On Inverse Adsorption Chromatography. 2 ... - ACS Publications

J. Phys. Chem. C 2007, 111, 7473-7486

7473

On Inverse Adsorption Chromatography. 2. Determination of Isotherms and Heats of Adsorption as well as Energy Distributions of Adsorption Sites V. A. Bakaev,* T. I. Bakaeva, and C. G. Pantano Department of Materials Science and Engineering, Materials Research Institute, The PennsylVania State UniVersity, UniVersity Park, PennsylVania ReceiVed: NoVember 21, 2006; In Final Form: March 6, 2007

The isotherms and heats of adsorption, as well as the energy distributions of adsorption sites for pyridine at 100 °C and benzene at 40 °C on the silica surface, were determined by the method of inverse gas chromatography (IGC). These characteristics are usually obtained from IGC after integration of its direct result: the derivative of the adsorption isotherm. It is shown that the constant of integration plays a critical role in obtaining the isosteric heats of adsorption and the energy distributions of adsorption sites. The method of extrapolation is shown herein to be most appropriate for obtaining this constant. The modified ECP method, which takes account of longitudinal diffusion and kinetics of adsorption, was developed in the companion paper (Bakaev, V. A. J. Phys. Chem. C 2007, 111, 7463-7472) and was applied here to the pyridine/silica and benzene/silica systems. It is shown how to experimentally determine the longitudinal diffusion coefficient and the kinetics of adsorption coefficient. The IGC method probes only a small fraction of all of the adsorption sites on a surface. These are the strongest adsorption sites whose adsorption energy in the case of pyridine is almost 6 times larger in magnitude than the heat of liquefaction.

1. Introduction This paper is the second part of a two part series concerned with inverse adsorption chromatography. The first part (I, see ref 29) considers the theory of inverse adsorption chromatography. Its main result is eq 4 (see below). In this second part (II), a method of obtaining of the eq 4 parameters is presented, along with other problems in adsorption. The majority of surfaces of high surface area solids are heterogeneous with respect to their adsorption properties. The theory of adsorption on heterogeneous surfaces originated in the first half of the previous century and is reviewed in many review papers and monographs.1-4 In particular, see the review about measuring the energy distributions of adsorption sites by the method of inverse gas chromatography (IGC) of finite concentrations.5 The adsorption heterogeneity of surfaces is a broad concept. One can consider a surface as a collection of different microscopic patches and call such patches adsorption sites. The energy distribution of these sites can be formally obtained from an adsorption isotherm (cf. section 3.2) and serve as a characteristic more useful than the original isotherm. It can show, for example, how a surface changes under some treatments. For example, the energy distribution of adsorption sites obtained from finite concentration IGC has been used to characterize changes to a muscovite surface in the process of grinding. Depending on the condition of grinding, either delamination of the muscovite crystals occurred or lateral surfaces were formed.5 Another approach to surface heterogeneity considers adsorption sites as minima of the adsorption potential (potential energy of a molecule) on a macroscopically homogeneous and flat surface. This approach can be visualized with the help of computer simulation of argon adsorption on some generic model of an amorphous oxide.6 The atomic configuration of the model

surface was simulated as a dense random packing of hard spheres (oxygen anions) and the local Henry constants for argon were calculated over such a model surface. These local Henry constants can be taken as measures of local adsorption activities and display sharp peaks at 150 K (see Figure 4b in ref 6). This means that adsorption activity for argon on the model amorphous surface is not evenly distributed over the surface but concentrated at specific sites. These can be called adsorption sites. Since their adsorption activities are different, the model amorphous surface is heterogeneous with respect to argon adsorption. The maxima of these peaks are located at the minima of the adsorption potential which are the loci of adsorption sites. These minima are so narrow that they can adsorb no more than one argon atom. Besides, the interaction between argon atoms adsorbed on different sites (lateral interaction) is much weaker than their interaction with the oxide surface. These two preconditions, only one molecule per site and negligibly small lateral interaction, are the foundation of the Langmuir model. This is the basis for the statistical mechanics derivation of the Langmuir equation7 and the reason why the Langmuir model plays such an important role in the theory of adsorption on heterogeneous surfaces (cf. section 3.2). The number of molecules adsorbed on the surface depends upon the chemical potential. At small values of the chemical potential, adsorbed molecules occupy the strongest adsorption sites (deepest potential wells). As the chemical potential and, correspondingly, coverage (the ratio of the number of adsorbed molecules to that in the monolayer) increase, the adsorbed molecules (argon atoms in the case considered) occupy weaker adsorption sites and even sites which are not the minima of the adsorption potential. It has been shown that when coverage increases new adsorption sites can be created on the surface. These are minima in the full potential energy of an adsorbed argon atom which consists of the normal (atom-surface) and lateral components.8 In this situation, one should not expect the

10.1021/jp067735w CCC: $37.00 © 2007 American Chemical Society Published on Web 04/27/2007

