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J. Phys. Chem. C 2010, 114, 14482–14488
Exponents for the Pressure Dependence of Hydrogen Permeation through Pd and Pd-Ag Alloy Membranes Ted B. Flanagan* and D. Wang Department of Chemistry, UniVersity of Vermont, Burlington, Vermont 05405 ReceiVed: February 12, 2010; ReVised Manuscript ReceiVed: July 12, 2010
A frequently employed boundary condition for H permeation of membranes is pdown ≈ 0 and, for this case, the flux is given by J ) (KDH/d)pnup, where K is a constant and d is the membrane thickness. Values of n can then be obtained from slopes of plots of ln J against ln pup. When the H solution is ideal, n ) 0.5, and DH is concentration-independent, but when the H solution is nonideal, i.e., when the H atoms interact, n * 0.5 and DH is concentration-dependent. An equation is derived for n, which includes the effects of nonideality of H on both DH and the solubility. The terms in this equation for n can be obtained from the appropriate equilibrium H2 isotherms for Pd or its alloys. Good agreement is found between the derived values of n and the experimental values for the pressure dependence of H2 permeation through Pd, Pd0.77Ag0.23, and Pd0.50Ag0.50 membranes. By comparing experimental n * 0.50 values to those calculated from the equilibrium isotherms, a decision can be made about whether bulk diffusion is the slow step. Introduction The purification of H2 is important for both mobile and stationary H2/O2 fuel cells. The most commonly employed membranes for H2 purification are those based on Pd alloys and, more specifically, Pd/Ag alloys. Pd and its alloy membranes are highly selective for permeation of H2 and its isotopes. A fundamental understanding of H diffusion in these membranes is therefore of considerable importance. A very comprehensive review on H permeation in these membranes has been given recently, which contains 96 pages of references (1411 references!) dramatically demonstrating the interest in this field.1 At low H concentrations, where H forms an ideal solution, H2 permeation through membranes such as Pd depends on the 1/2 1/2 - pdown ), i.e., Sieverts’ law of ideal solubility difference (pup holds, r ) KspH1/22 , where up and down refer to the up- and downstream sides of the membrane, r is the H-to-metal atom ratio, and Ks is a constant at a given temperature. Thus plots of 1/2 flux or permeability against (p1/2 up - pdown) for a given membrane thickness and temperature should be linear under conditions of ideal solubility, which are obtained at low pH2 and low r, and the measured diffusion constant should be the concentrationindependent Einstein diffusion constant. An H-H attractive interaction leads to nonideality, which is believed to arise from a long-ranged elastic H-H interaction.2,3 1/2 1/2 Plots of flux against (pup - pdown ) may no longer be linear at higher pH2 and H contents, and are, instead, frequently plotted n n - pdown ) where n is selected to give the best as flux against (pup 4-6 linear plot. In their studies of H permeation through Pd, Hurlbert and Konecky4 employed upstream H2 pressures from 1.54 to 7.18 bar (623 K) with pdown ) 0. They found n ) 0.68 for this temperature and pH2 range. DeRosset5 found n ) 0.80 using pup from 1 to 21.4 bar at 727 K. Morreale et al.6 carried out permeation experiments from 623 to 1173 K using pure Pd membranes and, although pdown * 0, it was quite small (∼0.1 bar) compared to pup, where the latter * To whom correspondence should be addressed.
