5750
J. Phys. Chem. B 2000, 104, 5750-5761
Diffusion in Porous Solids: The Parallel Contribution of Gas and Surface Diffusion Processes in Pores Extending from the Mesoporous Region into the Microporous Region Sebastia´ n C. Reyes,* John H. Sinfelt,† and Gregory J. DeMartin Corporate Research Laboratories, Exxon Research and Engineering Company, Annandale, New Jersey 08801 ReceiVed: September 3, 1999; In Final Form: March 10, 2000
When a species diffusing in a porous solid is adsorbed to some extent, the ratio of adsorbed molecules to gas molecules in the pores may be very high if there is a large surface area associated with the pores. In addition to transport of molecules through the pores by gaseous diffusion, transport by surface diffusion may then become important. Good examples have been reported in the literature. However, there are many cases in which the mode of transport of an adsorbing species is limited to gaseous diffusion. Although the overall process is simpler when gaseous diffusion alone must be considered, the experimental diffusion coefficient determined directly by classical nonequilibrium methods is still not a true diffusion coefficient. Because of the complication of the “adsorption capacity factor”, the experimental value is lower than the true value, and may be as much as orders of magnitude lower. For mesoporous solids this has been recognized by various investigators for some time. For certain microporous solids, namely zeolites with pore diameters of the order of a nanometer and frequently approaching the dimensions of very small molecules, there have been a number of reported examples where experimental values of diffusion coefficients obtained under nonequilibrium conditions are much lower than values obtained under equilibrium conditions by NMR. We suggest that the discrepancies in such examples can be understood if one attributes much of the total transport of molecules to a gaslike diffusion process in the pores of the zeolite. An adsorption capacity factor is then operative. For cases where experimental diffusion coefficients obtained under equilibrium and nonequilibrium conditions exhibit good agreement, which frequently involve zeolites with the smallest dimensions, we suggest that gas-phase diffusion no longer makes a contribution.
Introduction Porous solids are widely applied as adsorbents, membranes, and catalysts. Understanding the diffusion of molecules in such materials is important for analyzing the factors limiting their performance. Achieving such an understanding requires knowledge of the adsorptive properties of the materials and the extent to which transport of molecules within the pores involves surface diffusion as well as gas-phase diffusion.1-16 The dimensions of the pores and of the diffusing molecules, and the interactions of the latter with the pore surfaces, are clearly important issues. The experimental determination of diffusion coefficients is also an important issue. Traditionally, diffusion coefficients have been determined by macroscopic methods in which concentration gradients are present within a sample, under either steadystate or unsteady-state conditions.17-20 A microscopic method, nuclear magnetic resonance, has the feature that diffusional information can be obtained in the absence of concentration gradients, i.e., under equilibrium conditions.21-25 The differences frequently observed in diffusion parameters obtained by different methods are not clearly understood. The main features of macroscopic and microscopic methods as well as the discrepancies in diffusivity parameters have been discussed extensively by leading investigators in the field.26- 31 In this paper, we examine the issues in question and make some suggestions that we hope will be helpful in improving the present understanding of the issues. We begin by considering * Corresponding author. Voice, (908) 730-2533; fax, (908) 730-3198; e-mail,
[email protected]. † Senior Scientific Advisor Emeritus.