7474 J. Phys. Chem. C, Vol. 111, No. 20, 2007 Langmuir model to be valid. This means that the theory of adsorption on heterogeneous surfaces based on the Langmuir model is theoretically substantiated only at sufficiently small coverage. The most reliable experimental substantiation of the validity of the Langmuir model (localized adsorption with no lateral interactions) on a heterogeneous surface at small coverage was obtained by Drain and Morrison for adsorption of argon, nitrogen, and oxygen on TiO2 (rutile). They applied the standard method of chemical thermodynamics, the study of heat capacities at low temperatures, to this system and obtained the energy distribution of adsorption sites as the isosteric heat of adsorption extrapolated to the absolute zero of temperature.9,10 This allowed them to calculate the adsorption entropies for argon at 85 K and to show that they agree with those of the Langmuir model at small coverage (less than 60% of the monolayer) and drastically disagree at higher coverage (cf. Figures 5 and 6 in ref 9). A convenient experimental technique for studying adsorption at small coverage is IGC. We applied IGC to the study of adsorption heterogeneity of silicate glass (glass fiber) surfaces.11-13 A method has been developed that allowed one to obtain the energy and entropy distributions of adsorption sites.11,13-15 Application of the method gives reasonable energy and entropy distributions in the middle range of the experimentally accessible coverage, but unrealistic, double valued distributions were obtained at smaller and larger coverage (see, e.g., Figure 1 in ref 13). The purpose of this, and the companion paper (I, see ref 29), is to understand the reason for these artifacts and to eliminate them by modifying the IGC method. To that end, we consider here at first the isotherms, and especially differential (isosteric) heats of adsorption. This is because, on one hand, these adsorption characteristics are directly connected with IGC experimental data, and more so are needed to obtain the adsorption site distributions (cf. section 3.2). On the other hand, the adsorption site distributions are closely connected to the differential heats of adsorption as mentioned above. Thus, one may expect every experimental artifact observed in the energy distributions to have its counterpart in differential heats of adsorption and vice versa. As shown in section 3.1, this is really the case. Initially, it was assumed that the reason for the unrealistic energy distributions was the method of processing experimental data used in IGC.11,13 The method is called ECP (see the companion paper I). The ECP method is rather simple, but its simplicity is achieved at the expense of sweeping approximations. Our initial assumption was that these approximations of the ECP method are responsible for the problem.11,13 The approximations of the ECP method were partly removed (by the method developed in ref 29) which changed isotherms and isosteric heats of adsorption obtained from IGC considerably, but the unrealistic behavior of the isosteric heats (which reflects the unrealistic behavior of the site distributions mentioned above) at small coverage did not disappear. Finally, it was found that the primary reason for these artifacts is not so much the deficiency of the ECP method itself but rather some experimental peculiarity of obtaining isotherm of adsorption from IGC. This peculiarity and the method of its correction is described in section 2.1. This paper is organized as follows. The next section describes mainly the procedures by which the raw chromatographic data, elution profiles (EP), can be converted into the dependence of the derivative of the isotherm of adsorption with respect to concentration and how the effective diffusion coefficient can

Bakaev et al. be obtained from the EP of methane. Section 2.1 describes the method of obtaining the isotherms from their derivative. The main problem here is not the numerical integration but the determination of the integration constant. Section 3.1 discusses the dependence of the isotherms and isosteric heats of adsorption on the methods of processing EPs. It also shows how the parameter determining the kinetics of adsorption can be chosen in such a way that the isotherm of adsorption does not depend on the flow rate of the carrier gas. Finally, section 3.2 describes the method of obtaining the energy and entropy distributions from isotherms of adsorption, and then elucidates the meaning of these distributions by discussing the entropy-energy relationship. 2. Experimental Section The gas chromatograph (GC) used in these experiments was an Agilent 6890N with a flame ionization detector (FID) and a split/splitless injector. The chromatographic column was a glass tube 23 cm long, with an external diameter of 6.35 mm and an internal diameter of 3.9 mm. It was packed with a mixture of deactivated fused silica beads (Restek) and fumed silica (SigmaAldrich). Fumed silica is a nonporous silica adsorbent with the specific surface area of 200 ( 20 m2/g. The average size of its particle evaluated from the specific surface area is about 10 nm. A chromatographic column packed with such a fine powder would have prohibitively high resistance to the carrier gas (He) flow. To diminish that resistance, we made a 3.6 wt % mixture of fumed silica with deactivated fused silica beads (which have, in fact, an irregular shape). To make the mixture homogeneous, it was mixed on a shaker for about 10 h. The weight of the mixture in the column was 2.8 g, and the total surface area of fumed silica in the column was 20.2 m2. The drop of pressure on the column at the flow rate of He 10.2 mL/min at 100 °C was 4.1 kPa (0.60 psi). The pressure drop is too small for normal operation of the Agilent 6890N split injector. This problem was resolved by connecting the packed column with the injector through a guard capillary column. The guard column was a deactivated fused silica lined stainless steel tubing about 6 m long, 0.28 mm internal diameter (Silkosteel Tubing from Restek). Due to the presence of the guard column, the pressure at the input of the packed column cannot be directly measured but should be calculated by the following formula:

p1 )

x

p22 -

L1 16η poFo 4 π R

(1)

Here po, p1 , and p2 are pressures at the outlet of the packed column, at the inlet of the packed column (end of the guard column), and at the beginning of the guard column (at the injector), respectively; L1and R are the length and the internal radius of the guard column; Fo is the flow rate of the carrier gas (He) at the outlet of the packed column reduced to the temperature of the latter; η is the viscosity of He at the column temperature. Equation 1 is, in fact, the solution of problem 6 to §17 in ref 16 (it can also be obtained from eqs 1, 2, and 17 of Chapter 2 of ref 17). The value of p2 in eq 1 is directly measured as a sum of po and the head pressure measured by the electronic system of the GC. The flow rate of the carrier gas at the outlet of the column at room temperature was accurately measured (with water vapor correction) by a soap flow meter. Another important parameter of the guard column is the time lag δt between the time of injection and that of solute appearing at the inlet of the packed column which is

On Inverse Adsorption Chromatography. 2.

δt )

L1 ju1

J. Phys. Chem. C, Vol. 111, No. 20, 2007 7475

(2)

where

j)

p2 R2 p1 3 P2 - 1 ; u (P2 - 1); P ) ) 2 P3 - 1 1 L1 16η p1

Equation 2 follows from eqs 3, 10, and 11 in Chapter 2 of ref 17. This time lag makes a considerable contribution in the moments of the elution profile (EP) of nonadsorbing gas (see discussion of eqs 6). Parameters of the guard column introduced into the GC allowed the latter to automatically maintain the prescribed flow of the carrier gas (He). However, due to additional resistance of the packed column, the actual flow rate of the carrier gas deviated somewhat from the prescribed one. The probe molecules used in this work were pyridine and benzene from Sigma-Aldrich. A typical response of FID to the injection of pyridine and benzene observed in these experiments is presented in Figure 1. At the first moment after injection, the detector current is at the background level which in this case was about 6.4 ( 0.1 pA. Then (after about 8.2 min for pyridine) a sharp increase (shock) is observed. The reason for the shock is explained in the companion paper I (see ref 29). The shock is followed by the long tail. The tail is the main source of information on the thermodynamic properties of the silica surface with respect to pyridine or benzene as discussed in section 3. For pyridine, the column temperature is maintained at 100 °C for 50 min, then it was raised to the level of 230 °C, maintained at that level for 15 min, and returned back to 100 °C. The temporary increase of temperature gave rise to the second thermodesorption peak on the pyridine EP in Figure 1. The area under the thermodesorption peak can by used for evaluation of the integration constant as explained in section 2.1. For benzene, the column temperature was maintained at 40 °C for 15 min, then raised to 180 °C, kept at this level for 15 min, and returned to the initial temperature. The thermodesorption peak on the benzene EP also exists, but it is unnoticeable on the scale of Figure 1. The general conclusion one can draw from Figure 1 is that the interaction of benzene with the silica surface is much weaker than that of pyridine. This conclusion will be quantitatively confirmed by the isosteric heat of adsorption plots in section 3.1, and the energy distributions of adsorption sites considered in section 3.2. The total area between the background level and the elution profile in Figure 1 is proportional to the total amount of solute injected in the column. The background level was determined by linear interpolation between the beginning and the end of the plot. The constant of proportionality was used to convert the FID current into concentration of probe molecules in the carrier gas (concentration of solute) as follows. The concentration of the solute C in the carrier gas at the outlet of the column was calculated by the following equation:

C)

const I CeffFo

(3)

Here I is the FID current andCeff is an effective number of carbon atoms in a probe molecule which can be evaluated with the help of Table 10.6 on p 446 of ref 17. For pyridine and benzene, Ceff ) 5 and 6, respectively. For FID in our setting, const = 4 × 10-3 nmol pA-1 s-1. This value is determined as a ratio between the total quantity of carbon injected (injected amount of organic solute in nmol multiplied by Ceff) and the

Figure 1. Experimental elution profiles (EPs) from the column packed with fumed silica (see text). Open squares, pyridine at 100 °C; solid line, benzene at 40 °C (see text).

total area of the EP in pA s. The crude estimate of Ceff, does not influence the accuracy of calibration because Ceff enters implicitly (through const) in the numerator of eq 3. The convenience of Ceff in eq 3 is that it makes the value of const approximately equal for all organic solutes. The accuracy of const is determined by the accuracy of the injected amount because that of the EP area is much better. In our case, injection was performed with a 1 µL Hamilton syringe. We evaluate its accuracy to be approximately 10% which is the accuracy of our scale of concentration or the scale of pressure for adsorption isotherms. However, this inaccuracy of calibration does not depend on the temperature and does not influence the accuracy of isosteric heats of adsorption (see the next section). Besides, the value of const once calibrated, although not very accurately, stays the same in all of the experiments and can be steadily improved. Equation 3 allows one to convert the plot detector current vs time into the plot time vs concentration. The latter is EP. We designate EP as t exp(C) which is determined by the following equations (see eq 58 in ref 29):

f(C h 0) ) ht exp(C h 0 + τjkinC1(C h 0) + D h Cd(C h 0) + δCδ(C h 0)) + h 0) (4a) τj0ln(C where (see eq 22 in ref 29)

dN h0 ) f(C h 0) - 1 dC h0

(4b)

These equations are almost the same as eqs 22 and 58 in ref 29 except that the variables in eq 4 are marked by bars. This is because the overwhelming majority of variables in ref 29 are dimensionless, the smaller number of dimensional variables being marked by stars. In contrast to that, all of the variables in this paper, except those in eq 4, are dimensional. This is the reason that the dimensionless variables in this paper are marked by bars. In eq 4b, N h 0 is the dimensionless adsorption, C h 0 is the dimensionless concentration, and ht exp is the dimensionless time. According to eqs 7 in ref 29, the time is measured in units of tm: the retention time of a nonadsorbing gas (hold-up time). The concentration is measured in the units of Cmax ) Nin/Vin where Nin is the total injected amount measured as the area of the detector signal (see above) and Vin is the volume of the injector determined as explained below (see eq 6). The

7476 J. Phys. Chem. C, Vol. 111, No. 20, 2007

Bakaev et al.

dimensionless adsorption is determined per total mass of the adsorbent in the column and is measured in the units of VmCmax where Vm ) jFotm (see eqs 6 and 7 in ref 29). Here j is the same factor as in eq 2 with the only difference that now P ) p1/p0; Fo is the same as in eq 1. If D ) 0, τkin ) 0, δ ) 0, and τ0 ) 0, eq 4b reduces to

dN ) jFo[t(C) - tm] dC

(5)

which is the basic ECP equation of IGC (cf. ref 29). Here in distinction from the dimensionless eqs 4, all of the variables are in their natural dimensions: N is adsorption per the total mass of adsorbent in the column; C is the concentration of solute in the gas phase connected with its pressure by the ideal gas equation (values of C are very small); t(C) is EP. In reality, D h and τjkin in eq 4 are not zero. The determination of D is described below (see eq 6) and determination of τkin is described in section 3.1. The parameter δ equals p1/po - 1 (see eq 51 in ref 29) and the value of τ0 equals Vin/(jFotm) (see eq 10a in ref 29). The functions C1, Cd, and Cδ in eq 4a are described by eqs 42, 47, and 55 of ref 29, respectively. Finally, the value f(C h 0) can be found from the eq 4a by iterations as described in Section 4 of ref 29. The values of Vin and D can be determined from the first 3 moments of the EP of nonadsorbing gas, methane in our case. The formulas obtained as eqs 15 and 19 in ref 12 (in the present notations) are

m1 )

( )

Vin 2 12(D)2L L Vin 2DL + ; µ2 ) + ; µ ) + 3 u jFo jFo (u)3 (u)5 2

( )

( )

Vin 3 Vin 120(D)3L ; µ4 - 3µ22 ) +6 7 jFo jF (u) o

4

(6)

These equations were obtained for a noncompressible fluid. To take account of compressibility of the carrier gas, we substituted the flow rate υ in eq 19 of ref 12 for jFowhich is a fairly good approximation for our case where j deviates from unity only by 2-3%. One can exclude Vin/(jFo) and D/L2 from the first three of eqs 6 to obtain an eq for tm ) L/u

1 [µ2 - (m1 - tm)2]2 ) tm[µ3 - 2(m1 - tm)3] 3

(7)

