, 2.Naturforsch., 22a, 130 (1967).
(8) H. Brodowsky and H. Husemann, Ber. Bunsenges. Physik. Chem., 70, 626 (1966). (9) A. Maeland and T. B. Flanagan, J . Phys. Chem., 69, 3575 (1965). (10) J. W. Simons and T. B. Flanagan, Can. J . Chem., 43, 1665 (1965). (11) C. A. Mackleit and A. I. Schindler, Phys. Rev., 146, A463 (1966). (12) (a) P. 8. Perminov, A. A. Orlov, and A. N. Frumkin, Dokl. A k a d . N a u k S S S R , 84, 749 (1952); (b) P. L. Levine and K. E. Weale, Trans. Faraday Soc., 56, 357 (1960); (c) E. Wicke and G. H. Nernst, Z. EZektrochem., 68, 224 (1964). (13) T. R. P. Gibb, J. MacMillan, and R. J. Roy, J.Phys. Chem., 70, 3024 (1966). (14) H. Brodowsky, 2. Physik. Chem., 44, 129 (1965). (15) M. von Stackelberg and P. Ludwig, Z. Naturforsch., 19a, 93 (1964).
Volume 78, Number 1
January 1968
K. ALLARD,A. MAELAND,J. W. SIMONS, AND T. B. FLANAGAN
138
studies of the two-phase free-energy changes,181’7but such an inference is ambiguous. This surprising finding has introduced the necessity for a further interaction to be introduced into the statistical model, an interaction between the absorbed protons and the silver atoms. It is our opinion that the general features of the treatment of Brodowsky and co-workers are valid ; it represents the first detailed interpretation of absorption by palladium-rich alloys. Two details deserve comment, however: firstly, the assumption of n, = 1 is implicit in their treatment, and this choice may not be correct; secondly, the assumption that A p e - ( n ) is not significant as n --t 0 may be examined f~rther.~b?o The treatment given below discusses these points, and, in addition, the method outlined here has certain virtues, because the entropy and enthalpy terms are evaluated and discussed separately. It should also be pointed out that it makes little difference whether the BraggWilliams2 or quasi-chemical approximations7 are employed in this research, because for the small values of n employed here the two treatments have an identical functional dependence upon n. The extended Lacher model for pure palladiumhydrogen was given recently aslo
-x +d - - 2WHH
2RT
n
RT ns
+-We(n) RT
(2)
(3)
where E (ev) is the energy of the band relative t o the Fermi level of pure palladium; this approximation is quite good t o E > 0.3 ev (Figure 1). This is a better and more convenient approximation to the band shape than that employed previously.1° The rigid-band model upon which the band shape is based,ls has been shown to be erroneous.1g However, since both addition of hydrogen and silver or gold result in an apparent value of 0.6 hole in the d band, the use of the band established using silver and the rigid-band approximation may not be inappropriate for the case of hydrogen absorption. For any gold-palladium alloy we have from eq 3 12 =
0.584e-6.85E~u11 -
e-6.85AE~
1
(4)
where EAu is the energy of the band corresponding to a The Journal of Physical Chemistry
01
02
03
0.4
I
E(ev)
Figure 1. Density-of-states for the combined s and d bands of pure palladium for one spin direction base on specific-heat measurements of silver-palladium: 0, data of ref 18; 0 , approximate fit to data of ref 18 given by 2e-8.86E. The arrows represent the energy corresponding to the gold alloys used here, if gold donates its electron to the combined s-d bands of palladium.
given fraction of gold and AER is the energy increment corresponding to a given value of n ; i.e., AEH = E EAu where E is the actual energy of the band a t any value of n. For small values of n, AEH is also small and eq 4 can be approximated by AEH
where the terms are defined as before.1° This equation accounts for the alteration in the energy of electron donation with band filling by use of the We(n) term. The combined s-d density-of-states relationship for palladium established by Hoare and co-workersls from electronic heat capacities of silver-palladium, empirically, can be approximated by N(E)s+d = 4e-6.s5E
00
N
e6-S6EA~n/4.0
(5)
This shows that for increasing gold contents, the energy of electron donation will be altered by the e 6 . 8 5 E A u term, and, in addition, this should be directly proportional to n for small n. In this research, the poorest case of the approximation is for the 44.8% Au-Pd alloy and eq 4 is a good approximation to eq 3 for this alloy, to n 6 0.04. This allows eq 2 to be rewritten in a more general form In p’l’(1 - 69/19 = CI
+ (C;/RT) + (Ca/RT)8 = B(T,O) (6)
where 8 is n/n,, C1, Cz, and CI are determined from the experimental data, and CBincludes the H-H interaction and the effect given by eq 5 ; these two effects operate in different directions. Equation 6 has been employed before for pure palladium-hydrogen,6 and as in the treatments given earlier,lZcB(T,O) is plotted against 8, and Cl (Cz/RT)is obtained as theintercept and Ca/RT
+
(16) F. A. Lewis and W. H. Sohurter, Naturwissenschaften, 47, 177 (1960). (17) A. C.Makrides, J . Phys. Chem., 68, 2160 (1964). (18) F. E. Hoare and B.Yates, Proc. Roy. SOC.(London), A240, 42 (1957); F.E.Hoare, J. C. Mathews, and J. C. Walling, ibid., A216, 502 (1953). (19) J. J. Vuillemin and M. G. Priestley, Phgs. Rev. Letters, 14, 307 (1965).
