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Isosteric Heat for Monolayer Adsorption Obtained from Two-Dimensional Equations of State F. Cuadros,*,† A. Mulero,† L. Morala,† and V. Go´mez-Serrano‡ Departamento de Fı´sica and Departamento de Quı´mica Inorga´ nica, Universidad de Extremadura, 06071 Badajoz, Spain Received October 4, 2000. In Final Form: December 5, 2000 The Reddy-O’Shea (RO) and Cuadros-Mulero (CM) equations of state for two-dimensional LennardJones fluids have been used to obtain the isosteric heat of adsorption of simple fluids onto flat surfaces. The results are compared with a set of available experimental data, observing a large degree of agreement, especially when the more simple CM equation of state is used. Another advantage of the CM equation is that it is based on the Weeks-Chandler-Andersen theory for simple fluids. In light of these results, we propose, as a first approach, the application of the CM model to describe the thermodynamics of the adsorption of simple fluids onto flat surfaces.
1. Introduction 1,2
The two-dimensional approximation of Steele allows one to use the equation of state of two-dimensional fluids to describe the monolayer physisorption. Thus, the treatment of statistical mechanics developed by Steele reduces the three-dimensional problem to a purely two-dimensional one. Moreover, Steele’s model takes into account the periodical nature of the adsorbate-adsorbent interaction when the adsorbed molecules are moving parallel to an energetically homogeneous solid surface. It is also supposed that the adsorbed phase is composed of nonpolar molecules, with quasi-spherical shape, obeying the laws of the classical statistical mechanics, with pairwise additive and central molecular interactions. Provided that the adsorbate is a simple substance (rare gas, methane, etc.), the adsorbate-adsorbate interaction can be then modeled by means of the Lennard-Jones (LJ) potential function3
ULJ(r) ) 4[(σ/r)12 - (σ/r)6]
(1)
where r is the intermolecular distance in the twodimensional space, is the minimum value of the potential, and σ is the distance at which the potential is zero. The so-called LJ parameters, and σ, are different for each substance,4 characterizing this from a molecular point of view. Some of the equations of state (EOS) for two-dimensional LJ fluids reported in the literature are those of Henderson,5 Reddy-O’Shea (RO),6 and Cuadros-Mulero (CM).7 In this work, because the coefficients of the Henderson EOS are not known as a function of temperature, for comparison purposes we only use the RO and CM equations to calculate the isosteric heat of adsorption. The RO semiempirical EOS was obtained by fitting computer simulation data for LJ fluids, giving the pressure † ‡
Departamento de Fı´sica. Departamento de Quı´mica Inorga´nica.
(1) Steele, W. A. The interaction of gases with solid surfaces; Pergamon Press: New York, 1974. (2) Steele, W. A. J. Chem. Phys. 1976, 65 (12), 5256. (3) Hansen, J. P.; McDonald, I. R. Theory of simple liquids; Academic Press: New York, 1976. (4) Cuadros, F.; Cachadin˜a, I.; Ahumada, W. Mol. Eng.. 1996, 6, 319. (5) Henderson, D. Mol. Phys. 1977, 34 (2), 301. (6) Reddy, F. A.; O’Shea, S. F. Can. J. Phys. 1986, 64, 677. (7) Cuadros, F.; Mulero, A. Chem. Phys. 1993, 177 (1), 53.