Then one can use the first and the second equations in eq 6 to obtain the excluded variables and finally Vin and D. The EP of methane on our column was obtained by injection of the gas from the methane kit (Restek). It is shown in Figure 2. The left peak was obtained from the guard column connecting the injector and FID. The right peak was obtained from the full column consisting of the guard column followed by the packed column. 2.1. Extrapolation of Adsorption Isotherm to Lower Pressures. In contrast to conventional adsorption techniques, inverse adsorption chromatography does not give isotherms of adsorption directly. Instead it provides their first derivatives (cf. eq 5). An isotherm derivative does not allow one to obtain isosteric heats of adsorption or to separate the free energy distribution of adsorption sites into the energy and entropy distribution as explained in section 3.2. To do so, one has to get isotherms of adsorption which requires integration of the derivative obtained from IGC. This gives rise to the problem of constants of integration. In other words, one has to know in advance the value of adsorption at some fixed pressure to obtain

Figure 2. Elution profiles of methane from the column of Figure 1 at 100 °C. Dots, full column (guard+packed); solid line, guard column.

an isotherm from IGC data. As seen from the pyridine EP in Figure 1, after elution for 50 min, the detector signal is still 550 pA which is about 100 times larger than the background and the signal-to-noise ratio is about 1000. According to eq 3, this value of the detector current gives the concentration of pyridine in the carrier gas that corresponds to a partial vapor pressure of 0.042 Torr. This is the abscissa of the initial point of the isotherm that corresponds to the EP in Figure 1. The ordinate of that initial point can be obtained by thermodesorption as the area of the second (smaller) peak in Figure 1. This method of thermodesorption was employed in our previous publications.11,13 In fact, it has been always implicitly employed in inverse adsorption chromatography to evaluate adsorption at the lowest accessible pressure (concentration): Often one just neglects adsorption at the end of EP even without obtaining a thermodesorption peak. Here, we consider another method for obtaining the initial point of an isotherm. Isotherms of adsorption at very small pressure (concentration) can be described by a small number of well-known equations. If a surface is homogeneous, then its isotherm is almost linear in the region of very small concentrations (the Henry isotherm). Isotherms of adsorption on heterogeneous surfaces in the submonolayer region also obey several empirical equations which are reviewed, e.g., in Chapter 2 of ref 3. The most widely used of them is the Freundlich equation (F-equation) and the Dubinin-Radushkevich (DR) equation. These can be obtained as particular cases of the virial adsorption isotherm18

N ) exp[a0 + a1 lnC/Cs + a2(lnC/Cs)2 + a3(lnC/Cs)3 + ‚‚‚] (8) Here N is adsorption in appropriate units and C and Cs are the concentration of solute and concentration of its saturated vapor, respectively. If one preserves in the right-hand side of eq 8 only the 0th and the first terms, one obtains the F-equation. If only the 0th and the second terms are preserved, the DR equation is obtained. Differentiate eq 8 and take logarithm from the both sides to obtain

dN ( dC ) ) a + a ln C/C + a (lnC/C ) + ‚‚‚ + ln[a +

ln C

2

0

1

s

2

s

1

2a2 ln C/Cs + ‚‚‚] (9) Since the direct result of inverse adsorption chromatography is the plot of dN/dC vs C, one can use eq 9 to obtain the


Figure 3. Graphical representation of eq 9. Diamonds, experimental points for pyridine on the silica surface at 100 °C; solid line, fit of the half of those points by eq 9 where only the first three coefficients are preserved.

coefficients a0, a1, a2, ... directly from EP. In principle, these coefficients can be substituted in eq 8 to obtain the isotherm of adsorption without numerical integration. The practical procedure used below is somewhat different and can be explained with the help of Figure 3. The experimental points were obtained by eq 5 (ECP approximation) from the EP in Figure 1 corresponding to pyridine. (The points in Figure 3 are not the real experimental points from Figure 1 but their interpolated representation.) The half of these points corresponding to the lower values of concentrations were fitted by eq 9 where only the first three coefficients were preserved. In other words, the argument of the exponent in eq 8 is a quadratic (the reason for that is explained at the end of the section). The coefficients a0, a1, and a2 were determined by the nonlinear least-square fit (Marquardt method19). The initial guess was obtained from the linear least-square fit which neglected the logarithmic term on the right-hand side of eq 9. One can see from Figure 3 that the solid line fits very well the points corresponding to the lower concentrations but deviates from the experimental points at higher concentrations. Thus, we did not use coefficients a0, a1, and a2 obtained from this fit to calculate the whole isotherm by eq 8 (neglecting higher coefficients) but only the initial adsorption at the minimal concentration in Figure 3. Then the whole isotherm can be obtained by numerical integration of its derivative obtained from EP with the integration constant corresponding to that minimal concentration. Finally, it should be emphasized that the reliability of the extrapolation method considered in this Section is based on the assumption that the F and DR equations, or their combination, can fit the isotherm of adsorption of pyridine and benzene on the silica surface up to very low coverage where the initial adsorption (constant of integration for IGC) is so small that it can be safely neglected. This is known to be the case for some rare gases and nitrogen on pyrex, as well as for some other adsorption systems (see Chapter 2 in ref 3). However, the main argument in favor of the method considered in this section is that its application qualitatively improves the isosteric heats of adsorption and the energy distribution of adsorption sites in comparison to those obtained by the thermodesorption method. In fact, using the extrapolation method instead of the thermodesorption method for obtaining the constant of integration in IGC


Figure 4. Dependence of adsorption isotherms of pyridine on the silica surface at 100C on the flow rate and the IGC parameters. D, longitudinal diffusion factor in cm2/s; τ, kinetics of adsorption factor (τkin) in min-1; F, flow rate of the carrier gas in mL/min.

resolves the problem of unrealistic energy and entropy distributions of adsorption sites which arose in our previous publications.11,13 3. Results and Discussion 3.1. Isotherms and Isosteric Heats of Adsorption. One can calculate the derivative of an adsorption isotherm from an experimental EP using eq 4 which explicitly depends on four parameters: longitudinal diffusion factor D, kinetics of adsorption factor τkin, carrier gas compressibility factor δ, and the injector relaxation time factor τ0. (Here and below these factors are presented in their natural dimensions in distinction from eq 4 where they are dimensionless.) Of these four factors, only δ can be determined simultaneously with EP; the other factors have to be determined by independent methods as described below. If one nullifies all of these factors, eq 4 reduces to eq 5 which allows one to determine the derivative of an adsorption isotherm in the ECP approximation. One can obtain the adsorption isotherm by integrating this derivative and determining the constant of integration by the method described in section 2.1. Two such isotherms are shown in Figure 4 by the solid line and open squares. The constant of integration for all of the isotherms in Figure 4 was obtained by the method of extrapolation. We do not show here either isotherms with constant of integration obtained by the thermodesorption method or isotherms for benzene because they look rather similar in the graphical representation. The importance of the method of determining the constant of integration described in section 2.1 reveals itself when one considers the isosteric heats of adsorption. An isoseric heat of adsorption qst can be obtained from two isotherms corresponding to different temperatures by the equation

qst ) RT2

(∂ ∂Tln P)