ranged from about 0.8 to 24 bar. They found n ) 0.62 for the best fit of their data at temperatures from 623 to 1173 K. Zhang et al.7 studied the permeation of a 2.5 µm Pd membrane supported by microchanneled Ni, which should not offer any resistance to H2 flow within the channels. They employed pdown ) 0 and pup from about 6.5 kPa to about 35 kPa over the temperature range from 473 to 673 K. Over this pH2 range, they found n values ranging from 0.86 (473 K) to 0.73 (673 K). Li et al.8 determined n values for very thin Pd membranes deposited on porous alumina by varying pup with pdown ) 1 bar. They found n values of >0.5, which varied with temperature from 0.77 (363 K) to 0.69 (773 K) over a pup range of 1.4-3.8 bar. Hurlbert and Konecky4 explained their n ) 0.68 exponent as being due to deviations of the solubility from Sieverts’ law of ideal solubility, r ) KspH1/22 , although they did not explore this quantitatively. DeRosset5 realized that Sieverts’ law was not applicable at the high pH2 that he employed and converted his pH2 data to a dependence on H concentration rather than pH1/22 using an isotherm equation given by Lacher9 in 1937 which is not, however, very accurate in view of more recent results.10-12 Morreale et al.6 commented that their pH2 exponent of n ) 0.62 was due either to “an invalid assumption of a diffusionlimited mechanism and/or to changes in the Sieverts’ constant and diffusion coefficient with increasing pressure”, i.e., both possibilities were proposed for the deviation from 0.5. Very recently, McLeod et al.13 analyzed nonideal effects on H permeation through a Pd0.75Ag0.25 alloy membrane as being due to deviations of the solubilities from ideal behavior using data from the literature;14,15 however, they neglected the influence of the nonideality on the diffusion constant. As recently as 2008, it was pointed out by Li et al.8 that “it is still a challenge to understand the variation of n values at different temperatures and pH2, especially for Pd membranes with thickness of several micrometers”. It is the purpose of the present research to attempt to rectify this by deriving an equation to explain pressure exponents n * 0.5 based on effects of nonideality on both H solubility and DH. The derived equation will be tested using H2 permeation results obtained here and
10.1021/jp101364j 2010 American Chemical Society Published on Web 08/10/2010
Pressure Dependence of H Permeation of Membranes
J. Phys. Chem. C, Vol. 114, No. 34, 2010 14483
from the literature for Pd and also using permeation results for two Pd-Ag alloy membranes.16 The permeation experiments to be employed will be those with pdown ≈ 0, which is a common experimental situation for fundamental permeation studies, e.g., it is the condition employed by Hurlbert and Konecny,4 Zhang et al.7 and Yamakawa, et al.;17 however, an equation for n with pdown * 0 will also be given. It will be assumed in the following that bulk diffusion is the slow step and not, e.g., desorption of H2 from the downstream side as discussed by Ward and Dao18 or mass transport through porous supports, which can lead to n > 0.5.19,20 For the Pd and Pd-Ag membranes employed here, it has been shown that bulk diffusion is the slow step because of the direct proportionality found between the fluxes and 1/d where d is the membrane thickness.21,6
Permeation data for the Pd-H system at 423 K(120 µm), 473 K(140 µm), 523 K(122 µm) are from the present research where the membrane thicknesses are given in parentheses. The membranes were prepared by rolling pure Pd and then annealing the foil. The thicknesses of the Pd-Ag membranes, whose results will be described below, are Pd0.77Ag0.23 (193 µm) and Pd0.50Ag0.50 (100 µm). The membranes were cleaned mechanically and then ultrasonically in acetone. They were then thoroughly dried before being inserted into the apparatus. The membranes were sealed in the metal apparatus with a Cajon fitting and surrounded by an air furnace. A thermocouple in contact with the membrane measured the temperature, and this was removed before the permeation experiments. The downstream pH2 was kept at ∼0, and the flux was measured from the small pH2 decreases on the upstream side of the membrane, which included a large volume reservoir so that the pH2 fall was generally small during permeation. The fall depended upon the membrane thickness, temperature, and material, e.g., for a 150 µm Pd membrane, the fall was about 1.6% in the first 10 m at 423 K, and for a 150 µm Pd0.77Ag0.23 alloy membrane at 523 K, the corresponding fall was 5%. In any case, the flux at time t, J(t), can be approximately corrected for the fall using [p(0)/p(t)]1/2 × J(t), where p(t) is the pH2 at time t. Most values given in this paper are the initial fluxes (0-5 m) where the falloff is minimal. The apparatus has been described in more detail elsewhere.21 Data for the Pd-Ag alloy membranes were obtained previously16 but are treated here from a different perspective, i.e., plots of ln J versus ln p were not given previously. Theoretical Background Equation for n. A flux of H atoms, J, through a flat membrane closely approximates one-dimensional diffusion and can be described by Fick’s first law as
(1)
