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J. Phys. Chem. C 2009, 113, 2806–2815
Adsorption Entropies and Enthalpies and Their Implications for Adsorbate Dynamics Aditya Savara, Catherine M. Schmidt, Franz M. Geiger, and Eric Weitz* Department of Chemistry and the Institute for Catalysis in Energy Processes, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208 ReceiVed: July 14, 2008; ReVised Manuscript ReceiVed: NoVember 14, 2008
Experimental adsorption data is typically analyzed within the context of a Langmuir adsorption model. Such an analysis can yield values for adsorption entropies which are far too large based on a comparison with predictions from statistical mechanics. This is due to the fact that the Langmuir model does not always adequately replicate the dynamics of the adsorbate, and thus other prototypical models incorporating different dynamical assumptions should be considered. We extend prior work on adsorption entropies by providing a framework for the interpretation of adsorbate dynamics based on a comparison of experimentally determined adsorption entropies, saturation coverages, and enthalpies to those that are predicted for prototypical models of surface dynamics. Models that have been considered in this work include various Langmuir-type adsorption and 2D gas models. We demonstrate that a two-dimensional gas can display a low apparent saturation coverage when adsorption is facilitated by a subset of surface sites. Additionally, we consider a Langmuir replacement reaction and show that a large apparent adsorption entropy can arise from such a situation. Finally, we discuss how the choice of a model affects the range of plausible values for the adsorption entropy and equilibrium constant. Adsorption of acetone on Degussa P25 TiO2 is used as an illustrative example. I. Introduction One of the primary objectives in studying gas adsorption on surfaces and interfaces is to develop a detailed molecular-level understanding of the complex phenomena that can take place during or subsequent to adsorption. The adsorption enthalpy plays an important (and often dominant) role in determining ∆G°ads.1 However, for weakly adsorbed species, which are associated with a small adsorption enthalpy, the adsorption entropy can be a significant factor in determining the free energy of adsorption. As adsorbate motions partially compensate for the loss of degrees of freedom that occurs upon adsorption,2 a quantitative assessment of the entropy and enthalpy of adsorption yields information about the dynamics of the adsorbate, particularly in the case of weakly adsorbed species.2-7 The enthalpy and entropy for adsorption are normally extracted from experimental data that is analyzed in the context of a specific model. However, each model involves specific a priori assumptions for adsorbate dynamics. For example, the classic Langmuir model assumes that adsorption leads to translationally immobile molecules bound to specific sites, while the two-dimensional (2D) gas model assumes that molecules translate across the available surface. The experimental entropy calculated from a given set of data consequently depends on the choice of a model for the adsorption process, which is then used to extract thermodynamic parameters from the experimental data. Thus, a challenge is to determine the most appropriate modelforanalyzingexperimentaldataforaspecificadsorbate-surface system. Here, we show that a comparison of experimentally obtained adsorption entropies with statistical mechanical calculations, along with data on adsorption enthalpies and absolute saturation coverages, can indicate which model (e.g., Langmuir, 2D gas) is most appropriate for a given system and consequently provide * To whom correspondence should be addressed. E-mail: weitz@ northwestern.edu.