N

= -R

lnP2 - ln P1 1/T2 - 1/T1

(10)

Here P ) CRT is pressure of ideal gas corresponding to concentration C. (The IGC method usually gives isotherms in the domain of small concentration so that the equilibrium gas can be considered as ideal.) The points (P2,T2) and (P1,T1) lay


Figure 5. Dependence of the isosteric heats of adsorption of pyridine on the silica surface at 100-110 oC on the method of calculation: D ) 0 corresponds to the ECP approximation; D ) 0.4 and D ) 0.7 designate corrections to ECP (see text); inscriptions “thermodesorption” and “extrapolation” in the insert designate the method of determination of initial adsorption described in section 2.1. Heat of liquefaction of pyridine at 100 °C is about 37 kJ/mol.

Figure 6. Dependence of the isosteric heats of adsorption of benzene on the silica surface at 40-50 °C on the method of calculation: D ) 0 corresponds to the ECP approximation; D ) 0.4 and D ) 0.7 designate corrections to ECP (see text); inscriptions “thermodesorption” and “extrapolation” in the insert designate the method of determination of initial adsorption described in section 2.1. Heat of liquefaction of benzene at 40 °C is about 33 kJ/mol.

on the same isostere (whence the term “isosteric”), i.e., correspond to the same value of adsorption N. An isostere of adsorption is usually close to a straight line in the coordinates ln P vs 1/Twhich alleviates the numerical differentiation in eq 10. Isosteric heats of adsorption qst are shown in Figures 5 and 6. Compare first the plots designated by the open squares and diamonds in Figures 5 and 6. Both of them were obtained from the isotherm derivatives calculated from EPs by eq 5, i.e., in the ECP approximation, the difference being only in the integration constants. For pyridine, the integration constant (adsorption at the minimal measured concentration) determined by the method of thermodesorption (see section 2.1) is 3-5 times less than that determined by the method of extrapolation.

Bakaev et al. For benzene the former is almost 10 times less than the latter. Despite this large relative difference in the initial values of adsorption determined by the two methods, the overall shift of the isotherm due to the shift of its initial point is very small and almost indiscernible in plots like those in Figure 4. This is because the initial value of adsorption itself is very small. For example, the initial value of adsorption of benzene at 40 °C and 0.013 Torr is only 0.03% of monolayer. However those small differences in the initial values of adsorption give rise to the dramatic differences in the isosteric heats of adsorption in Figures 5 and 6. The isosteric heat of adsorption can be considered as the differential enthalpy of adsorption which is the molar enthalpy of gas minus the differential energy of adsorbate. It shows how the magnitude of the energy of a molecule adsorbed on a heterogeneous surface changes with coverage. For example, the plot designated by open squares in Figure 5 shows that at the minimal coverage of about 0.02 pyridine molecules occupy the sites on the silica surface whose energy is more than 120 kJ/ mol in magnitude (the adsorption energy is negative). Roughly speaking, this means that all the sites whose adsorption energy is less than -120 kJ/mol are occupied by adsorbed molecules at the above coverage. These are the strongest sites on the silica surface, all of the vacant sites being characterized by lower magnitudes of adsorption energy. If the coverage increases, those vacant sites will be occupied by the newly arrived molecules, which is the reason why the isosteric heat of adsorption on a heterogeneous surface is a monotonously decreasing function of coverage. Moreover, this function approximately (in the condensation approximation, see section 3.2) determines distribution of adsorption sites in energy. The above consideration is based on the assumption that each new molecule that appears on a heterogeneous surface always occupies the strongest vacant site. Strictly speaking, the assumption is true only at the absolute zero of temperature (and if one neglects the interaction between molecules on neighboring sites). Thus, to obtain the energy distribution of adsorption sites from the isosteric heat, one has to extrapolate the latter to the absolute zero of temperature. This procedure was experimentally carried out by Drain and Morrison (see the Introduction). Both benzene and pyridine are organic bases. They belong to the group B of molecules according to the classification on p 13 of ref 20. One can see from Figure 26 on p 43 of ref 20 that the molecules of group B display monotonously decreasing isosteric heats of adsorption vs coverage on the silica surface which is typical for adsorption on heterogeneous surfaces. This means that the plots of isosteric heats of adsorption vs coverage in Figures 5 and 6 designated by diamonds and obtained from the isotherms whose initial points were determined by the method of thermodesorption are qualitatively wrong. The same method was used in our previous studies of surface heterogeneity by IGC.11,13 Since the isosteric heat of adsorption vs coverage is so close to the energy distribution of adsorption sites, one may conclude that the main reason for the inconsistencies in the energy distributions of adsorption sites discovered in those studies (and mentioned in the Introduction) is the method of thermodesorption that was used in those studies for determination of initial points of isotherms. Now we consider the influence of other factors on an isotherm of adsorption obtained from inverse adsorption chromatography. As mentioned in the Introduction (see also ref 29), the ECP eq 5 is conceptually a rather crude approximation of inverse adsorption chromatography. A more refined method based on eq 4 requires the numerical values of the longitudinal diffusion