where cH is the H concentration and DH is the concentrationdependent Fick’s diffusion constant. When the steady state is reached with cH,down ) 0, eq 1 reduces to
J ) DHcH,up /d ) DHK′rup /d
DH ) DH/
(2)
where r is the H-to-metal atom ratio, K′ is a constant needed to convert cH to r, d is the membrane thickness, and the quantities
( )
cH ∂µH RT ∂cH
(3) T
where µH is the chemical potential of the dissolved H, and a multiplicative factor, the correlation factor or Haven’s ratio, is omitted because it is assumed to be 1.0, which is consistent with the results of Seymour et al.23 for Pd-H. The term multiplying D*H in eq 3 can be written as
( ) (
cH ∂µH RT ∂cH
Experimental Section
J ) -DH(dcH /dx)
are defined so that J is positive. Fick’s diffusion constant is related to Einstein’s concentration-independent constant, D*H, by22
)
T
∂ ln p1/2 ∂ ln r
)
T
) f(r)
(4)
which follows from the relationship µH ) (1/2)µ°H2 + RT ln p1/2, where f(r) is referred to as the thermodynamic factor.22,24,25 Values of f(r) at different r values can be determined from the slopes of ln p1/2 versus ln r isotherms. During steady-state permeation, there will generally be a significant gradient of r from the up to the down side of the membranes and, consequently, the thermodynamic factor will vary with distance through the membrane. The following equation has been employed to allow for the variation of f(r) with distance through the membrane for pdown ≈ 0:21,26
DH ) DH/ (
∫0r
up
f(r) dr)/rup ) DH/ F(r)/rup
(5)
where the integral in eq 5 has been abbreviated as F(r). F(r) can be evaluated by integrating the experimental f(r) versus r plots from r ) 0 to rup.16 For finite pdown, the lower limit of the integral of eq 5 must, of course, be changed to the value of rdown employed. If eq 2 is modified to incorporate the result in eq 5, eq 6 is obtained:
J ) K′DH/ F(rup)/d or ln J ) ln K′ + ln DH/ + ln F(rup) ln d (6) Taking the derivative of ln J with respect to ln p in eq 6, gives an equation for n:
n)
( ∂∂lnln pJ ) ) ( ∂ ∂lnlnF(r)r ) × ( ∂∂ lnln pr ) T
T
T
(7)
where the up subscript for r and p should be understood. As far as we are aware, this is the first time an explicit expression for n has been given that is applicable to nonideal conditions. Equation 7 will be employed to calculate n values for Pd-H and for two Pd-Ag alloys for comparison with experimental values of n with pdown ) 0. Values of n can be calculated from thermodynamic data using eq 7 and then compared to experimental n values from the slopes of ln J versus ln pup plots and, if the values agree, it can be concluded that bulk diffusion is the slow step, i.e., eq 7 provides a means to determine whether or not bulk diffusion is the slow step under conditions where n * 0.5. It can be seen that eq 7 gives n )
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Flanagan and Wang
0.5 in the limit of ideal behavior, i.e., as r f 0, F(r) f r and (∂ ln F(r)/∂ ln r)T f 1.0 and (∂ ln r/∂ ln p)T f 0.5, or n f 0.5. Approximate Treatment for n for Pd-H. An approximate expression for n for Pd-H can be obtained from eq 7 using the regular interstitial solution model, which is a good approximation for µH at small r,27 i.e.,
∆µH ) RT ln p1/2 ) ∆µHo + RT ln(r/(1 - r)) + g1r
(8) where ∆µ°H ) µ°H - (1/2)µ°H2, and ∆µH is defined similarly without the standard designation for H; the standard value, µ°, H is that as r f 0 without the configurational term, RT ln(r/(1 r)). g1 is the first-order term in a polynomial expansion of µHE, the excess chemical potential, i.e., µHE ≈ g1r. Using the expression for ln p1/2 from eq 8 and the definition of f(r) (eq 4) we obtain
f(r) )
g1r 1 + (1 - r) RT
(9)
F(r) can be obtained by integration of f(r) from 0 to rup, which gives
F(r) ) -ln(1 - r) +
g1r2 2RT
(10)
which was derived previously by the present workers in order to derive D*H from DH values.21 Using this approximation, n can be obtained from eq 7. The first term in eq 7 is obtained by taking the derivative of ln F(r) with respect to ln r from eq 10 and, the second term is obtained from eq 9 since (∂ ln(r)/∂ ln p)T ) (1/2)f(r), i.e.,
n≈
r -2 ln(1 - r) + g1r2 /RT
(11)
which, at small r, reduces to
n≈
1 2 + r(1 + g1 /RT)
(12)
Since, for Pd-H, g1 is negative at moderate temperatures and small r,10 it follows that n will be >0.5 for Pd membranes, and it can be seen from eq 12 that, as r f 0, n ) 0.5. This limiting behavior may not be seen at moderate temperatures where measurements are generally made at pH2, where nonideality may already be a factor. Results and Discussion H Permeation of Pd at Moderate pH2 and Temperatures (pdown ) 0). It has been shown that bulk diffusion is the slow step for Pd membranes for the experimental setup employed in this work21 because there is a very linear relationship between J and 1/d obtain over a wide range of membrane thicknesses, i.e., from d ) 500 to 85 µm, at both 423 and 523 K.21 In these experiments pup, and therefore rup, is held constant at all the thicknesses and therefore the degree of nonideality will be similar, whereas, this is not the case when the permeability is
Figure 1. Experimental plots of ln Psp-ln pup for Pd-H, where Psp is the specific permeability, which is proportional to J. ∆, 423 K; O, 473 K; 0, 523 K.
