5722
Langmuir 1999, 15, 5722-5732
Looking for Gold in Langmuir’s Data: Surface Heterogeneity Identification through Pressure Derivatives† S. D. Prasad* Physical Chemistry Division, National Chemical Laboratory, Pune-411008, India Received October 2, 1998. In Final Form: March 30, 1999 Two methods of determining surface heterogeneity from experimental adsorption data are suggested in this work. The first one, which is a statistical method involving a diagnostic parameter, has been employed to discriminate between the generalized Langmuir model (GLM) and the Langmuir model (LM). For three adsorbate systems, viz. (a) CO2/mica, (b) CH4/mica, and (c) N2/mica, the method yields a value of -0.5 for the diagnostic parameter, implying that the LM is the correct model. A more sensitive second procedure based on the concept of a pressure derivative product (PDP), which is the product of surface coverage and its pressure derivative, is employed to identify surface heterogeneity. This PDP when plotted against the surface coverage gives a maximum. For the N2/mica and CH4/mica systems, these values are given by θmax ) 0.333 and θmax ) 0.25, respectively. This means that the former surface is homogeneous and the latter is heterogeneous with respect to the adsorption behavior of these adsorbates. It is also shown that the error variance of the PDP is minimal at its maximum. A thermodynamic interpretation of the PDP is made. The PDP is shown to be related to the gradient of the differential molar entropy, and also a qualitative interpretation is made of the PDP peaking at intermediate surface coverages. This originates mainly from the configurational part of the differential molar entropy.
Introduction Irvin Langmuir inaugurated a significant event in adsorption theory by proposing the Langmuir isotherm1,2 through a series of classic papers. Since then, the ideal Langmuir adsorption theory, which pictured an ideal checkerboard of sites on which the adsorbed molecules are nailed, formed the core of subsequent developments, like the BET3 theory, the Fowler-Guggenheim4 model, and its variants.5-7 Incorporation of nonidealities such as surface heterogeneity, adsorbate interactions, mobility,8,9 and so forth began almost immediately afterward, and still the story is not yet over.5-7 A classic diagnostic method for detecting nonideality is to follow the variation of the isosteric heat with surface coverage.10,11 If the variation is too steep at lower surface * E-mail:
[email protected]. † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, August 9-16, 1998. (1) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (2) Langmuir, I. J. Am. Chem. Soc. 1916, 38, 2221; 1917, 39, 1848. (3) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 66, 309. (4) Fowler, R. H.; Guggenheim, E. A. Stastical Thermodynamics; Cambridge University Press: London, 1949. (5) Steele, W. A. The Interaction of Gases With Solid Surfaces; Pergamon Press: Oxford, 1974. (6) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (7) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Surfaces; Elesevier: New York, 1988. (8) Hill, T. L. Introduction to statistical Thermodynamics; Cambridge University Press: London, 1949. (9) De Boer, J. H. The Dynamical Character of Adsorption; Clarendon Press: Oxford, 1962. Somorjai, G. A. The Chemistry in Two Dimensions: Surface; Cornell University Press: Ithaca, NY. (10) Wedler, G. Chemisorption: An Experimental Approach; Butterworths: London, 1976; p 39. Brenman, D.; Graham, M. J. Discuss. Faraday Soc. 1966, 41, 95. (11) Beeck, O. Adv. Catal. 1951, 2, 151. Beeck, O. Discuss. Faraday Soc. 1950, 8, 118.
coverage, it often points to a high degree of surface heterogeneity. If it occurs at appreciable surface coverage, adsorbate interactions are probably important. When these two effects act in opposition, maxima in the isosteric plots can occur, especially in the high-coverage region.10-12 A differential quantity like the derivative of the surface coverage with respect to pressure and its product with the coverage could also be an effective tool for identifying the presence of surface heterogeneity.13-15 For the ideal Langmuir model (LM), this pressure derivative product (PDP) displays a maximum at the coverage θ ) 1/3.13 The deviation of this surface coverage at the maximum of the PDP from the ideal value means the presence of surface heterogeneity and or adsorbate interactions.14,15 While the utility of the PDP has been illustrated, the problem of sensitivity to experimental coverage determinations of the former has not been analyzed. In this paper, a proposal is made to analyze this problem and it will be proved that the error variance is a minimum at the maximum of the PDP. The precision of the experimental data is of paramount importance in the estimation of the PDP, and it will be shown that Langmuir’s original data contain a veritable gold mine in this regard, hence the justification of the title of this article. In the previous work, it has been shown that the pressure derivative can be used to compute a thermodynamic quantity like the isosteric heat,16 but a detailed thermodynamic analysis of the PDP itself was not done. In this paper, the PDP will be related to the gradient of the adsorbed-phase chemical potential with respect to the (12) Hill, T. L. J. Chem. Phys. 1950, 18, 246. (13) Prasad, S. D.; Doraiswamy, L. K. Chem. Phys. Lett. 1984, 104 (4), 315. (14) Datar, A. S.; Prasad, S. D.; Doraiswamy, L. K. Chem. Phys. Lett. 1989, 159, 337. (15) Datar, A. S.; Prasad, S. D.; Langmuir 1991, 7, 1310. (16) Prasad, S. D. Langmuir 1997, 13 (8), 1307.