TABLE 1: Deviations (eq 11) between Two Adsorption Isotherms Calculated with the Same Values of the Longitudinal Diffusion Factor D and the Adsorption Kinetics Factor τkin but Different Values of the Carrier Gas Flow Rate (about 5 and 10 mL/min) pyridine

benzene

D, cm2/s

τkin, min-1

dev, %

D, cm2/s

τkin, min-1

dev, %

0 0.4 0.4 0.4 0.4 0.7 0.7 0.7 0.7 0.7

0 0 0.006 0.008 0.010 0 0.0100 0.0115 0.0120 0.0121

2.31 9.6 1.79 1.17 1.83 12.5 2.46 1.56 1.33 1.30

0 0.4 0.4 0.4 0.4 0.7 0.7 0.7 0.7 0.7

0 0 0.010 0.015 0.020 0 0.010 0.015 0.020 0.025

2.63 7.31 1.81 1.72 3.14 9.43 3.67 2.13 1.43 1.92

factor D, kinetics of adsorption factor τkin, as well as factors τ0 and δ. The last factor can be directly measured simultaneously with EP (see eq 51 in ref 29). To determine τ0, one has to know the volume of the injector Vin (see eq 10a in ref 29). That volume can be determined together with D from the elution peak of methane such as that in Figure 2 using eq 6. In the paper where eq 6 was obtained, we determined D ) 0.88 cm2/s and Vin ) 0.42cm3.12 These data were obtained with a column of the same dimensions as in this study but packed not with a powder but with glass fibers. The GC was also the same but without the guard column. With the new column, we obtained D ) 0.4 cm2/s and Vin ) 0.35 cm3. The discrepancy in Vin is tolerable, and we accept Vin ) 0.4 cm3 in what follows. However, the discrepancy in D seems to be considerable. As explained in section 2 of ref 29, D is not a molecular diffusivity in the carrier gas but a lump parameter which takes account of the fluctuation of the carrier gas velocity in a packed column and other stochastic factors. As a molecular diffusivity, D would depend on a molecule and could be different for pyridine, benzene or methane. It also could depend on the temperature of the column. It will be shown below that, at the present level of refinement, all of these differences are not very important. Now, we compare two results of calculations using eq 4: one with D ) 0.4 cm2/s and another with D ) 0.7 cm2/s. The latter value of D was chosen because at higher values the iterations used in calculations do not converge (see below). The isotherms obtained using eq 4 are shown in Figure 4. The initial points for all of them are obtained by the method of extrapolation. The isotherms in Figure 4 were obtained for two different flow rates: Those corresponding to the flow rate of about 5 mL/min are designated by lines, whereas those designated by points correspond to the flow rate of about 10 mL/min. A consistent IGC method should give an isotherm that does not depend on the flow rate. As seen from Figure 4, the ECP isotherm (open squares, solid line) reasonably satisfies this consistency condition. However, the isotherm obtained from eq 4 with D ) 0.7 cm2/s and τkin ) 0 clearly violates the consistency condition. To compare isotherms obtained by different methods more accurately, we introduce the following measure of deviation between two curves y1(x)and y2(x):

dev )

100 n

n

2 ∑ i)1

|y1(xi) - y2(xi)| y1(xi) + y2(xi)

(11)

To use this formula one has to interpolate both of the curves to the same values of xi and then use eq 11 for n abscissas mutual to both functions. The results of calculations using eq 11 are presented in Table 1. One can see from the first row of Table

1 that the deviation between two adsorption isotherms of pyridine obtained by ECP approximation of IGC at two different flow rates of the carrier gas (squares and solid line in Figure 4) is 2.31%. That deviation rises to 12.5% when one tries to refine ECP by using eq 4 with D ) 0.7 cm2/s and τkin ) 0 (diamonds and broken line in Figure 4). However, one can bring down the deviation by increasing the value of τkin. As one increases that value, the deviation goes through a minimum which for pyridine at D ) 0.4 in Table 1 corresponds approximately to τkin ) 0.008 min-1 with dev ) 1.17%. For pyridine and D ) 0.7 in Table 1, the deviation decreases up to the value τkin ) 0.0121 (triangles and dotted line in Figure 4). The minimum deviation could not be reached because the method of iterations which was used for the solution of eq 4 did not converge at larger values of τkin. To find f(C h 0) from eq 4a, the first step for an isotherm calculation, one has to solve an equation because the right-hand side of eq 4a implicitly depends on f(C h 0) as well as on its first and second derivatives (see eqs 42, 47, and 55 in ref 29). We solve this equation by iterations with the acceleration parameter λk ) 0.25 (see eq 60 in ref 29). The initial value of f(C h 0) was obtained by nullifying all of the parameters of eq 4a which makes the latter a simple formula and corresponds to the ECP approximation. Usually the solution was obtained in 15-25 iterations. The condition for termination of iterations was that the deviation eq 11 between the arrays of points representing f(C h 0) obtained in the last iteration and that obtained in the previous one was more than 1000 times smaller than the deviation between the initial f(C h 0) and that obtained in the first iteration. An important condition for convergence of the iterations is thorough smoothing of the arrays of points representing f(C h 0). It has been shown that for a model isotherm of adsorption approximately corresponding to that of pyridine on the silica surface this method gives the true function f(C h 0) (see Figure 2 in ref 29). However, eq 4 cannot be solved by this method for any values of parameters. As mentioned above in the case of pyridine, the iterations do not converge with D ) 0.7 cm2/s and τkin > 0.0121 min-1 or for D > 0.8 cm2/s. The method described above allows one to evaluate all the parameters of eq 4a. The longitudinal diffusion factor D was determined, in fact, for methane and transferred without any correction to pyridine and benzene. After that, the kinetics of adsorption factor can be obtained as the value of τkin that provides the minimal deviation eq 11 between the isotherms obtained from EP corresponding to the different flow rates of the carrier gas (see Table 1). Thus it turns out that D and τkin are not independent parameters. This can be elucidated as follows. In fact eq 4a depends not on D and τkin which in this paper are dimensional values but on their dimensionless counterparts D h and τjkin. (In ref 29, D and τkin are dimensionless, whereas their dimensional counterparts are D* and τ/kin.) The dimensionless counterparts of D and τkin are (see eq 7 in ref 29)

D h)

τkin Vm D t and τjkin ) 2 m tm Vs L

(12)

Here Vm and Vs are the volumes of the mobile and stationary phases in the column, respectively. When the flow rate of the carrier gas increases, the value of tm decreases as L/u where u is the linear velocity of the carrier gas and L is the length of the packed column. Thus, the flow rate variation exerts an opposite influence on the two main parameters of eq 4a: When one of them increases, the other