and the potential energy with a relatively good precision. However, as seen below, it contains a large number of adjustable parameters and thus it is difficult its mathematical handling. In contrast, the CM semitheoretical EOS is based on the Weeks-Chandler-Andersen8 (WCA) perturbation theory, and it has a simple analytical expression. In a previous work,9 Mulero and Cuadros used the CM equation to obtain a theoretical expression for the isosteric heat. Good agreement was found with the Steele theoretical approximation2 (based on a modified WCA theory), the Percus-Yevick (PY) theoretical values,10 and the Henderson equation.5 In the aforementioned work, a first attempt was also made to compare the theoretical results given by the CM EOS with the experimental isosteric heat data from Putnam and Fort.11 To complete this previous analysis, in the present work we have considered seven adsorption systems (see Table 2), at different temperatures and coverages, the corresponding experimental isosteric heat being compared with the predictions of both the RO and CM models. Because of the complexity of the adsorption process in which not only the adsorbate-adsorbate lateral interactions but also the adsorbate-adsorbent transversal interactions must contribute, with multibody effects, and especially when the adsorbate molecule has dipolar or cuadrupolar moment, and because of the heterogeneity of the adsorption surface, to represent the molecular behavior of the adsorption of real fluids, the RO and CM equations require the use of effective two-dimensional (2D, hereafter) LJ parameters for the adsorbed gases, instead of their 3D corresponding values.12 It must be noted that the choice of the adequate 2D LJ parameters to connect the microscopic and macroscopic worlds is very important, because the isosteric heat (as well as all thermodynamic properties) is very sensitive to the value of these parameters. These effective 2D LJ parameters incorporate in the RO and CM models, on an average at least, the main characteristics of the adsorption processes described above. (8) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237. (9) Mulero, A.; Cuadros, F. Chem. Phys. 1996, 205, 379. (10) Glandt, E. D. and Fitts, D. D. J. Chem. Phys. 1977, 66 (10), 4503. (11) Putnam, F. A.; Fort, T. J. Chem. Phys. 1975, 79 (5), 459. (12) Jiang, S.; Gubbins, K. E. Mol. Phys. 1995, 86 (4), 599.
10.1021/la0014048 CCC: $20.00 © 2001 American Chemical Society Published on Web 02/07/2001
Isosteric Heat for Monolayer Adsorption
Langmuir, Vol. 17, No. 5, 2001 1577
Table 1. Coefficients of the Reddy-O’Shea Equation of State6 i
bi
i
bi
i
bi
1 2 3 4 5 6 7 8 9 10 11
0.7465941518D + 00 0.2170202387D + 01 -0.3867162453D + 01 0.1955514819D + 00 -0.3080792027D + 00 0.7254286978D + 00 -0.8449806323D + 00 0.1312097237D + 01 0.1270817291D + 00 0.2906037677D + 01 -0.6390601256D + 01
12 13 14 15 16 17 18 19 20 21 22
-0.4654184040D + 00 0.2060578013D + 02 0.2778370707D + 01 -0.3788766473D + 03 -0.1227538308D + 03 0.1276939405D + 03 -0.3730636670D + 03 0.1354795604D + 04 -0.6993741337D - 01 -0.4066552651D - 01 0.6531068158D + 02
23 24 25 26 27 28 29 30 31 32 g
0.7948290821D - 01 0.8494574507D + 03 -0.4038923642D + 01 -0.1388504769D + 04 0.1001438224D + 01 -0.1962817948D + 02 -0.1844924653D + 02 -0.7164817351D + 03 0.2630515216D + 02 0.4773207984D + 00 0.6682623864D + 00
Table 2. Application of the RO and CM Theoretical Equations: Adsorption Systems and Temperatures, Effective 2D-LJ Parameters, and Absolute Average Deviations (AAD in %) of the RO and CM Values with Respect to Experimental Results adsorbate
adsorbent
T (K)
σ2D (nm)
(/k)2D (K)
qst(F ) 0) (kJ/mol)
RO AAD (%)
CM AAD (%)
ref
Ar Kr Xe CH4
graphite P-33(2700) Sterling FT grafoil graphite grafoil
84 104.49 195.5 84.5 92.5 84.5 119 84 79.3 120 112 108 105 98
0.330 0.355 0.390
94 128 183
0.350
112
9.6 12.6 17.3 13.6 13.3
76
11.0
1.20 1.42 0.58 1.01 1.48 1.16 1.03 1.81 0.84 1.27 2.08 2.28 2.99 2.91
26 11 27 28 22
0.355
3.23 5.58 1.38 8.35 6.36 8.64 3.19 3.21 2.08 6.97 7.08 9.50 8.60 13.96
CO C2H4
grafoil grafoil (MAT)
0.415
150
This paper has been arranged as follows. The next section is devoted to describe the two equations used in this work. The results are discussed in section 3, and finally, in the section 4 we present the main conclusions. 2. Theoretical Models In a perfect mobile monolayer adsorption, the twodimensional Steele’s theoretical equation for the isosteric heat, qst*, is
(
qst*(F*) ) qst*(F* ) 0) + T*2