10.1021/la981375p CCC: $18.00 © 1999 American Chemical Society Published on Web 06/15/1999
Surface Heterogeneity Identification
pair probability of the adsorbed molecules, and its thermodynamic meaning will be explored. Very often a scientist encounters the problem of choosing between two rival theoretical models, both of which fit the experimental data equally well. Many times recourse is made to mathematical statistics. In this paper we consider two models, one of which accounts for surface heterogeneity. We propose a novel statistical method using a diagnostic parameter17,18 to make a choice between two rival models of adsorption, viz. the Langmuir model (LM) and the generalized Langmuir model (GLM). Even though this statistical method has been somewhat used in chemical kinetics,19 to our knowledge, this will be the first time it will be employed in affecting a choice between the two isotherms. The latter also takes into account surface heterogeneity in a simple way and fits the experimental data equally well! The power of this method will be shown using the classic adsorption data of Langmuir’s original papers. Since these seminal papers of Langmuir painted a picture of molecules nailed to a checkerboard of lattice sites, viz. localized adsorption, we do not consider other models of mobile adsorption such as the Volmer or HilldeBoer models. It will also be shown that the precision of Langmuir’s data is unmatched for quantitative comparison with the theoretical predictions. Some justification should be made as to why we choose surface heterogeneity as the only added complication. It may be mentioned that preliminary screening of the data eliminates a plethora of models as not very relevant to the experimental systems at hand. Thus, mobile adsorption models with and without interactions such as Volmer and Hill-deBoer isotherms were found to provide a poor fit to the experimental data and are not considered. A localized adsorption model with interaction, such as the Fowler-Guggenheim model, when tried gave a zero value for the interaction parameter, suggesting that the interaction effects are not important for the systems at hand. This is not surprising, as the same set of data was used by Langmuir in postulating and validating the now famous Langmuir Model (LM). This leaves us only with the possibility of surface heterogeneity as an intruder, as mentioned earlier. The paper is arranged as follows: First, Langmuir’s data are fitted to common models, viz. the Langmuir model (LM) and the generalized Langmuir model (GLM); the latter corresponds to the simplest model of a site energy distribution, namely the Sip’s distribution.20 The diagnostic parameter is now introduced,17-19 and the statistical procedure is fully exemplified. Then the power of the pressure derivative method (PDP) is demonstrated and the question of determining the error variance of the PDP21 is addressed. The thermodynamic significance of the PDP and its relation to the gradient of the chemical potential of the adsorbed phase with respect to the adsorbed molecular pair probability is demonstrated. Finally, the question as to whether Langmuir’s original data (where he inaugurated the Langmuir adsorption theory)1,2 also contain latent surface heterogeneity is answered. (17) Kittrell, J. R. In Advances In Chemical Engineering; Drew, T. B., Cokelet, G. R., Hoopy , J. W., Vermeulen, T., Ed.; Academic Press: New York, 1970; Vol. 8. (18) Cox, D. R. J. R. Stat. Soc., Part B. 1962, 24, 406. (19) Kittrell, J. R.; Mezaki, R.; Watson, C. C. Ind. Eng. Chem. 1965, 57 (12), 19. (20) Sips, R. J. Chem. Phys. 1950, 18, 1024. (21) Draper, N. R.; Smith, H. Applied Regression Analysis; John Wiley & Sons: New York, 1967.
