ARTICLE pubs.acs.org/JPCC
Elucidation of the Local Character of Chemical Reactivity through the Time-Resolved Chromatographic Analysis of Local Molecular Properties of Gaseous Molecules Adsorbed on Solid Surfaces Eleni D. Metaxa* National Technical University of Athens, School of Chemical Engineering, 9 Iroon Polytechniou St., 157 80 Zografou, Athens, Greece
bS Supporting Information ABSTRACT: The basic objective of this research work is to contribute to the understanding of “how adsorption processes could be used in the determination of Lennard-Jones parameters for gaseous molecules adsorbed on a solid surface, by means of the time-resolved analysis of molecular properties, such as polarizabilities, ionization energies, electron affinities, electronegativities and hardness, which mirror the local character of chemical reactivity”. To this end, the well-known methodology of reversed-flow inverse gas chromatography, which provides us with real experimental values for significant physicochemical quantities illustrating the adsorptiondesorption phenomenon step by step, is appropriately associated with quantitative structureproperty relationship model and density functional theory. Finally, local molecular properties and Lennard-Jones parameters are determined for nine gassolid systems at 323.2 K, namely: C2H6(g)/TiO2(s), C2H4(g)/TiO2(s), C2H2(g)/TiO2(s), C2H6(g)/Fe2O3(s), C2H4(g)/Fe2O3(s), C2H2(g)/Fe2O3(s), C2H6(g)/ZnO(s), C2H4(g)/ZnO(s), and C2H2(g)/ZnO(s).
1. INTRODUCTION Adsorption occurs, at least partially, as a result of (and likewise influences and alters) forces active within phase boundaries or surface boundaries: these forces result in characteristic boundary energies. Lateral molecular interactions between neighbor adsorbates greatly affect the kinetics of primary surface processes. Adsorbate aggregation, island formation, the appearance of different overlayer structures, and phase transitions between them are among the “mesoscopic” manifestations of adsorbate lateral interactions. All of these phenomena affect the local environment of the adsorbed particles and are thus expected to affect significantly the kinetics of the various surface processes such as diffusion, desorption, adsorption, and chemical reaction. Different techniques can be used to quantify lateral interactions. Until recently, most of experimental research into the ordering of adsorbates on surfaces has provided us with only statistical averages of correlation functions, convoluted with some instrument response function. By calculating the total energy of different coverages using density functional theory (DFT), it is possible to derive some values of the interactions. The method has the inherent drawback that as the interactions are small and are obtained by subtracting total energies, the inaccuracy is large. In diffraction measurements, one can measure the long-range order parameter below the transition and its fluctuations above it, or in a different limit, an ill-defined sum over short-range correlation functions. Vibrational probes similarly give information about long-range order parameter but with a far shorter range instrument response.1 The experimental results obtained from methods like calorimetry and temperature-programmed desorption (TPD) have been simulated by Monte Carlo calculations to obtain the lateral interactions by fitting the simulated curve with the experimental curve.2,3 r 2011 American Chemical Society
Field ion microscopy (FIM) and later on scanning tunneling microscopy (STM) were used to determine directly pairwise lateral interactions.48 Recently, it was shown that pairwise interactions are not always sufficient to describe the formed patterns and manybody effects have to be taken into account.9,10 The full power of STM and FIM as quantitative probes of atomic positions is that they allow experimental observation of specific (not just combinations of) short-range correlation functions. However, neither of these schemes has yet been applied to actual experimental data. The use of gas chromatography (GC) for physicochemical measurements based on the traditional techniques of elution, frontal analysis, and displacement development is not a new field of research. However, the novel method of reversed-flow inverse GC, which is based on perturbations of the carrier gas flow made by the reversing of its direction from time to time, offers the possibility of determination of physicochemical quantities that are not easily or accurately measured by the aforementioned traditional chromatographic methods.11,12 In the present work, local molecular properties and LennardJones parameters are determined, for first time so far, directly from real experimental adsorption data, as they are supplied from the RF-IGC measurements in various adsorption systems gas/solid. It is worth noting, at this point, that Lennard-Jones parameters or, in other words, collision cross-sectional parameters, have also been determined in the past by using RF-IGC but through the experimentally measured mutual diffusion coefficients in binary gas mixtures in the absence of any solid material, namely, without Received: September 16, 2011 Revised: November 28, 2011 Published: November 28, 2011 25389
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Figure 1. Schematic representation of the RF-IGC experimental setup.
adsorption phenomena taking place.13,14 However, the research presented in this manuscript forms a different approach of the problem for the determination of Lennard-Jones parameters without ignoring adsorption phenomena but with the adsorption to govern the whole phenomenon. In addition, this research paper supplies us with local values of molecular properties in a timeresolved way, namely, as a function of the experimental time, which involves only the adsorption energy sites that are active at this time and not all adsorption energy sites. By considering that the adsorbed species on a solid surface are “van der Waals molecules”, which are trapped together by van der Waals forces forming either islands or patches and by using the tools offered by the reversedflow inverse GC in combination with further derived from the application of quantitative structureproperty relationship model and DFT, the description of lateral molecular interactions developed in the adsorption of gaseous C2-hydrocarbons onto solid powdered metaloxides was attempted through the experimental determination of local molecular properties polarizabilities, ionization energies, electronegativities, electron affinities, and hardness for the reason that these properties reflect the strengths and character of interacting molecules, with or without the presence of a solid substrate, namely, their reactivity, which has essentially a local character. The method was applied to nine gassolid systems, at 323.2 K, namely, at the systems: C2H6(g)/TiO2(s), C2H4(g)/TiO2(s),
C2H2(g)/TiO2(s), C2H6(g)/Fe2O3(s), C2H4(g)/Fe2O3(s), C2H2(g)/ Fe2O3(s), C2H6(g)/ZnO(s), C2H4(g)/ZnO(s), and C2H2(g)/ZnO(s).
2. EXPERIMENTAL METHODS 2.1. Instruments and Materials. Reversed-flow inverse GC is a differential technique for studying adsorption and catalysis, consisting of continuously switching the system under examination from a flow dynamic system to a static system and vice versa. This permits diffusion and other related phenomena to come into play when the flow is reversed. A schematic representation of the columns and gas connections for the application of RF-IGC is given in Figure 1. It is a simple apparatus, which consists of a conventional gas chromatograph (Shimadzu 8A) supplied with a flame ionization detector capable of detecting the vapor(s) contained in the carrier gas. Inside the oven of the gas chromatograph, two columns are placed into: (i) the sampling column, constructed from stainless steel with a 4.0 mm i.d. and a total length of 1.0 m, in which the carrier gas flows, and (ii) the diffusion column, perpendicular placed into the former, which is constructed from glass, has a 3.5 mm i.d., and consists of two distinct regions: an empty part (z-region, with a length of 22.4 cm) and a filled part (y-region, filled with a solid powder covering a length of 5.2 cm). The diffusion column forms the “reactor” of the whole system, and no carrier gas flows inside it. 25390
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Ultra high-purity nitrogen was used as carrier gas, supplied from AirLiquide Hellas Company. The gaseous analytes used for the adsorption studies were ethane, ethane, and ethyne supplied from AirLiquide as well. The purities of these hydrocarbons were 99.95, 99.5, and 99.6%, respectively. The inorganic pigments TiO2, Fe2O3, and ZnO used as adsorbents were Merck pro-analysi products, with external porosity values being 0.5459, 0.7988, and 0.3840, respectively; their specific surface areas were 31 100, 23 400, and 17 316 cm2 3 g1, correspondingly. The external porosities of these solids were determined by the mercury-penetration method with a Porosimeter 2000 instrument, Milestone 200, and the specific surface areas of them were determined by the nitrogen-desorption method, with a Sorptomatic 1900 instrument, Milestone 200. 2.2. Procedure. Before each adsorption experiment, the solid bed was conditioned at 473.2 K for 24 h under a constant carrier gas flow. The experiments were carried out at 323.2 K under a carrier gas flow rate of 26.1 cm3 3 min1. At the working temperature, 1 cm3 of gaseous hydrocarbon was injected into the end of the column L2, and the sampling procedure, namely, the double reversing of the carrier gas flow direction, was starting when a continuously rising baseline appeared. Each (double) reversal lasted 10 s, namely, shorter than the gas hold-up time in section (l + l0 ) of the sampling column (Figure 1). Following each sampling procedure, narrow fairly symmetrical sample peaks superimposed on the continuous baseline were created and recorded in a PC-computer desktop by the CLASS VP chromatography data system, supplied from Shimadzu. A representative chromatogram is also depicted in Figure 1. 2.3. Mathematical Model and Calculations. The mathematical model describing the physicochemical phenomena taking place inside the diffusion column is quite complicated, and its circumstantial description can be searched elsewhere.1518 Anyway, the following equation forms the core of the experiment and mathematical model H 1=M ¼ g 3 cðl0 , tÞ ¼
∑i Ai expðBi tÞ
ð1Þ
Equation 1 describes the so-called diffusion band of the RF-IGC method, with M denoting the response factor of the chromatographic detector (for FID, M = 1) and Ai, and Bi being functions of the physicochemical quantities pertaining to the adsorption phenomena taking place in the filled region y of the diffusion column (Figure 1). By using nonlinear least-squares regression analysis,1518 one can calculate from the values of the experimental pairs (H, t) the values of Ai and Bi of eq 1. Furthermore, g [cm/(mol 3 cm3)] is the calibration factor19 and c(l0 ,t) is the measured sampling concentration of the gaseous analyte at x = l0 or z = 0, expressed in mol 3 cm3. The form of eq 1 is not an a priori assumption but the solution of a system of three partial differential equations and of a local adsorption isotherm by using double Laplace transformations with respect to time and length coordinates, under the appropriate initial and boundary conditions. The entire manipulations and procedure for the solution finding are quite complicated and can be searched for elsewhere.1523 It is noteworthy that all of the calculated local adsorption parameters, ε, θ, β, cy, c/smax, and j(ε,t), are hidden under the coefficients Ai and Bi of eq 124, and they are not referred to here but placed in the Supporting Information (eqs 210 therein). Because in adsorption phenomena the activation of various active sites on the surface of the solid adsorbent does not happen at the same moment for the entirety of them but only various and different groups of them are activated for adsorption at different time intervals, the adsorption phenomenon is considered to be