)
∂(µ*/T*) ∂T*
F*
(2)
where T*, F*, and µ* are respectively the temperature, the numerical density (molecules per unit of area), and the excess over the ideal gas of the 2D chemical potential/ molecule. In the previous and the following equations, all the quantities marked with an asterisk are expressed in reduced Lennard-Jones units: the lengths are reduced by σ (the 2D LJ diameter of the adsorbed molecules), and the energies are reduced by (the effective 2D LJ depth of the lateral interactions). The chemical potential can be calculated from a 2D EOS which takes into account only the lateral interactions between the adsorbed molecules. If we define the 2D compressibility factor Z as
P* Z) F*T*
(3)
P* being the 2D pressure, then the excess over the ideal gas of the chemical potential can be obtained using the following expression:
[∫(Z G*- 1) dG* + Z - 1]
µ* ) Τ*
(4)
So, if we know the 2D EOS of the adsorbed substances, it is possible to obtain the isosteric heat of adsorption
20.6
23 24
inside the Steele scheme. As pointed out above, the RO and CM equations of state used here are valid to reproduce the thermodynamic behavior of 2D LJ fluids. 2.1. Reddy-O’Shea EOS. The RO equation6 is one of the best available EOS for the entire 2D LJ fluid region; it is valid on a wide range of temperatures and densities. The equation is of the Benedict-Webb-Rubin type, having 33 coefficients, which were obtained by Reddy-O’Shea by fitting computer simulation data. The analytical form of the RO equation for the two-dimensional pressure is
P* ) F*T* + F*2[b1T*1/2 + b3 + b4T*-1 + b5T*-2] + F*3[b6T* + b7 + b8T*-1 + b9T*-2] + F*3[b20T*-2 + b21T*-3] exp(-γF*2) + F*4[b10T* + b11 + b12T*-1] + F*5b13 + F*5[b22T*-2 + b23T*-4] exp(-γF*2) + F*6[b14T*-1 + b15T*-2] + F*7b16T*-1 + F*7[b24T*-2 + b25T*-3] exp(-γF*2) + F*8[b17T*-1 + b18T*-2] + F*9b19T*-2 + F*9[b26T*-2 + b27T*-4] exp(-γF*2) + F*11[b28T*-2 + b29T*-3] exp(-γF*2) + F*13[b30T*-2 + b31T*-3 + b32T*-4] exp(-γF*2) (5) where the parameters bi and γ are listed in Table 1. Obviously, the final expression for the chemical potential obtained from the RO EOS by using eq 4 will have a very complex analytical expression, and this fact seems to have dissuaded researchers from its application. Despite its complexity, Mulero and Cuadros13 have used the RO equation to obtain theoretical adsorption isotherms and spreading pressure in order to predict experimental results of Ar and Kr adsorbed onto graphite and Sterling FT, respectively, obtaining average deviations less than or equal to 10%. Moreover, Mulero et al.14 have calculated (13) Mulero, A.; Cuadros, F. J. Colloid Interface Sci. 1997, 186, 110.
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Langmuir, Vol. 17, No. 5, 2001
Cuadros et al.
the vapor-liquid equilibrium properties for 2D LJ fluids from the RO equation, finding a good agreement with the results obtained through different available computer simulations. Another disadvantage of the RO EOS is that it does not separate the contributions associated with the repulsive and attractive intermolecular forces. 2.2. Cuadros-Mulero EOS. The Weeks-ChandlerAndersen (WCA) perturbation theory gives an EOS with appropiate separation between the repulsive and attractive contributions of the intermolecular forces,7,8 its analytical form being
(
P* ) P0* - F*2R(T*, F*) - F*3
)
∂R(T*, F*) ∂F*
T,N
0.3837T* + 1.068 0.4293T* + 1
(8)
C1(T*) ) 3.8574 - 3.39554T* + 2.20294T*2 0.46687T*3 (9) and
C2(T*) ) -1.00985 + 4.83792T* - 3.19864T*2 + 0.68183T*3 (10) The final expression for the CM EOS is then cubic on density, as the classical van der Waals equation is, and cubic on temperature, being clearly simpler than the RO EOS (eq 5). From eqs 3 and 4, and if one further makes use of eq 6, the 2D potential (µ*) can be easily obtained:
( )
SPT
) -ln(1 - y) +
(3y - 2y2) (1 - y)2
d* being given by eq 7. The final analytical CM equation of the isosteric heat has the following expression:
(
)
∂(µ0*/T*)SPT ∂T* 2 ∂ 1 ∂ [F* RCM(T*, F*)] T*2 (14) ∂T* T* ∂F*
qst*(F*) ) qst/(F* ) 0) + T*2
(
)
Obviously, eq 14 also separates between the repulsive and attractive contributions of the intermolecular forces. As mentioned above, the CM equation for the isosteric heat of adsorption gives good results when it is compared with some theoretical models as the Steele, PercusYevick, and Henderson ones, as well as with the experimental values of the isosteric heat reported by Putnam and Fort11 for Kr on Sterling FT.