TABLE 1: Contributions to n from (D ln F(rup)/D ln r)T and (D ln r/D ln pup)T for Pd-H at 473 K (Present Data) r
(∂ ln F(r)/∂ ln r)T
(∂ ln r/∂ ln p)T
na, eq 7
0.010 0.015 0.020 0.025 0.03 0.04
1.23 0.98 0.88 0.88 0.84 0.82
0.53 0.56 0.58 0.61 0.63 0.72
(0.65) 0.55 0.51 0.54 0.53 0.59
a The parentheses for n in the first row of column 4 indicates that the value is rather uncertain.
TABLE 2: Values of n at r ) 0.02 for Pd-H Calculated at 423, 473, and 523 K from Eqs 11 and 12 Where g1 Values Are from Isotherms Measured in This Laboratory T/K
g1/kJ/mol H
n (eq 11)
n (eq 12)
423 473 523
-34.6 -32.8 -32.3
0.55 0.54 0.53
0.55 0.54 0.53
determined as a function of pup at constant d as in Figure 1, which shows data for Pd-H at 423, 473, and 523 K plotted as ln Psp against ln pup (pdown ) 0), where Psp is the specific permeability in (mol H/s)cm/cm2.28 Other quantities proportional to the permeability, such as the flux, can be substituted for Psp for such ln-ln plots, as shown in Figure 1. The observed slopes are n ) 0.58 at 423 and 473 K and 0.55 at 523 K. Table 1 shows calculated values of n at 473 K from eq 7, and the corresponding (∂ ln F(r)/∂ ln r)T and (∂ ln r/∂ ln p)T values needed for n are given in columns 2 and 3, respectively, which have been evaluated from equilibrium isotherms for Pd-H2.21 The average n calculated from eq 7 over the range r ) 0.01-0.04 is 0.56, and that from the approximate eq 12 is 0.57; both of these values are close to the experimental value of 0.58. It can be seen from Table 1 that (∂ ln F(r)/∂ ln r)T decreases with r, while (∂ ln r/∂ ln pup)T increases, tending to keep n nearly constant in this pH2 range. The experimental n ) 0.55 at 523 K is slightly smaller than that for the two lower temperatures. Values of n have been calculated using eqs 11 and 12 at different temperatures at an average value of r ) 0.02 (Table 2). There is no difference between the two approximations (eq 11 or 12) for this particular data. It is clear that n tends to approach 0.5 with increase of temperature; however, the dependence is weak. The smaller value of n found at 523 K compared to 473 and 423 K (Figure 1) is consistent with this trend. Although the tabulation of g1 values for Pd-H based on experimental data given by Kuji et al.11 does not extend to
Pressure Dependence of H Permeation of Membranes
Figure 2. ln p-ln r isotherm for Pd-H at 613 K from the data of Blaurock.12 The dashed line represents the ideal slope of 2.0.