both site- and time-dependent. The activation of different groups of active sites in different time intervals implies the necessity of the use of local physicochemical quantities for the realistic description of the adsorption phenomenon. In fact, a local quantity, which is site-dependent, is also time-dependent because of there permanently being dependence among site and time. Therefore the time-resolved analysis of local parameters describing chemical reactivity of molecules that is local owing to its variation from one site in a molecule to another and determines the physical and chemical character of adsorption, such as local polarizability, local ionization energy, local hardness, local electron affinity (EA), and local electronegativity, is essential. The target of our research is the determination of local molecular properties and LJ parameters by using real adsorption data supplied from the reversed-flow inverse GC methodology in association with appropriate relations derived either from the quantitative structureproperty relationship model or DFT as well as hard soft acidbase (HSAB) theory.2527 2.3.1. Interrelation between Molecular Polarizabilities and Ionization Energies for the Adsorbates with Their Vapor Pressure above the Solid Surface. The vapor pressure of a substance not only is a measure of the maximum possible concentration of the substance in the gas phase at a given temperature but also provides important quantitative information on the attractive forces among the substance’s molecules in the condensed phase.28 If, now, we consider as “condensed phase” the “adsorbed concentration” of a gaseous substance onto a solid surface, then we could obtain information about lateral molecular interactions in the adsorbed phase by correlating in a way vapor pressure over the solid surface with intermolecular attractions, namely, with dispersion forces, which are a function of the molecule’s polarizability.29 Polarizability (α) is an operator that describes the distortion induced by an external electric field at the electric charge distribution in a molecule. It is an essential physicochemical quantity concerning both covalent and noncovalent interactions, although sometimes different names are used to describe it. Because the stronger the intermolecular forces are, the more tightly the molecules are held together onto the solid adsorbent surface, the lower the vapor pressure will be. Furthermore, if we consider an “ideal gas” behavior for the gas phase over the solid, then we can substitute nonadsorbed concentration of the gaseous analyte for its vapor pressure over the solid, that is, p = RTCy. Liang and Gallagher (1998) used quantitative structure property relationship model (QSPR) to predict vapor-pressure from only computational-derived molecular descriptors, such as molecular polarizability, aiming, on the one hand, at the study of hypothetical molecular structures, without the need for their prior synthesis and testing and, on the other hand, at the design of new molecules with more desirable properties. Finally, they formulated various equations to relate vapor pressure (log p) with molecular polarizability (α) for a set of 479 compounds, for which all subcooled vapor pressure (log p) data used in this model development were obtained directly from the literature. Therefore, they found that polarizability produced relatively high r2 values within many of the individual compound classes and did especially well for the nonpolar hydrocarbons with r2 = 0.997.25 In this research manuscript, we adopt two of these correlation equations, in particular: (i) concerning nonpolar hydrocarbons, as eq 11, represents log½pðTorrÞ ¼ 0:541a þ 5:600
ð11Þ
(n = 70 compounds in the database, r2 = 0.997, standard error = 0.185) 25391
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(ii) the general model equation, eq 12, for all compounds in the database log½pðTorrÞ ¼ 0:401a þ 3:940
ð12Þ
(n = 479 compounds in the database, r2 = 0.922, standard error = 0.745) The physical meaning of eqs 11 and 12 is indicated by the negative sign on the polarizability term. Increased polarizability would increase the dispersion type intermolecular interactions and thus decrease the vapor pressure.26 Because the values of p, slopes, and constants in the above equations are in Torr, taking into account the following relationships and interconversions
where α is the molecular polarizability (in cubic angstroms), V is the molecular volume (in cubic angstroms), I is the first ionization energy (in electronvolts), and Is,ave is the average local ionization energy (in electronvolts).27 By solving eq 15 as for V and substituting it in each one of eqs 1619 and finally by rearranging the so-resultant equations as for I or Is,ave, we obtain the following equations I ¼
I ¼
p ¼ RTcy , R ¼ 62 548 Torr 3 cm3 3 K 1 3 mol1 ,
aðÅ Þ ¼ 3:153 1:848 log½cy ðtÞðmol 3 cm Þ
ð13Þ
aðÅ Þ ¼ 8:393 2:494 log½cy ðtÞðmol 3 cm3 Þ
ð14Þ
3
The significance of eqs 13 and 14 consists of providing us with the capability of determining local polarizability values through the existing experimental local values for nonadsorbed gaseous concentration, cy, namely, as functions of experimental time, t, which is measured from the moment of introduction of the gas under examination into the solid bed and continued as long as the chromatographic detector records substance concentrations. Perturbation theory links polarizability, α, to the size of a system and its excitation energies.30,31 In practice, however, it has been found that α correlates directly with volume, V, alone, for both of atoms and molecules and inversely with the first ionization energy, I, alone, only for atoms.3133 For molecules, I tends to vary over a smaller range than either the volume or polarizability and does not show any consistent pattern with respect to each other.32,34 Interrelations between molecular polarizability, α, molecular volume, V, ionization energy, I, and average local ionization energy, Is,ave, have been extensively investigated by Jin et al. (2004), the latter being a measure of the energy required to remove an electron from any point Br in the space of an atom and molecule. As the same r Þ, or researchers support, the average local ionization energy, Ið! Is,ave, can be viewed as indicative of local polarizability.27,35,36 The relationships extracted from Jin et al. (2004) investigation on a series of 29 molecules of different chemical types are reproduced here (eqs 1519) because they will form the basis for the development other different model equations for the determination of local molecular polarizabilities and ionization potentials as functions of experimental time, t, as before in the case of eqs 13 and 14. a ¼ 0:09360V 1:504
ð15Þ
a ¼ 0:7343ðV =IÞ 0:105
ð16Þ
a ¼ 0:2637ðV =I 1=2 Þ 0:812
ð17Þ
a ¼ 1:450ðV =Is, ave Þ 1:276
ð18Þ
a ¼ 0:3713ðV =Is,1=2 ave Þ 1:453
ð19Þ
Is, ave
ð16aÞ 2 ð17aÞ
1:450ð1:504 þ aÞ 0:09360ð1:276 þ aÞ
followed by rearrangement as for molecular polarizability, α, the above eqs 11 and 12 are converted into the eqs 13 and 14, respectively 3
0:2637ð1:504 þ aÞ 0:09360ð0:812 þ aÞ
Is, ave ¼
T ¼ 323:2 K, cy ½¼mol 3 cm3
3
0:7343ð1:504 þ aÞ 0:09360ð0:105 þ aÞ
0:3713ð1:504 þ aÞ ¼ 0:09360ð1:453 þ aÞ
ð18aÞ 2 ð19aÞ
Now it becomes obvious that if α in each one of eqs 16a19a is expressed by eq 13 or 14, then I or Is,ave will finally be transformed into a time expression because cy is a time-resolved quantity. Another attempt of approach of the problem of the local molecular polarizabilities calculation as a function of time t comprises as “starting-point” eq 15, followed by replacing of term of molecular volume, V, by the term “log{1024 (Å3 3 cm3) 3 [NA (molecules 3 mol1)] 3 [cy (mol 3 cm3)]}” and repealing of the coefficient 0.09360, namely, replacing it with 1. The use of logarithm instead of the net quantity representing the molecular volume according to RF-IGC data is not arbitrary, but it has resulted from the fact that the application of experimental values of cy in this relation with no use of logarithm produces extremely high values for α. In addition, the unit substituted for 0.09360 because of negative values arising in the opposite case. Therefore, the new relation resulting from the previous procedure, which calculates local molecular polarizabilities by using local nonadsorbed concentration values for the gaseous analyte under examination as they are determined by RF-IGC as a function of time, is described by the following equation (NA = 6.022 1023 molecules 3 mol1) " # 3 1024 ðÅ cm3 Þ 3 aðÅ Þ log 1:504 NA ðmolecules 3 mol1 Þ 3 cy ðmol 3 cm3 Þ ¼ 1:284 log½cy ðmol 3 cm3 Þ
ð20Þ
It is noteworthy that eqs 13 and 14 are very similar to eq 20 having only different scale factors. This similarity validates, further, the form of dependence of molecular polarizability on the nonadsorbed concentration of the gaseous analyte and by extension on the experimental time. Afterward, we are able to determine local molecular polarizabilities by means of eqs 13, 14 and 20 and the corresponding local ionization energies by using any of eqs 16a19a, all of them as functions of experimental time, by incorporating in them real experimental data resulted from RF-IGC adsorption experiments. For this purpose, the above equations are applied to nine gas/solid systems at 323.2 K, namely, at the systems: C2H6(g)/ TiO2(s), C2H4(g)/TiO2(s), C2H2(g)/TiO2(s), C2H6(g)/Fe2O3(s), 25392
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Table 1. RF-IGC Experimental Values of Local NonAdsorbed Concentrations and Local Molecular Polarizabilities for the Adsorption System C2H6(g)/TiO2(s) at 323.2 K α (Å3) cy (μmol 3 cm3)
eq 13
eq 14
eq 20
2
5.849
6.520
4.657
3.949
4
9.648
6.118
4.115
3.732
t (min)
6 8
12.05 13.50
5.940 5.848
3.874 3.751
3.635 3.586
10
14.30
5.802
3.688
3.561
12
14.67
5.781
3.661
3.550
14
14.75
5.777
3.655
3.547
16
14.63
5.784
3.664
3.551
18
14.39
5.797
3.682
3.558
20
14.06
5.816
3.707
3.568
22 24
13.68 13.27
5.838 5.862
3.737 3.770
3.580 3.593
26
12.84
5.889
3.805
3.608
28
12.41
5.916
3.842
3.623
30
11.98
5.944
3.880
3.638
32
11.55
5.973
3.920
3.654
34
11.14
6.003
3.959
3.670
36
10.73
6.033
4.000
3.686
38 40
10.33 9.949
6.063 6.093
4.040 4.081
3.702 3.718
42
9.578
6.124
4.123
3.735
44
9.220
6.155
4.164
3.752
46
8.874
6.185
4.205
3.768
48
8.540
6.216
4.247
3.785
50
8.219
6.247
4.288
3.801
52
7.909
6.278
4.330
3.818
54 56
7.611 7.323
6.308 6.339
4.372 4.413
3.835 3.852
58
7.047
6.370
4.455
3.868
60
6.781
6.401
4.497
3.885
62
6.525
6.432
4.538
3.902
64
6.279
6.463
4.580
3.918
66
6.042
6.494
4.622
3.935
68
5.814
6.525
4.663
3.952
70 72
5.595 5.384
6.555 6.586
4.705 4.746
3.968 3.985
74
5.181
6.617
4.788
4.002
76
4.986
6.648
4.830
4.018
78
4.799
6.679
4.871
4.035
80
4.618
6.710
4.913
4.052
82
4.444
6.740
4.954
4.068
84
4.277
6.771
4.996
4.085
86
4.117
6.802
5.037
4.102
88
3.962
6.833
5.079
4.118
90
3.813
6.863
5.120
4.135
92
3.670
6.894
5.162
4.152
94
3.532
6.925
5.203
4.168
96
3.400
6.955
5.244
4.185
98
3.273
6.986
5.286
4.201
100
3.150
7.017
5.327
4.218
Table 1. Continued α (Å3) t (min)
cy (μmol 3 cm3)
eq 13
eq 14
eq 20
102 104
3.032 2.918
7.047 7.078
5.368 5.410
4.235 4.251
106
2.809
7.109
5.451
4.268
108
2.704
7.139
5.492
4.284
110
2.603
7.170
5.534
4.301
112
2.506
7.200
5.575
4.317
114
2.412
7.231
5.616
4.334
116
2.322
7.261
5.657
4.350
118 120
2.235 2.152
7.292 7.323
5.699 5.740
4.367 4.383
122
2.071
7.353
5.781
4.400
124
1.994
7.384
5.822
4.417
126
1.920
7.414
5.863
4.433
128
1.848
7.445
5.905
4.450
130
1.779
7.475
5.946
4.466
132
1.713
7.506
5.987
4.483
134 136
1.649 1.587
7.536 7.567
6.028 6.069
4.499 4.516
138
1.528
7.597
6.111
4.532
140
1.471
7.628
6.152
4.549
142
1.416
7.658
6.193
4.565
144
1.363
7.689
6.234
4.582
146
1.313
7.719
6.275
4.598
148
1.264
7.750
6.316
4.615
150
1.217
7.780
6.357
4.631