In the comparison with experimental data it must be taken into account the influence of the approximations used here. That is, the theoretical results are obtained for 2D fluids, whereas in experiments the fluid is not strictly 2D. Furthermore, for real fluids at least, the surface is never perfecly flat. It always contains imperfections such as scratches, bumps, and pits, which render a rough surface. If we consider the LJ interaction as a valid representation of the true intermolecular potential, we must limit our comparison to classical inert simple fluids and to other fluids with approximately spherical molecules and with dipolar moments close to zero. Then, however, one encounters another difficulty. It is dealt with how to choose the 2D LJ parameters (/k and σ) for a given fluid. This is of vital importance since the 2D LJ parameters influence not only the results but also the thermodynamical states (T and F) to be considered. Typically, the experimental data are shown by plotting the values of isosteric heat qst in J/mol versus the number of adsorbed moles, n. Then, to compare experimental and theoretical (in reduced LJ units) values we must use the following relations:
T ) T*(/k)
(15)
∆qst ) ∆qst*(/k)R
(16)
and
(11) n)
where µ0* is the chemical potential due to the repulsive part of the LJ potential. The µ0* value can be derived from the SPT expression
µ0* T*
(13)
3. Comparison with Experimental Results
where
2 µ* µ0* ∂ (F* R(T*, F*)) ) T* T* T*∂F*
πF*d*2 4
(7)
The other terms of the WCA EOS containing the function R(T*, F*) represent the contribution of the attractive forces. To obtain an analytical expression for R(T*, F*), an extensive molecular dynamics simulation of 2D LJ system was carried out by Cuadros and Mulero.7 The proposed function has the analytical form
RCM(T*, F*) ) C1(T*) + F*C2(T*)
y)
(6)
where P0* is the repulsive contribution to the pressure, which can be approximated15 by the EOS for hard disk obtained through the scaled particle theory (SPT),16 with an effective molecular diameter given by the Verlet-Weis expresion:17
d* )
where y is the 2D packing fraction,
(12)
(14) Mulero, A.; Cuadros, F.; Fau´ndez, C. A. Aust. J. Phys. 1999, 52, 101. (15) Cuadros, F.; Mulero, A. Chem. Phys. 1991, 156, 33. (16) Heldfand, E.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1961, 34, 1037. (17) Verlet, L.; Weis, J. J. Phys. Rev. A 1972, 5 (2), 939.
F*A Naσ2
(17)
where T is the temperature expressed in Kelvin, R is the ideal gas constant, A is the adsorption surface area, and Na is Avogadro’s number. As shown previously by Mulero and Cuadros,9 the choice of the appropriate LJ parameters meaningfully affects the reproduction of the experimental data. Accordingly, some authors have suggested the use of 3D LJ parameters.18 However, as described above, the presence of the solid may perturb the intensity of the interactions, and (18) Toxvaerd, S. Mol. Phys. 1975, 29 (2), 373.