temperatures >650 K, their results will, nonetheless, be employed for a rough estimate that, at 1000 K, g1 ≈ 0, which means that n ≈ 0.5 at this temperature according to eq 12. Equation 12 can be employed to estimate the error in the values of n calculated from these approximate expressions. From Blaurock’s data12 we obtain g1 ) -38.9 kJ/mol H at 473 K, whereas the value obtained from isotherms measured in this laboratory is -32.8 kJ/mol H. Using Blaurock’s value in eq 11, at 473 K, we obtain n ) 0.55 instead of the value of 0.54 in Table 2, which employed g1 ) -32.8 kJ/mol H. Kuji et al.11 gave g1 ) -36.4 kJ/mol H, which is about half way between the two, so it seems that the error in n is (0.01. H Permeation of Pd Membranes up to High p with pdown ) 0 (613 K). This temperature is of interest because an isotherm was measured by Blaurock12 at 613 K, which extends to very high pH2 (15.1 MPa), and therefore n can be calculated from eq 7 over a large range. This high temperature, high pH2 region is of interest for H2 purification. Pd is a good membrane for H2 purification, provided that it is not cooled in H2 to temperatures where the hydride phase forms. Figure 2 shows Blaurock’s data at 613 K12 plotted as ln p against ln r. It can be seen that, at low r, the slope is the ideal one of 2.0 (dashed line), but, at higher r values, (∂ ln p/∂ ln r)T decreases and then increases again at quite high r. In principle, H2 fugacities should be employed in, e.g., eq 7, instead of pH2; however, even at 15.1 MPa (613 K), there is only a 4% difference between the two, and therefore they will be not be employed. In order to evaluate n from eq 7, the values of (∂ ln r/∂ ln p)T and (∂ ln F(r)/∂ ln r)T must be known as functions of r, and these have been evaluated from the isotherm (Figure 2). Figure 3 shows 2f(r) ) (∂ ln p/∂ ln r)T against r, where the f(r) values are determined from slopes of the isotherm (Figure 2). The integrals, F(r) (eq 5), have been evaluated from the results in Figure 3 and are shown in Figure 4 as ln F(r) versus ln r. The slopes, (∂ ln F(r)/∂ ln r)T, of the ln F(r)-ln r relationship (Figure 4) with corresponding values of (∂ ln p/∂ ln r)T, from Figure 3, have been employed to calculate n as a function of r (eq 7) with results shown in Table 3. The estimated error in n arises principally from evaluating the slopes, i.e., the partial derivatives in eq 7, as noted above. The error depends in part on the value of r where the derivatives are evaluated, but a typical error in n from the product of the two derivatives in eq 7 is about (0.02 at r ) 0.20. It can be seen from Table 3 that (∂ ln F(r)/∂ ln r)T decreases with r while (∂ ln r/∂ ln p)T increases with r to r ≈ 0.26. The two terms in eq 7 tend to compensate each other in the range from r ≈ 0.04 to 0.10, leading to a nearly constant n, which is the r range employed by Morreale et al.6 The average value of
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Figure 3. (d ln p/d ln r) versus r for Pd-H at 613 K from the data shown in Figure 2.12
Figure 4. ln F(r) versus ln r for Pd-H at 613 K from integration of the data in Figure 3. The dashed line represents ideal behavior.
TABLE 3: (d ln F(rup)/d ln r)T and (d ln r/d ln pup)T for Pd-H at 613 K from the Data of Blaurock12 as a Function of r with Their Product Equal to n p/bar
r
(d ln F(r)/d lnr)T
(d ln r/d ln p)T
n (eq 7)
0.73 2.21 6.9 16.9 20.9 26.7 30.6 32.4 32.7 34.2 35.7 37.6 40.0 44.5 52.4 67.6
0.01 0.02 0.04 0.08 0.10 0.14 0.18 0.20 0.22 0.26 0.30 0.34 0.38 0.42 0.46 0.50
0.91 0.87 0.89 0.64 0.66 0.54 0.40 0.42 0.39 0.36 0.46 0.73 1.34 1.76 2.68 2.61
0.54 0.74 0.67 0.93 1.11 1.59 2.38 2.78 3.33 3.77 2.94 1.96 1.11 0.67 0.42 0.26
0.49 0.64 0.59 0.60 0.73 0.86 0.95 1.17 1.30 1.35 1.35 1.43 1.49 1.18 1.12 0.68
n over this region calculated from eq 7 is ∼0.60 (613 K), in good agreement with the 0.62 given by Morreale et al. for 613 K. It should be noted that it is assumed that pdown ≈ 0 in calculating n from eq 7, whereas in their experiments it is finite but small enough to not introduce significant error. It was shown above that there is only a weak temperature dependence of n so that the small temperature difference between the isotherm of Blaurock (613 K) and the data of Morreale (623 K) should not cause any significant difference in the calculated n. Morreale et al.6 indicate that their value of n > 0.5 could be due to “an
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Figure 5. Psp against p1/2 for a Pd0.77Ag0.23 membrane at 473 K from the data of Wang et al.16 The linear dashed line represents ideal behavior.
