Elucidation of the Local Character of Chemical Reactivity through the

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 adsorption
desorption phenomenon step by step, is appropriately associated with quantitative structure
property relationship model and density functional theory. Finally, local molecular properties and Lennard-Jones parameters are determined for nine gas
solid 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.4
8 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 structure
property 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 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), 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 Air
Liquide Hellas Company. The gaseous analytes used for the adsorption studies were ethane, ethane, and ethyne supplied from Air
Liquide 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 g
1, 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 min
1. 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.15
18 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,15
18 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 cm
3)] 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 cm
3. 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.15
23 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 2
10 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 structure
property relationship model or DFT as well as hard
soft acid
base (HSAB) theory.25
27 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 16
19 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 mol
1 ,

aðÅ Þ ¼
3:153
1:848 log½cy ðtÞðmol 3 cm Þ

ð13Þ

aðÅ Þ ¼
8:393
2:494 log½cy ðtÞðmol 3 cm
3 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.31
33 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 15
19) 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 cm
3

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 16a
19a 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 cm
3) 3 [NA (molecules 3 mol
1)] 3 [cy (mol 3 cm
3)]}” 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 mol
1) " # 3 1024 ðÅ cm
3 Þ 3 aðÅ Þ log
1:504 NA ðmolecules 3 mol
1 Þ 3 cy ðmol 3 cm
3 Þ ¼
1:284
log½cy ðmol 3 cm
3 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 16a
19a, 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 cm
3)

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 cm
3)

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 1
6, and the corresponding diagrams are represented in Figures 2
4. 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 Kohn
Sham (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.37
43 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 cm
3)

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 cm
3)

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 α
1/3), resulting in very good results in cases of atoms but not so good in cases of molecules.47
49 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 molecule41
43 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 cm
3)

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 cm
3)

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 27
29 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 21
30 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.52
54 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 2
1=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 10
12. 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 3.85
3.76
4.42
4.45
4.47
5.00
5.06 Å3, 3.52
3.65
4.19
4.25
4.36
4.69
4.71
4.76 Å3, and 3.16
3.33
3.60
3.74
3.93
3.94 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: 10.05
10.39
10.5
10.51
10.52 eV;57,61
63 and C2H2: 11.4 eV57 As one can see from Tables 1
6 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 4
6 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 Å
1)

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 Å
1)

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 Acid
Base 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. Acid
base reactions and interactions are of fundamental importance in a variety of chemical and physicochemical processes. The application of various acid
base theories in the investigation and assessment of the acid
base 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

dx.doi.org/10.1021/jp2089824 |J. Phys. Chem. C 2011, 115, 25389–25412

The Journal of Physical Chemistry C

ARTICLE

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 Å
1)

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 Å
1)

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 acid
base 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 acid
base 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

dx.doi.org/10.1021/jp2089824 |J. Phys. Chem. C 2011, 115, 25389–25412

The Journal of Physical Chemistry C

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

dx.doi.org/10.1021/jp2089824 |J. Phys. Chem. C 2011, 115, 25389–25412

The Journal of Physical Chemistry C

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 mol
1)

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 mol
1)

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

dx.doi.org/10.1021/jp2089824 |J. Phys. Chem. C 2011, 115, 25389–25412

The Journal of Physical Chemistry C

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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.72
74 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 acid
base 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 HOMO
LUMO gap and soft molecules have a small HOMO
LUMO 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,15
21,24 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 10
12. 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 10
12, 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 10
12). Finally, these values are also compared with corresponding ones found in literature,14,80
82 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 10
12) 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 10
12) agrees absolutely with literature data14,80
82 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  10
11 N/molecule • C2H4(g)/ZnO(s): Fmax = 1.386  10
11 N/molecule • C2H2(g)/ZnO(s): Fmax = 1.566  10
11 N/molecule It is worth noting that because the conventional laboratory balances can measure down to ∼10
9 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 adsorbate
adsorbate 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 structure
property 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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The Journal of Physical Chemistry C and ethane confirms the validity of our method because this finding agrees perfectly with the definition of van der Waals volume and radius, namely, that these quantities are intrinsic features of each particular molecule.

’ 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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