Isosteric Heat for Monolayer Adsorption
if so, the use of 3D LJ parameters to describe the monolayer adsorption would not be fully adequate. Although the validity of the LJ potential to characterize the physisorption of gases was emphasized by Jiang and Gubbins,12 it was also stated by these authors that, when this model is used to describe the monolayer adsorption, interactions other than the repulsive and attractive forces, such as higher-order dispersion terms, three-body forces, and induction interactions, should be allowed for. Nevertheless, because the difficulty in including all these last interactions in the LJ potential expression, a number of authors have pointed out19,20 the matter that their effects on the adsorption process may be taken into account roughly, but in a simpler maner, by using a smaller LJ energy parameter, i.e., the so-called effective . Thus, for example, a decrease of 7% was suggested in order to fit properly the simulated liquid-vapor coexistence curve to the experimental curve for methane on graphite.12 On the other hand, when studying the monolayer adsorption of rare gases (i.e., Ar, Kr, Xe) as well as methane, Wolfe and Sams21 used values that are between 12% and 17% lower than the corresponding 3D values. However, the σ values were only 1% higher than their respective 3D values. In this work, the effective 2D LJ parameters were estimated by fitting the experimental data of the isosteric heat of adsorption for simple fluids onto flat surfaces to those obtained for the same quantity by applying the CM and RO equations. The resultant values for such parameters are given in Table 2. It is found that, with respect to the 3D values,4 σ2D ranges between -8.5% for Xe and -12.8% for methane, whereas the percentage of variation of (/k)2D does between -14.5% for Xe and -25.3% for ethylene. The different variation of the effective 2D LJ parameters with respect to the 3D values depending on the system can be merely considered as a logical consequence of the assumed molecular interaction model, which is pairwise additive, nonpolar and with spherical symmetry. Table 2 lists the gas/solid adsorption systems used in these investigations, together with the adsorption temperatures and with the calculated σ2D and (/k)2D values obtained for each adsorption system. In Table 2 are also set out the values of the isosteric heat of adsorption at zero degree of surface coverage, qst(F ) 0), which was needed to be used in the curve fittings. The value of this heat of adsorption was estimated by the well-known standardized procedure of extrapolation of the linear portion of the experimental curve of qst against coverage. Nevertheless, to achieve the best fitting between the experimental and theoretical results for the isosteric heat of adsorption, some qst(F ) 0) values (especially for CH4,22 CO,23 and C2H424) have been modified slightly with respect to those reported previously.22-24 It must be noted that this procedure of qst(F ) 0) estimate is not very accurate and, accordingly, the differences between qst(F ) 0) values can be regarded as inside the “experimental” error. The values of the absolute average deviation (AAD) between the experimental data and the RO and CM theoretical values of the isosteric heat of adsorption are given in Table 2 (the sources for the experimental data are also indicated in column 9 of this table). It is shown (19) Nicholson, D.; Cracknell, R. F.; Parsonage, N. G. Mol. Simul. 1990, 5, 307. (20) Kim, H.-Y.; Steele, W. A. Phys. Rev. B 1992, 45, 6226. (21) Wolfe, R.; Sams, J. R. J. Chem. Phys. 1966, 44, 2181. (22) Piper, J.; Morrison, J. A. Phys. Rev. B 1984, 30 (6), 3486. (23) Piper, J.; Morrison, J. A.; Peters, C. Mol. Phys. 1984, 53 (6), 1463. (24) Inaba, A.; Morrison, J. A. Phys. Rev. B 1984, 34 (5), 3238.
Langmuir, Vol. 17, No. 5, 2001 1579
Figure 1. Isosteric heat of adsorption of Kr on Sterling FT at 104.49 K: squares, experimental data;11 continuous line, CM theoretical values; dashed line, RO theoretical values.
Figure 2. As in Figure 1 but for Xe on grafoil at 195.5 K.27
first that the CM equation provides very good results when a noble gas (i.e., Ar, Kr, or Xe) is used as the adsorbate, as expected in view of the used molecular interaction model. This is fulfilled in a wide range of n values (see Figures 1 and 2) while the formation of the adsorbate monolayer occurs. Second, the deviations between the qst values are usually more significant only in a narrow range of n values, which are further very high. At these n values it is expected, in accordance with the results reported previously of the calorimetric heat of adsorption of nitrogen on carbon black,25 that the adsorption process proceeds by completion of the monolayer and inception of the multilayer. In fact, the experimentally determined qst for Xe/grafoil shows a marked decrease at such n values, which is compatible with the aforesaid change in the monomultimolecular adsorption mechanism. For Kr/Sterling FT, however, qst increases noticeably. This behavior was plausibly explained earlier in terms of the lateral interactions of the adsorbed molecules as they become more tightly packed in the monolayer.25 In this connection it should be noted here that for Kr/Sterling FT qst was obtained at n values that are much lower than for the rest of adsorption systems (Figures 1-4). Accordingly, the studied values of qst for Kr/Sterling FT may correspond exclusively to the submonolayer adsorption. Neverthelees, the possibility of the fact that the smaller n values for this adsorption system are due merely to a more reduced adsorption ability of Sterling FT toward Kr than, for instance, of grafoil for Xe should not be ruled out. (25) Grillet, Y.; Rouquerol, F.; Rouquerol, J. J. Colloid Interface Sci. 1979, 70, 239.