invalid assumption of a diffusion-limited mechanism and/or to changes in the Sieverts’ constant and diffusion coefficient with increasing pressure”. The present results indicate that the latter possibility is correct, i.e., diffusion is the slow step in their permeation results because the calculated n is closely equal to their experimental n. This demonstrates that eq 7 is useful in determining whether or not bulk diffusion is the slow step. Morreale et al. found the same n, 0.62, from 623 to 1173 K. Although there is only a weak temperature dependence of n, it would, nonetheless, be expected to change over this 500 K temperature increase. Therefore their experimental results leading to the same n over this large temperature range are probably due to experimental error, which is reflected by the large error bars shown.6 Often values of n ≈ 1.0 are taken as evidence that permeation is surface-controlled;29 however, it is clear that this is not necessarily true because, for Pd-H at r > 0.18 (613 K), nonideality leads to n g 1.0 (Table 3). The maximum in n as a function of r is due to the increase of (∂ ln r/∂ ln p)T with r and its subsequent decrease. There are no permeation data at these high pH2 to compare with these predictions, but hopefully they may prompt such experimental data to be obtained. Pd0.77Ag0.23 Membrane (pdown ) 0). Permeation of H through Pd0.77Ag0.23 membranes is of particular interest because this alloy is commonly employed for industrial and laboratory H2 purification. The permeation of H2 through a Pd0.77Ag0.23 membrane is shown in Figure 5 as a function of p1/2, where there are large positive deviations from linearity, indicating that n > 0.5; the dashed line indicates ideal behavior obtained by correcting the experimental p1/2 to the values it would have if it obeyed Sieverts’ law, which is obtained by linear extrapolation of the p1/2-r isotherm from the low content, ideal range. Since bulk diffusion is the slow step,16 deviations from linearity in the permeability-p1/2 plot (Figure 5) must be due to nonideality. Figure 6 shows the same data as in Figure 5 plotted as ln Psp against ln p, where there appear to be two linear regions, one from r ) 0.018 to 0.07 with n ) 0.61 and the other from about r ) 0.095 to 0.19 with n ) 0.67. Table 4 shows values of n calculated from eq 7 for this alloy (473 K) using the appropriate isotherm.16 Values of n in the H content range investigated are g0.50, and they agree reasonably well with the experimental data shown in Figure 6. The calculated n values shown in Table 4 at high r values appear to level-off at about n ) 0.65 ( 0.05. Thus there should be a continuous, but small, increase in n at low H contents and then
Flanagan and Wang
Figure 6. ln Psp as a function of ln p for a Pd0.77Ag0.23 membrane (473 K) from the data of Wang et al.,16 where the numbers on the plots indicate the slopes, i.e., the n values.
TABLE 4: Contributions to n from (D ln F(rup)/D ln r)T and (D ln r/D ln pup)T using eq 8 for the Pd0.77Ag0.23 Alloy Membrane (473 K) p/kPa
r
(d ln F(r)/d lnr)
(d ln r/d ln p)
n
4.09 5.76 7.55 11.5 20.4 32.4 39.8 45.2 56.3 70.0 72.6
0.025 0.030 0.036 0.046 0.068 0.094 0.111 0.122 0.143 0.169 0.174
0.91 0.95 0.95 0.91 0.83 0.85 0.81 0.82 0.88 0.98 0.98
0.56 0.61 0.64 0.66 0.69 0.72 0.74 0.75 0.79 0.67 0.66
0.51 0.58 0.61 0.60 0.57 0.61 0.60 0.62 0.70 0.66 0.65
a leveling-off; however, at higher pH2, n may decrease according to the results of Grashoff et al.30 Pd0.50Ag0.50 Membrane (pdown ) 0). Because of its small permeability, the Pd0.50Ag0.50 membrane is not promising as a H2 purification membrane; however, its behavior is of fundamental interest with regards to its n values. H2 solubility in the Pd0.50Ag0.50 alloy is small with positive deviations from Sieverts’ law, even at low H contents as shown in Figure 7. The isotherm suggests that there is, in effect, a positive g1 (eq 8) and, consequently, from eq 12, n will be e0.5. The experimental results verify this as shown in Figure 8, where the average experimental n is 0.40 over the range from r ≈ 0.03 to ∼0.065. Using eq 7 and input data from the isotherm (Figure 7), n )
Figure 7. Isotherm for the Pd0.50Ag0.50 alloy at 473 K.16
Pressure Dependence of H Permeation of Membranes
J. Phys. Chem. C, Vol. 114, No. 34, 2010 14487 Since rup can be varied independently of rdown, and vice versa, it follows that
ln const + ln DH/ + ln F(rup) ) n ln pup + ln DH
(17) and
ln const + ln DH/ + ln F(rdown) ) n ln pdown + ln DH (18) Figure 8. ln Psp × 108 against ln p for the Pd0.50Ag0.50 alloy at 473 K. The solid line without data points represents ideal behavior with a slope of 0.50, and the dashed curve is drawn through the experimental data.