Langmuir, Vol. 15, No. 18, 1999 5723
Basic Models The Langmuir and generalized Langmuir models are well-known:
Langmuir Model (LM): θ(T,p) ) Kp/(1 + Kp)
(1)
Generalized Langmuir Model (GLM):20 θ(T,p) ) {K′p/(1 + K′p)}R
(2)
where T, p, K, and K′ have the usual meanings: temperature, pressure, and adsorption equilibrium constants, respectively. The R in eq 2 is the heterogeneity parameter which also specifies the site energy distribution and also parameterizes the isosteric heat variation. In deriving the GLM, the tacit assumption of the lower and upper limits of the heats of adsorption (0, ∞), respectively, is made. The R parameter characterizing the isotherm is related to the site energy distribution by the following expression, as shown by Sips20
δ(Q) ) CN/{exp(Q/RT) - 1}R
(3a)
where CN is a normalization constant given by
CN ) (sin ΠR/Π)/RT
(3b)
The rigorous mathematical procedure (using the theory of Stieltjes transforms) leading to the GLM of eq 2 is worked out in the original paper of Sips,20 to which the reader may refer for details. It may be mentioned that eq 3b, defining the normalization constant, is invalid at R ) 1. Thus, the Langmuir isotherm should not be interpreted as a special case of the GLM with R ) 1(see eq 2). Rather, it defines a heterogeneous surface with a Dirac δ function distribution. Using the theory of Stieltjes transforms, it can be shown that GLM results only when 0 < R < 1; otherwise, the overall isotherm expression itself becomes a divergent integral. Thus, from a rigorous mathematical point of view, the LM can never arise as a limiting case of the GLM, with R ) 1. There is a subtle difference between the GLM and the Freundlich isotherm,22,23 in the sense that the former predicts a saturation surface coverage of unity. This has a nontrivial consequence: almost all experimental isotherms predict a maximum in the PDP when we plot it against pressure or surface coverage. The Fruendlich isotherm does not predict a maximum in the PDP when plotted against the surface coverage. Further differences will become apparent when we discuss the pressure derivative product (PDP) in a subsequent section. In Figures 1 and 2 the Langmuir isotherm and the generalized Langmuir isotherm are plotted for three different sets of data, viz. for (a) CO2/mica at 155 K, (b) CH4/mica at 90 K, and (c) N2/mica at 90 K. In Figure 2 the GLM plots are shown. A look at Table 1 will reveal that the residuals are very close together; hence, it is very difficult to say which of the models are indeed the correct ones. Clearly, there is a need for some alternative (22) Halsey, G. D.; Taylor, H. S. J. Chem. Phys. 1947, 15, 624. (23) Clark, A. The Theory Of Adsorption and Catalysis; Academic Press: London, 1970.
5724 Langmuir, Vol. 15, No. 18, 1999
Prasad
Figure 1. Illustration of the validity of the Langmuir model (LM) for three experimental systems: (a) CO2/mica at 155 K; (b) CH4/mica at 90 K; (c) N2/mica at 90 K. The open circles are experimental data.
Figure 2. Fit of the generalized Langmuir model (GLM) to the same set of data as described in Figure 1.
procedure to identify the correct model. In the following section, we present a novel method of discriminating between rival models on the basis of a diagnostic parameter.
Model Discrimination: A Statistical Method In the following section, a novel statistical method based on a diagnostic parameter is outlined.17-19 If Y1 and Y2 are
Surface Heterogeneity Identification
Langmuir, Vol. 15, No. 18, 1999 5725
Figure 3. Least squares plot for determining the diagnostic parameter Λ for discriminating between the LM and the GLM. Notice the identical slope of -0.5 for all three systems. Table 1. Adsorption Parameters for the Langmuir Model (LM) and the Generalized Langmuir Model (GLM) Langmuir model
generalized Langmuir model
system
saturation capacity Vm, mL
K, the equilibrium constant
stand. dev of fit
saturation capacity Vm, mL
K′, the equilibrium constant
R, the exponent
stand. dev of fit
CO2/mica at 155 K N2/mica at 90 K CH4/mica at 90 K
57.438 38.89 108.299
0.0813 0.1584 0.1191
1.19 0.387 2.99
58.8087 38.794 113.869
0.055636 0.1637 0.04963
0.8188 1.0197 0.618
1.04 0.41 1.65
R ) Y - (Y1 + Y2)/2
(3)
S ) (Y2 - Y1)
(4)
experimental error. If |Λ| * 0.5, probably both the models are inadequate. We can also compute the 95% confidence limits for Λ, which include the values +0.5 and -0.5 for the diagnostic parameter. If the estimated value of Λ falls outside this range, the statement that model 1 or model 2 is not the correct one can be rejected with 95% confidence.