APPLICATION OF THE ELECTRON-DONATION MODELTO PALLADIUM-RICH ALLOYS
139
Table I : Hydrogen Absorption Data for Several Gold-Palladium Alloys
n
273.2OK
2Q8.0°K
P(atm) X
P(atm) X 104
104
323.2'K P(atm) X
348.0"K P(atm) X
104
104
21.97 44.42 71.37 105.21 141.22 182.89 225.28 261.98 302.46 344.17 388.91 489.38
42.21 93.94 149.83 211.95 282.29 361.31 480.22 535.47 641.05 732.56 837.09 1085.66 1308.38 1505.29
n
273.Z°K P(etm) X 104
3.06 6.59 10.97 16.36 22.59 29.15 24.55
8.71 18.11 29.35 42.98 57.78 73.55 90.05 106.05 122.95 140 35 157.74
44.7% Au 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.048 0.050 0.055 0.060 0.065 0.070
0.556 1.21 1.98 8.64 15.65 29.27 41.04 61.95 88.98 l25.75 176.33 245.27 341.14 473.58
0.986 3.79 9.53 19.19 33.66 56.73 89.68 135.67 192.54 263.57 362.26 501.85 684.24 942.31
323.2ex PWm) X
348.0'K PWm) X
IO'
104
18.8% AU
5.7% AU 0.0040 0.0062 0.0082 0.0102 0.0123 0.0144 0.0165 0.0185 0.0206 0.0227 0.0248 0.03OO 0.0380 0.0400
2Q8.0°K
P(atm) X 104
26.27 52.26 86.69 129.32 208.16 314.92 455.95 650.91 912.64 1282.2 1762.2 2422.0
8.48 37.13 90.59 178.46 305.62 496.89 751.69 1105.8 1553.6 2169.7 2893.8 3851.9 5096.5 6275.0
as the slope. The value of 2RC1 will be called A,!?' (the entropy change upon absorption when 0 = 0.5 and dCa/dT = 0) and 2Cz will be called A.R" (e -t 0). For palladium-hydrogen, Ca = - 11,038 cal/mole; therefore, the predominant effect is the H-H attractive interaction, and effects due to band filling may be present, but are not detectable because they have been obscured by the attractive interaction.