C2H4(g)/Fe2O3(s), C2H2(g)/Fe2O3(s), C2H6(g)/ZnO(s), C2H4(g)/ ZnO(s), and C2H2(g)/ZnO(s), which have been studied by RFIGC method. The local values of molecular polarizabilities and ionization energies determined experimentally, using RF-IGC method and eqs 13, 14, and 20, for the systems in question, at 323.2 K, are given in Tables 16, and the corresponding diagrams are represented in Figures 24. 2.3.2. Determination of Local Hardness, Local Electron Affinity, and Local Electronegativity for C2-Hydrocarbons through Their Adsorption on Solid Surfaces By Using Time-Resolved Analysis of Molecular Polarizabilities and Ionization Energies. The global chemical potential (μ) and the global absolute chemical hardness (n) are defined according to DFT as the first- and the second-order derivatives, respectively, of the KohnSham (KS) energy to the number of electrons at constant potential (of nuclei and any other external potential) μ ¼ χ ¼ ð∂E=∂NÞuð n ¼ ð∂2 E=∂N 2 Þuð r Þ B
Br Þ
ð21Þ ð22Þ
where E and N are the total energy and the total number of electrons in the system, respectively, and u(Br ) is the total external potential (of nuclei and any other external potential) and χ is the global absolute electronegativity.3743 The nonchemical meaning of the word “hardness” is “resistance to deformation or change”. In a similar way, “chemical hardness” measures the resistance of a chemical potential of the electrons to change in the number of electrons, or equivalently, it 25393
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Table 2. RF-IGC Experimental Values of Local NonAdsorbed Concentrations and Local Molecular Polarizabilities for the Adsorption System C2H4(g)/Fe2O3(s) at 323.2 K α (Å3) t (min) 3
cy (μmol 3 cm3)
eq 13
eq 14
eq 20
4.943
6.655
4.839
4.022
5
10.50
6.050
4.023
3.695
7
13.78
5.832
3.728
3.577
9
15.64
5.730
3.592
3.522
11 13
16.59 16.98
5.683 5.664
3.527 3.503
3.496 3.486
15
17.00
5.663
3.501
3.486
17
16.79
5.673
3.515
3.491
19
16.43
5.691
3.538
3.501
21
15.99
5.712
3.567
3.512
23
15.50
5.738
3.601
3.526
25
14.98
5.765
3.638
3.541
27 29
14.45 13.92
5.794 5.824
3.677 3.718
3.556 3.573
31
13.39
5.855
3.760
3.589
33
12.88
5.886
3.802
3.606
35
12.38
5.918
3.845
3.624
37
11.89
5.950
3.888
3.641
39
11.42
5.983
3.932
3.659
41
10.97
6.015
3.976
3.676
43 45
10.53 10.11
6.048 6.080
4.020 4.064
3.694 3.711
47
9.707
6.113
4.108
3.729
49
9.319
6.146
4.152
3.747
51
8.946
6.179
4.197
3.765
53
8.587
6.212
4.241
3.782
55
8.242
6.244
4.285
3.800
57
7.911
6.277
4.330
3.818
59 61
7.593 7.288
6.310 6.343
4.374 4.418
3.836 3.854
63
6.995
6.376
4.463
3.871
65
6.714
6.409
4.507
3.889
67
6.444
6.442
4.552
3.907
69
6.184
6.475
4.596
3.925
71
5.935
6.508
4.641
3.943
73
5.696
6.541
4.685
3.961
75
5.467
6.574
4.730
3.978
77
5.247
6.607
4.774
3.996
79
5.035
6.640
4.819
4.014
81
4.832
6.673
4.864
4.032
83
4.638
6.706
4.908
4.050
85
4.451
6.739
4.953
4.068
87
4.271
6.772
4.997
4.086
89
4.099
6.805
5.042
4.104
91
3.934
6.838
5.086
4.121
93
3.775
6.871
5.131
4.139
95
3.622
6.904
5.176
4.157
97
3.476
6.938
5.220
4.175
99
3.336
6.971
5.265
4.193
101
3.201
7.004
5.310
4.211
Table 2. Continued α (Å3) t (min)
cy (μmol 3 cm3)
eq 13
eq 14
eq 20
103 105
3.072 2.948
7.037 7.070
5.354 5.399
4.229 4.247
107
2.829
7.103
5.444
4.265
109
2.714
7.136
5.488
4.283
111
2.605
7.169
5.533
4.301
113
2.499
7.202
5.578
4.318
115
2.398
7.236
5.622
4.336
117
2.301
7.269
5.667
4.354
119 121
2.208 2.119
7.302 7.335
5.712 5.756
4.372 4.390
123
2.033
7.368
5.801
4.408
125
1.951
7.401
5.846
4.426
127
1.872
7.434
5.891
4.444
129
1.796
7.468
5.935
4.462
131
1.723
7.501
5.980
4.480
133
1.654
7.534
6.025
4.498
135 137
1.587 1.522
7.567 7.600
6.070 6.115
4.516 4.534
139
1.461
7.634
6.159
4.552
141
1.401
7.667
6.204
4.570
143
1.345
7.700
6.249
4.588
145
1.290
7.733
6.294
4.606
147
1.238
7.766
6.339
4.624
149
1.188
7.800
6.383
4.642
151 153
1.140 4.943
7.833 6.655
6.428 4.839
4.660 4.022
measures the sensitivity of electronegativity to change in the number of electrons.38,42 The global hardness is an indicator of the overall stability of the system. Among the most fundamental chemical reactivity principles is the maximum hardness principle (MHP), which asserts that “molecules arrange themselves so as to be as hard as possible”. Therefore, according to this HSAB theory introduced by Pearson in 1963, for the empirical description of the chemical reactivity of molecules,44,45 hardness and softness simply correspond to low and high polarizability, respectively. It has been observed that a state of minimum polarizability is usually associated with higher stability or maximum hardness.46 Hardness has been linked inversely to the cubic power of polarizability (n µ α1/3), resulting in very good results in cases of atoms but not so good in cases of molecules.4749 The most operational equations used for the calculation of global absolute chemical potential, electronegativity, and hardness are based on the three point finite difference approximation and express χ and n through the first vertical ionization potential (I) and the EA of the neutral molecule4143 μ ¼ χ ¼ 0:5ðI þ EAÞ
ð23Þ
n ¼ I EA
ð24Þ
Another DFT approximation originating from the Koopmans’ theorem is50 fI ≈ εHOMO , EA ≈ εLUMO g 25394
ð25Þ
dx.doi.org/10.1021/jp2089824 |J. Phys. Chem. C 2011, 115, 25389–25412
The Journal of Physical Chemistry C
ARTICLE
Table 3. RF-IGC Experimental Values of Local NonAdsorbed Concentrations and Local Molecular Polarizabilities for the Adsorption System C2H2(g)/ZnO(s) at 323.2 K α (Å3) t (min) 2
cy (μmol 3 cm3)
eq 13
eq 14
eq 20
3.638
6.901
5.171
4.155
4
13.81
5.830
3.727
3.576
6
19.26
5.563
3.366
3.432
8
21.90
5.460
3.227
3.376
10 12
22.85 22.83
5.426 5.427
3.181 3.182
3.357 3.358
14
22.26
5.447
3.209
3.369
16
21.39
5.479
3.253
3.386
18
20.37
5.518
3.305
3.407
20
19.31
5.561
3.363
3.430
22
18.24
5.607
3.425
3.455
24
17.20
5.654
3.488
3.481
26 28
16.20 15.26
5.702 5.750
3.553 3.618
3.507 3.533
30
14.36
5.799
3.684
3.559
32
13.52
5.847
3.749
3.585
34
12.73
5.896
3.815
3.612
36
11.98
5.944
3.880
3.638
38
11.28
5.993
3.945
3.664
40
10.62
6.041
4.010
3.690
42 44
10.01 9.426
6.089 6.137
4.075 4.140
3.716 3.742
46
8.880
6.185
4.205
3.768
48
8.367
6.232
4.269
3.794
50
7.883
6.280
4.333
3.820
52
7.429
6.328
4.398
3.845
54
7.000
6.376
4.462
3.871
56
6.597
6.423
4.526
3.897
58 60
6.217 5.859
6.471 6.518
4.591 4.655
3.923 3.948
62
5.522
6.566
4.719
3.974
64
5.204
6.614
4.783
4.000
66
4.904
6.661
4.848
4.026
68
4.621
6.709
4.912
4.051
70
4.355
6.757
4.976
4.077
72
4.104
6.804
5.040
4.103
74 76
3.868 3.644
6.852 6.900
5.105 5.169
4.129 4.155
78
3.434
6.947
5.233
4.180
80
3.236
6.995
5.298
4.206
82
3.049
7.043
5.362
4.232
84
2.873
7.091
5.427
4.258
86
2.706
7.138
5.491
4.284
88
2.550
7.186
5.556
4.310
90
2.402
7.234
5.621
4.336
92
2.263
7.282
5.685
4.362
94
2.132
7.330
5.750
4.388
96
2.008
7.378
5.815
4.413
98
1.892
7.426
5.879
4.439
100
1.782
7.474
5.944
4.465
Table 3. Continued α (Å3) t (min)
cy (μmol 3 cm3)
eq 13
eq 14
eq 20
102 104
1.678 1.581
7.522 7.570
6.009 6.074
4.491 4.517
106
1.489
7.618
6.139
4.543
108
1.402
7.666
6.204
4.569
110
1.320
7.715
6.269
4.596
112
1.243
7.763
6.334
4.622
114
1.171
7.811
6.399
4.648
116
1.103
7.859
6.464
4.674
118 120
1.038 0.9777
7.907 7.956
6.529 6.594
4.700 4.726
122
0.9206
8.004
6.659
4.752
124
0.8668
8.052
6.724
4.778
126
0.8161
8.101
6.790
4.804
128
0.7684
8.149
6.855
4.831
130
0.7234
8.198
6.920
4.857
132
0.6811
8.246
6.986
4.883
134 136
0.6412 0.6036
8.294 8.343
7.051 7.116
4.909 4.935
138
0.5682
8.391
7.182
4.962
140
0.5349
8.440
7.247
4.988
142
0.5035
8.488
7.313
5.014
144
0.4740
8.537
7.378
5.040
146
0.4462
8.586
7.444
5.067
148
0.4200
8.634
7.509
5.093
150
0.3953
8.683
7.575
5.119
For closed-shell species, it is nHL ¼ εLUMO εHOMO
ð26Þ
where εHOMO and εLUMO are the energies of the highest occupied molecular orbital (HOMO) and the lowest occupied molecular orbital (LUMO), respectively.41,42 A series of DFT calculations on a variety of 52 representative molecular/atomic systems, including various inorganic and organic molecules, with ionic and covalent bonds, by using a commonly used exchangecorrelation functional, B3LYP, finally produced satisfactory linear relationships that can be used to semiquantitatively estimate all of the above fundamental molecular properties, that is, I, EA, nHL and n, based on the calculated HOMO and LUMO energies41 IðeVÞ ¼ 1:3023ð εHOMO ÞðeVÞ þ 0:481
ð27Þ
EAðeVÞ ¼ 0:6091ð εLUMO ÞðeVÞ 0:475
ð28Þ
n ¼ 1:6112nHL þ 1:201
ð29Þ
By combining eqs 2729 with eq 26, a relationship between the calculated HOMO and LUMO energies arises εLUMO ¼ 0:3083εHOMO 1:192
ð30Þ
All of the above eqs 2130 determine the values of global absolute molecular parameters I, EA, χ, μ, n, εHOMO, and εLUMO. The use of local values of I supplied from its time-resolved analysis described in the subsection 3.2, instead of its global values in the above relationships, will give the local values of the 25395
dx.doi.org/10.1021/jp2089824 |J. Phys. Chem. C 2011, 115, 25389–25412
The Journal of Physical Chemistry C
ARTICLE
Table 4. RF-IGC Experimental Values of First Local Ionization Energies for the Adsorption System C2H6(g)/TiO2(s), at 323.2 K, According to Equations 16a and 13, 17a and 13, 18a and 13, 19a and 13, 16a and 14, and 17a and 14 from I1 to I6, Respectively t (min) 2 4 6 8
I1 (eV) 9.502 9.609 9.661 9.689
I2 (eV) 9.506 9.602 9.648 9.672
I3 (eV) 15.944 15.968 15.980 15.986
I4 (eV) 15.938 15.949 15.954 15.957
I5 (eV) 10.150 10.446 10.604 10.692
Table 4. Continued t (min)
I1 (eV)
I2 (eV)
I3 (eV)
I4 (eV)
I5 (eV)
I6 (eV)
102 104
9.380 9.373
9.396 9.391
15.915 15.913
15.926 15.925
9.851 9.836
9.814 9.801
106
9.367
9.385
15.912
15.924
9.821
9.788
I6 (eV)
108
9.360
9.379
15.910
15.924
9.806
9.775
10.073
110
9.354
9.373
15.909
15.923
9.792
9.763
10.324 10.455
112
9.348
9.367
15.907
15.922
9.778
9.750
114
9.341
9.362
15.906
15.922
9.764
9.738
10.527
116
9.335
9.356
15.904
15.921
9.750
9.726
9.329 9.323
9.351 9.345
15.903 15.901
15.920 15.920
9.736 9.723
9.714 9.702
10
9.703
9.685
15.989
15.958
10.739
10.566
12
9.710
9.691
15.991
15.959
10.760
10.583
118 120
14
9.711
9.692
15.991
15.959
10.764
10.587
122
9.317
9.340
15.900
15.919
9.710
9.691
10.581
124
9.311
9.334
15.898
15.918
9.697
9.679
9.305
9.329
15.897
15.918
9.684
9.668
16
9.709
9.690
15.991
15.959
10.758
18
9.705
9.686
15.990
15.958
10.744
10.570
126
20 22
9.699 9.692
9.681 9.675
15.989 15.987
15.958 15.957
10.725 10.702
10.554 10.536
128
9.299
9.323
15.895
15.917
9.672
9.657
130
9.293
9.318
15.894
15.916
9.659
9.646
24
9.685
9.668
15.985
15.956
10.678
10.516
132
9.287
9.313
15.893
15.916
9.647
9.635
26
9.677
9.661
15.983
15.956
10.652
10.495
28
9.668
9.654
15.982
15.955
10.626
10.473
134 136
9.282 9.276
9.308 9.302
15.891 15.890
15.915 15.915
9.635 9.623
9.624 9.614
30
9.660
9.646
15.980
15.954
10.599
10.451
138
9.270
9.297
15.889
15.914
9.611
9.603
32
9.651
9.639
15.978
15.953
10.572
10.429
140
9.265
9.292
15.887
15.913
9.600
9.593
34
9.642
9.631
15.976
15.952
10.546
10.406
142
9.259
9.287
15.886
15.913
9.588
9.583
36 38
9.634 9.625
9.623 9.615
15.974 15.972
15.951 15.950
10.519 10.493
10.384 10.362
144
9.254
9.282
15.884
15.912
9.577
9.573
146
9.248
9.277
15.883
15.912
9.566
9.563
40
9.616
9.608
15.970
15.950
10.467
10.341
148
9.243
9.272
15.882
15.911
9.555
9.553
42
9.607
9.600
15.968
15.949
10.442
10.319
150
9.237
9.267
15.880
15.910
9.544
9.543
44
9.599
9.592
15.966
15.948
10.416
10.298
46
9.590
9.585
15.964
15.947
10.392
10.278
48
9.582
9.577
15.962
15.946
10.367
10.257
50
9.573
9.570
15.960
15.945
10.344
10.237
52 54
9.565 9.557
9.562 9.555
15.958 15.956
15.944 15.944
10.320 10.297
10.217 10.198
56
9.548
9.548
15.954
15.943
10.274
10.179
58
9.540
9.540
15.952
15.942
10.252
10.160
60
9.532
9.533
15.951
15.941
10.230
10.141
62
9.524
9.526
15.949
15.940
10.209
10.123
64
9.516
9.519
15.947
15.940
10.188
10.105
66
9.509
9.512
15.945
15.939
10.167
10.088
68 70
9.501 9.493
9.505 9.498
15.943 15.941
15.938 15.937
10.147 10.127
10.070 10.053
72
9.486
9.491
15.940
15.936
10.108
10.037
74
9.478
9.485
15.938
15.936
10.088
10.020
76
9.471
9.478
15.936
15.935
10.070
10.004
78
9.463
9.471
15.934
15.934
10.051
9.988
80
9.456
9.465
15.933
15.933
10.033
9.972
82
9.449
9.458
15.931
15.933
10.015
9.957
84
9.442
9.452
15.929
15.932
9.997
9.941
86
9.434
9.446
15.928