7480 J. Phys. Chem. C, Vol. 111, No. 20, 2007 decreases. Since in eq 4a these parameters are multiplied by the functions C1(C h 0) and Cd(C h 0) which according to Figure 1 of ref 29 are close to one another, the variation of f(C h 0) in eq 4a and correspondingly an isotherm of adsorption in Figure 4 caused by the variation of D h due to a change of the flow rate is counterbalanced by the variation of τjkin. This situation is similar to that which is well-known for analytical gas chromatography: On one hand, the width of a chromatographic peak in a packed column depends on the diffusion factor which is inversely proportional to the carrier gas velocity u (see eq 5, in Chapter 4 of ref 17). On the other hand, it depends on the factors that are directly proportional tou (see eqs 8 and 12 in Chapter 4 of ref 17). The factors which are directly proportional to u are, in fact, those which determine the kinetics of adsorption in a packed column. In Chapter 4 of ref 17, the factors that determine the kinetics of adsorption are called “resistance to mass transfer” and are connected with the diffusion factor by the diffusion coefficient of an analyte in the mobile phase. In our approach, the kinetics of adsorption is determined by the value of τkin which is also connected to the diffusion factor by the experimental procedure described above. One can see from Figure 4 that taking account of the longitudinal diffusion and kinetics of adsorption factors make a considerable difference for the calculated isotherm of adsorption. Deviations in eq 11 between the isotherm in Figure 4 depicted by triangles, and that depicted by squares, is 35%. The influence of D and τkin values on the calculated isosteric heats in Figures 5 and 6 is less pronounced. When the value of D increases starting from D ) 0 (ECP approximation), the deviation between the isosteric heat plot eq 11 corresponding to D ) 0 and that corresponding to a given value of D at first sharply increases and then flattens. Thus, the isosteric heat plots of pyridine corresponding to D ) 0.1, 0.2, 0.4, and 0.7 (the last two are shown in Figure 5) with the respective values of τkin (which eliminates the dependence of an isotherm on the flow rate) deviate from that corresponding to D ) 0 in Figure 5 by 6, 10.8, 12.0, and 13.2 percents, respectively. One can see that the isosteric heats corresponding to D ) 0.4 and 0.7 in Figures 5 and 6 are very close. This justifies our rather crude evaluation of the D value. This value can be refined, but in our opinion, there is not much value. It would be highly desirable to compare our data with the results of independent measurements. Unfortunately, we found only one paper which allows us to compare the data in Figure 6 with independent calorimetric measurements.21 These are three filled diamond points in Figure 6. They are the extreme left points from the plot of differential heats of adsorption of benzene on the wide pore silica gel from the upper part of Figure 3 in ref 21. To obtain the points in Figure 6 from those in ref 21, we added to the latter points RT ) 2.4kJ/mol (the difference between the isosteric and differential heats) and reduced adsorption to coverage using the monolayer capacity indicated in Figure 3 of ref 21. The data from ref 21 are larger than our data. There may be at least three reasons for that. First, our data correspond to 40 °C, whereas those from ref 21 correspond to 20 °C. The heat of liquefaction of benzene at 40 °C is 33.4 kJ/mol, and at 20 °C, it is 34.2 kJ/mol (from vapor pressure,22 taking the ideal gas equation of state for saturated vapor). Since the heat of physical adsorption is comparable to the heat of condensation, one should expect the three filled diamond points in Figure 6 corresponding to 20 °C to lay somewhat higher than those corresponding to 40 °C. However, taking account of the temperature difference can decrease the discrepancy between our data and those from ref 21 only by about 1 kJ/mol while

Bakaev et al. that difference is about 10 kJ/mol. Second, one should take into account that the measurements in ref 21 were performed on a silica gel which often contains traces of alumina which produces strong Lewis centers that increase the interaction of benzene with the silica gel surface in comparison with fumed silica (aerosil) which is a pure nonporous silica (see ref 20, p 46). The third reason why the data from ref 21 could deviate from our results is that the former data were obtained on a porous solid. Although the three filled diamond points in Figure 6 correspond to a silica gel with a relatively homogeneous porous structure and diameter of pores about 10 nm, it could contain a small number of narrower pores. The heats of adsorption of benzene in those narrow pores should be considerably higher than on a plane surface. This is seen from the differential heats of adsorption of benzene on a narrow pore silica gel (2.5 nm diameter) which were also measured in ref 21 and were about 30% higher in the small coverage area than those for the wide pore silica gel. Finally, comparison of our results obtained by IGC with those obtained by conventional adsorption techniques21 shows that usually the IGC data are shifted to the area of lower surface coverage in comparison to those obtained by the conventional adsorption techniques. As explained above, the region of low surface coverage in a plot of an isotherm or isosteric heat of adsorption on a heterogeneous surface corresponds to adsorption on the strongest adsorption sites. Thus, the IGC is a method which is especially convenient for probing the strongest adsorption sites on heterogeneous surfaces. Now finally, we consider how the energy and entropy distribution of these sites can be obtained from the experimental data described above. 3.2. Distributions of Adsorption Sites in Energy and Entropy. For the reasons explained in section 1, the Langmuir equation plays a very important role in the theory of adsorption on heterogeneous surfaces at low coverage. It can be obtained by the method of grand canonical ensemble in the following form:7

θ(η) ) [1 + exp(ξ - η)]-1

(13)

where

ξ)

( )

min µ P 3 - ln js(T); η ) ) ln Λ - ln jg(T); kBT kBT kBT h Λ) (14) x2πmkBT

Here T is the absolute temperature and P is the pressure of gas in equilibrium with adsorbate at that temperature; kB is the Boltzman constant; h is the Plank constant; m is the mass of a molecule; min is the minimal potential energy of the molecule on an adsorption site; js(T) is the partition function of the molecule adsorbed on that site; jg(T) is the partition function of the same molecule in the gas phase; µ is the chemical potential (the same for the gas and the adsorbate). A partition function ja(T) where subscript a is either g or s is connected with the basic thermodynamic functions as follows:

Ga ) Ea - TSa ) - kBT ln ja(T)

(15)