Since the constant term is the same in eqs 17 and 18, the equations can be subtracted to give an expression for n:
n) 0.40 at r ≈ 0.036 and about 0.41 at r ≈ 0.053. The calculated results are close to the experimental ones and demonstrate the validity of eq 7 and the fact that n < 0.5 for this alloy. The only experimental report of n < 0.5 appears to be the results for Pd0.80Ag0.20 and Pd0.70Ag0.30 alloy membranes at 473 and 573 K for the former and 673 K for the latter at pH2 > 0.68 MPa.30 The present data for the Pd0.77Ag0.23 alloy did not reach these pH2 values (Table 4). n for pdown * 0. This section is concerned with a thermodynamic expression for n when pdown * 0. The experimental value of n needed for the equality to hold in eq 13 n J ) DHK(pnup - pdown )/d
(13)
is an average value reflecting nonideality for both pup and pdown over the pH2 range employed. After eq 13 has been made a near equality by choosing the appropriate n, DH will be an apparent n n - pdown )/d) is constant because the slope of J against ((pup constant in the steady state. Since DH varies through the membrane during steady-state permeation, the experimental value from eq 13 will be an average over the membrane and not, of course, equal to D*H except when pup and pdown are small enough for the solution to be ideal. The flux can be written as
J ) DH(cH,up - cH,down)/d ) KDH(rup - rdown)/d
(14) where r has been substituted for cH in the second equality and K ) F/M where F and M are the density and molar mass of the metal-H solution, respectively. For pdown * 0, eq 5 must be rewritten as
DH ) DH/ (
∫rr
up
down
f(r) dr)/(rup - rdown) ) DH/ (F(rup) F(rdown))/(rup - rdown)
(15)
Substituting this into eq 14, eq 16 is obtained: n J ) KDH/ (F(rup) - F(rdown))/d ) K′DH(pnup - pdown )/d (16)
(
ln F(rup) - ln F(rdown) ln pup - ln pdown
)
(19)
If (ln pup - ln pdown) is varied over a range, eq 19 can be employed to calculate an n value, which will be an average for the pH2 range investigated. Therefore n can be calculated from thermodynamic data to compare to n determined in the usual way, i.e., by choosing the best n value to make eq 13 an equality. Thin Membranes. Equations 7 and 19 may be of especial use for the understanding of permeation through thin membranes where often n * 0.50. The observation that n > 0.50 has been postulated to be due to the relative importance of surface steps when bulk permeation becomes very fast. If the experimental n values are greater than the theoretical ones calculated from eqs 7 or 19, it can be concluded that there is a surface contribution, assuming that mass transport through a porous support is not rate controlling. For example, in a recent study of thin Pd membranes by Zhang et al.,7 n was found to be >0.5 and attributed to some surface control. They found that n was 0.86 (473 K) for a microchanneled Ni-supported 2.5 µm Pd membrane. Their experiments were made with pdown ) 0 so that eq 7 can be employed. In a similar pH2 range, the calculated n (eq 7) is about 0.55 (Table 1) and therefore n ) 0.86 is too large for bulk diffusion control, supporting their assumption that there is a surface component. A quantitative estimate of the role of the surface step can be made. Following Ward and Dao,18 it will be assumed that, for a purely surface-controlled reaction, n ) 1.0, and then the observed value of n at 473 K found by Zhang et al.7 of 0.86 can be equated to fs × 1.0 +(1 - fs)0.55 or fs ) 0.69, which is the fraction occurring as the slow surface reaction. At 673 K, from their data and the calculated n ) 0.52 (Table 1), fs ) 0.44. These estimates do not seem unreasonable. Conclusions The common practice of determining n by choosing it to n obtain a linear plot of J against pup (pdown ) 0) or else against n n (pup - pdown), where n * 0.50, seems to be useful only if a theoretical value of n based on bulk diffusion in a nonideal solution (eqs 7 or 19) is available, because then it can be decided whether bulk diffusion is the slow step. Otherwise, when n * 0.5, it cannot be concluded that steps other than diffusion are the slow steps. In this paper, equations for n are given in terms of the thermodynamic properties of the system for both pdown ) 0 and * 0. These equations for n include the effects of nonideality due to r * KspH1/22 , Sievert’s law, and to the concentration dependence of DH, Fick’s diffusion constant.