the phenomenological description of diffusion in porous solids. Next, we consider the matter of apparent diffusion coefficients and how they are related to true diffusion coefficients, with emphasis on adsorption effects. The relevance of this in providing some understanding of the differences in situations leading to agreement vs disagreement of diffusion parameters, determined on one hand by traditional methods and on the other hand by NMR, is then examined. Well-substantiated examples of agreement in some cases and of huge discrepancies in others are drawn from the literature in making this examination. Phenomenological Description of Diffusion in Porous Solids In general, diffusion in a porous material can be considered as the parallel occurrence of two processes, a gas-phase diffusion process within the pores and a surface diffusion process along the pore walls.3-9,11-16 If we adopt the gradient of the chemical potential µ as the driving force for diffusion,32-35 the total net flux B J of molecules through a unit cross-sectional area of the pore voids in a particle is given by the equation
B J ) BgC∇ B µg/NA + βBsn∇ B µs/NA
(1)
where the first term is the contribution of gas-phase diffusion to the flux and the second term is the contribution of surface diffusion. The gradients ∇ B µg and ∇ B µs of the chemical potentials of the gas-phase and adsorbed molecules, respectively, may be regarded as forces responsible for the two diffusion processes.36 The force on a single molecule, driving its transport in either
10.1021/jp9931354 CCC: $19.00 © 2000 American Chemical Society Published on Web 05/31/2000
Diffusion Processes in Porous Solids
J. Phys. Chem. B, Vol. 104, No. 24, 2000 5751
the gas phase or on the surface, is obtained by dividing the appropriate gradient by Avogadro’s number NA, as is done in both terms in eq 1. The quantities Bg for the gas-phase molecules and Bs for the adsorbed molecules are the mobilities, where mobility is defined as the diffusion velocity per unit force,37 similar to the definition of ionic mobility in an electrolytic solution as the velocity per unit electric field.38 The quantity C in eq 1 is the concentration of molecules in the gas phase in the pores, expressed as molecules per cm3 of pore volume, and n is the number of adsorbed molecules per cm2 of surface. In the second term of eq 1, the factor β is the average surface-tovolume ratio of the pores. The chemical potentials µg and µs are given by the equations
µg ) µg0 + RT ln fg
(2)
µs ) µs0 + RT ln fs
(3)
where fg and fs are the fugacities of the molecules in the gas and adsorbed phases, respectively, and R and T are the gas constant and the absolute temperature. The quantities µg0 and µs0 are the values of µg and µs in the standard states of unit fugacity. For the molecules in both the gas and adsorbed phases, the standard state is taken to be the molecules in the gas at a pressure of one atmosphere. Applying eqs 2 and 3, we can write eq 1 as follows:
B ln fg + βBskTn∇ B ln fs ) B J ) BgkTC∇ B fg + βDs(n/fs)∇ B fs (4) D(C/fg)∇ The quantities D ) BgkT and Ds ) BskT are the diffusion coefficients37 of the gas-phase and adsorbed molecules, respectively, and k, equal to R/NA, is the familiar Boltzman constant. If the gas-phase behaves ideally, fg is simply equal to the gas pressure p, which in turn is equal to CkT. When these substitutions are made in the first term of the right-hand member of eq 4, and when the second term is written in a slightly different way, the equation becomes
B J ) D∇ B C + βDs(d ln fs/d ln n)∇ Bn
(5)
Equation 5 is an expression of Fick’s first law of diffusion, as applied to parallel diffusion processes in which part of the total flux is associated with the concentration gradient ∇ B C in the gas phase and part with the gradient ∇ B n on the surface. The time dependence of the total concentration of molecules (those in the gas phase in the pores plus those on the surface) at a given position in the particle, expressed as molecules per cm3 of pore volume per sec, is given by the equation
∂ (C + βn) ) ∇ B ‚D∇ BC+∇ B ‚[βDs(d ln fs/d ln n)∇ B n] (6) ∂t which is identified with Fick’s second law of diffusion. The idea that diffusion of a species is due to a difference in its chemical potential from one point to another in a system is an old one, as pointed out in the classical treatise of Glasstone, Laidler, and Eyring in 1941 on the theory of rate processes.35 In that work, reference is made to a seminal paper of Onsager and Fuoss32 in 1932 and to related papers of other workers which appeared later in the 1930s.33,34 Moreover, long before the 1930s, a famous paper by Nernst39 in 1888 contained the substance of the idea: Nernst related diffusion in solution to osmotic pressure, which in turn is related to the chemical potential.40 The well-