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Figure 3. As in Figure 1 but for CO on grafoil at 104.49 K.23
Figure 4. As in Figure 1, but for ethylene on grafoil (MAT) at 120 K.24
For the Kr/Sterling FT system (Figure 1), the RO equation yields qst values that are significantly larger than the ones obtained either experimentally or by the CM equation. The discrepancies in the qst values are greater in a very wide range of n values. Notice however that they are smaller at the lowest n values. In the case of Xe/grafoil (Figure 2), the agreement between the qst values obtained by the two methods is better than for Kr/Sterling FT in the entire range of n values. It is so in particular at intermediate n values. Thus, the RO curve cuts the experimental and CM curves at n values between 1.0 × 10-3 and 1.2 × 10-3 mol. In relation to the comparative analysis of the adsorption for methane, it should be pointed out first that the possible influence of the adsorbent on the molecular structure of this adsorbate in the adsorbed state has not been taken into account in this study. In other words, the same LJ parameters have been used irrespective of whether the adsorbent is grafoil or graphite. As shown in Table 2, the AADs obtained from the CM isosteric heat equation are as high as 1.5% at the most, whereas the corresponding AADs for RO equation range between the 6.4% and 8.6%. In the case of the CM equation also with methane, similarly to the above-described results for the noble gases, the greatest deviations between the experimental and calculated qst values arise at the largest n values (i.e., n > 2.0 × 10-3 mol). On the other hand, the application of the RO equation gives rise to qst values that are always greater than the experimental ones in the whole range of n values for the two adsorbents and adsorption temperatures. For the carbon monoxide/grafoil system, the AADs are close to 1-2% for the CM equation and to 2-3% for the RO equation. Furthermore, as shown in Figure 3, the CM
Cuadros et al.
qst values fit fairly well to the experimental qst data, except for at the highest n values. At these n values, qst decrease with increasing n, and therefore, the behavior is similar to the one exhibited by the Xe/grafoil system (Figure 2). Moreover it is worth noting that for the carbon monoxide/ grafoil system the agreement between the experimental and theoretical (both RO and CM) qst values is reasonably good in a somewhat wide range of low n values. The cut points between different qst curves are situated at n values below n ≈ 1.3. At higher n values, the agreement is better for the CM qst than for the RO qst as compared to the experimental qst. The RO qst deviates significantly downward from the linearity with increasing n. Concerning the ethylene/grafoil(MAT) system, the AADs vary between 1.3% and 3% for the CM equation and between 7% (at the highest temperature) and 14% (at the lowest temperature) for the RO equation. The CM qst curve (Figure 4) fits better than the RO qst curve to the experimental qst curve. The RO curve is similarly shaped to the CM curve (the former nearly parallels the latter in a wide range of n values), but the RO curve shifts to higher qst values with respect to the experimental and CM curves. This occurs also for the Kr/Sterling FT system, as shown in Figure 1. In view of the results obtained in this study for the RO equation, the possibility of making them better has also been investigated for the ethylene/grafoil (MAT) by using other values for the 2D LJ parameters. We have verified that the values of σ2D and (/k)2D that should be used so that the RO equation fit the experimental data are very different (>100%) from the corresponding 3D values. It is clear that the suitable 2D LJ parameters for the RO equation do not correspond to a “real” situation and that they deviate greatly from those predicted by other authors.12,21 Moreover, as a final comment, the RO equation is very sensitive to the values used for σ2D and (/k)2D. Furthermore, different values of these parameters need to be used at each temperature in order to get a satisfactory agreement with the experimental results. In connection with the above results it should be borne in mind that any study based on the application of the LJ potential function to model the adsorbate-adsorbate interaction in mobile adsorbed monolayers for diverse adsorption systems, even for adsorbents with an energetically homogeneous surface, is a rough approximation to the actual adsorption situation. Thus, the LJ 12-6 relation only considers the attractive dispersion forces together with short-range repulsive forces. The electrostatic (Coulombic) forces, which occur if the gas (or the solid) is polar in nature, are not allowed for. Furthermore, for adsorption of Ar, Kr, Xe, and CH4 (i.e., they are spherically symmetrical molecules and nonpolar in nature, which possess the merit of involving dispersion forces only), the degree of surface localization depends on the adsorbate/ adsorbent interaction energy, which increases with the molecular size of the adsorptive. Moreover, the shape of the experimentally determined isotherm is also a function of the adsorptive. Whereas for Kr and CH4 steplike (type VI) isotherms were reported,29 for Ar the steps almost disappeared from the isotherm. The different isotherm shapes were associated with the higher potential energy for Kr than for Ar by virtue of the greater polarizability (26) McAlpin, J. J. and Pierotti, R. A. J. Chem. Phys. 1964, 41 (1), 68. (27) Piper, J.; Morrison, J. A. Chem. Phys. Lett. 1984, 103 (4), 323. (28) Jiang, S.; Gubbins, K. E.; Zollweg, J. A. Mol. Phys. 1993, 80 (1), 103. (29) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982.