15.931
9.980
9.926
88
9.427
9.439
15.926
15.930
9.963
9.912
90
9.420
9.433
15.924
15.930
9.946
9.897
92
9.414
9.427
15.923
15.929
9.929
9.883
94
9.407
9.421
15.921
15.928
9.913
9.869
96
9.400
9.414
15.920
15.928
9.897
9.855
98
9.393
9.408
15.918
15.927
9.881
9.841
100
9.386
9.402
15.916
15.926
9.866
9.827
rest molecular parameters. Moreover, the local EA combined with the local first ionization energy (I) can result in the calculation of the local electronegativity (χ) by means of the following relationship χ ¼ 0:7450ð εHΟΜΟ Þ þ 0:3660
ð31Þ
51
As it has been discussed elsewhere, the local EA resulting from the aforementioned travelogue does not actually represent an EA, even within the definition of Koopmans’ theorem, but rather it indicates the local acceptor properties of the molecule. The proposed methodology concerning the determination of local values for EA, n, nrel, nHL, χ, εHOMO, and εLUMO is applied for the adsorption of C2-hydrocarbons on powdered solids of ZnO, Fe2O3, and TiO2 (Tables 7 and 8 and Figure 5). An alternative and more operational relation used for the determination of hardness in terms of the average local ionization energy is by means of eq 32, which bypasses the limitations occurred in the application of eq 22 as well as those used from others36,49 nrel ¼ ½Is, ave =V 1=3
ð32Þ
By combining eqs 18 and 32, we finally correlate nrel with local molecular polarizability (α) as follows 1=3 1:450 ð33Þ nrel ¼ α þ 1:276 By replacing α in eq 24 with any of eqs 13, 14 or 20, the local relative hardness (nrel) is finally transformed into a time-expression because cy is a time-resolved quantity. The so-described application of the appropriate equations for the time-resolved 25396
dx.doi.org/10.1021/jp2089824 |J. Phys. Chem. C 2011, 115, 25389–25412
The Journal of Physical Chemistry C
ARTICLE
Table 5. RF-IGC Experimental Values of First Local Ionization Energies for the Adsorption System C2H4(g)/Fe2O3(s), at 323.2 K, According to Equations 18a and 14, 19a and 14, 16a and 20, 17a and 20, 18a and 20, and 19a and 20, from I7 to I12, Respectively t (min) 3 5 7 9
I7 (eV) 16.068 16.157 16.196 16.216
I8 (eV) 15.992 16.031 16.047 16.056
I9 (eV) 10.504 10.733 10.826 10.871
I10 (eV) 10.372 10.562 10.637 10.674
I11 (eV) 16.158 16.202 16.219 16.228
Table 5. Continued t (min)
I7 (eV)
I8 (eV)
I9 (eV)
I10 (eV)
I11 (eV)
I12 (eV)
103 105
16.023 16.020
15.973 15.971
10.378 10.367
10.266 10.257
16.133 16.131
16.020 16.019
107
16.016
15.970
10.357
10.249
16.129
16.018
I12 (eV)
109
16.013
15.968
10.347
10.240
16.127
16.017
16.031
111
16.009
15.967
10.336
10.231
16.125
16.016
16.049 16.057
113
16.006
15.965
10.326
10.223
16.123
16.015
115
16.002
15.964
10.316
10.214
16.121
16.015
16.060
117
15.999
15.962
10.306
10.206
16.119
16.014
15.996 15.993
15.961 15.960
10.296 10.287
10.198 10.189
16.117 16.115
16.013 16.012
11
16.226
16.060
10.893
10.692
16.232
16.062
13
16.230
16.062
10.901
10.699
16.233
16.063
119 121
15
16.230
16.062
10.902
10.699
16.233
16.063
123
15.989
15.958
10.277
10.181
16.113
16.011
16.062
125
15.986
15.957
10.267
10.173
16.111
16.010
15.983
15.955
10.258
10.165
16.109
16.010
17
16.228
16.061
10.897
10.695
16.232
19
16.224
16.059
10.889
10.689
16.231
16.062
127
21 23
16.220 16.215
16.057 16.055
10.879 10.868
10.681 10.672
16.229 16.227
16.061 16.060
129
15.980
15.954
10.248
10.157
16.107
16.009
131
15.977
15.953
10.239
10.149
16.105
16.008
25
16.209
16.053
10.855
10.662
16.225
16.059
133
15.974
15.952
10.230
10.141
16.103
16.007
27
16.203
16.051
10.843
10.651
16.222
16.058
29
16.198
16.048
10.829
10.640
16.220
16.057
135 137
15.971 15.968
15.950 15.949
10.220 10.211
10.133 10.125
16.101 16.099
16.006 16.005
31
16.192
16.046
10.816
10.629
16.217
16.056
139
15.965
15.948
10.202
10.117
16.098
16.005
33
16.186
16.043
10.802
10.618
16.215
16.055
141
15.963
15.946
10.193
10.110
16.096
16.004
35
16.180
16.041
10.789
10.607
16.212
16.054
143
15.960
15.945
10.184
10.102
16.094
16.003
37 39
16.174 16.169
16.038 16.036
10.775 10.761
10.596 10.585
16.210 16.207
16.053 16.052
145
15.957
15.944
10.175
10.094
16.092
16.002
147
15.954
15.943
10.166
10.087
16.090
16.001
41
16.163
16.033
10.748
10.574
16.205
16.051
149
15.952
15.942
10.157
10.079
16.088
16.001
43
16.157
16.031
10.734
10.562
16.202
16.050
151
15.949
15.940
10.149
10.072
16.087
16.000
45
16.152
16.028
10.721
10.551
16.200
16.048
47
16.146
16.026
10.708
10.541
16.197
16.047
49
16.141
16.024
10.694
10.530
16.195
16.046
51
16.136
16.022
10.681
10.519
16.192
16.045
53 55
16.131 16.126
16.019 16.017
10.668 10.656
10.508 10.498
16.190 16.187
16.044 16.043
57
16.120
16.015
10.643
10.487
16.185
16.042
59
16.116
16.013
10.630
10.477
16.182
16.041
61
16.111
16.011
10.618
10.466
16.180
16.040
63
16.106
16.009
10.605
10.456
16.178
16.039
65
16.101
16.007
10.593
10.446
16.175
16.038
67
16.096
16.005
10.581
10.436
16.173
16.037
69 71
16.092 16.087
16.003 16.001
10.569 10.557
10.426 10.416
16.171 16.168
16.036 16.035
73
16.083
15.999
10.545
10.406
16.166
16.034
75
16.079
15.997
10.533
10.396
16.164
16.033
77
16.074
15.995
10.521
10.386
16.161
16.032
79
16.070
15.993
10.509
10.377
16.159
16.031
81
16.066
15.991
10.498
10.367
16.157
16.030
83
16.062
15.989
10.487
10.357
16.155
16.029
85
16.057
15.988
10.475
10.348
16.152
16.028
87
16.053
15.986
10.464
10.339
16.150
16.027
89
16.049
15.984
10.453
10.329
16.148
16.026
91
16.046
15.983
10.442
10.320
16.146
16.025
93
16.042
15.981
10.431
10.311
16.144
16.024
95
16.038
15.979
10.420
10.302
16.142
16.024
97
16.034
15.978
10.409
10.293
16.139
16.023
99
16.030
15.976
10.399
10.284
16.137
16.022
101
16.027
15.974
10.388
10.275
16.135
16.021
analysis of local relative hardness (nrel) in the case of adsorption of C2-hydrocarbons on powdered ZnO is also reported in Tables 7 and 8. 2.3.3. Determination of Lennard-Jones Parameters for C2Hydrocarbons through Their Adsorption on Solid Surfaces, By Using Time-Resolved Analysis of Molecular Lateral Interaction Energies, Molecular Polarizabilities and Ionization Energies. In a simplified way, the lateral molecular interactions between adsorbed molecules could be described quantitatively by means of a Lennard-Jones potential as follows εlat ¼ VLJ
ð34Þ
VLJ ¼ Vattr þ Vrep ¼ ð Aattr =r 6 Þ þ ðBrep =r 12 Þ
ð35Þ
The numerators Aattr and Brep in eq 35 are positive quantities, which are related with the attractive and repulsive contributions to the LJ potential, and they are given by the relations Aattr ¼ 4εο σ6
ð36aÞ
Brep ¼ 4εο σ12
ð36bÞ
where σ is the molecular collision diameter and εο is equal to the depth of the LJ potential well.5254 Furthermore, the attractive parameter A is correlated with molecular polarizability and first ionization potential via the relation Aattr ¼ ð3=4Þα2 I
ð37Þ
Therefore, the attractive parameter, Aattr, is another one time-dependent quantity due to the same type dependence of α and I. Consequently, the repulsive parameter, Brep, is also 25397
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Table 6. RF-IGC Experimental Values of First Local Ionization Energies for the Adsorption System C2H2(g)/ZnO(s), at 323.2 K, according to eqs 16a and 13, 18a and 13, 16a and 14, 18a and 14, 16a and 20, 18a and 20, for I1, I3, I5, I7, I9, and I11, Respectively t (min) 2 4 6 8
I1 (eV) 9.412 9.694 9.782 9.818
I3 (eV) 15.922 15.987 16.007 16.015
I5 (eV) 9.926 10.71 11.01 11.14
I7 (eV) 16.038 16.196 16.251 16.275
I9 (eV) 10.421 10.827 10.949 10.998
Table 6. Continued t (min)
I1 (eV)
I3 (eV)
I5 (eV)
I7 (eV)
I9 (eV)
I11 (eV)
102 104
9.284 9.275
15.892 15.890
9.640 9.622
15.975 15.971
10.233 10.219
16.104 16.101
106
9.266
15.888
9.603
15.967
10.206
16.098
I11 (eV)
108
9.258
15.885
9.585
15.963
10.193
16.096
16.142
110
9.249
15.883
9.567
15.959
10.180
16.093
16.219 16.242
112
9.240
15.881
9.550
15.955
10.167
16.090
114
9.232
15.879
9.533
15.951
10.154
16.088
16.251
116
9.223
15.877
9.516
15.947
10.142
16.085
9.215 9.207
15.875 15.873
9.500 9.484
15.943 15.939
10.129 10.117
16.082 16.080
10
9.830
16.017
11.19
16.283
11.015
16.254
12
9.829
16.017
11.18
16.283
11.015
16.254
118 120
14
9.822
16.016
11.16
16.278
11.005
16.252
122
9.199
15.871
9.468
15.936
10.105
16.077
16.249
124
9.191
15.869
9.452
15.932
10.093
16.075
9.183
15.867
9.437
15.928
10.081
16.072
16
9.811
16.013
11.11
16.270
10.989
18
9.797
16.010
11.06
16.261
10.970
16.246
126
20 22
9.782 9.767
16.007 16.004
11.01 10.95
16.252 16.242
10.949 10.928
16.242 16.238
128
9.175
15.865
9.422
15.925
10.069
16.070
130
9.167
15.863
9.408
15.921
10.057
16.067
24
9.751
16.000
10.90
16.232
10.906
16.234
132
9.160
15.861
9.393
15.918
10.045
16.065
26
9.735
15.997
10.85
16.222
10.884
16.230
28
9.720
15.993
10.79
16.212
10.862
16.226
134 136
9.152 9.145
15.859 15.858
9.379 9.365
15.915 15.911
10.034 10.023
16.062 16.060
30
9.704
15.990
10.74
16.203
10.840
16.222
138
9.137
15.856
9.352
15.908
10.011
16.058
32
9.689
15.986
10.69
16.193
10.819
16.218
140
9.130
15.854
9.338
15.905
10.000
16.055
34
9.674
15.983
10.65
16.184
10.798
16.214
142
9.123
15.852
9.325
15.902
9.989
16.053
36 38
9.660 9.645
15.980 15.976
10.60 10.56
16.175 16.167
10.777 10.757
16.210 16.206
144
9.115
15.850
9.312
15.899
9.978
16.051
146
9.108
15.849
9.299
15.895
9.967
16.048
40
9.631
15.973
10.51
16.159
10.737
16.203
148
9.101
15.847
9.287
15.892
9.957
16.046
42
9.617
15.970
10.47
16.150
10.717
16.199
150
9.094
15.845
9.274
15.889
9.946
16.044
44
9.604
15.967
10.43
16.143
10.698
16.195
46
9.590
15.964
10.39
16.135
10.679
16.192
48
9.577
15.961
10.35
16.127
10.660
16.188
50
9.564
15.958
10.32
16.120
10.642
16.185
52 54
9.551 9.539
15.955 15.952
10.28 10.25
16.113 16.106
10.623 10.605
16.181 16.178
56
9.527
15.949
10.22
16.099
10.588
16.174
58
9.514
15.946
10.18
16.092
10.570
16.171
60
9.502
15.944
10.15
16.086
10.553
16.168
62
9.491
15.941
10.12
16.080
10.536
16.164
64
9.479
15.938
10.09
16.073
10.519
16.161
66
9.467
15.935
10.06
16.067
10.502
16.158
68 70
9.456 9.445
15.933 15.930
10.03 10.01
16.061 16.055
10.486 10.469
16.154 16.151
72
9.434
15.928
9.978
16.050
10.453
16.148
74
9.423
15.925
9.952
16.044
10.437
16.145
76
9.412
15.922
9.926
16.038
10.422
16.142
78
9.402
15.920
9.901
16.033
10.406
16.139
80
9.391
15.917
9.877
16.028
10.391
16.136
82
9.381
15.915
9.853
16.022
10.376
16.133
84
9.371
15.913
9.829
16.017
10.361
16.130
86
9.361
15.910
9.807
16.012
10.346
16.127
88
9.351
15.908
9.784
16.007
10.331
16.124
90
9.341
15.905
9.762
16.003
10.317
16.121
92
9.331
15.903
9.741
15.998
10.302
16.118
94
9.322
15.901
9.720
15.993
10.288
16.115
96
9.312
15.899
9.699
15.989
10.274
16.112
98
9.303
15.896
9.679
15.984
10.260
16.109
100
9.293
15.894
9.660
15.980
10.246
16.107
time-dependent, as Brep ¼ Aσ6
ð38Þ
An example of the time dependence of both Aattr and Brep is given in Table 9 and Figure 6. The combination of eqs 34 and 35 and eq 8 in the Supporting Information leads to the following equation ðβθRTÞ 3 r 12 þ ð Aattr Þ 3 r 6 þ Brep ¼ 0
ð39Þ
By definition, Lennard-Jones potential becomes minimum at the point called the van der Waals radius (rVDW), where the first derivative of eq 35 or 39 becomes minimum, namely, dVLJ/dr =0. Therefore rVDW is given by the relation rVDW ¼ ½Aattr =2ðβθRTÞmin 1=6
ð40Þ
where the quantity (βθRT)min represents the minimum value of LJ potential, namely, the aforementioned “LJ-potential well depth, εο”, namely, (βθRT)min εο. It is necessary to note that the molecular collision diameter σ is related with rVDW by means of the relation σ ¼ 21=6 rVDW
ð41Þ
Because of the time dependence of lateral molecular interaction energy, that is, εlat = εlat(t), the intermolecular distance, r, among the adsorbed species is also dependent on the experimental time, t, that is, r = r(t). To find the type of this dependence, we simply define the apparent van der Waals radius (rapp VDW) as follows app
rVDW ¼ ½Aattr =2ðβθRTÞ1=6 25398
ð42Þ
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Figure 2. (a) Time-resolved analysis of gaseous nonadsorbed concentration, cy = cy(t), and (b) the variation of local molecular polarizability against gaseous nonadsorbed concentration, α = α(cy), for the adsorption systems C2H6(g)/MxOy(s) at 323.2 K.