known treatment of diffusion by Einstein in the early 1900s37,41,42 incorporated the earlier ideas of Nernst. The thermodynamic quantity (d ln fs/d ln n) appearing in the term for surface diffusion of the adsorbed molecules in eqs 5 and 6 also appears in expressions for rates of diffusion in solution, as was shown by Onsager and Fuoss32 in 1932. Shortly after the work of Onsager and Fuoss was published, this quantity became an important consideration in investigations of diffusion in ionic solutions.33,34,43,44 More than a decade later, it also found application by L. S. Darken45 in a widely cited paper on diffusion in binary metallic solutions. Darken’s paper subsequently came to the attention of investigators concerned with diffusion in porous solids, who then began to consider the quantity in their interpretations of diffusion data in these systems. Reference to this quantity as the “Darken correction” is common in many papers on diffusion in zeolites,23,26-29,31,46-70 despite the fact that others before Darken had considered the relevance of this quantity in diffusion phenomena. The quantity arises naturally as soon as one adopts the gradient of chemical potential as the driving force for diffusion. Our comments here are made in the interest of historical perspective and accuracy. They are not intended to detract from the valuable contributions of either Darken or the scientists investigating diffusion in zeolites who frequently make reference to Darken’s work. Returning to a further consideration of eqs 5 and 6, we note that experimental studies of diffusion in porous solids require some knowledge of the adsorptive properties of the materials for interpretation of the data, as is evident from the appearance of the quantities n and (d ln fs/d ln n) in these equations. For purposes of discussion, it is useful to consider the physical situation in which adsorption-desorption equilibrium is established in the pores. The fugacity fs of the adsorbed molecules is then equal to the fugacity fg of the gas molecules, which in turn is equal to the gas pressure p when the gas behaves ideally. Equations 5 and 6 then reduce to eqs 7 and 8:
B J ) [D + β(n/C)Ds]∇ BC)D h∇ BC
(7)
∂ ∂C (C + βn) ) [1 + β(dn/dC)] ) ∇ B ‚D h∇ BC ∂t ∂t
(8)
The quantity D h is commonly called the permeability and is given by the expression
D h ) D + β(n/C)Ds
(9)
The adsorption-desorption equilibrium is characterized by an adsorption isotherm. The quantity (n/C) is the ratio of the concentration of adsorbed molecules to the concentration of gas molecules at a given point on the isotherm, and (dn/dC) is the correponding slope at that point. At sufficiently low concentrations, the two quantities approach the same constant value, commonly known as the Henry’s law constant. We note that the quantity (n/C) ∇ B C is equal to (d ln C/d ln n) ∇ B n, which contains the derivative (d ln C/d ln n). Investigators routinely divide experimental diffusion coefficients by this derivative to obtain “corrected diffusivities” in cases (e.g., diffusion in zeolites) where it is perceived by the investigators that a single mode of transport (surface diffusion) is operative in the pores. 23,26-29,31,46-70
If the quantity D h can be treated approximately as a constant during a diffusion experiment, eq 8 is simplified to the expression
5752 J. Phys. Chem. B, Vol. 104, No. 24, 2000
∂C ) D*∇2C ∂t
Reyes et al.
(10)
where ∇2 is the familiar Laplacian operator and D* is the apparent diffusion coefficient defined by the expression
D* )
D + β(n/C)Ds D h ) 1 + β(dn/dC) 1 + β(dn/dC)
(11)
We see that D* depends on the quantities (n/C) and (dn/dC) obtainable from the adsorption isotherm for the gas-solid system under investigation. Macroscopic Measurements of Diffusion Coefficientss The Effects of Adsorption Traditionally, diffusion coefficients for the transport of gases in porous solids have been obtained from direct measurements of diffusion rates of molecules through a sample (steady-state experiments) or from concentration measurements as a function of time at a given point (unsteady-state experiments). In general, the diffusion coefficients obtained from such measurements are apparent, rather than true diffusion coefficients. Thus, a steady-state experiment based on eq 7 can be used to obtain data on D h . If both gas-phase and surface diffusion contribute to transport in the pores, both the true diffusion coefficient D and Ds will make contributions to D h . Identification of D h with a true diffusion coefficient is possible only if surface diffusion