R ) ΛS
(5)
Results
the model predictions, and if Y is the observed response, we define two transformed variables R, S and a diagnostic parameter Λ by the following set of equations:
where Λ is the diagnostic parameter and R and S are linearly related. The value of Λ and its error are determined by a linear least squares procedure.21 It is also assumed that both models not only are valid from a numerical point of view but also have physical significance. If Λ ) - 0.5, then it implies that the first model is the right one:
Y ) Y1 +
(6)
On the other hand, if Λ ) 0.5, then it means that the second model Y2 is the correct one:
Y ) Y2 +
(7)
Y, as before, denotes the set of experimental observations, Y1 and Y2 are the model 1 and 2 predictions, and is the
In Figure 3 the Λ values are determined for the three sets of data which are mentioned in Figure 1. The hypothesis that Λ * -0.5 is rejected with 95% confidence. Hence, the method of the diagnostic parameter suggests that the Langmuir model is the correct one for each of the subsystems. In Table 1, the results are tabulated. As is obvious, the saturation capacities for the GLM and the LM are almost identical, as they should be. For the nitrogen adsorbate, the R parameter = 1, implying that the uniform surface LM may be valid. For the methane adsorption system, R ) 0.62, implying a heterogeneous surface, quantified by the Sip’s distribution. This also resembles a negativeexponential distribution22 which gives rise to the Freundlich isotherm. But there are subtle differences between these two distributions and their corresponding isotherms.
5726 Langmuir, Vol. 15, No. 18, 1999
Prasad
Figure 4. Pressure derivative product (PDP) plot for the Langmuir model with CO2 adsorbate: (a) PDP versus p; (b) PDP versus θ.
The estimates of residuals (or standard deviation) also indicate that for two systems, viz., CO2/mica and N2/mica, the residuals are almost identical for the LM and GLM models, making the choice between them difficult. On the other hand, for the methane/mica adsorption data, the standard deviation of the fit is twice as good for the GLM, in comparison with the LM. So, it is interesting to try to apply the diagnostic parameter method in the present case. The value of Λ ) -0.5 shows that the Langmuir model is the correct one! We now employ the pressure derivative method to answer the question of whether the surface is heterogeneous or not. Pressure Derivative Product (PDP) For any isotherm which predicts saturation coverage, and for which the pressure derivative decreases with increasing pressure, the PDP displays a maximum, when plotted against the pressure or surface coverage.14 This statement is invalid at the phase transition point, for the pressure derivative can become infinite. The utility of this method (even in the two-phase region) has been amply illustrated in previous publications. Here we merely state the conclusions: For an ideal Langmuir model, the maximum in the PDP always occurs at the surface coverage θ ) 0.333 33.14,15 Positive and negative deviations from this value indicate the presence of attractive interactions and/or surface heterogeneity. In the present work, we ignore the role of interactions and focus on surface heterogeneity. We also bear in mind that there can be uncertainty in the determination of monolayer capacity, and the surface coverage is very much dependent on this. The pmax and
θmax for the LM and the GLM are given as
Langmuir Model: θmax ) 1/3 ) 0.3333 33
(8)
pmax ) 1/2K
(9)
Generalized Langmuir Model: θmax ) [(2R - 1)/(2R + 1)]R
(10)
pmax ) (2R - 1)/2K′
(11)
Equation 10 clearly shows that when R e 0.5, no maximum is possible for the pressure derivative plot, and hence, this is an added check of the values of the pressure exponent R determined from fitting the isotherm values. Besides, when R ) 1, eqs 10 and 11 reduce to that of the values deduced for the Langmuir model. However, it should be clearly kept in mind that this only proves the correctness of the analysis and the numerical results. In general the GLM, by definition, is strictly valid only for values of R in the range 0 < R < 1. The LM corresponds to a Dirac δ function distribution, and the GLM can never be reduced to the LM. Certain comments are in order regarding eqs 8-11. For the Langmuir model θmax is identically equal to the value 1/ , irrespective of the heat of adsorption and the tem3 perature at which the isotherm is measured. pmax, on the other hand, depends sharply on the heat of adsorption Q and T through the Langmuir adsorption equilibrium constant. In contrast to this, for the GLM, θmax and pmax depend slightly on temperature. This is evident from eqs 10 and
Surface Heterogeneity Identification
Langmuir, Vol. 15, No. 18, 1999 5727
Figure 5. PDP plot with the LM for CH4/mica.
11, which show that both the above quantities depend on R.