Isosteric Heat for Monolayer Adsorption
of the larger molecule of Kr.29 On the other hand, for CO and C2H4, which are gas molecules with a quadrupole moment, the polar interaction must influence both the monolayer formation and the overall adsorption energy. Finally, it should be mentioned that nitrogen has not been used in this study, despite being the most commonly used adsorptive, as it was initially focused on various gas nobles and subsequently on their comparison with polar molecules. Furthermore it should be stated, as a guide, that argon and nitrogen are a pair of adsorptives that possess a similar polarizability and molecular size. As a result, the heat of adsorption for nitrogen is almost the same as for argon on nonpolar adsorbents. For adsorbents that can interact with the quadrupole of nitrogen, however, the heat of adsorption is significantly higher for nitrogen.29,30 Using a number of this class of substrates, an extensive study is being carried out on the adsorption of nitrogen, which includes experimental results obtained by ourself, and that will be matter for a future work. 4. Conclusions This work represents a preliminary attempt to use 2D equations of state for LJ fluids in order to reproduce a set of experimental data for the isosteric heat of simple fluids adsorbed on homogeneous nonporous surfaces. A deeper study of this subject would need to carry out a systematic and extensive experimental work consisting of basically measuring the adsorption isotherms at various temperatures by using a number of previously selected adsorption systems. This would enable us to investigate the probable influence of a series of factors (i.e., the adsorbate, adsorbent, and so on) on the experimental adsorption isotherms and on the isosteric heat of adsorption. In the present study, despite the difficulties encountered regarding repects such as the limited information available in the literature, it has been analyzed the connection in being between the microscopic and macroscopic adsorption (30) Bottani, E. J.; Llanos, J. L.; Cascarini de Torre, L. E. Carbon 1989, 27 (4), 531.
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worlds, inside the two-dimensional approximation. From the above results, the following main conclusions can be drawn: (1) The CM equation of state, and its corresponding isosteric heat equation (eq 14), are an excellent model to study the physisorption of simple gases onto flat surfaces. (2) The CM model has a very simple mathematical form and is based on the WCA theory, with the repulsive and the attractive contributions separated in both equation of state and the expression of the isosteric heat of adsorption. (3) Despite the simplicity of the CM model, the AADs between the CM results and the experimental data are always similar or lower than for the RO equation. Another characteristic of the CM isosteric heat equation is its nonsensitivity to the variation of both the 2D LJ interaction parameters and the qst(F ) 0) values. The opposite just applies to the RO equation as in this case the isosteric heat of adsorption varies greatly even if such parameters are only slightly modified. (4) A set of 2D LJ parameters are put forward for the adsorbed gases, the values of which are always lower than the corresponding 3D values (i.e., around 10% for the 3D diameter, σ, and between 15 and the 25% in the 3D energy parameter, (/k)3D). These results are in accord with those reported previosly in the literature.12,21 (5) Both RO and CM models work better at intermediate coverages. Usually, the desviations from the experimental data are greater for the RO equation than for the CM equation. This equation for more adsorption systems deviates to a larger extent at the highest coverages. At these coverages, depending on the adsorption system, it is possible either that the lateral interactions between the adsorbate molecules in the adsorbed state are already significant or the commencement of the formation of the second adsorbate layer, which may be even incipient. Acknowledgment. This work was supported by the Junta de Extremadura and by the Fondo Social Europeo under Project IPR98B004. LA0014048