makes a negligible contribution, in which case D h is equal to the diffusion coefficient D for gas-phase diffusion in the pores. If surface diffusion dominates the transport, D h approaches the quantity β(n/C)Ds in value rather than the surface diffusion coefficient Ds itself. In the determination of D h from a steady-state diffusion experiment, one simply imposes a difference in concentration across a plug or pellet of the porous material and measures the flux of molecules through the sample. For the porous solids of interest here, the pores are too small for Poiseuille flow to be of any importance. Unfortunately, steady-state flux measurements present difficulties for certain types of important materials, for which it is difficult to prepare a sample of suitable thickness which truly reflects the diffusional resistance that one wishes to measure. Examples of such materials are microporous zeolite crystals. For these materials the diffusional process of interest occurs within the zeolite crystal, rather than in pores between such crystals in a polycrystalline aggregate or pellet of the material. A desirable sample for a steady-state flux measurement might conceivably be a thin single-crystal specimen with a thickness of the order of 1 mm and a cross-sectional area of perhaps 1 cm2. Unfortunately, the sizes of zeolite crystals that can be obtained are generally a couple of orders of magnitude too small for such a sample. There are only few reports in the literature where diffusion measurements have been carried out in single-crystal specimens.71-74 Unsteady-state diffusion measurements based on eq 10 do not involve the direct determination of a diffusion flux through a sample and are not limited by a difficulty of the type mentioned in the previous paragraph. The diffusion coefficient obtained in a measurement of this kind differs from the coefficient D h obtained in a steady-state experiment by the factor [1 + β(dn/dC)]-1 of eq 11. When the quantity β(dn/dC) is large compared to unity, D* is much smaller than D h . In an unsteadystate diffusion experiment, these two quantities reflect the time dependences of two different concentrations. The quantity D* relates to the time dependence of the concentration of only the
gas-phase molecules in the pores in a given sample element, whereas D h relates to the time dependence of the concentration of both the gas phase and the adsorbed molecules in the sample element. If a porous solid lends itself to a study of its diffusional properties by a steady-state method, a direct comparison of a value of D h from an experiment with a value of D* from an unsteady-state experiment is not meaningful when adsorption of the diffusing species is significant, i.e., when the quantity β(dn/dC) cannot be neglected in comparison with unity in eq 11. The values of D h and D* are equal only when the diffusing species is not adsorbed, in which case both quantities are identified with the true diffusion coefficient D for gas-phase transport in the pores. The temperature dependencies of D h and D* can sometimes be very different. Consider a situation in which the diffusing species is adsorbed and β(dn/dC) . 1, but transport in the pores occurs almost exclusively by gas-phase diffusion (due to a very small value of the surface diffusion coefficient Ds). In this case, D h ) D and D* ) D/β(dn/dC). The temperature dependence of D h is simply that of D, which is weak (∼T1/2 for Knudsen diffusion). In contrast, the temperature dependence of D* may be relatively strong because the adverse effect of temperature on (dn/dC) imparts an additional positive effect of temperature on D*. If the Henry’s law region of the adsorption isotherm is involved in a particular instance, the ratio D*/D is proportional to exp(-q/RT), where q is the heat of adsorption. Thus, D*/D may on occasion increase exponentially with temperature as if D* represented an activated diffusion process, when in fact it does not. Clearly, if information is not available on the adsorptive properties of a porous solid, one does not have a sound basis for rationalizing any discrepancies in magnitudes or temperature dependencies that may be observed when experimental diffusion coefficients obtained by steady-state and unsteady-state methods are compared. Failure to account properly for adsorption effects can lead to erroneous conclusions regarding the diffusion processes in porous solids. Consequently, measurements of diffusion coefficients must be accompanied by measurements of adsorption isotherms. A departure from traditional macroscopic methods of determining diffusion coefficients, with certain important features in its favor, is based