Here it should be pointed out that the apparent rapp VDW does not coincide with the real rVDW because the dominators of eqs 40 and 42, although appearing to be identical, are differentiated in that they involve the lateral molecular energy at time t and the minimum lateral molecular energy, respectively. Moreover, rapp VDW is also a timedependent quantity. Some representative examples of the time dependence of rapp VDW and VLJ and of the dependence of VLJ upon rapp VDW as well, are given at Figures 7 and 8. In addition, the experimentally determined values for the real rVDW for all adsorption systems under examination and by using all combinations of equations referred above are given in Tables 1012. It should be noted that the accuracy of the so-determined value of real rVDW can also be confirmed if we try to calculate it alternatively by using in the numerator of the relation 40 the experimental value of Amin, namely ðAattr Þmin ¼ 0:75ðαmin Þ2 Imax
ð43Þ
and in the dominator, the value of the first local maximum of the diagram εlat = εlat(t). Finally, a comparison of our experimental values for the real rVDW of C2-hydrocarbons, which have been determined with the one or the other way, with corresponding ones from literature is also provided in Table 13.
3. RESULTS AND DISCUSSION 3.1. Time-Resolved Analysis of Local Molecular Polarizabilities and Local Ionization Energies. The polarizability, α, is
a measure of how readily the overall electronic charge distribution can be distorted by an external electric field. In general, molecular polarizability is related to the number of electrons of molecule, their distribution, and the shape of the adsorbate molecule. In specific, polarizability increases with the number of electrons in a molecule. For a given class of adsorbate molecules, α increases with increasing adsorbate size or increasing number of carbon-atoms in the adsorbate molecule. The shape of molecules is another factor: elongated molecules are more easily polarized than compact, symmetrical molecules. Moreover, hybridizm influences polarizability; namely, among hydrocarbons with the same number of carbon atoms, alkanes are the most polarizable molecules. For instance, indicative literature values for ethane, ethene, and ethyne are 3.853.764.424.454.475.005.06 Å3, 3.523.654.19 4.254.364.694.714.76 Å3, and 3.163.333.603.74 3.933.94 Å3, respectively.5560 To identify the most reactive sites in a molecule, namely, these sites where electrons are least strongly held, it is necessary to focus not upon a particular electronic orbital, which is usually delocalized to some extent, but rather upon specific points in the space of the molecule, even though electrons from several different orbital may have a significant probability of being at each such point. By definition, the first ionization energy, I, of an atom or molecule indicates how tightly bound the most energetic electron is; namely, it focuses upon a particular electronic orbital. Contrary to this, the average local ionization energy, Is,ave, is 25399
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Figure 3. Time-resolved analysis of local molecular polarizability, α = α(t), for various adsorption systems C2Hy(g)/MxOy(s) at 323.2 K.
defined in terms of the ionization energies of the electrons in individual molecular orbitals,27,35,36 which means it focuses upon the point, B r in space, rather than upon a particular orbital as the common ionization energy, I, does. In other words, Is,ave indicates how strongly bound, on average, an electron at B r is, whereas I measures the extraction energy of the outermost electron of the molecule and thus indirectly reveals how tightly an electron is bound within the nuclear attractive field of the system. Although polarizability and ionization energy pertain to different numbers of electrons, an inverse relationship among them has already been established.54Some indicative values of first ionization energies for C2-hydrocarbons, obtained from literature, are the following ones C2H6: 10.4 or 11.5 eV;57,60 C2H4: 10.0510.39 10.510.5110.52 eV;57,6163 and C2H2: 11.4 eV57 As one can see from Tables 16 and Figures 3 and 4, things are not so unambiguous in the case of adsorption of C2-hydrocarbons on the surfaces of solid powdered metaloxides, at least at first sight, but there is an evident variability in both local polarizabilities and ionization energies. The latter raises serious issues that need to be clarified. It is worth noting, at first, that the inverse relationship among molecular polarizability and ionization energy54 is confirmed. Second, the same shape of curves representing the time-variation of molecular polarizability has already been recorded in the literature by Chattaraj and Sengupta in 1996.64 Concerning the interaction of C2-hydrocarbons with the same oxide, there is the
following variation in values of local molecular polarizabilities and the corresponding local ionization potentials • CXHY(g)/TiO2(s): α(C2H2) > α(C2H6) > α(C2H4), I(C2H2) < I(C2H6) < I(C2H4) • CXHY(g)/Fe2O3(s): α(C2H6) > α(C2H2) > α(C2H4), I(C2H6) < I(C2H2) < I(C2H4) • CXHY(g)/ZnO(s): α(C2H4) > α(C2H6) > α(C2H2), I(C2H4) < I(C2H6) < I(C2H2) The observed variation of the above experimental molecular polarizability values from the expected ones, especially concerning the expected trend from ethane to ethyne, and the observed differentiation concerning the metal-oxide used, could be attributed obviously to the different local environment of each hydrocarbon molecule or to the induced different orientation of it on the concrete solid substrate.65 Maybe the different acidic or basic behavior of both adsorbate and adsorbent is one reason for this fact. In particular, ethyne is more acidic according to Br€onstedLowry theory than ethene and ethane, because the hydrogens sharing an sp orbital are more acidic than those sharing an sp2 orbital and even more acidic than those sharing an sp3 orbital.66 As regards solid oxides used in the experiments of this investigation, all of them are amphoteric, with basic character to be predominant in a greater or lesser extent over the acidic going from Fe2O3 to TiO2 and ZnO.67 Somehow or other, the issue of acidity and basicity concerning the above solids and gases is going 25400
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Figure 4. Time-resolved analysis of local first ionization energy, I = I(t), and local average surface ionization energy, Is,ave = Is,ave(t), for various adsorption systems C2Hy(g)/MxOy(s) at 323.2 K.
to be investigated in more detail afterward and from the aspect of HSAB-theory, which introduces the use of separate scales for acidity and basicity. It is worth noting that HSAB theory classifies compounds as acids or bases but does not accept them as having amphoteric character.68 A qualitatively different and novel explanation for the above observed variability in molecular polarizabilities and ionization energies might be that the same molecule of hydrocarbon is part of an ordered supramolecular structure.69 Furthermore, for the same adsorption system, eq 13, gives the highest α values whereas eq 20 gives the lowest ones, with the only exception of system C2H4(g)/TiO2(s) for which the lowest minimum value for α is obtained from eq 14. Independently of the equation used, the highest α values are obtained for the adsorption system C2H4(g)/ZnO(s). In general, the experimental local α values change according to the following sequence C2 H4ðgÞ =TiO2ðsÞ < C2 H2ðgÞ =ZnOðsÞ < C2 H6ðgÞ =ZnOðsÞ ≈ C2 H4ðgÞ = Fe2 O3ðsÞ < C2 H6ðgÞ =TiO2ðsÞ e C2 H2ðgÞ =Fe2 O3ðsÞ =C2 H2ðgÞ = TiO2ðsÞ e C2 H6ðgÞ =Fe2 O3ðsÞ < C2 H4ðgÞ =ZnOðsÞ
Regarding Figure 2, where the nonadsorbed concentration of ethane over the various solid adsorbents against time, cy=cy(t)
(Figure 2a), and the corresponding molecular polarizabilities against nonadsorbed concentration, a = a(cy) (Figure 2b), are depicted, one can see that as polarizability increases the nonadsorbed concentration of probe gas decreases. This observation simply forms a confirmation of the aforementioned fact that an increase of polarizability causes an increase in the dispersion type intermolecular interactions and thus the adsorbed molecules of hydrocarbon are held together more tightly onto the solid adsorbent surface; as a consequence, the nonadsorbed concentration of the gaseous adsorbate decreases. A thorough observation of Tables 46 and Figure 4 leads to the following remarks: (1) The separation of I values for each system C2HY(g)/solid oxide is greater in the case of I1and I2, namely, when local ionization energy is determined by means of eqs 16a and 13 or 17a and 13, respectively. (2) The Is,ave values for each system C2HY(g)/solid oxide are greater than the corresponding I values. In the author’s opinion, a possible explanation for this fact could deduce from the definition of Is,ave, in the sense that Is,ave represents not only the first ionization energy but also the second or even the third ionization energy. (The latter one has the least possibility of happening.) 25401
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Table 7. RF-IGC Experimental Values of Local Electron Affinities (EA), Local Electronegativities (χ), Local Hardness (n), and Local Relative Hardness (nrel) for the Adsorption System C2H2(g)/ZnO(s), at 323.2 K, according to eqs 16a and 20
Table 7. Continued t (min)
εHOMO
εLUMO
EA
(eV)
(eV)
(eV)
nrel χ (eV)
n (eV)
(eV1/3 3 Å1)
100
7.499
3.504
1.659
5.953
8.587
0.544
102
7.489
3.501
1.657
5.945
8.576
0.543
εHOMO
εLUMO
EA
7.478
3.498
1.655
5.937
8.564
0.542
(eV)
(eV)
χ (eV)
n (eV)
(eV1/3 3 Å1)
104
(eV)
106
7.468
3.494
1.653
5.930
8.553
0.541
2 4
7.633 7.944
3.545 3.641
1.684 1.743
6.053 6.285
8.737 9.084
0.557 0.584
108
7.458
3.491
1.652
5.922
8.541
0.540
110
7.448
3.488
1.650
5.915
8.530
0.539
6
8.038
3.670
1.760
6.354
9.188
0.592
112
7.438
3.485
1.648
5.907
8.519
0.538
114 116
7.428 7.419
3.482 3.479
1.646 1.644
5.900 5.893
8.508 8.498
0.537 0.536
t (min)
nrel
8
8.076
3.682
1.768
6.383
9.230
0.595
10
8.089
3.686
1.770
6.393
9.245
0.596
12
8.089
3.686
1.770
6.392
9.245
0.596
118
7.409
3.476
1.642
5.886
8.487
0.535
120
7.399
3.473
1.641
5.879
8.476
0.534
14
8.081
3.683
1.769
6.387
9.236
0.596
16
8.069
3.680
1.766
6.378
9.223
0.595
122
7.390
3.470
1.639
5.872
8.466
0.533
18 20
8.055 8.039
3.675 3.670
1.764 1.761
6.367 6.355
9.206 9.189
0.594 0.592
124
7.381
3.468
1.637
5.865
8.456
0.532
126
7.372
3.465
1.635
5.858
8.445
0.531
22
8.022
3.665
1.758
6.343
9.170
0.591
128
7.362
3.462
1.634
5.851
8.435
0.530
130 132
7.353 7.345
3.459 3.456
1.632 1.630
5.844 5.838
8.425 8.415
0.529 0.528
24
8.005
3.660
1.754
6.330
9.152
0.589
26
7.988
3.655
1.751
6.318
9.133
0.588
28
7.972
3.650
1.748
6.305
9.114
0.587
134
7.336
3.454
1.629
5.831
8.405
0.527
30
7.955
3.645
1.745
6.293
9.096
0.585
136
7.327
3.451
1.627
5.825
8.396
0.526 0.525
32
7.939
3.640
1.742
6.280
9.077
0.584
138
7.318
3.448
1.625
5.818
8.386
34 36
7.923 7.907
3.635 3.630
1.739 1.736
6.268 6.257
9.059 9.042
0.582 0.581
140
7.310
3.446
1.624
5.812
8.376
0.524
142
7.301
3.443
1.622
5.806
8.367
0.523
38
7.891
3.625
1.733
6.245
9.024
0.580
144
7.293
3.440
1.621
5.799
8.358
0.523
40
7.876
3.620
1.730
6.234
9.007
0.578
7.285 7.276
3.438 3.435
1.619 1.617
5.793 5.787
8.348 8.339
0.522 0.521
7.268
3.433
1.616
5.781
8.330
0.520
42
7.861
3.615
1.727
6.222
8.990
0.577
146 148
44
7.846
3.611
1.724
6.211
8.974
0.576
150
46
7.831
3.606
1.722
6.200
8.957
0.574
48
7.817
3.602
1.719
6.190
8.941
0.573
50 52
7.802 7.788
3.597 3.593
1.716 1.714
6.179 6.168
8.925 8.910
0.572 0.571
54
7.775
3.589
1.711
6.158
8.894
0.569
56
7.761
3.585
1.708
6.148
8.879
0.568
58
7.747
3.581
1.706
6.138
8.864
0.567
60
7.734
3.576
1.703
6.128
8.849
0.566
62
7.721
3.572
1.701
6.118
8.835
0.565
64
7.708
3.568
1.699
6.109
8.820
0.563
66 68
7.695 7.683
3.564 3.561
1.696 1.694
6.099 6.090
8.806 8.792
0.562 0.561
70
7.670
3.557
1.691
6.080
8.778
0.560
72
7.658
3.553
1.689
6.071
8.764
0.559
74
7.646
3.549
1.687
6.062
8.751
0.558
76
7.634
3.545
1.685
6.053
8.737
0.557
78
7.622
3.542
1.682
6.044
8.724
0.555
80
7.610
3.538
1.680
6.035
8.711
0.554
82
7.598
3.535
1.678
6.027
8.698
0.553
84
7.587
3.531
1.676
6.018
8.685
0.552
86
7.575
3.527
1.674
6.010
8.672
0.551
88
7.564
3.524
1.671
6.001
8.660
0.550
90
7.553
3.521
1.669
5.993
8.647
0.549
92
7.542
3.517
1.667
5.985
8.635
0.548
94
7.531
3.514
1.665
5.977
8.623
0.547
96
7.520
3.510
1.663
5.969
8.611
0.546
98
7.510
3.507
1.661
5.961
8.599
0.545