R ) RT/Qm
(12)
where Qm is a parameter, the value of which decides the extrapolated value of the isosteric heat of adsorption observed at surface coverage tending to zero. K′, the entropy change factor for each of the patches, is assumed to be the same for all the patches and is independent of the heat of adsorption. K′ also has only a weak dependence on temperature. The influence of the heat of adsorption is not very evident, when we integrate over a large spread of (0, ∞) of heat values, which are mainly chosen for mathematical convenience, and not on the basis of any physical significance. The choice of these limits has been criticized in the past; nevertheless, it has the advantage of avoiding two arbitrary parameters, viz. the finite limits of heats of adsorption in the overall isotherm. In Figures 4-6 the PDP plots versus pressure and surface coverage are given for the (1) CO2/mica, (2) CH4/ mica, and (3) N2/mica adsorption systems, respectively. In these graphs, the validity of the Langmuir model (LM) is investigated. At the very outset, it may be thought that the LM is the correct one, as the original Langmuir Model was proposed using the same data. It is a worthwhile exercise to verify this truth. In Figure 4 the plot for the CO2/mica system is shown, and the PDP plots reveal that there are systematic differences between the computed and experimental values. In parts a and b of Figure 4, the regions of divergence are shown on a slightly expanded scale (especially in Figure 4b, where all the points near the maximum are used), on both the pressure and surface coverage axes, respectively. Figure 4b indicates that the PDPmax peaks around a lower θ value, instead of the LM
Table 2. θmax and pmax for the Langmuir Model system
K, Torr-1
θmax
pmax, Torr
CO2/mica at 155 K CH4/mica at 90 K N2/mica at 90 K
0.0813 0.1191 0.1584
0.333 33 0.333 33 0.333 33
6.15 4.198 3.16
Table 3. pmax and θmax for the Generalized Langmuir Model system
K′, Torr-1
R, exponent
θmax
pmax
CO2/mica at 155 K CH4/mica at 90 K N2/mica at 90 K
0.055636 0.04963 0.1687
0.8188 0.618 1.0197
0.31 0.249 0.335
5.73 2.378 3.175
prediction of θ ) 0.3333. The exact values are listed in Tables 2 and 3. In Figure 5, similarly, the CH4/mica adsorption sytem is analyzed. Unfortunately, unlike in the previous graph, we do not have too many points near the maximum; nevertheless, the inadequacy of the LM is apparent, especially from Figure 5b. Perhaps the closest agreement with LM predictions is seen for the N2/mica system, depicted in Figure 6. The location of θmax is somewhat hampered by the availability of only a few experimental points near θmax ) 0.3333; nevertheless, the excellent agreement of the PDP predictions using the LM and the experimental values over the entire range of pressure and θ is quite striking. In Figures 7 and 8 the generalized Langmuir model is tested for applicability to the CO2/mica and CH4/mica systems. As is evident, the GLM provides a very good fit, especially for θmax. The value displayed by the GLM for this quantity is much smaller than the LM value of θmax ) 1/3. This is clearly seen in Figure 7b for the CO2/mica system. On the other hand, for the CH4 adsorbate, once again we are somewhat limited by the availability of only a few
5728 Langmuir, Vol. 15, No. 18, 1999
Prasad
Figure 6. PDP plot with the LM for N2/mica.
Figure 7. PDP plot with the generalized Langmuir model (GLM) for the CO2/mica system: (a) PDP versus p; (b) PDP versus θ.
Surface Heterogeneity Identification
Langmuir, Vol. 15, No. 18, 1999 5729
Figure 8. PDP plot as in Figure 7 for the CH4/mica System with the GLM. Table 4. Illustration of Precision of Langmuir’s Original Data system
Λ, diagnostic parameter
t distribution, calc
t distribution, tabulated values
N2/mica CH4/mica CO2/mica
-0.500 ( 0.0063 -0.484 ( 0.020 -0.493 ( 0.011
0.021 0.801 0.667
2.26 2.36 2.306
points near the maximum of the PDP. Nevertheless, the excellent agreement between calculated and experimental values gives enough confidence in the pressure derivative product method, in choosing the GLM as the correct model. For the N2 adsorbate, the R exponent is very close to unity (see also Table 2), indicating that the LM is indeed the correct model, as has already been shown in Figure 6. It is really pointless to look for the validity of the GLM in this instance, as strictly speaking GLM is valid only for 0 < R < 1. It is illustrative to tabulate the values of θmax and pmax, so that readers can understand the problems in pursuing the PDP method. A cursory glance at the values of θmax