on the use of frequency response techniques. The utility of such techniques is illustrated by a recent paper from this laboratory.75 In a frequency response experiment, the system under investigation is perturbed very slightly from a state of equilibrium. From a measurement of an appropriate response to the perturbation, one can obtain information on diffusion (and adsorption) parameters of the kind of system under consideration here. Consider, for example, a conventional unsteady-state type of experiment for obtaining information on the apparent diffusion coefficient D*. Earlier, we considered the physical situation in which adsorption-desorption equilibrium is established in the pores throughout the experiment. The assumption of adsorption-desorption equilibrium is common practice in the analysis of data on diffusion of gases in porous solids. Although this may indeed be a good assumption in many cases, we must nevertheless recognize that it may not always be applicable. In the application of the frequency-response approach, there is no need to assume adsorption-desorption equilibrium in the analysis of the data. The experiment is started from an equilibrium state and perturbed very slightly (typically (1%) from such a state. The quantity (dn/dC) characterizing the slope of the adsorption isotherm for the particular concentra-
Diffusion Processes in Porous Solids tion defining the initial equilibrium state is determined directly from the frequency response data. From experiments at different concentrations, one can construct the adsorption isotherm for the system and also have the necessary data to determine the diffusion coefficients D and Ds for the system. Consequently, there is no need to do separate experiments to measure adsorption isotherms. All the necessary diffusion and adsorption data are obtained in the frequency-response experiments.64,70,75-77 Diffusion Measurements Under Equilibrium vs Nonequilibrium Conditions Up to this point, our attention has been centered on macroscopic measurements of diffusion coefficients by either steady-state or unsteady-state methods. Because concentration gradients are present in the samples being studied, the measurements are said to occur under nonequilibrium conditions. Measurements of diffusion coefficients can also be made on systems in an equilibrium state, in which there are no net fluxes, no concentration gradients, or no variations of concentrations with time. In such measurements, the molecular motion underlying diffusion is probed directly at the molecular level. Various investigators have employed NMR experiments for this purpose.20-23,25,27-29,56,68,78-88 For studies of diffusion in porous solids, a diffusion coefficient obtained from NMR experiments is a true diffusion coefficient to be identified with either D or Ds in equations such as 9 or 11. Thus, in general, one should not expect to identify a diffusion coefficient from NMR experiments with an apparent diffusion coefficient such as D* in eqs 10 and 11. Such an identification is valid only for limiting cases when D* is a good approximation of either D or Ds. Experimental (apparent) diffusion coefficients, obtained from unsteady-state diffusion experiments (i.e., D* values) on one hand and from NMR studies on the other hand, have been compared for zeolites.26-29,31 Some comparisons show reasonably good agreement with regard to both the magnitudes of the experimental diffusion coefficients and their temperature dependencies. However, other comparisons have revealed major discrepancies, in which diffusion coefficients from unsteadystate measurements are as much as several orders of magnitude smaller than the coefficients derived from NMR experiments. Why do comparisons exhibit good agreement in some cases, but huge discrepancies in others? Equation 11 provides a basis for rationalizing this, despite a predisposition that one may have to abandon a model for diffusion through zeolite crystals in which gas-phase diffusion can still make a substantial contribution to the total transport through the pores. We expect agreement when D* is a good approximation of either D or Ds, because it is then a true diffusion coefficient of the kind obtained in an NMR experiment. A discrepancy is expected when this is not the case. We now examine the physical basis for each of these possible results. Consider first the simplest situation, that in which the diffusing species would not be adsorbed at all. In such a case, the parameter (dn/dC) in eq 11 could be taken as zero, and the apparent (experimental) diffusion coefficient D* from the unsteady-state measurement would simply be equal to D h and also to D. Because D* in this limiting case would be a direct measure of the true gas-phase diffusion coefficient D, one would expect it to agree with a diffusion coefficient from an NMR measurement and to have the same temperature dependence. In the more common situation in which the diffusing species is adsorbed, good agreement can still be observed, provided that surface diffusion is the predominant mode of transport in the pores. The gas-phase diffusion coefficient D is then small