(3) The higher ionization energy values determined in this research relate to I7 and Iaa, namely, those given by eqs 18a and 14 and 18a and 20, respectively. (4) An overall estimation of all of the experimental findings concerning local ionization energy values finally leads to the conclusion that there is a distinct separation of them depending both on the nature of the hydrocarbon and the solid oxide used only in case of I1, I2, I5, and I6, namely, of those determined by means of eqs 16a and 13, 17a and 13, 16a and 14 and 17a and 14, respectively. For all of the rest of the equations used for the determination of local ionization energy values, the results are almost the same and depend only on the nature of solid oxide used and not on the hydrocarbon. (5) Concerning the order of chemical bond in C2-hydrocarbons, the separation of I or Is,ave values is, in any case, much smaller than that of α values; this observation agrees with the one made by Jin et al. elsewhere.27 3.2. Describing AcidBase Interactions, According to HSAB Theory, in the Adsorption Systems C2HY(g)/Solid Oxide, by Means of Time-Resolved Analysis of All of the Local Molecular Properties: I, α, n, x. Acidbase reactions and interactions are of fundamental importance in a variety of chemical and physicochemical processes. The application of various acidbase theories in the investigation and assessment of the acidbase character of surfaces does not always lead to consistent conclusions, probably because of the nature of the scale chosen, the choice of reference points, and the use of 25402
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Table 8. RF-IGC Experimental Values of Local Electron Affinities (EA), Local Electronegativities (χ), Local Hardness (n), and Local Relative Hardness (nrel) for the Adsorption System C2H2(g)/ZnO(s), at 323.2 K, According to Equations 18a and 20 t (min) 2 4 6
εHOMO
εLUMO
EA
(eV)
(eV)
(eV)
12.026 12.086 12.103
4.900 4.918 4.923
2.509 2.521 2.524
Table 8. Continued εHOMO
εLUMO
EA
t (min)
(eV)
(eV)
(eV)
χ (eV)
n (eV)
(eV1/3 3 Å1)
100
11.999
4.891
2.504
9.305
13.602
0.632
102
11.997
4.891
2.504
9.304
13.600
0.631
104
11.995
4.890
2.503
9.302
13.598
0.630
106
11.993
4.889
2.503
9.301
13.595
0.629
0.644 0.669
108
11.991
4.889
2.503
9.299
13.593
0.628
110
11.989
4.888
2.502
9.298
13.591
0.627
0.675
112
11.986
4.887
2.502
9.296
13.588
0.626
0.678
11.984 11.982
4.887 4.886
2.502 2.501
9.295 9.293
13.586 13.584
0.626 0.625
nrel χ (eV) 9.326 9.370 9.383
n (eV) 13.632 13.699 13.718
(eV1/3 3 Å1)
nrel
10
12.112
4.926
2.525
9.390
13.728
0.679
114 116
12
12.112
4.926
2.525
9.390
13.728
0.679
118
11.980
4.886
2.501
9.292
13.582
0.624
0.678
120
11.978
4.885
2.500
9.290
13.579
0.623
11.977
4.884
2.500
9.289
13.577
0.622
8
14
12.110
12.111
4.925
4.926
2.525
2.525
9.388
9.389
13.726
13.727
16
12.108
4.925
2.525
9.387
13.724
0.678
122
18 20
12.106 12.103
4.924 4.923
2.524 2.524
9.385 9.383
13.721 13.718
0.677 0.675
124
11.975
4.884
2.500
9.287
13.575
0.621
126
11.973
4.883
2.499
9.286
13.573
0.620
22
12.100
4.922
2.523
9.381
13.715
0.674
128
11.971
4.883
2.499
9.284
13.571
0.619
24
12.097
4.921
2.523
9.378
13.711
0.673
26
12.094
4.920
2.522
9.376
13.708
0.672
130 132
11.969 11.967
4.882 4.881
2.499 2.498
9.283 9.282
13.569 13.567
0.618 0.617
28
12.091
4.920
2.521
9.374
13.704
0.671
134
11.965
4.881
2.498
9.280
13.565
0.617
30
12.088
4.919
2.521
9.371
13.701
0.669
136
11.963
4.880
2.498
9.279
13.563
0.616
32
12.085
4.918
2.520
9.369
13.698
0.668
138
11.961
4.880
2.497
9.277
13.560
0.615
34 36
12.082 12.079
4.917 4.916
2.520 2.519
9.367 9.365
13.694 13.691
0.667 0.666
140
11.960
4.879
2.497
9.276
13.558
0.614
142
11.958
4.879
2.497
9.275
13.556
0.613
38
12.076
4.915
2.519
9.363
13.688
0.665
144
11.956
4.878
2.496
9.273
13.554
0.612
40
12.073
4.914
2.518
9.360
13.685
0.663
42
12.070
4.913
2.518
9.358
13.681
0.662
146 148
11.954 11.952
4.877 4.877
2.496 2.496
9.272 9.271
13.552 13.550
0.611 0.611
44
12.067
4.912
2.517
9.356
13.678
0.661
150
11.951
4.876
2.495
9.269
13.549
0.610
46
12.064
4.911
2.517
9.354
13.675
0.660
48
12.062
4.911
2.516
9.352
13.672
0.659
50 52
12.059 12.056
4.910 4.909
2.516 2.515
9.350 9.348
13.669 13.666
0.658 0.657
54
12.054
4.908
2.515
9.346
13.663
0.656
56
12.051
4.907
2.514
9.344
13.660
0.654
58
12.048
4.906
2.514
9.342
13.657
0.653
60
12.046
4.906
2.513
9.340
13.654
0.652
62
12.043
4.905
2.513
9.338
13.652
0.651
64
12.041
4.904
2.512
9.337
13.649
0.650
66 68
12.038 12.036
4.903 4.903
2.512 2.511
9.335 9.333
13.646 13.643
0.649 0.648
70
12.033
4.902
2.511
9.331
13.641
0.647
72
12.031
4.901
2.510
9.329
13.638
0.646
74
12.028
4.900
2.510
9.327
13.635
0.645
76
12.026
4.900
2.509
9.326
13.632
0.644 0.643
78
12.024
4.899
2.509
9.324
13.630
80
12.021
4.898
2.508
9.322
13.627
0.642
82
12.019
4.897
2.508
9.320
13.625
0.641
84
12.017
4.897
2.508
9.319
13.622
0.640
86
12.014
4.896
2.507
9.317
13.620
0.639
88
12.012
4.895
2.507
9.315
13.617
0.638
90
12.010
4.895
2.506
9.314
13.615
0.637
92
12.008
4.894
2.506
9.312
13.612
0.636
94
12.005
4.893
2.506
9.310
13.610
0.635
96
12.003
4.893
2.505
9.309
13.607
0.634
98
12.001
4.892
2.505
9.307
13.605
0.633
either poor or inconsistent statistical procedures in addition to experimental difficulties and the limited experimental results already obtained. Concerning the various theories that have been occasionally formulated, the Arrhenius model requires an acidic or basic species to possess ionizable H+ or HO ions. Br€onsted-Lowry theory requires only the transfer of a proton from an acid to a base, thus broadening the available bases to include ammonia and other species that do not have hydroxide groups. In addition, Lewis extends the theory to include acids other than proton sources, breaking the second barrier to a broader definition of acidic and basic compounds. In modern usage, a Lewis acid would be defined as any substance capable of accepting electron-density and a Lewis base as any substance capable of donating electron density. Many substances are capable of being either one or the other, and some materials (e.g., H2O) are capable of being both. A Lewis acidbase interaction requires coordination of the two so that the bonding electron density is shared by both the acid (acceptor) and the base (donor). In the context of this definition, it becomes difficult to find chemical reactions that do not involve Lewis acidbase interactions in at least some step of the whole process; outsphere electron transfer quickly comes to mind as one of the few reactions that does not qualify. Although Lewis’ concept is genuinely useful, it has not completely replaced the Br€onsted-Lowry interpretation, for many reasons, and thus Lewis theory mostly remains a qualitative theory. Even it is possible to classify qualitatively the strength of Lewis acids by comparing them with reference to their reactions 25403
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ARTICLE
Figure 5. Time-resolved analysis of local orbital energy gap, nHL = nHL(t), local electronegativity, χ = χ(t), and local electron-affinity, EA = EA(t), for various adsorption systems C2Hy(g)/MxOy(s), at 323.2 K.
with the same base, it is still impossible to find a single scale encompassing all possible Lewis acids. This is primarily because the origin of protic acid strength is essentially electrostatic and a pure electrostatic model cannot be used when all Lewis acids are
considered. It must be noted that most Lewis bases are also Br€onsted-Lowry bases, although a good Lewis base can be a very poor Br€onsted-Lowry base and vice versa; only the H+ ion is both a Lewis and a Br€onsted-Lowry acid. 25404
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ARTICLE
Table 9. RF-IGC Experimental Values of Local Attraction (Aattr) and Repulsion (Brep) Parameters for the Adsorption Systems C2Hy(g)/ZnO(s), at 323.2 K, According to Equations 18a and 20 4
6
1
Aattr (10 , Å 3 kJ 3 mol ) t
7
12
1
Brep (10 , Å 3 kJ 3 mol )
Table 9. Continued Aattr (104, Å6 3 kJ 3 mol1)
t C2H6(g)/ C2H4(g)/ C2H2(g)/ C2H6(g)/ C2H4(g)/ C2H2(g)/ ZnO(s) ZnO(s) ZnO(s) ZnO(s) ZnO(s) (min) ZnO(s)
C2H6(g)/ C2H4(g)/ C2H2(g)/ C2H6(g)/ C2H4(g)/ C2H2(g)/
(min) ZnO(s) 2 4 6 8
2.315 1.684 1.535 1.471
ZnO(s) 2.275 1.785 1.663 1.616
ZnO(s) 2.017 1.501 1.384 1.340
ZnO(s) 5.495 3.995 3.644 3.491
ZnO(s) 6.874 5.394 5.025 4.884
ZnO(s) 4.672 3.477 3.206 3.104
10
1.441
1.600
1.326
3.419
4.836
3.071
12 14
1.428 1.426
1.600 1.608
1.326 1.335
3.389 3.383
4.834 4.859
3.072 3.092
16 18 20 22 24
1.430 1.439 1.450 1.464 1.479
1.621 1.638 1.656 1.676 1.696
1.348 1.365 1.383 1.403 1.423
3.394 3.414 3.442 3.475 3.511
4.899 4.949 5.005 5.064 5.125
3.123 3.161 3.204 3.249 3.297
26
1.495
1.717
1.444
3.549
5.188
3.345
28 30
1.512 1.530
1.738 1.759
1.465 1.487
3.589 3.631
5.252 5.316
3.395 3.444
32 34 36 38 40
1.548 1.566 1.584 1.603 1.621
1.781 1.802 1.824 1.845 1.867
1.509 1.530 1.552 1.574 1.596
3.673 3.716 3.759 3.803 3.848
5.381 5.445 5.510 5.575 5.640
3.494 3.545 3.596 3.647 3.698
42
1.640
1.888
1.619
3.892
5.706
3.749
44 46
1.659 1.678
1.910 1.932
1.641 1.663
3.937 3.981
5.771 5.837
3.801 3.853
48
1.697
1.953
1.686
4.026
5.903
3.905
50
1.716
1.975
1.709
4.071
5.969
3.958
52
1.735
1.997
1.731
4.116
6.035
4.011
54
1.754
2.019
1.754
4.162
6.101
4.064
56
1.773
2.041
1.777
4.207
6.168
4.117
58
1.792
2.064
1.801
4.253
6.235
4.171
60 62
1.811 1.830
2.086 2.108
1.824 1.847
4.298 4.344
6.303 6.370
4.225 4.279
64
1.850
2.131
1.871
4.390
6.438
4.334
66
1.869
2.153
1.895
4.436
6.507
4.389
68
1.889
2.176
1.919
4.482
6.575
4.445
70
1.908
2.199
1.943
4.529
6.644
4.501
72
1.928
2.222
1.967
4.575
6.714
4.557
74
1.947
2.245
1.992
4.622
6.783
4.613
76
1.967
2.268
2.016
4.669
6.853
4.670
78
1.987
2.291
2.041
4.715
6.924
4.728
80
2.007
2.315
2.066
4.763
6.994
4.785
82
2.027
2.338
2.091
4.810
7.065
4.843
84
2.047
2.362
2.116
4.857
7.137
4.902
86
2.067
2.386
2.142
4.905
7.209
4.961
88
2.087
2.410
2.167
4.953
7.281
5.020
90
2.107
2.434
2.193
5.001
7.353
5.080
92
2.127
2.458
2.219
5.049
7.426
5.140
94
2.148
2.482
2.245
5.097
7.499
5.200
96
2.168
2.506
2.271
5.146
7.573
5.261
Brep (107, Å12 3 kJ 3 mol1)
98
2.189
2.531
2.297
5.194
7.647
5.322
100
2.209
2.555
2.324
5.243
7.721
5.383
102
2.230
2.580
2.351
5.292
7.796
5.445
104
2.251
2.605
2.378
5.341
7.871
5.508
106
2.271
2.630
2.405
5.391
7.947
5.570
108
2.292
2.655
2.432
5.440
8.022
5.633
110
2.313
2.680
2.459
5.490
8.099
5.697
112 114
2.334 2.356
2.706 2.731
2.487 2.515
5.540 5.590
8.175 8.252
5.761 5.825
116
2.377
2.757
2.543
5.641
8.329
5.890
118
2.398
2.782
2.571
5.691
8.407
5.955
120
2.420
2.808
2.599
5.742
8.485
6.020
122
2.441
2.834
2.627
5.793
8.563
6.086
124
2.463
2.860
2.656
5.845
8.642
6.152
126
2.484
2.886
2.685
5.896
8.721
6.219
128 130
2.506 2.528
2.912 2.939
2.714 2.743
5.948 5.999
8.800 8.880
6.286 6.353
132
2.550
2.965
2.772
6.051
8.960
6.421
134
2.572
2.992
2.801
6.104
9.041
6.489
136
2.594
3.019
2.831
6.156
9.122
6.557
138
2.616
3.046
2.861
6.209
9.203
6.626
140
2.638
3.073
2.891
6.262
9.285
6.696
142
2.661
3.100
2.921
6.315
9.367
6.765
144 146
2.683 2.706
3.127 3.154
2.951 2.981
6.368 6.422
9.449 9.532
6.835 6.906
148
2.729
3.182
3.012
6.475
9.615
6.977
150
2.751
3.210
3.043
6.529
9.698
7.048
Many species react with bases using a significant amount of covalent bonding, and these cannot be directly related to the others that are compatible with an electrostatic model. In response to this discovery, a simple but extremely useful qualitative scheme has been developed; this is the Pearson’s HSAB theory. According to this theory, Lewis acids and bases are divided into two groups, along with a less well-defined “borderline” category. The HSAB model is originated from the consideration of the thermodynamic strength of the interaction of acids with halides. For a molecule to be “hard”, it must have large ionization energy; this is because the ionization energy of a molecule is customarily much greater than its EA so that the former is the dominant contribution to the chemical hardness of the molecule.70 It should be noted that the HSAB principle applies only when the acids or bases have similar strengths.71 When acids and bases do not have similar strengths, the tendency of strong acids to displace weak acids (or strong bases to displace weak bases) dominates. Strong acids have high electronegativity, and high electronegativity is associated with hard reagents. Similarly, strong bases have low electronegativity and thus tend to be soft reagents.70 Furthermore, the EA of molecules is a complicated function of their electronic structure. Because EA measures the change in energy of an atom or molecule when an electron is added to the outer energy level of it to form a negative ion, an 25405
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ARTICLE
Figure 6. Time-resolved analysis of attractive and repulsive parameters, Aattr= Aattr(t) and Brep = Brep(t), respectively, for the adsorption systems C2Hy(g)/ZnO(s), at 323.2 K.