listed in Tables 2 and 3 shows clearly that the θmax values for the GLM are identical to the Langmuir model value of θmax ) 1/3, except for the CH4/mica system, wherein it shows θmax ) 0.25. This is clearly discernible within the precision of the experiments. This shows that the CH4/ mica system is strongly heterogeneous. The validity of all the conclusions drawn above depends to a large extent on the precision of the data analyzed. A cursory glance at Table 4 will convince us about this. For all the experimental systems, the value of the diagnostic parameter Λ ) - 0.5 and the standard errors are very small. Further, we test the hypothesis that the 95% confidence intervals estimated for Λ include the value Λ ) 0. A set of student t distribution values estimated from
experiment are compared with the tabulated values of the t distribution with appropriate degrees of freedom. The estimated values are much smaller than the tabulated values, and the hypothesis Λ ) 0 is rejected with 95% confidence. Even with limited degrees of freedom, the small values of the estimated t distribution reflect the inherent precision and limited variation in Langmuir’s original data. In conclusion, it may be said that the PDP method validates the LM for the N2/mica system, whereas for the CH4/mica system, the GLM is the more correct one. For the CO2/mica system it is hard to eliminate either model as incorrect; nevertheless, the pressure exponent value R ) 0.82 suggests that GLM may be more appropriate. How much sensitivity to experimental errors can be expected in the estimates of PDP at θmax? In the following section, we show that when we estimate the PDP at its maximum, the error variance is minimal. Error Variance of the PDP First assume that there is very little error in the measurement of the independent variable, namely pressure, and that there is error in determining the surface coverage. We are interested in relating the uncertainty in determining θ to that of the PDP. We also bear in mind the statistical formula for the variance of an arbitrary function (Var being a standard notation for variance), and the prime denotes differentiation with respect to the independent variable x.21
Y ) g(x)
(13)
Var(Y) ) |g′(x)|2 Var(x)
(14)
5730 Langmuir, Vol. 15, No. 18, 1999
Prasad
Figure 9. Plot illustrating the minimum error variance for the pressure derivative product (PDP) at the maximum of the PDP, shown as a function of θ for the CH4/mica system: (a) the Langmuir model; (b) the generalized Langmuir model.
For the Langmuir model in particular
given by
θ dθ/dp ) g(θ)
(15a)
2
(15b)
g(θ) ) θ(1 - θ) K g(θ)
max
2
) (1/3)(2/3) K
(15c)
g(θ)max = 0.148K
(15d)
Var{g(θ)} ) |g′(θ)|2 Var(θ)
(16)
Since at the maximum of the PDP
g′(θ) ) 0
(17)
It is obvious that the error variance of the PDP is minimal in a small neighborhood ∆ of θmax. Thus, the error of measurement of the PDP is minimal at the maximum of the PDP. In other words, a min-max theorem exists for the PDP of error variance. The foregoing discussion shows that the PDP measured at the maximum may be useful for discriminating between the rival distributions.
For the Langmuir Model (δ Function Distribution): g′(θ) ) (1 - θ)(1 - 3θ)K
θ ) (1/3) - ∆
(18.2)
Var[g(θ)] ) [(2/3 + ∆)∆] K
(18.4)
b0 refers to the exponential entropy change factor exp(∆S/R), and Q is the heat of adsorption. It is understood that the standard state of the adsorbed phase corresponds to that of half surface coverage. Retaining only the lower order terms in ∆, we get
Var(θ dθ/dp) ) (2/3)2∆4K2
(18.5)
For ∆ ) 0.03, the error variance is of the order of 10-6. The error variance for an illustrative system, viz. CH4/ mica, is plotted in Figure 9a with the equilibrium constant deduced from the LM. Notice the computed curve predicting a minimum for the error variance at θ ) 0.333. We, however do know that the LM is not the correct model for the CH4/mica system; Figure 9a is shown just for illustrative purposes. This will be the error variance behavior of a hypothetical LM. We now turn to the GLM, which provides a better description of the above adsorption system. A similar analysis is done for the error variance of the GLM, and we merely tabulate the final results:
(18.1)
In a Small Neighborhood near the Maximum Given by
2
K ) 1/[b0 exp(-Q/RT)]
2
(18.3)
where K, as before, is the adsorption equilibrium constant
θ dθ/dp ) g(θ) ) Rθ2/[p(1 + pK′)]
(19)
p ) [θ1/R/(1 - θ1/R)]/K′
(20)
where
Differentiating with respect to θ, and proceeding as before, we get
Surface Heterogeneity Identification
Langmuir, Vol. 15, No. 18, 1999 5731