insights into adsorbate dynamics.2,6 In addition, we discuss how the dynamics inherent in certain adsorption models can lead to large apparent entropies of adsorption. The current work is expected to be most relevant for weakly bound systems (∆H°ads > -40 kJ) for which entropies of adsorption have a greater influence on the magnitude of ∆G°ads than in chemisorbed systems. Thus, this work does not encompass systems which have significant adsorbate-adsorbate interactions. II. Background A. Adsorption Isotherms. The most common model for reversible gas adsorption is the well-known Langmuir model. The Langmuir isotherm is given by2,7-9
KL )
θ σ ) (1 - θ)P (σmax - σ)P
(1)
Here, KL is the Langmuir equilibrium constant with units of inverse pressure, P is the partial pressure of the adsorbing gas, and θ is the relative surface coverage, which is given by the ratio of the absolute surface coverage of the adsorbate (e.g., molecules cm-2), σ, to the absolute surface coverage of the adsorbate at saturation, σmax. When adsorbates are capable of translating across the surface with negligible activation energies (Ea , RT), they have been approximated as a 2D gas.2,7,8,10 The standard 2D gas model is a Tonks 2D gas (hard disk),11 which includes the finite area occupied by each adsorbate and has an equilibrium constant, K2D, given by eq 1 modified by a coefficient of eθ/(1-θ).2,8 In this work, we use the Tonks 2D gas by convention and for simplicity. The Henderson 2D gas7,12,13 is expected to be a more accurate representation of a 2D gas,7 and we provide the configurational entropy term for a Henderson 2D gas in the Supporting Information. B. Saturation Coverages and Shapes of Isotherms. Saturation of adsorption can occur either when all accessible sites in a repeating lattice are occupied or when adsorption saturates
10.1021/jp806221j CCC: $40.75 2009 American Chemical Society Published on Web 01/26/2009
Adsorption Entropies and Enthalpies due to the close packing of adsorbates with finite size. In this work, we will call the former case a lattice-saturated monolayer, MLL, and the latter case a geometric monolayer, MLG. This distinction is not always well appreciated but can be important in modeling and discriminating between adsorption models. When adsorption occurs on dispersed sites such as defects, saturation can occur at much lower coverage than either 1 MLL or 1 MLG. The Langmuir isotherm, eq 1, can describe all three of the above scenarios.7,14-18 The standard Langmuir model is defined as saturating at 1 MLL.19 Here we use the more general term, “Langmuir-type adsorptions,” to refer to immobile adsorbates which lead to a Langmuir isotherm. For Langmuirtype adsorptions, σmax may then correspond to 1 MLL, 1 MLG, or less than either type of monolayer. While for Langmuir-type adsorption processes low saturation coverages ( -800 kJ is based on the extreme limits for chemisorption enthalpy.85 In most cases, ∆H° > -400 kJ for chemisorption7 and >-40 kJ for physisorption.7,85 c Ea,diff must be ∼eRT for a 2D gas.2,6 Typically, Ea,diff for diffusion is 1/10 to 1/2 ∆H.6,19,86 d These models can additionally include an adsorbate-surface vibration, the entropy of which can be added to the entropy shown in this table. e For a site-confined 2D/3D gas, D is the dimensionality of the box and A is the length of one side of the box. For a standard 2D gas and a 2D gas confined to patches, D is the fractal dimension of the surface. The entropy expression for a 2D gas on patches is identical to that of the standard 2D gas, only with the adsorbate density on the patches used rather than the adsorbate density relative to the whole geometric surface. f A 2D gas confined to patches saturates σmax e 0.5 MLG on the patches, which corresponds to a lower absolute coverage relative to the full BET surface area. g See sections IV.D, IV.E, and the Supporting Information. The guideline of ∆H° for the two-well 2D gas model should not be considered a true upper limit.