Figure 7. Time-resolved analysis of the apparent van der Waals radius, (rVDW)app = (rVDW)app(t) for the adsorption system: (a) C2H2(g)/ZnO(s) and (b) C2H4(g)/Fe2O3(s) at 323.2 K.
atom or molecule having a positive EA is often called an “electronacceptor” and may undergo “charge-transfer” reactions, whereas if it has a negative EA, then it is called “electron-donor”. For instance, the EA for benzene is negative, as is that of naphthalene, whereas those of anthracene, phenanthrene, and pyrene are positive. In addition, the EA measured from a material’s surface is a function of the bulk material as well as the surface condition. Often negative EA is desired to obtain efficient cathodes that can supply electrons to the vacuum with little energy loss.7274 In Tables 7 and 8 and in Figure 5, some representative experimental results are reported and depicted, respectively,
concerning the time-resolved analysis of local EAs, local electronegativities (χ), local hardness (n), local relative hardness (nrel), and local orbital energy gaps (nHL) for one or more adsorption systems of the type C2HY(g)/solid metal-oxide, and useful observations and remarks on them are done. It should be noted that the choice of one or two equations concerning local ionization energy from all of the aforementioned in subsection 2.3.1 is totally random without this affecting the qualitative slant of the conclusions deduced. Therefore, a holistic approach is attempted to elucidate the acidbase behavior of various adsorption systems under consideration by coestimating the time variation of 25406
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Figure 8. (a) Time-resolved analysis of the apparent van der Waals radius, (rVDW)app = (rVDW)app(t), (b) time-resolved analysis of the experimental LJ potential, VLJ = VLJ(t), and (c) the variation of LJ potential versus the apparent van der Waals radius, VLJ = VLJ((rVDW)app), for the adsorption system C2H6(g)/TiO2(s) at 323.2 K.
the local molecular properties in discussion. This way of treatment conceptually resembles the procedure implemented from the general interaction properties function (GIPF); the latter has been developed by J. S. Murray et al. and depends on noncovalent interactions and uses computed quantities as statistical measures of the variation of a single physical observable, the electrostatic potential, over a well-defined molecular surface.75,76 (1) The somewhat higher experimental values found with this methodology (Tables 7 and 8), in comparison with corresponding ones found in literature,27,41 are justified in the reminder that equations used for the determinations in question, that is, eqs 16a and 20 or eqs 18a and 20, led to the higher values for the local (surface) ionization energy, as it has already been mentioned in the previous subsection 3.1. (2) In general, the experimental local nHL, n, χ, and EA values change according to the following sequence, at least for the first 40 min of the adsorption process C2 H4ðgÞ =TiO2ðsÞ > C2 H2ðgÞ =ZnOðsÞ < C2 H4ðgÞ =Fe2 OðsÞ ≈ C2 H6ðgÞ = ZnOðsÞ > C2 H6ðgÞ =TiO2ðsÞ g C2 H2ðgÞ =Fe2 O3ðsÞ g C2 H2ðgÞ = TiO2ðsÞ g C2 H6ðgÞ =Fe2 O3ðsÞ > C2 H4ðgÞ =ZnOðsÞ
(3) Taking account on the one hand that “hard molecules have a large HOMOLUMO gap and soft molecules have a small HOMOLUMO gap”66,77 and on the other
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hand that “hard acids tend to bind hard bases and soft acids tend to bind soft bases”, the latter being the HSAB principle,71 in addition to the aforementioned experimentally determined values for local molecular polarizabilities and local ionization energies (Section 3.1) and finally the fact that RF-IGC determines high adsorption energies compatible to chemisorption at least in the early stages of adsorption,1521,24 one can easily formulate the following remarks: Adsorption systems C2H2(g)/MxOy(s): All of the above findings advocate that acetylene (or ethyne, according to IUPAC rules) presents the hardest and the most Lewis-acidic behavior when it is adsorbed onto the surface of the solid powdered ZnO, whereas it presents the softest and most Lewis-basic behavior toward the solid powdered TiO2. An intermediate behavior is indicated when acetylene is adsorbed onto the solid powdered Fe2O3. Therefore, the experimental evidence of chemisorption initially in the adsorption process and the HSAB principle could compose the conclusion that solid ZnO acts as hard base toward C2H2, TiO2 acts as soft acid toward C2H2, whereas Fe2O3 behaves as a moderate soft and weak Lewis acid toward acetylene’s adsorption. Adsorption Systems C2H4(g)/MxOy(s) The maximum hardness and EA are observed for ethylene (or ethane, according to IUPAC-rules) when it is adsorbed onto the surface of the solid powdered TiO2; that means ethylene acts as a very hard and strong Lewis acid toward TiO2. Contrary to this, the minimum hardness and EA are observed during the adsorption of ethylene onto the surface of the solid powdered ZnO. In comparison with the adsorption of acetylene on these oxides, ethylene manifests an absolutely opposite behavior. However, in the case of the solid powdered Fe2O3, ethylene seems to be more hard and acidic than acetylene and ethane. (It should be noted that the quantitative adverbs used here by author, such as very hard or weak acid and so on, concern only relative comparisons between substances and systems of the present study). In a similar way as in (a), solid TiO2 behaves as a hard Lewis-base, ZnO behaves as a soft Lewis acid, and Fe2O3 behaves as a quite hard and basic oxide toward ethylene’s adsorption. Adsorption systems C2H6(g)/MxOy(s): The adsorptive behavior of ethane over the solid powdered ZnO is similar to this of ethylene over Fe2O3, as all of the above findings imply. That means ethane is a quite hard Lewis acid, which in turns means the solid powdered ZnO acts as a quite hard Lewis base for ethane. Furthermore, the solid powdered TiO2 behaves as a softer and weaker Lewis base than solid ZnO toward ethane. When solid powdered Fe2O3 is used as an adsorbent for gaseous ethane, it acts as a soft Lewis acid for the soft Lewis base being ethane. (4) The above observations relating to the hardness/softness of C2-hydrocarbons over metaloxides are confirmed from the experimentally determined values for their rVDW , as they are reported in Tables 1012. These determined values for rVDW of each probe hydrocarbon over each individual metaloxide, which have been determined as analytically described in subsection 2.3.3, are predicted to be smaller when hardness increases and it does occur, as the corresponding Tables show. 3.3. Determination of Lennard-Jones Parameters for C2Hydrocarbons by Using Time-Resolved Analysis of Local Molecular Parameters Determined from Adsorption Studies. The time-resolved analysis of the attractive and repulsive 25407
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Table 10. Experimentally Determined Values by RF-IGC Method as Analytically Described in this Manuscript for the Various LJ-Parameters, rVDW, σ, and εο of C2H6, Extracted from Adsorption Data Concerning the Systems C2H6(g)/MxOy(s), at 323.2 K σ (Å)
rVDM (Å) experimental-2 (by using experimental-1 (by using eq 40) calculation
average
eqs 40 and 43)
average
experimental value experimental value
C2H6(g)/ C2H6(g)/ C2H6(g)/ C2H6(g)/ C2H6(g)/ C2H6(g)/
(ε0/kB) (K) C2H6(g)/ C2H6(g)/ C2H6(g)/
equations used
TiO2(s)
Fe2O3(s)
ZnO(s)
TiO2(s)
Fe2O3(s)
ZnO(s)
16a and 13 16a and 14
4.50 3.94
4.56 4.02
4.43 3.85
4.49 3.92
4.55 3.99
4.41 3.83
4.49 3.92
16a and 20
3.89
3.94
3.84
3.88
3.93
3.83
17a and 13
4.50
4.56
4.43
4.48
4.54
4.41
17a and 14
3.93
4.01
3.84
3.91
3.98
17a and 20
3.88
3.93
3.82
3.87
18a and 13
4.89
4.96
4.81
18a and 14
4.22
4.31
18a and 20 19a and 13
4.17 4.89
19a and 14 19a and 20
TiO2(s)
Fe2O3(s)
ZnO(s)
4.00 3.49
173.3 173.3
165.4 165.4
183.8 183.8
3.88
3.46
173.3
165.4
183.8
4.49
4.00
173.3
165.4
183.8
3.82
3.92
3.49
173.3
165.4
183.8
3.92
3.81
3.87
3.45
173.3
165.4
183.8
4.87
4.94
4.79
4.88
4.35
173.3
165.4
183.8
4.12
4.19
4.28
4.09
4.20
3.74
173.3
165.4
183.8
4.22 4.96
4.10 4.81
4.15 4.87
4.21 4.94
4.09 4.79
4.16 4.88
3.71 4.35
173.3 173.3
165.4 165.4
183.8 183.8
4.22
4.30
4.11
4.19
4.28
4.09
4.20
3.74
173.3
165.4
183.8
4.16
4.21
4.09
4.15
4.20
4.08
4.15
3.70
173.3
165.4
183.8
174.2 (average value)
Table 11. Experimentally Determined Values by RF-IGC Method as Analytically Described in this Manuscript for the Various LJ-Parameters, rVDW, σ, and εο of C2H4, Extracted from Adsorption Data Concerning the Systems C2H4(g)/MxOy(s) at 323.2 K σ (Å)
rVDM (Å)
experimental-2 (by using eqs 40 experimental-1 (by using eq 40) and 43) calculation equations used
average
average
experimental value
experimental value
C2H4(g)/ C2H4(g)/ C2H4(g)/ C2H4(g)/ C2H4(g)/ C2H4(g)/ TiO2(s)
Fe2O3(s)
ZnO(s)
TiO2(s)
Fe2O3(s)
(ε0/kB) (K) C2H4(g)/ C2H4(g)/ C2H4(g)/
ZnO(s)
TiO2(s)
Fe2O3(s)
ZnO(s)
16a and 13
4.32
4.51
4.61
4.31
4.49
4.59
4.47
3.98
167.6
165.1
162.9
16a and 14
3.58
3.93
4.09
3.55
3.90
4.07
3.85
3.43
167.6
165.1
162.9
16a and 20
3.77
3.91
3.98
3.76
3.89
3.97
3.88
3.46
167.6
165.1
162.9
17a and 13
4.32
4.51
4.61
4.30
4.49
4.59
4.47
3.98
167.6
165.1
162.9
17a and 14
3.56
3.92
4.09
3.53
3.89
4.06
3.84
3.42
167.6
165.1
162.9
17a and 20
3.75
3.90
3.97
3.74
3.88
3.96
3.87
3.45
167.6
165.1
162.9
18a and 13 18a and 14
4.68 3.78
4.90 4.21
5.02 4.40
4.66 3.75
4.88 4.17
5.00 4.37
4.86 4.11
4.33 3.66
167.6 167.6
165.1 165.1
162.9 162.9
18a and 20
4.01
4.18
4.27
4.00
4.17
4.25
4.15
3.70
167.6
165.1
162.9
19a and 13
4.67
4.90
5.02
4.66
4.88
5.00
4.86
4.33
167.6
165.1
162.9
19a and 14
3.77
4.20
4.40
3.73
4.16
4.37
4.10
3.65
167.6
165.1
162.9
19a and 20
4.00
4.17
4.26
3.99
4.15
4.24
4.14
3.69
167.6
165.1
162.9
165.2 (average value)
parameters, Aattr and Brep, respectively, for the various adsorption systems C2HY(g)/metal-oxide(s), is given at Figure 6 and Table 9; the so-depicted results have been produced by the combination of eqs 18a and 20. As we can see, there is a following variation for both of them versus time, which is reasonable and predictable because these parameters are both positive on the one hand and competitive on the other hand. Furthermore, the local minimum observed in each curve corresponds to the depth of the LJpotential well, namely, to the real rVDW.