Results and Discussion Heat of Absorption. When palladium is alloyed with gold, extensive single-phase (a)absorption of hydrogen can occur in certain of these alloys.9 One goal of this research was to utilize this fact in order to show directly from the isosteric heats of absorption, determined from the Clausius-Clapeyron equation, that the heat increases with n. This was not possible for pure palladium in the temperature range studied because the aphase content is so small that any increase in the heat was within experimental error.6 Previously, the increase in the heat was shown to exist with the aid of eq 1 or its equivalent, eq 6.6,7b,12cAdmittedly, these equations are very general, but they do assume localized absorption and an interaction which depends linearly
0.0031 0.0040 0.0057 0.0066 0.0075 0.0083 0.0090 0,0099 0.0107 0.0116 0.0123 0.0133 0.0140 0.0147 0.0156 0.0164 0.0180 0.0197 0.0214 0.0230 0.0247 0.0288 0.0329 0.0370 0.0412 0.0494 0.0576 0.0658 0.0743
0.406 0.568 0.761 1.04 1.29 1.53 1.89 2.18 2.48 2.79 3.12 3.45 3.79 4.20 4.57 4.98 5.80 6.70 7.55 8.43 9.42 11.55 13.76 15.54 17.21 20.40 23.38 25.45 27.01
1.18 1.75 2.42 2.96 3.79 4.44 5.23 6.01 6.65 7.77 8.67 9.52 10.54 11.57 12.60 13.31 14.62 17.76 20.11 22.78 25.21 31.85 40.24 48.13 56.24 72.18 87.02 98.74 108.22
2.06 2.87 3.99 5.11 6.45 7.60 8.96 10.33 11.65 13.13 14.81 16.44 18.12 20.13 22.36 23.92 28.19 32.51 36.92 41.94 45.88 57.42 69.78 82.89 99.85 121.45 147.54 170.12 188.92
3.30 4.57 6.38 7.88 9.51 11.41 13.36 15.70 18.02 20.68 23.23 26.09 29.51 32.66 36.16 39.45 46.62 54.28 62.76 71.00 79.74 102.79 125.03 145.61 165.92 201.81 240.17 277.70 316.44
upon e, and it is reassuring that the isosteric heats generally agree with those predicted by the equations (see below). A series of slow runs with H2-He mixtures were made and are illustrated in Figure 2 . The data are a composite of runs from different specimens, and a variety of surface activation procedures. It is believed that these data, which can be converted to P vs. n,6 utilizing relationships between R/Ro and n determined elsewhere for these alloys (Table I),correspond closely to equilibrium. The most important fact showing that these data closely approach equilibrium is that any further reduction in the Hz partial pressure, and its accompanying slowing of the absorption run, did not significantly alter the relationships obtained. Plots of p"' vs. n show straight-line behavior at small values of n, and these generally pass through the origin; Le., the data follow Sieverts' law as expected for equilibrium data. It was not possible to use this experimental technique for alloys with gold contents of 44.7%. This is, in all probability, due to the fact that diffusion is too slow for the attainment of equilibrium using this technique (0-60°). The isosteric heats of absorption are shown as a Volume 79,Number 1 January 1968
140
K. ALLARD, A. MAELAND, J. W. SIMONS, AND
ToB. FLANAGAN
Table 11: Thermodynamic Parameters for Hydrogen Absorption in the a Phase by Au-Pd Alloys a t 25"
- AS', % Au
0 5.7
Hz
-2C2,
oal/rnole of H?
cal/mole of Ha
(4623)" 5980 7000 7540 9040 9340
4623" 6440 8300 7 140 8480 9440
25.9" 29.3 11.2 32.2 31.6 32.2
15.3 18.8 26.5 44.7 a
-AB.,
eu/rnole of
dAB -
zcab
2CsC
dT
...
- 22,075 -21,258 -18,750 -12,540 -6,600 f14,388
-22,075 - 19,886 12,366 - 10,542
37.4 32.0 10.5
-43 ,400
- 37,600 -20,600 - 14,400 f 2 1 , 160
+
IO0
I03
I06
R
+
IO9
R,
Figure 2. Electrode potentials of 18.8% Au-Pd alloy os. standard hydrogen electrode, plotted against relative resistance of the specimen: A, 273°K; A, 298°K; 0, 310°K; and 0, 320'K.
function of n for various alloys (Figure 3). It may be seen that is approximately a linear function of n for the small values of n studied here. Scatter of the data is considerable at very small values of n and these have not been weighted as much as larger values. The slope bAR/bn becomes less negative with gold content and, in fact, the 44.7% alloy shows an increase in AH with n (Table 11). This latter observation is in keeping with the electron-donation model, ie., the density-ofstates has declined to a value low enough so that the filling in of the band predominates over the H-H interaction. The important quantities, AH", can also be obtained by extrapolation to n = 0 (Figure 3). These values are plotted in Figure 4, where they are compared to the data of silver-palladium.7b It is of interest that the differences between the added gold or added silver content are less marked when the fundamental quantity AB" is plotted against per cent of added metal rather than AHa+. The reason for this is that in the
An
The Journal of Physical Chemistry
-101
000
'
1
...
-
...
...
$7,874
' Calculated using na = 1.