and no change in the molecular vibrational and rotational modes of the molecule (see section IV.A). The large difference between the experimentally determined ∆S°ads and the statistical mechanical calculation implies that the Langmuir model provides, at best, an incomplete description of adsorption in this system, and that another adsorption model may provide a better description. With large entropies of adsorption, one common explanation is that the molecule adsorbs to a 2D gas state. Fitting the Tonks 2D gas isotherm to the experimental data does not appreciably change the equilibrium constants or saturation coverages from those obtained by fits of eq 1 to the experimental data in Table 1. Since the Tonks 2D gas model assumes a saturation coverage of ∼1 MLG (section II.A), the size of an acetone molecule can be used to estimate the adsorbate density at saturation. Taking the molecular area of an acetone molecule as 28 Å2,25 the adsorbate density which corresponds to 1 MLG is 5.93(5) × 10-6 mol m-2. The experimentally observed saturation coverage of 2.7 × 10-8 mol m-2 (obtained again by dividing by the BET surface area) thus corresponds to a coverage of ∼0.005 MLG, violating the assumptions of the Tonks 2D gas model. Thus, the standard 2D gas model (a Tonks 2D gas) cannot apply. However, the assumptions of a 2D gas model can be satisfied if it is assumed there are regions on the surface where a Tonks 2D gas exists, while the neighboring regions of the surface do not accommodate the 2D gas. This scenario is called patchwise adsorption,29,30 the physical basis for which is described in section IV.C. To apply a patchwise adsorption model to the data in Table 1, the adsorbate density on the patches at saturation
is set to that of a single geometric monolayer, σmax ) σML-G, which assumes that the 2D gas only resides on patches that correspond to 5/1000 of the BET surface area. Using the calculated adsorbate density for a geometric monolayer of adsorbed acetone of σML-G ) 5.93(5) × 10-6 mol m-2, the equilibrium constants from Table 1, and σ° ) 1.3928 × 10-7 mol m-2 for the 2D gas standard state, we obtain an adsorption enthalpy, ∆H°ads, of -16 ( 7 kJ mol-1 and an adsorption entropy, ∆S°ads, of +140 ( 20 J mol-1 K-1, via a van’t Hoff analysis. Once again, the adsorption entropy is at variance with the negative value expected from statistical mechanics, -19.6 J mol-1 K-1 (See Table 2 and section V). The failure of a Langmuir-type adsorption model, the standard 2D gas model, and the 2D gas adsorbed on patches model to describe the apparent entropy calculated from the equilibrium constants listed in Table 1 motivated us to investigate other simple models for adsorption. In the following sections we present the models we have investigated as possible options when large apparent adsorption entropies are observed. Because these discussions involve prototypical models, we also report the maximum equilibrium constants for the different prototypical models (Table 3). Finally, we provide a flowchart to show the procedure we followed in choosing an appropriate model (Chart 1). IV. Discussion and Derivations of Adsorption Models The models for adsorption discussed in this work are grouped into two categories: Langmuir isotherm-based models are
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TABLE 3: Maximum Equilibrium Constants for Different Adsorption Models at 298.15 K for the Limits Indicated in Table 2a model
m (au) b
Langmuir physisorption w/o vibration
400 kJ Langmuir chemisorption w/o vibrationb 800 kJ Langmuir chemisorption w/o vibrationb Langmuir w/ vibration and site-confined gas Langmuir replacementc Tonks 2D gasd two-well 2D gas low pressure two-well 2D gas high pressure
∆H°ads,min (kJ)
∆S°obs,max (J/mol K)
∆G°ads,min (kJ/mol)
K°max (bar-1)
Kmax (bar-1)
-1
10 ∼-40 ∼-138 ∼+1.02 6.64 × 10 6.64 × 10-1 50 ∼-40 ∼-158 ∼+7.00 5.94 × 10-1 5.94 × 10-1 10 ∼-400 ∼-138 ∼-359 7.85 × 1062 7.85 × 1062 50 ∼-400 ∼-158 ∼-353 7.02 × 1061 7.02 × 1061 10 ∼-800 ∼-138 ∼-759 9.45 × 10132 9.45 × 10132 50 ∼-800 ∼-158 ∼-753 8.46 × 10131 8.46 × 10131 Indistinguishable from “Langmuir adsorption w/o vibration” by K°max. However, an empirical correlation is provided in the literature of -∆S°ads e 51 J mol-1 K-1 - 0.0014∆H°ads, where ∆H° is in kJ mol-1.3,4,6 10 ∼-800 -∞ to ∞ -∞ to ∞ 0 to ∞ 0 to ∞ 50 ∼-800 -∞ to ∞ -∞ to ∞ 0 to ∞ 0 to ∞ 10 ∼-25 ∼-24e ∼-17.9 1.34 × 103 1.23 × 102 50 ∼-25 -31e ∼-15.6 6.00 × 102 5.50 × 101 10 ∼-50 f See Supporting Information. 50 ∼-50f See Supporting Information. 10 ∼-25 ∼-24e ∼-18.4 1.34 × 103 1.50 × 102 50 ∼-25 ∼-31e ∼-16.4 6.00 × 102 5.50 × 101
a Note: the contribution from the translational entropy depends on molecular weight. b In most cases, ∆H° > -400 kJ for chemisorption7 and > -40 kJ for physisorption.7,85 ∆H° > -800 kJ is based on the extreme limits for chemisorption enthalpy.85 c The other thermodynamic limits for the Langmuir replacement model are independent of which value is chosen for ∆H°. d The limits are the same for a Tonks 2D gas confined to patches. e Includes configurational entropy and assumes that the area of coverage of the molecule is 100 Å2. f The guideline for ∆H° for the two-well 2D gas model is not a true upper limit. The true limit is dependent on parameters discussed in the Supporting Information, particularly the ratio of the number of deep well sites to shallow well sites.