J. Phys. Chem. B, Vol. 104, No. 24, 2000 5753 compared with the quantity β(n/C)Ds in eq 11. If, in addition, the quantity β(dn/dC) in eq 11 is large compared to unity and the applicable values of (n/C) and (dn/dC) are for the Henry’s law region of the adsorption isotherm, the experimental diffusion coefficient D* becomes a good approximation of the true surface diffusion coefficient Ds. Agreement of D* with a diffusion coefficient from an NMR experiment would therefore not be surprising, and again one would expect the same kind of temperature dependence. For cases in which the applicable values of (n/C) and (dn/dC) do not correspond to the Henry’s law region of the isotherm, division of D* by the quantity (n/ C)/(dn/dC), which yields the quantity D*(d ln n/d ln C), should improve the agreement. The latter correction is routinely employed in studies of diffusion in zeolites by investigators who assume that surface diffusion is always the only mode of transport in such materials.23,26-29,31,46-70 This issue will be examined later in the paper. Let us now consider a different type of situation, i.e., one in which the contribution of surface diffusion is small compared to that of gas-phase diffusion in the pores (D . β(n/C)Ds), but where the quantity β(dn/dC) for the system is still large compared to unity. In contrast to the first two situations considered, D* in this situation would not be a reasonable approximation of either of the true diffusion coefficients (D or Ds) for the system, being small compared to D and large compared to Ds. Consequently, one would not expect D* to have a value corresponding to the value of a true diffusion coefficient obtained from an NMR experiment. In fact, its value could be orders of magnitude smaller provided that the quantity β(dn/dC) has a high enough value while the condition D . β(n/C)Ds is still maintained. Moreover, it would exhibit a different dependence on temperature, because it would incorporate the relatively strong temperature dependence of the quantity (dn/dC) from the adsorption isotherm. This matter was treated earlier when we considered the effect of adsorption on the temperature dependence of the ratio D*/D in relation to comparisons of steady-state with unsteady-state measurements of apparent diffusion coefficients. As a result of the foregoing considerations of the problem of diffusion in porous solids, we conclude that many varied types of observations with regard to experimental diffusion coefficients can be understood in a rational manner. The understanding stems from a proper accounting for adsorption in the analysis of diffusion data, and from the simple extension of a sound phenomenological description of diffusion in mesoporous solids to microporous solids. To provide support for our conclusion, we consider in the next section of this paper a number of examples of measurements of diffusion coefficients taken from the literature. They include data obtained in both equilibrium and nonequilibrium measurements. We consider examples in which agreement between the two types of measurements is good, and also examples in which the agreement is very poor. Explanations for differences in the degree of agreement observed in the various examples are proposed on the basis of the discussion that has been presented here. Analysis of Literature Data on Experimental Diffusivities This section considers the literature on diffusion coefficients, with emphasis on the role of adsorption during unsteady-state diffusion measurements. We focus on a selected number of publications that provide the necessary data for examining the applicability of eq 11 in rationalizing why the diffusion parameters determined under equilibrium and nonequilibrium conditions agree in some cases but disagree in others. For a
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Reyes et al.
TABLE 1: Adsorption Capacity Factors as a Function of Temperature for Nitrogen, Xenon, and Isobutane in Mesoporous Silica at 2.13 KPa (r ) 3.5 nm)a adsorption capacity factor: 1 + β(dn/dC) temperature (K)
nitrogen
xenon
isobutane
308 348 398 573
1.27 1.23 1.16 1.04
2.04 1.52 1.39 1.21
8.40 3.85 2.19 1.66
a
Adsorption capacity factors deduced from data in ref 75.