g′(θ) ) (K/β)(1 - θβ)[-2βθ + (1 - θβ)(2 - β)θ1-β] (21) where β ) 1/R and
Var[g(θ)] ) |g′(θ)|2∆2
(22)
Unfortunately, in the present case, development in terms of small powers of ∆ is not possible, in a small neighborhood of θmax, θ*, given by
θ* ) θmax - ∆
(23)
where θ* is a small neighborhood of θmax as before. Nevertheless, once again we arrive at a quantity of the order of ∆,6 as before. Figure 9b gives the actual behavior of the GLM, with parameters deduced for the CH4/mica system. In comparison with LM predictions (Figure 9a), the minimum error variance occurs at a lower value of θ ) 0.25, which also corresponds to the value of the PDP maximum (see Table 3). Thus, we conclude that the error variance is minimal if we measure the PDP at a region near its maximum. Thermodynamic Interpretation of the Pressure Derivative Product It is common knowledge that when an adsorption equilibrium exists between the gas and surface phases, the chemical potentials of both the gas and adsorbed phases are equal,8,9 that is
µ a ) µg
(24)
variation of the chemical potential of the adsorbed phase with respect to the pair probability p11. In the mean field approach, the pair probability is given by
p11 ) 2(θ2/2)
The factor 2 is to account for the fact that the 11 pair can be counted in two ways (say ij pair and ji are spatially distinct) for enumerating the total number of molecular pairs of all types. Now the chemical potential of the adsorbed phase can be split into entropy and enthalpy factors, as follows:
µs ) Hs - TSs
(31)
where the underlined enthalpy and entropy terms are to be understood as partial differential molar quantities. For a homogeneous surface, the enthalpy term is independent of surface coverage, but the entropy term is not. This differential entropy term is mainly configurational in origin. Within the framework of the Langmuir model (LM), the configurational part of the differential molar entropy can be written as23
Scs ) -R ln(θ/1 - θ)
(32)
By differentiating the above equation with respect to θ and multiplying with -T, it is easily verified that it is identical to the denominator in eq 26, namely the gradient of the adsorbed phase chemical potential with respect to the coverage. For the LM, the chemical potential is given by a very simple expression:
µs ) -RT ln[θ/(1 - θ)qs]
Assuming ideal gas behavior
(30)
(33)
where µ0 is the chemical potential of the standard state of the ideal gas, that is, of unit fugacity. We first write an expression for the pressure derivative of the surface coverage at constant T:
As is well-known, qs represents the molecular partition function. It is a very straightforward excercise to verify the importance of the configurational entropy term. We can derive eq 15b by substituting the above expression for the chemical potential in eq 29. If we work in the linear region of the isotherm (Henry’s law region), then we have
(∂θ/∂p)T ) (∂µg/∂p)T/(∂µs/∂θ)T
p ) θ/Kh
µg ) µ0 + RT ln p
(25)
(26)
where we have used eq 24 in expressing the pressure derivative as a ratio of two gradients of the chemical potentials of the gas and surface phases with respect to the appropriate concentrations. For most of the adsorption isotherms, this ratio will monotonically decrease with pressure, except when a phase-transition occurs in the adsorbate phase. We also keep in mind that
dµa ) dµg
(27)
in deriving eq 26. A simple expression results for the gradient of the chemical potential of the gas phase with respect to pressure:
(∂µg/∂p)T ) RT/p
(28)
The PDP can now be simply written as
θ(∂θ/∂p)T ) (RT/p)/[∂µs/∂(θ2/2)]
(29)
The interesting part of the right hand side of eq 29 is the denominator derivative term. This represents the
(34)
and the PDP will be entirely determined by the denominator of eq 29, that is, by the gradient of the configurational differential entropy with respect to the surface coverage. If this alone mattered, the PDP would have peaked at exactly θ ) 0.5. The actual peaking of the PDP at θ ) 0.333 33 shows that additional factors, such as the nonlinearity of the isotherm, do play some role; nevertheless, the importance of the configurational entropy is obvious. When surface heterogeneity is important, the isosteric heat of adsorption is dependent on surface coverage. And the differential molar entropy can be computed using
Ss ) Sg° - R ln p - Qiso/T
(35)
In the present case also, the differential molar entropy can be separated into configurational and thermal entropy parts. The isosteric heat behavior of the GLM can be computed using the Clasius-Clapeyron equation, viz.