described in sections IV.A and IV.B, while 2D gas-based models are described in sections IV.C and IV.D. We discuss the requirements of these models and the dynamics of the adsorbate for these models. The constraints inherent in the models and the equations for the entropies that arise from the different models are listed in Table 2. A comparison of equilibrium constants and entropy limits (Table 3) are provided in section V. As discussed below, a comparison of the adsorption entropy derived from fitting experimental data with a given model and that from statistical mechanical calculations is essential to the evaluation of the plausibility of a model. A. Expected Change in Entropy for Langmuir-Type Adsorptions. The largest change in entropy upon adsorption is generally the result of constraints on the translational degrees of freedom.3 The loss of molar entropy associated with the gas phase translational degrees of freedom can be calculated from statistical mechanics using the Sackur-Tetrode equation (shown in Table 2),2,31 and is typically on the order of 100 J mol-1 K-1 for small molecules. However, formation of a new vibrational mode between the adsorbate and the surface offsets some of this loss in entropy. The gain in entropy from this vibrational mode of an immobile molecule is typically e12.5 J mol-1 K-1,3 which is quite small by comparison to the change in entropy associated with the loss of translational motion. Experimental data that yield an adsorption entropy ∆S°ads > -S°gas,trans + 12.5 J mol-1 K-1 should evoke questions regarding the source of the apparent “extra” entropy. Here, such situations will be referred to as having “large apparent adsorption entropies,” which refers to the arithmetic value of the adsorption entropy and not the absolute magnitude of the change in entropy (i.e., -10 is larger than -15). In such cases, it is instructive to evaluate the applicability of the model being used in the analysis and to consider related models. In a Langmuir-type adsorption, the rotations of a polyatomic molecule may be hindered, and when they are, the entropy of the adsorbate is expected to decrease2,32-35 by up to tens of J mol-1 K-1.35-37 The minimum loss of entropy due to these factors will occur when there are no changes in the number of rotational degrees of freedom and intramolecular vibrations subsequent to adsorption.2,5,33,34 The assumption that there are no changes in the internal modes has been found to successfully describe a variety of systems, including some systems with
polyatomic adsorbates.5,8 In addition, there is also a possible entropy gain from the newly formed adsorbate-surface vibration; the expression for this entropy gain is shown in Table 2.3,8,38 While this vibration typically provides an entropy contribution of e12.5 J mol-1 K-1,3 for weakly bound species8 this contribution can be as large as tens of J mol-1 K-1. For nondissociative molecular adsorption, ν is often approximated as the Arrhenius pre-exponential for desorption, with a typical range of 109-1019 s-1, and ∼1013 s-1 cited as the most common value.3,22,39,40 The frequency of the adsorbate-surface vibration correlates with the strength of the adsorbate-surface bond40 and thus with the adsorption enthalpy.41 An exception exists if the molecule is so weakly bound that its motion approaches free translations within the confines of the distance the bond can stretch. In this case, the motions of the adsorbate are better approximated as a translation within a box.22 This scenario has been referred to as a “site-confined” 2D/3D gas,2 or a “hybrid” model,25 the latter term arising because the model combines discrete Langmuir adsorption sites with translational motion. The entropy for a site-confined 2D/3D gas is provided in Table 2.2,3 Such a box is typically estimated to have an area the size of the adsorbate, typically 10-20 Å2.2,3,7,22 We point out that that such an estimate often exceeds the square of the sum of the atomic van der Waals radii42 and thus can lead to an overestimate of the entropy of the adsorbate. Since most weakly bound species are expected to have motions on the surface that fall between the two cases of binding described above (vibration versus box), the entropy from adsorbate motions would also be expected to fall between the stated limits.22 B. Langmuir Replacement Reactions. A Langmuir replacement reaction is another possible explanation for a system that exhibits a Langmuir-type isotherm but for which a van’t Hoff analysis can yield adsorption entropies on the order of positive hundreds of J mol-1 K-1 when modeled as a simple Langmuir adsorption. Langmuir replacement reactions are special cases of competitive adsorption in which (a) the impinging molecule initiates the desorption of a previously adsorbed molecule of another compound and replaces it (an associative Langmuir replacement) or (b) the impinging molecule adsorbs at a site that has become unoccupied due to desorption of a different molecule, and thereby prevents readsorption of the previously
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adsorbed molecule (a dissociative Langmuir replacement). The terms “associative” and “dissociative” are analogous to those for ligand exchange in coordination chemistry. Within this mechanism, the entropy loss associated with the adsorption of the impinging molecule is offset by the entropy gain from desorption of the previously adsorbed molecule. Due to the different dimensionality of the standard states between a Langmuir replacement reaction and a Langmuir adsorption (described in the Supporting Information), the replacement reaction can yield a large apparent entropy of adsorption if the process is erroneously modeled as a simple Langmuir adsorption. To our knowledge this explanation for a large apparent entropy of adsorption has not been presented in the literature. In the Supporting Information, expressions are derived showing how a Langmuir replacement mechanism can give rise to a large apparent entropy of adsorption. The Langmuir replacement model is then applied to the data in Table 1 with the assumption that acetone must replace another adsorbate previously bound to the surface, such as water. We conclude that the observed adsorption entropy obtained from the basic Langmuir model, ∆S°obs, is related to the adsorption entropy obtained from the Langmuir replacement reaction, ∆S°rep, by