number of materials, we quantify the contributions of gas-phase diffusion and surface diffusion to transport in the pores at various conditions. Because adsorption effects and the contribution of surface diffusion increase in importance as the sizes of the pores in the materials decrease, we consider the materials with the largest pore sizes first. Thus, we begin with an analysis of the diffusion characteristics of macroporous solids. We then proceed to materials with progressively smaller and smaller pores, from mesoporous, to microporous, to the extreme examples of microporous systems embodied by zeolites. The pore sizes we associate with each class of porous materials are intended only to emphasize the concept of surface-to-volume ratio that increases gradually with decreasing pore size. They are not intended to redefine the ranges usually attributed to macro-, meso-, and micropores. The specific pore sizes in each example are explicitly noted in the tables that summarize the respective results. Macroporous Systems (Pore Radii Greater Than 3.0 nm). In a previous publication75 we described the application of a frequency-response technique to the measurement of diffusion coefficients of nitrogen, xenon, and isobutane in silica particles. The key findings of that study were that diffusion occurred by a Knudsen mechanism in the pores and was accompanied by equilibrated adsorption on the silica surfaces. This situation can be seen as a special case of eq 11 where the surface diffusion term vanishes and the magnitude of the denominator achieves values larger than unity (i.e., D* ) D/(1 + β(dn/dC))). Given the central importance of the term (1 + β(dn/dC)) in the discussion that follows, we will refer to the term as the adsorption capacity factor. This factor measures the total rate of change of the number of molecules residing in the gas phase and on the pore surfaces in a given element of the porous solid relative to the rate of change of the number of molecules in the gas phase in that element. Clearly, this factor reduces to unity only when the molecules do not adsorb on the material. Because the frequency-response technique allows for the simultaneous measurement of the diffusivity and the slope of the adsorption isotherm, the adsorption capacity factor can be directly calculated from the experimental data. The only other quantity needed is the average surface-to-volume ratio of the pores (β), which is determined from a standard physical characterization of the material (i.e., specific surface area and density). If the pores in the material are approximately cylindrical in shape, the parameter β may be estimated as 2/r, where r is the mean pore radius. Table 1 summarizes the calculated adsorption capacity factors for nitrogen, xenon, and isobutane at 2.13 kPa and various temperatures. Consistent with the expected adsorption characteristics of the molecules on silica, the adsorption capacity factors are small for nitrogen, moderate for xenon, and large for isobutane. The adsorption capacity factors decrease with temperature, an effect that is consistent with the experimentally measured heats of adsorption of the various gases on the silica material (2.2 kcal/ mol for nitrogen, 3.4 kcal/mol for xenon, and 5.6 kcal/mol for
isobutane). More importantly, the results of Table 1 provide a measure of the extent to which adsorption effects can influence the diffusivity parameters obtained under nonequilibrium conditions. The adsorption capacity factor is the factor by which the true diffusion coefficient (D) is underestimated when an apparent diffusion coefficient (D*) is reported due to the lack of adsorption equilibrium information. This table shows that even for a very weakly adsorbing molecule such as nitrogen, the effect of adsorption should not be neglected in the analysis if accurate diffusion coefficients are desired. This is particularly the case in situations such as the one analyzed here, where the specific surface area of the material is high (∼204 m2/g). In the case of isobutane, the adsorption capacity factor can be large, especially at the lower temperatures where the slope of the adsorption isotherm becomes large. An additional benefit of properly incorporating adsorption information in the analysis is that one can extract the true temperature dependence of the diffusion process rather than a dependence confounded by the adsorption process. We have addressed this issue in detail in another publication.89 In the present examples, the diffusion process exhibited the expected T1/2 dependence characteristic of Knudsen diffusion.75 The previous example emphasizes the importance of accounting for adsorption effects during diffusion measurements under nonequilibrium conditions. Failure to account for the adsorption capacity factor can lead to apparent diffusion coefficients that are much lower than the true values and that exhibit much stronger temperature dependencies. In general, as the defining equation implies, the adsorption capacity factor is a function of pressure, temperature, and the nature of the porous material and of the diffusing molecule. Its magnitude increases as pressure and temperature are decreased. High-surface-area materials that have a high affinity for the diffusing molecules can lead to very large values. The effects of material properties and experimental conditions on the adsorption capacity factors will now be illustrated by an analysis of data from several investigations in the literature. We show that adsorption capacity factors of the order of hundreds or even thousands are fairly common at conditions used in the diffusion measurements. Theoretical90,91 and experimental92 papers that also illustrate well the role of adsorption during transient diffusion measurements are those of Weisz and co-workers. In their experimental investigation, Weisz and Zollinger92 studied the diffusion dynamics of dye-solvent pairs in various silica-alumina particles. They obtained apparent diffusivities by optically tracking the penetration depth of the dye as a function of time. They complemented their dynamic experiments with independent sorption equilibrium measurements to extract the true diffusivities from their apparent values. Even though the materials they used had relatively low surface areas (