Reference 6. gowas not determined directly in ref 6, but was obtained from 2192. Pd).
ns = Pd/(Au
-
dCsC
dn
1
0.03
'
1
n
'
-8.0 Calculated using
"
I
0.08
Figure 3. Typical plots of isosteric heats of absorption against n, H/M: e, 5.7Y0; A, 18.8%; and A, 44.770 Au-Pd.
case of, for example, the 10% Ag and Au alloys Pmin is ~ 0 . 4 4 and ' ~ 0.37, respectively (the latter value is obtained by interpolation between the 8.7 and 11.90% Au alloysQ). Accompanying the transformation in the silver alloy, there is a larger contribution from H-H interaction because &in corresponds to a larger H content than in the comparable gold alloy. This illustrates the caution which must be used in employing only the thermodynamic parameters characteristic of the a + P phase change when generalizing about the properties of the alloy systems toward hydrogen absorption. Figure 4 shows that AR" increases with percentage of added metal; the gold data are linear to approximately 26.5% and extrapolate back t o the value for pure palladium.6 The value for the 44.7% alloy shows a marked deviation from this linearity; this is in accordance with the electron-donation model. The fact that the heat of absorption increases with added metal means that gold in palladium, analogously to silver in p a l l a d i ~ m , ~ exhibits ~ ~ ' ~ ~ 'an ~ apparent interaction between the added metal and the hydrogen, and
141
APPLICATION OF'THE ELECTRON-DONATION MODELTO PALLADIUM-RICH ALLOYS
k
I
\
- 6 1
\
IX-IO
1 0-
'\
Q
\
'\
0 0
0
IO
20
70 A U
30
40
Figure 4. Heats of absorption a t infinite dilution ( n -.c 0) compared to comparable values for silver-palladium alloys (ref 7b): 0 , from extrapolated values of AT? (where two values are given, these represent values determined by two different workers at different times); A, calculated values of 2Cz; 0, values for silver-palladi~im.~b
:
I 50
0 00
0 03
e
0 OE
0.m
Figure 5. Representative B( T,e) us. e plot,s for several gold contents: open symbols represent n. chosen as 1; filled symbols represent ns chosen as fraction of palladium; V, v, 5.7%; 0, 0, 18.8%; and A, A, 44.7% Au-Pd.
allowance mur;t be made for this in the models for hydrogen absorption by alloys. The origin of this interaction is subject to c o n t r ~ v e r s y , ' ~but ~ ' ~ recent 0.552 (fraction of palladium) for the 44.7% alloy. results by Wioke, Brodowsky, and their c o - w o r k e r ~ ~ ~There ~ is no basis for choice between n, = 1 or n, = offer strong evidence in favor of an interaction arising fraction of palladium, on the basis of the B plots; both from the relief of strain by the congregation of interexhibit approximately the same degree of linearity. stitial protons, the energy of which is perturbed by the The values of CI, Cz, and C3 are determined from these nature of the added metal; Le., the presence of large plots and the heats can be calculated from compressible added atoms increases the apparent interaction. Gold fits into the picture in a qualitative way, Le., its compressibility and size are greater than that of palladium. There are, therefore, at least two 2Cz 2C3e - 2 T B ( g ) (7) effects governing the relationship between Al?O and percentage of added gold, the H-Au interaction and the The last term on the right-hand side of this equation effect of filling the combined s-d bands of palladium must be included in order to get reasonable agreement (these, of course, operate in different directions). It with the more directly determined values. It is also can be shown from the approximate band shape (eq 3) that the initial energy of electron donation increases in theoretically expected that C3 should have a significant a linear way with fraction of gold in the alloy to include temperature dependen~e.~In earlier work with pure at least the 18.8% gold alloy; therefore, a nearly linear palladium,6 Simons and Flanagan found good agreerelationship observed between AH" and percentage ment between an average isosteric heat and that evalugold (0 to 18.8%, Table I) is not inconsistent with the ated via eq 7 without including the dC3/dT term. The electron-donation model. Both the H-Au interaction reasons for this is that while dCs/dT may be large for and the electron-donation energy change in direct pure palladium (its value may be estimated by extrapoproportion to the gold content (Figure 4), and the H-Au lation of the data for the alloys given in Table II), the interaction is the most important factor. limiting concentration of hydrogen in the (Y phase of It is useful, in order to test the model of absorption, pure palladium is so small that on an average -2T& to calculate the heats of absorption from eq 6 and to (dCa/dT) contributes less than 200 cal; whereas, in the compare them with the isosteric heats. Some typical alloys, which have larger values of e, this term becomes plots of B (eq 6) us. 0 are shown in Figure 5 . It may more important. The remarkable agreement between be seen that the plots are reasonably linear; C3 is the average isosteric heat of 4780 f 100 cal/mole with obtained from the slope, and CI and Cz are obtained the average value of 4777 determined from eq 4 in ref 6 from the temperature dependence of the intercepts of (which has an incorrect sign-it should read, 2Cz the B plots (Table 11). Equation 6 may be plotted 2C8n = -4623 -22,075n) must be regarded as forusing various values for n,; Figure 5 shows n, = 1 and tuitous, although, coincidentally, ref 12c gives 4777
+
+
Volume 78, Number 1
January 1068
K. ALLARD,A. MAELAND, J. W. SIMONS, AND T. B. FLANAGAN
142
for this value too, again taking n = 0.007 for theaverage value of n. The temperature dependence of C1 has been omitted because it cannot be evaluated directly from the data; estimates based on the Einstein model give the value of RT2(dCl/dT) of the order of 300 cal/mole. Values of dC3/dT can be determined directly from the slopes of the B plots for the alloys, or, alternatively, dC3/dT can be obtained by differentiation of eq 7 with respect to 8, i.e.