∆S◦obs ) ∆S◦rep + R ln(P ° /PW)
(2)
where Pw is the pressure of the previously adsorbed gas. The adsorbate being replaced is associated with a gas phase pressure given by
PW ) P ° exp[(∆S◦rep - ∆S◦obs)/R]]
(3)
We note that adsorption enthalpies which are small in magnitude and accompanying large apparent adsorption entropies are reasons to consider the possibility of a Langmuir replacement reaction. Defect sites are common sites of electronic unsaturation43 which may stabilize and/or be stabilized by adsorbed species,44-49 and thus are candidates for Langmuir replacement reactions. In the Supporting Information, we show that based on the water partial pressure that would have to be present to account for replacement reactions in our acetone adsorption studies involving Degussa P25, ∼1.54 × 10-4 torr, a Langmuir replacement reaction is unlikely for the data in Table 1. Although we have presented this model because it can give rise to an extremely positive ∆S°obs when Pw , P°, we note that a Langmuir replacement reaction does not guarantee a large apparent adsorption entropy, and in fact, the model can also give rise to an extremely negative ∆S°obs when Pw . P°, as can be seen from eq 2. C. 2D Gases and 2D Gases Confined to Patches. Due to the entropy associated with 2D gas models relative to an immobile adsorbate, 2D gas models have sometimes been invoked to explain large apparent adsorption entropies for weakly adsorbed species.2,5,8,50-52 As alluded to in section II.A, for adsorbates to be characterized as a 2D gas requires an activation energy for diffusion, Ea,diff, of less than ∼RT.2,6 In this case, the entropy associated with translational motion is proportional to the 2D gas density.2,6,53,54 The consequences of Ea,diff > RT are discussed in the Supporting Information.6,8 The surfaces of materials are not perfect planes and can have a characteristic roughness at the molecular scale which can be quantified by a nonintegral fractal dimension.55-57 We previously pointed out25 that surface roughness could result in a larger (or smaller) adsorption entropy for a 2D gas if the 2D gas accesses a nonintegral dimensionality higher (or lower) than two.2,58 As shown in Table 2, the entropy of a 2D gas is then a function of
Figure 2. Schematic diagram of the potential on a surface for the twowell adsorption model. The surface is composed of two different types of sites (potential energy wells), and the number of shallow wells greatly outnumbers the number of deep wells. Species adsorbed in deep wells are termed A, and species adsorbed in shallow wells are termed B.
both the 2D gas density and the dimensionality accessible to the adsorbate. The packing and volume occupied by the molecules do not affect the translational partition function of a 2D gas and are included in the expression for the configurational entropy described in section V. For the hybrid model (section IV.A), fractal dimensions due to surface roughness are not applicable because the adsorbates are confined to sites that are too small to sample surface fractal dimensionality.58,59 As mentioned in section II.B, a 2D gas is not expected to saturate until a coverage of >0.5 MLG. However, as indicated in section III, lower absolute saturation coverages are possible if the 2D gas adsorbs only on specific regions of a surface such as when adsorbates are confined to either a particular crystal face2,7,19,60-62 or segregated regions on the surface consisting of discrete sites referred to as patches.29,30 These patches may be large enough for the adsorbate to sample the fractal dimensionality of the surface. While patchwise adsorption is possible, most cases of adsorption heterogeneity are due to adsorption on separate crystal faces19 or on a distribution of sites with differing adsorption energies.29,30,63 The treatment for adsorption confined to distinct crystal faces is the same as that for adsorption on patches, while a detailed consideration of sites with many differing adsorption energies63 is beyond the scope of this paper. D. Two-Well 2D Gas Model. We now consider whether it is possible to have a low apparent saturation coverage (,0.5 MLG, section II.B) while retaining the entropy associated with a low-density 2D gas that accesses the full geometric surface: Such a model could explain the data in Table 1.25 In this section, we introduce the isotherm for such a 2D gas and show that such a situation could occur with physically realistic kinetic and thermodynamic parameters (see Supporting Information). In this model, isolated sites facilitate adsorption to the 2D gas statesin effect, a type of “portal model.”64,65 We start by introducing a two-well model, where the surface is composed of two types of adsorption sites or potential energy wells (Figure 2). In this model the vast majority of the surface (e.g., 0.99 MLG) is composed of shallow adsorption wells, while the minority (e.g., 0.01 MLG) is composed of deep adsorption wells. The adsorbates are freely mobile between shallow wells, behaving as a 2D gas since RT is larger than the well depth. However, adsorbates in the deep wells are only mobile through an activated hopping mechanism, since the well depth for these sites is larger than RT. The reactions needed to describe this model are
P + SA h A
(4)
P + SB h B
(5)
B + SA h A + SB
(6)
where P represents the species in the gas phase, SA and SB represent the adsorption sites for the deep and shallow wells, respectively, and A and B represent the same molecule adsorbed at the deep and shallow wells, respectively.