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Although predictions from eq 7 have been compared to experimental n values for relatively thick membranes, eq 7 should also be valid for thin membranes, which are of most recent interest for H2 purification. Reasonably good agreement is found here between calculated and experimental values of n (eq 7) for Pd membranes. For pure Pd membranes, at 613 K, n ≈ 0.5 up to about 2 bar but at 473 K, deviations from 0.5 occur at pH2 > 0.5 bar. It is shown for two Pd-Ag alloys of different compositions that calculated n values also agree reasonably well with the experimental values. For the Pd0.50Ag0.50 alloy, n < 0.5, which differs from the usual case where n g 0.50. With reference to H diffusion through Pd-Ag membranes, Hou et al.31 noted recently that “the value of 0.614 for the H2 pressure exponent, rather than the value of 0.5 expected from a purely diffusive flux through the Pd/Ag film, as predicted by Sievert’s law, indicates that processes other than the diffusion of H atoms play a role in the overall control of the permeation rate”. The present work has shown that it does not necessarily follow that, if n > 0.5, processes other than diffusion are a factor. Acknowledgment. The authors are indebted to one of the reviewers pointing out the work of Grashoff et al.30 References and Notes (1) Paglieri, S.; Way, J. Sep. Purif. Methods 2002, 31, 1. (2) Eshelby, J. Solid State Physics 1956, 3, 79. (3) Alefeld, G. Ber. Bunsen-Ges. Phys. Chem. 1972, 76, 746. (4) Hurlbert, R.; Konecny, J. J. Chem. Phys. 1961, 34, 655. (5) deRosset, A. Ind. Eng. Chem. 1960, 52, 525. (6) Morreale, B.; Ciocco, M.; Enick, R.; Morsi, B.; Howard, B.; Cugini, A.; Rothenberger, K. J. Membr. Sci. 2003, 212, 87. (7) Zhang, Y.; Gwak, J.; Murakoshi, Y.; Ikehara, T.; Maeda, R.; Nishimura, C. J. Membr. Sci. 2006, 277, 203. (8) Li, H.; Xu, H.; Li, W. J. Membr. Sci. 2008, 324, 44.
Flanagan and Wang (9) Lacher, J. Proc. R. Soc. London, Ser. A 1937, 161, 525. (10) Wicke, E.; Nernst, G. Ber. Bunsen-Ges. Phys. Chem. 1964, 68, 224. (11) Kuji, T.; Oates, W.; Bowerman, B.; Flanagan, T. J. Phys. F: Met. Phys. 1983, 13, 1785. (12) Blaurock, J. Ph.D. Dissertation, Universita¨t Mu¨nster, 1985. (13) McLeod, L.; Degertkin, F.; Fedorov, A. J. Membr. Sci. 2009, 339, 109. (14) Darling, A. Proceedings of the Symposium on Less Common Means of Separation, Birmingham, England, April 1963; Institute of Chemical Engineers: London, 1963. (15) Yoshida, H.; Konishi, S.; Naruse, Y. Nucl. Technol. Fusion 1983, 3, 471. (16) Wang, D.; Flanagan, T.; Shanahan, K. J. Phys. Chem. B 2008, 112, 1135. (17) Yamakawa, K.; Ege, M.; Ludescher, B.; Hirscher, M.; Kronmu¨ller, H. J. Alloys Compd. 2001, 321, 17. (18) Ward, T.; Dao, T. J. Membr. Sci. 1996, 153, 211. (19) Collins, J.; Way, J. Ind. Eng. Chem., Res. 1999, 32, 3006. (20) Huang, T.-C.; Wei, M.-C.; Chen, H.-I. Sep. Sci. Technol. 2001, 36, 199. (21) Flanagan, T.; Wang, D.; Shanahan, K. J. Membr. Sci. 2007, 306, 66. (22) Wicke, E. Brodowsky, H. Hydrogen in Metals; Alefeld, G., Vo¨lkl, J., Eds.; Springer-Verlag: Berlin, 1978. (23) Seymour, E.; Cotts, R.; Williams, W. Phys. ReV. Lett. 1975, 35, 165. (24) Wipf, H. Hydrogen in Metals, III, Wipf, H. Ed.; Springer-Verlag: Berlin, 1997. (25) Fukai, Y. The Metal-Hydrogen System, 2nd ed., Springer-Verlag: Berlin, 2005. (26) Flanagan, T.; Wang, D. Scr. Mater. 2007, 56, 261. (27) Oates, W.; Flanagan, T. J. Mater. Sci. 1981, 16, 3235. (28) Koffler, S.; Hudson, J.; Ansell, G. Transition Met. Soc. AIME 1969, 245, 1735. (29) Amandusson, H.; Ekedahl, L.; Danneun, H. J. Membr. Sci. 2001, 193, 35. (30) Grashoff, G.; Pilkington, C.; Corti, C. Platinum Met. ReV. 1983, 27, 157. (31) Hou, K.; Hughes, R. J. Membr. Sci. 2003, 214, 43.
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