upon substitution of this value for dC3/dT into eq 7, we obtain A H @ ) = 2C2
+ e(baR(e)/be)
(9)
This procedure now becomes essentially an expansion of about 0 = 0 since 2C2 = AR" and the dependence of AR on 0 essentially duplicates the results of the directly determined isosteric heats, except that this procedure does give two methods to obtain AR", and it allows us to determine dCa/dT (Table 11) for use in the entropy calculations (see below). It can be seen from the form of eq 0 that in the relationships between AR(n) vs. n, it makes no difference which value of n, is chosen. To show the validity of eq 7 then, we must use an independent value of dC3/dT; for the 44.76% Au alloy, these data appear good and the AH(n) values agree reasonably well with those determined from the Clausius-Clapeyron equation. It should also be mentioned that the value of 2C2 agrees quite well with that of A P (Table 11). There is no basis for judgment in the choice between n, = 1 and n, = fraction of palladium from these heats of absorption. Entropy of Absorption. The entropy of absorption is determined experimentally from 2F [ (bE)/(bT),] and is shown as a function of n for several alloys in Figure 6. The entropy of absorption also can be determined from eq 6, i.e.
An(e)
AS = 2RC1
+ 2e(dC3/dT) + 2R In [e/(l - e ) ]
(10)
and in addition AS can be calculated from the Einstein model, i.e.
+
AS = 2R In [e/(l - e ) ] - SoHl 25,
- 2e(dC3/dT)
(11)
where S O H % = 31.21 (298OK) and Sais the entropy of a proton treated as a localized three-dimensional isotropic oscillator. In earlier work by the authors on a-phase palladium-H2,6 a value of V H was employed corresponding to the /3 phase, since the a-phase value of VH was then unavailable. Recently, a determination of VH for the a phase has been made using neutron scattering.20 The value of Y H for the alloys may be somewhat different from this, but in the absence of The Journal of Physical Chemistry
0 00
0 03
0 08
009
Figure 6. Representative plots of isosteric entropy of absorption w n : ( a ) 18.8% Au-Pd and (b) 44.7% Au-Pd; and A, experimental values; 0 , calculated values using nl = fraction of palladium; and 0 , calculated values using = 1.
specific data the value for a Pd-H2 will be employed. One of the authorsz1has found that the hydrogen is located in the octahedral holes in the a phase of the 15.3% gold-palladium alloy. It was also found that VH changed with n, but this effect is believed not to alter seriously the value of S, for the small values of n employed in this research. Again in the earlier work,6 the last term in eq 6 was omitted, but its contribution is small in the a-Pd-H2 system because 0 is small. The value of dCl/dT is a relatively small term (eq 9) and has been omitted. If n, is chosen as fraction of palladium, the agreement of AS(%)with the experimental values is somewhat better than for n, = 1 for the 44.7% Au alloy, where the differences between the two values of n, are the greatest. This is the only evidence in favor of this choice for n,, and further work must be done to allow a choice t o be made, e.g., determination of VH in the alloys by neutron scattering. Values of A,!? (e = 0.5, dC3/dT = 0) are equal to 2RC1 and are shown in Table 11. The variation in the entropy of absorption is small enough in the change from the 5.7 to 44.7% Au to be accounted for by a change in vH with gold content. The changes in A T are comparable to those observed for the silver-palladium system.?b The equilibrium value of n is, of course, attained when A H = TAS and, for example, the 44.7% Au-Pd alloy (20) W. Kley, J. Peretty, R. Rubin, and G. Verson, Symposium on Inelastic Scattering of Neutrons by Condensed Systems, Brookhaven National Laboratory, Upton, N. Y . , 1965. (21) A. Maeland, presented at the 153rd National Meeting of the American Chemical Society, Miami Beach, Fla., April 1967.