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CHART 1: Flowchart for Adsorption Dynamics
The derivation of this model is presented in Appendix 1. The enthalpic advantage of the occupancy of the smaller number of deep well sites is overcome by the entropic advantage obtained by the large number of configurations resulting from occupancy of the large excess of shallow well sites. Under these circumstances, adsorbates in the shallow well (the 2D gas state) are the dominant species. The derivation also reveals that this situation can give rise to a 2D gas with the functional form of a Langmuir isotherm at very low coverage. This case occurs provided that (1) adsorption into the deep well has the form of a Langmuir isotherm, (2) the concentration of the species in the shallow wells scales linearly with the concentration of the species adsorbed in the deep wells ([B] ) [A]*constant, thereby also giving an apparent Langmuir isotherm), and (3) adsorbates in the shallow well constitute >90% of the adsorbed molecules even at low surface coverages. Thus, the experimentally observed isotherm for all adsorbed molecules (i.e., [A] + [B]), which we will call the “total isotherm,” reflects the population of shallow well species with a negligible contribution from adsorbates in the deep wells. This result can functionally be described as a 2D gas with a Langmuir isotherm and apparent saturation at very low coverages (Appendix 1), followed by an approach to complete saturation of adsorption (1 MLG) at higher pressures described by a 2D gas isotherm. Figure 3 shows an isotherm for the two-well 2D gas model using physically realistic parameters that are discussed in the
Supporting Information. The data shown in Figure 3 exhibits apparent saturation at very low coverages, observed as a flat plateau that covers several orders of magnitude of pressure, before the isotherm begins increasing at higher pressures. This
Figure 3. Two-well 2D gas isotherm. Filled circles: Population of deep well species, A, normalized to the full surface coverage (eq A13a). Open circles: Isotherm for a standard 2D gas, shown for comparison. Solid line: Two-well 2D gas total isotherm is composed of the sum of both deep well species (filled circles) and shallow well species (not shown for clarity) normalized to the full surface area. The total isotherm is effectively a reflection of the shallow well species, B, which are >90% of the species at all pressures (eq A1-3b). The total isotherm resembles a Langmuir isotherm at low pressures (compare to filled circles) and a regular 2D gas isotherm at high pressures (compare to open circles). The parameters used and methods for solving eqs A13b and A1-3a are included in the Supporting Information.