(∂ ln p/∂T)θ ) Qiso/RT2 Qiso in turn is given by the following expression:
(36)
5732 Langmuir, Vol. 15, No. 18, 1999
Qiso ) FQe
Prasad
(37)
where F and Qe are given by
Qe ) -Qm ln(θ)
(38)
F ) exp(Qe/RT)/[exp(Qe/RT) - 1]
(39)
In the limit of low coverages, F = 1, and we recover the well-known approximate (neverthless adequate) isosteric heat relationship for the GLM:
Qiso ) Qe ) -Qm ln(θ)
(40)
Here we also notice that Qm is a parameter related to the isosteric heat observed at the limit of zero coverage. At this juncture, we may mention that Langmuir’s original paper does not contain data at different temperatures, so that eq 36 cannot be used to map the isosteric heat behavior as a function of the coverage. Besides, an entropy calculation using eq 35 will be uncertain as to the extent of an arbitrary constant which requires the knowledge of vibrational frequencies of the adsorbed phase. In addition, this may also depend on the heat of adsorption of the local patch. We do not undertake to do the adsorbed-phase entropy calculations for the LM and GLM. Nevertheless, the statement that the configurational entropy has a predominant role to play in the PDP values is true. At this juncture, we may add that the separation of the overall differential molar entropy determined using eq 35 is not straightforward for the GLM, as in the case of the LM. Some qualitative considerations are also in order, as to the nature of the differential entropy of a heterogeneous surface. It should also be noted that for a random patch model of the surface, for each of the patches, we get
µg ) µia
(41)
as the necessary condition of equilibrium. Since
µia ) Hia - TSia
(42)
The differential entropy Sia has to decrease when Hia decreases with the patch number, so as to make the chemical potential of the ith patch µia constant. This statement is true irrespective of the external potential to which the adsorbed molecule is subjected and whether the adsorption varies drastically from patch to patch or not. We also notice that, unlike isosteric heat, the differential entropy variation is more noticeable at an intermediate range of mean surface coverages. Concluding Remarks Before we conclude this paper, a few words outlining the limitations of the work are perhaps appropriate. Thus, we exclusively concern ourselves with surface heterogeneity, and complications arising out of adsorbate interactions and mobility are neglected. Some justification has been given earlier, as preliminary screening of Langmuir’s data eliminated many models as not very plausible. We are unable to furnish any molecular level interpretation as to why for the same adsorbent, viz. freshly cleaved mica, the N2 adsorbate shows a homogeneous surface, where as CH4 reveals a heterogeneous surface! It is true that CH4 is a spherical top molecule and therefore should show more uniform surface behavior than N2 and CO2 molecules, which have quadrupole moments
and which interact with surface electric field gradients. We suspect that even the freshly cleaved mica may have a superficial layer of water molecules adsorbed (Langmuir himself mentioned this in his original paper1), which can mask the true clean-surface behavior. The surface electric field will be much reduced as a result of this. We should bear in mind that, in 1918, techniques to maintain atomically clean surfaces were rather absent. Langmuir also speculates two types of surface sites on the surface of mica and predicts that the uptake can occur in steps (in other words surface heterogeneity). Since the saturation capacity of CH4 is much larger, probably heterogeneity effects are seen only for this molecule. If data were available at many temperatures, we could have gotten some clues on the nature of binding and the adsorption energies and so forth and could have tried to separate the dipole and quadrupole components and so forth; unfortunately that is not possible. Still a molecular picture eludes us, and the data interpretation is therefore limited in scope. Conclusions We have shown the power of a statistical method, using a diagnostic parameter, in discriminating between the Langmuir and generalized Langmuir models. Even though this has been used only for a pair of localized adsorption models, it is applicable also to models taking into account adsorbate mobility. Also, nonidealities such as surface heterogeneity and adsorbate interactions do not limit its usefulness. The precision of Langmuir’s data is such that all the estimates of the diagnostic parameter fall within 95% confidence limits computed using a suitable t distribution. This value is equal to -0.5, and it means that if statistics is to be believed, then the Langmuir model is the correct one for all three adsorption systems. On the other hand, the method of the pressure derivative product (PDP) shows that the Langmuir model is incorrect at least for the methane system, where θmax ) 0.25. This shows that the system is strongly heterogeneous. As for the question of estimating the error involved in determining the PDP, we have shown that a min-max theorem is valid; that is, the error is minimum at the maximum of the PDP. A thermodynamic interpretation of the PDP is made and a relationship is established between the PDP and the gradient of the adsorbate chemical potential with respect to the pair probability. A qualitative interpretation of the PDP peaking at intermediate surface coverages is given. It is suggested that the configurational part of the entropy of the adsorbed phase is the main cause. This is also in accordance with the observation that the PDP for a heterogeneous surface is smaller than that of the ideal surface. This is also evident from the observation that PDPmax occurs at θmax ) 0.333 for the Langmuir model, which is independent of temperature (the configurational part of the entropy is independent of temperature for the LM). For the generalized Langmuir model (GLM), PDPmax is smaller than that of the LM and shows only a weak temperature dependence. This observation is in good agreement with the well-known result that the entropy of adsorption for a heterogeneous surface is smaller than that of a homogeneous surface, at less than half surface coverage. The validity of the conclusions drawn above depends on the precision of the experimental data. In this sense, Langmuir’s data provide a gold mine, as illustrated in Table 4, and hence the ample justification of the title of the paper. LA981375P