APPLICATION OF
THE
143
ELECTRON-DOKATION MODELTO PALLADIUM-RICH ALLOYS
reaches n = 0.107 at 2 6 O , 1atm of Hz.No unexpected effects operate rand the main reason that absorption ceases in this alloy at a rather low value of n is that the heat of absorption declines with n, so that TAS = AR at n = 0.107. The internal consistency of the data for this alloy can be shown from the fact that experimentally determined plots of T A S vs. AB intersect at values of n close to those determined by direct analysis of the hydrogen content of samples at equilibrium. Plots of A B vs. n for the 44.7% alloy are quite linear from n = 0.06.-0.10; this is expected if the logarithmic relationship holds, i.e., In pl’%(atm) = A ( T ) B(T)n. Wicke, Brodowsky, and co-workers have shown that this relationship holds for silver-palladium alloys.’ Flanagan and Simonszz have shown recently that this previously empirical relationship follows from the effect of filling in of the combined s-d bands of palladium by hydrogen. Thus, the fact that the data of the 44.7% Au-Pd alloy follow the logarithmic relationship at relatively low values of added hydrogen supports the electron-donation model, i.e., both gold and hydrogen act as electron donors to the empty s-d bands of palladium. Role of the Added Gold i n the Filling i n of Ihe Combined s-d Bands of Palladium. Equation 2 can be rewritten in a form suitable for alloys as
Table 111: Some Parameters for Absorption of Hydrogen by Au-Pd Alloys Derived from the Density-of-States Curve for Palladium Alloy, % Au
0 5.7 15.3 18.8 26.6 44.7
2 WHH*U, cal
WAu!
oal
0
- 1232 - 1662 - 2873 - 3959 -7319
- 16,807 -15,398 - 12,809 - 12,187 -10,970 - 10,378
EL, cal
8403 7699 6404 6093 5485 5189
+
In p = 2CI0
6 2c,o + 2 In + -RT +1 - e
2WM
+
RT
where CIO, Czo are the values corresponding to pure palladium-hydrogen, WM and WeM are the changes in the energy relative to pure palladium introduced by the added metal for the H-lattice interaction and the shift of the Fermi energy by the added metal at n 4 0 and W H M is the H-H interaction as perturbed by the added metal and the last term accounts for the filling in of the s-d bands of the alloy by hydrogen using eq 4. Expressing Cz and C3 in terms of the parameters of eq 12, we have Cz = Czo W M WeM and C3 = ~ W H H 5770n8e6.85w6v.Now if the approximate band shape is employed, we may obtain WM and W H H ~ This . is, admittedly, a questionable procedure, but at the present stage of development there is no other possibility for separation of these effects and this procedure yields results which serve as a semiquantitative test for the electron-donation model. For example, by employing the experimental values of Czo,Cz and the calculated values of WeM for the various gold alloys
+
+
+
(Table III), where W Arepresents ~ W Mfor the specific case of gold-palladium alloys, etc., n, has been taken as the fraction of palladium in the alloy in the calculations shown in Table 111, since this value appears to give more reasonable results. It can be seen that the energy of the inclusion of the proton into the lattice decreases almost linearly (more favorable entry into the lattice) with gold content; this holds true even for the high gold-content alloy (44.7% Au). This effect could not have been seen from the values of C3 (Table H ~ ~ with gold 11). Similarly values of ~ W H decrease content. These approximate calculations show that the electron-donation model allows the values of CZand C3 to be broken down into terms which show quite reasonable trends with gold content. It is concluded that the data of the present research are compatible with the electron-donation model in which both gold and hydrogen donate electrons to the combined s-d band of palladium. In addition, the results are in qualitative agreement with the strain model introduced to account for the H-H14315and H-M7 interaction. For example, Brodowsky14 gives EL ‘v Cz, which may be modified for the density-of-states effect so that in our terminology EL% - W H H , where EL is the energy needed to insert an isolated proton into the lattice. Now WA” and EL should be related, Le., if the strain effect lowers the energy necessary to incorporate a proton into the lattice, then the decrease ~ in W A should ~ be equal to the decrease in EL. This is roughly seen to be the case.
Acknowledgmeni. Financial support of this research by the U. S. Atomic Energy Commission is gratefully acknowledged. The authors are indebted to Englehard Industries, Inc., for the loan of the gold-palladium alloys used in this research. (22) T. B. Flanagan and J. W. Simons, J. Phys. Chem., 70, 3750 (1966).
Volume 79,Number 1
January 1968