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apparent asymptote at very low coverages has a Langmuir form described by (see Appendix 1)
KB-obs )
(
)
k1 [B] ) k-1 + k-3[SB] ([B]max-app - [B])[P]
(7)
where the variables are as defined in Appendix 1. In this verylow-coverage regime, adsorption to the deep wells dominates and is followed by diffusion to the shallow wells. The second asymptote at higher pressures is due to the formation of a geometric monolayer coverage (1 MLG), which occurs when the gas phase pressure is high enough for adsorption directly into the shallow wells. In this higher pressure regime, the surface coverage and its approach to 1 MLG is given by the 2D gas isotherm (eq 4), with an equilibrium constant of KB (Appendix 1). Note that even in the very-low-coverage regime the number of species bound in the deep wells contributes less than 10% to the total isotherm. At higher coverages, adsorption to the shallow wells is dominated by direct adsorption from the gas phase, and the two-well 2D gas isotherm approaches the mathematical form of the 2D gas isotherm. As a result, the total isotherm for this system at low coverages is characterized by an asymptote which could be mistaken as saturation of all available sites (1 MLL) if the experiments were performed only at low pressures. In the two-well 2D gas model, the deep wells are more energetically favored and proportionally more populated at all pressures than the shallow wells. However, we note that the absolute number of shallow wells greatly outnumbers the absolute number of deep wells, and as a result a greater absolute number of the shallow wells are populated. In this situation, the molecules populating the shallow wells are the major contributors to the total isotherm, even at low pressures, if the adsorbate population is dominated by more than 90% of species being bound in shallow wells,. E. Physical Interpretation and Applicability of the TwoWell 2D Gas Model. The two-well 2D gas model requires that at low coverage adsorption first occurs in isolated deep wells, and the adsorbate then migrates to the shallow wells. There is precedence for such a mechanism. It is known that kinks, steps, and other forms of crystal roughness can, in conjunction with adsorbate mobility, affect the extent to which adsorption occurs66,67 and facilitate adsorption to the remainder of the surface,7 as is the case for the portal model.64,65 Several examples of adsorption which may meet the conditions for the two-well 2D gas model are H2O on TiO2,68 O2 on Au,69 CO on Cu,70,71 O2 on graphite,72,73 hydrogen adsorption on silica-supported transition metals,74 O2 adsorption on transition metal surfaces with alkali metal adatoms,75,76 and acetone adsorption on TiO2.77-79 The two-well 2D gas model may play a role in catalysis, where defects, steps, kinks, etc., are commonly the most active sites.7,19,80 The thermodynamics of the two-well 2D gas model differ from the other models discussed in this work in that ∆S°ads primarily reflects the entropy of the shallow well species, which, within the assumptions of the model, constitute more than 90% of the adsorbates. The upper limit of ∆S°ads corresponds to that of a 2D gas, while the value of ∆H°ads ranges between the adsorption enthalpies for the shallow and deep well sites. We note that a van’t Hoff analysis of the experimentally observed Kobs at low pressures will not yield the entropy or enthalpy of adsorption for either type of site, as the magnitude of Kobs depends on the ratio of the rates for population exchange between the deep and shallow wells (see Appendix 1). A more detailed discussion of the thermodynamics of this model and
how ∆H°ads and ∆S°ads can be obtained from isotherms is included as Supporting Information. Provided that the other requirements described in section IV.D are met, we note that the functional form of the isotherm expected for the two-well 2D gas model can also arise when weakly adsorbed species rapidly hop between shallow wells. In this scenario, the high-pressure region of Figure 3 will assume the form of a Langmuir isotherm rather than that of a 2D gas isotherm. A rapidly hopping adsorbate will not exhibit the entropy associated with a 2D gas, because in such a case the height of the diffusion barrier is greater than RT, as described in the Supporting Information. V. Application to Experimental Data In practice, one uses the functional forms and standard states listed in Appendix 2 to obtain model-specific values for adsorption isotherms and equilibrium constants. The values obtained from these models for ∆H°ads, ∆S°ads, and σmax can then be compared to the expected limits for a given model. The theoretical entropy of adsorption (∆S°ads) is obtained by adding the standard configurational entropy of the adsorbate (∆S°config) to the molecular entropy of adsorption given in Table 2 (∆S°mol), prior to comparison with experimentally obtained values for ∆S°ads. The standard configurational entropies are given by -R ln[θads°/(1 - θads°)] for immobile species (Langmuir-type adsorption)2 and by -R(ln[σads°/σs°] + [σads°/σs°]) for the Tonks 2D gas model (see Supporting Information). In addition to the adsorption entropy, the experimentally determined adsorption enthalpy and saturation coverage can be compared to those expected for a given model. The physically expected limits are included in Table 2. There is also an empirical correlation between ∆H°ads and ∆S°ads for immobile adsorbates (section IV.A), according to -∆S°ads e 51 J mol-1 K-1 - 0.0014∆H°ads,3,4,6 where ∆H°ads is in kJ mol-1. For the 2D gas models, ∆H°ads and Ea,diff are correlated as described in the Supporting Information and the footnotes of Table 2. By using the limits in Table 2, models can be evaluated and excluded for a set of adsorption data. Applying these limits to the adsorption equilibrium constants obtained in Table 1 for the adsorption of acetone to TiO2 in Table 1 at low partial pressures (