The time-resolved analysis of the apparent rVDW (Figures 7 and 8a) supplies us with a curve that includes two local extrema, at least, following the “cut-off range”. The first one of them is a minimum and corresponds to the real Van de Waals radius, rVDW that, in turn, corresponds to the first local extremum maximum of the diagram εlat = εlat(t) or, equivalently, to the first local extremum minimum of the diagram VLJ = VLJ(t) (Figure 8b). The “cut-off range” corresponds to the strong repulsive forces’ area of the VLJ potential. The second or more (Figures 7a and 8) local 25408
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Table 12. Experimentally Determined Values by RF-IGC Method as Analytically Described in this Manuscript for the Various LJ-Parameters, rVDW, σ, and εο of C2H4, Extracted from Adsorption Data Concerning the Systems C2H2(g)/MxOy(s) at 323.2 K σ (Å)
rVDM (Å) experimental-2 (by using eqs 40 experimental-1 (by using eq 40) calculation
and 43)
average
average
experimental value experimental value
C2H2(g)/ C2H2(g)/ C2H2(g)/ C2H2(g)/ C2H2(g)/ C2H2(g)/
(ε0/kB) (K) C2H2(g)/ C2H2(g)/ C2H2(g)/
equations used
TiO2(s)
Fe2O3(s)
ZnO(s)
TiO2(s)
Fe2O3(s)
ZnO(s)
16a and 13 16a and 14
4.47 3.93
4.41 3.88
4.41 3.78
4.45 3.90
4.39 3.85
4.39 3.76
4.42 3.85
16a and 20
3.87
3.81
3.873
3.85
3.80
3.82
17a and 13
4.47
4.41
4.40
4.45
4.39
4.39
17a and 14
3.92
3.87
3.77
3.89
3.84
17a and 20
3.86
3.80
3.81
3.84
18a and 13
4.86
4.79
4.78
18a and 14
4.22
4.16
18a and 20 19a and 13
4.14 4.86
19a and 14 19a and 20
TiO2(s)
Fe2O3(s)
ZnO(s)
3.94 3.43
184.3 184.3
200.6 200.6
175.0 175.0
3.83
3.41
184.3
200.6
175.0
4.42
3.94
184.3
200.6
175.0
3.74
3.84
3.42
184.3
200.6
175.0
3.79
3.80
3.82
3.40
184.3
200.6
175.0
4.84
4.77
4.77
4.80
4.28
184.3
200.6
175.0
4.03
4.18
4.12
4.00
4.12
3.67
184.3
200.6
175.0
4.08 4.79
4.08 4.78
4.12 4.84
4.06 4.77
4.07 4.76
4.09 4.80
3.64 4.28
184.3 184.3
200.6 200.6
175.0 175.0
4.21
4.15
4.02
4.18
4.11
3.99
4.11
3.66
184.3
200.6
175.0
4.13
4.07
4.08
4.12
4.06
4.06
4.09
3.64
184.3
200.6
175.0
186.6 (average value)
extrema relate to other layers of adsorbates after the first monolayer, which is considered to have completed when the second local extremum minimum of the diagram εlat = εlat(t) or, equivalently, the second local extremum maximum of the diagram VLJ = VLJ(t) (Figure 8b) is reached. In Figure 8c, an example of the dependence of LJ-potential toward the rapp VDW is represented for the adsorption system C2H6(g)/TiO2(s) by using various combinations of equations. However, a care review of Figures 7 and 8 raises some justifiable queries: Why is the first local extremum minimum after the “cut-off range” in the diagram of (rVDW)app = (rVDW)app(t) the real value of van der Waals radius? The answer to this equation arises from the physical interpretation of rVDW because the latter determines the so-called “van der Waals volume” of a molecule of a particular substance, namely, the volume occupied by a molecule of this substance, which is impenetrable for other molecules with thermal energies at ordinary temperatures.78,79 Alternatively, the rVDW is the intermolecular distance at which the repulsion forces between interacting molecules are balanced from their attraction forces. In Tables 1012, the experimentally determined values for the real rVDW of each C2-hydrocarbon according to the methodology described in this manuscript are analytically given. The accuracy among these values found by using the one or the other way proposed in this manuscript is so-confirmed (Tables 1012). Finally, these values are also compared with corresponding ones found in literature,14,8082 especially concerning ethane and ethene (Table 13), because analogous literature data for ethyne have not been found by the author of this manuscript. As we can easily ascertain, there is a good agreement between them in any case. A further annotation relatively to both of our experimentally determined values for the (real) rVDW of each C2-hydrocarbon molecule (Tables 1012) concerns the fact that the first local maximum value of εlat does not occur at the same time point occurring for the minimum value of the attractive parameter, Aattr, but they differ from one each other at ∼10 min. This time- variation could be attributed to the fact that the
exact description of lateral molecular interaction energy includes more terms than contained in the expression of LJ potential. In fact, the intermolecular potential is composed of isotropic contributions, such as the LJ potential, and anisotropic contributions as well, which are dependent on the relative orientation of interacting van der Waals molecules.79 Is there any possible explanation for the values observed in the diagram VLJ = VLJ(t) and in the diagram εlat = εlat(t) that correspond to the “cut-off range” of the diagram (rVDW)app = (rVDW)app(t)? A possible explanation for this experimental finding could be that previously mentioned, namely, that the expression of LJ potential used here for our determinations is not the most accurate to explain all intermolecular interactions that occurred in adsorption phenomena.79 Is there any possible explanation for the other local extremum values observed in the diagram VLJ = VLJ(t) or in the diagram εlat = εlat(t) that correspond to values of apparent van der Waals radius coming after the “real van der waals radius” of the diagram (rVDW)app = (rVDW)app(t)? As above, the coexistence of anisotropic contributions,79 which did not taken into account in the present study, which could induce more strong intermolecular forces of other type, is a possible explanation for this experimental finding. The situation is especially complicated in the case of the adsorption of C2H2 onto solid powdered ZnO (Figure 7b). The fact that the experimentally determined values of LJ parameters, by using the methodology presented in this research work, are in a very good agreement with corresponding ones found in various literature sources (Table 13) and especially the fact that the order of increase of our experimental (average or not) values for the (real) rVDW and the collision-diameter, σ, going from ethyne to ethene and ethane (Tables 1012) agrees absolutely with literature data14,8082 and is independent of the nature of the solid adsorbent used. The latter confirms the definition of “what exactly represents the van der Waals radius and volume”, namely, the fact these quantities are intrinsic features of the particular hydrocarbon molecule in the sense of 25409
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Table 13. Comparison of LJ Parameters Determined in this Research with Literature Data σ (Å)
rVDW (Å) C2H6
C2H4
C2H2
C2H6
3.87 (lower)
3.84 (lower)
3.82 (lower)
3.45 (lower)
4.88 (higher)
4.86 (higher)
4.80 (higher)
4.35 (higher)
4.85
4.60
(ε0/kB) (K)
C2H4
C2H2
C2H6
C2H4
C2H2
3.42 (lower)
3.40 (lower)
174.2 (lower)
165.2 (lower)
186.6 (lower)
4.33 (higher)
4.28 (higher)
(average experimental value found using RF-IGC in this research work) 4.32
100.8
113.7
146.8
148.6
(experimental value found using RF-IGC in ref 14) 4.52
4.36
4.03
3.88
(estimated by LB combining rules and use of the extended principle of corresponding-states (refs 14 and 78)) 4.58
4.48
4.08 3.99 145.1 (estimated by LB combining rules and use of viscosity measurements (refs 14 and 79)) 3.77
127.9
98.1 (ref 80)
their invariance under the most drastic environmental changes, that is, irrespective of the chemical combination of the molecule and of its nearest nonbonded neighbors as well as of the phase state in which it is found.78 All of these findings confirm the validity of our method because they approve that although the local character of the adsorption phenomenon really exists, emerges, and finally settles on the results, it does not repeal the intrinsic features in which the substances participate. It could be said that “adsorption” is the name of a global physicochemical phenomenon and its “local character” determines the mechanism of this phenomenon. Another one physicochemical parameter that could be determined by the present methodology and mainly could be used as a further verification for the correctness of the method is the “maximum attractive force, Fmax”, which occurs at (d2VLJ/dr2) = 0. If we apply the latter condition to eq 35, then we finally take the relation Fmax = (126A2/169B) 3 (7A/26B)1/6 (the negative “sign” has the physical meaning of attraction). By using in this relation the minimum values of Aattr and Brep found with our method, we determine values for Fmax that are in best agreement with corresponding ones from literature. Concerning, for instance, the adsorption of C2-hydrocarbons onto solid powdered ZnO, we take the following results (according to Table 9): • C2H6(g)/ZnO(s): Fmax = 1.637 1011 N/molecule • C2H4(g)/ZnO(s): Fmax = 1.386 1011 N/molecule • C2H2(g)/ZnO(s): Fmax = 1.566 1011 N/molecule It is worth noting that because the conventional laboratory balances can measure down to ∼109 N, to measure attractive forces in the order of 10 to 11 N, we need specialize equipment, such as the surface force apparatus (SFA) or atomic force microscopy (AFM).53
4. CONCLUSIONS Lateral interactions between adsorbed species play a crucial role in both equilibrium and nonequilibrium ordering behavior of the adsorbates on the adsorbent surface and significantly influence the surface function and properties in important applications like heterogeneous catalysis. By gaining more insight into adsorbateadsorbate or lateral interactions, it might even become feasible to make use of them because they can change reaction routes that would otherwise be unfavorable. In virtue of the consideration that the adsorbed species on a solid surface are “van der Waals molecules”, which are trapped
together by van der Waals forces forming either islands or patches, and by using the tools offered by the reversed-flow inverse GC in combination with further derived from the application of quantitative structureproperty relationship model and DFT, the description of lateral molecular interactions developed in the adsorption of gaseous C2-hydrocarbons onto solid powdered metaloxides was attempted through the experimental determination of local molecular properties (polarizabilities, ionization energies, electronegativities, electron affinities, and hardness) for the reason that these properties reflect the strengths and character of interacting molecules in or without the presence of a solid substrate, namely, their reactivity, which has essentially a local character. To summarize the observations and remarks reported in the previous section, we can say that: • An inverse relationship among molecular polarizability and ionization energy is confirmed. • Local molecular polarizabilities and ionization energies of C2-hydrocarbons adsorbed on the same metaloxide change in a way does not always agree with the reported in the literature values for the corresponding global values of these quantities. This fact was attributed to the one or more of the following reasons: (i) the different local environment of each hydrocarbon molecule over each metaloxide (ii) the induced different orientation of each adsorbate hydrocarbon over each solid surface (iii) the possibility of being the same hydrocarbon mole cule a part of an ordered supramolecular structure (iv) the different acidic or basic behavior of both the adsorbate hydrocarbon molecule and the solid adsorbent The last one reason was further investigated through the timeresolved analysis of local molecular properties and under the view of the HSAB principle. Thus, although HSAB-theory does not predict for the amphoteric character of (some) substances, in the present study and through adsorption studies involving various systems adsorbent/adsorbate, the amphoterism of substances is revealed. Indeed, the local version of HSAB theory as structured here affords a feasible solution to the investigation of amphoterism. Finally, Lennard-Jones parameters were determined for all the C2-hydrocarbons, the values of which are in a very good agreement with corresponding ones from various literature sources. In particular, the fact that they increase going from ethyne to ethene 25410
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’ ASSOCIATED CONTENT
bS
Supporting Information. Final forms of all of the calculated local adsorption parameters by means of RF-IGC methodology. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Tel: 0030 210 772 4030. Fax: 0030 210 772 3184.
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