Here Gg, Eg, and Sg designate the Gibbs free energy, the enthalpy, and the entropy per gas molecule, respectively. These thermodynamic functions refer only to the vibration and rotational (other than translational) degrees of freedom of a single gas molecule. Accordingly, Gs, Es, and Ss refer to the vibration and rotation degrees of freedom of a single molecule



adsorbed on a site. They designate the so-called thermal free energy, energy, and entropy of an adsorbed molecule. For brevity, below we understand by the energy and entropy of a site the energy and entropy of a molecule adsorbed on the site. The right-hand side of eq 13 depends only on the difference between the thermodynamic function of the adsorbate and the gas. Therefore, one can subtract from the thermodynamic functions determined by eqs 14 and 15 any value provided that this is done to the functions referring both to the adsorbed and to the gas molecules simultaneously. For example, one can subtract from the chemical potential µ and kBTξ in eq 14 the chemical potential of the ideal gas at some pressure P0 and temperature T. Then in eq 14 one will have

µ ) kBTη ) kBT ln P/P0

(16)

and

G ) kBTξ ) min + Gs - Gg - kBT ln(P0Λ3/kBT) (17) where G is the reduced free energy of a site (a molecule on the site) and correspondingly

E ) min + Es - Eg - (5/2)kBT and S ) Ss - Sg kB ln(P0Λ3/kBT) + (5/2)kB (18) are the reduced energy and the entropy of a site. These are connected by the equation

G ) E - TS

(19)

Another form of the Langmuir equation which is frequently used in the theory of adsorption on heterogeneous surfaces (see, for example, ref 18) is

[

θ(P) ) 1 +

(

K exp P kBT

)]

-1

(20)

The left-hand side of eq 22 is an experimental isotherm of adsorption expressed as the ratio of the number adsorbed molecules to the number of sites (coverage) vs the dimensionless chemical potential of adsorbate eq 16. The limits of integration are discussed below. Temkin and Levich were the first who gave a remarkably simple solution of eq 22 with infinite limits of integration - ∞ < ξ < ∞1,23

φ(ξ) )

1 [θ (ξ + iπ) - θt(ξ - iπ)] 2πi t

(23)

One can easily check that eq 23 is really a solution of eq 22 with infinite limits of integration by inserting the former into the latter and making the substitution z ) ξ + 2iπ in the first term in the brackets of eq 23 and z ) ξ in the second term. One then obtains a closed contour of integration with a simple pole of the integrand at z ) η + iπ whose residue gives the left-hand side of eq 22.23 The solution of eq 23 gives the distribution density of sites some of which (those where ξ > 0) have no physical meaning in the Langmuir model because the sites with positive values ξ cannot be considered as adsorption sites since they repel molecules instead of attracting them. Probably, the inclusion of these fictitious adsorption sites in eq 22 was considered to be such a serious drawback that eq 23 even was not mentioned in section 3.6 of ref 4 that reviews the so-called exact methods of solution of eq 22. The solution of eq 22 with the negative limits of integration (without introduction of fictitious sites) can be obtained by the Wiener and Hopf method which is considerably more complex than eq 23.24 In fact, the inclusion of fictitious adsorption sites in eq 22 which makes its solution so simple may turn out to be even an advantage before the methods similar to that of ref 24 which considers only real adsorption sites. The reason for that will be explained below (see discussion of Figure 8). Another argument in favor of eq 23 is that its right-hand side can be expanded in powers of iπ to obtain the equation1,23

where

) - E and K ) P0 exp(-S/kB)

(21)

and E and S are determined by eq 18. It can be obtained from eq 13 using eqs 16-19. Finally, it should be mentioned that the Langmuir equation can be derived not only on the basis of the statistical mechanics considerations but also on the basis of the thermodynamics or the kinetics of adsorption. The importance of the statistical mechanics derivation for the theory of adsorption on heterogeneous surfaces is that the former is based on the model of adsorption sites described in the Introduction (we call it the Langmuir model). It is this model that plays the most important role in the theory of adsorption on heterogeneous surfaces at small coverage. The basic equation of adsorption on heterogeneous surfaces in the Langmuir model is a convolution of the Langmuir equation with the distribution density of adsorption sites φ(ξ).1-4

θt(η) )

∫ dξφ(ξ)[1 + exp(ξ - η)]-1

(22)

It should be emphasized that we use here the terminology of the probability theory and that of ref 1. In refs 2-5 as well as in the majority of papers on the subject, the distribution density φ(ξ) in eq 22 is called a distribution function. In what follows, a distribution function corresponding to the distribution density φ(ξ) is the left-hand side of eq 27.

∞

φ(ξ) )

∑ n)0

( - 1)n

π2n (2n + 1)!

θt(2n+1)(ξ)

(24)

which is basically the same as that obtained by Jagiello and Rudzinski (see ref 3, p 471, and ref 4, p 22) by an independent method. Equation 24 is widely used in the theory of adsorption on heterogeneous surfaces, especially its approximate form known as the condensation approximation where only the first term on the right-hand side of eq 24 is used. To use eqs 22 and 23, one needs to extrapolate an experimental isotherm of adsorption which is always obtained in some limited range of chemical potentials to the infinite range of chemical potentials. One way of such an extrapolation is the virial adsorption isotherm eq 8. To convert the left-hand side of eq 8 into that of eq 22, one only needs to change the coefficient a0 in eq 8 (renormalize N) and take into account eq 16. The latter equation shows that lnC/ Cs in eq 8 is, in fact, η in eq 22 if P0 in eq 16 is the saturation vapor pressure. Thus eq 8 takes the form

θt(η) ) exp(a0 + a1η + a2η2 + a3η3 + ‚‚‚ + anηn) (25) This equation can be inserted in eq 23 and the argument of the exponents in the right-hand side of the resulting equation can be presented as A + iB and A - iB:


φ(ξ) ) π-1 exp(A) sin(B); A )

n

Bakaev et al. n

∑ akAk; B ) k)1 ∑ akBk k)0

(26)

n-2kg0

(ξ + iπ)n ) An + iBn; An )

∑

( - 1)kπ2k(n2k)ξn-2k;

k)0 n-2k-1g0

(n > 0); Bn )

∑ k)0

n ( - 1)kπ2k+1(2k+1 )ξn-2k-1

Experimental and extrapolated isotherms are shown in Figure 7. The deviation of experimental points eq 11 from the extrapolated isotherm that fitted only experimental points (extrapolation 2 in Figure 7) is 0.02%. Despite the highly accurate fit of the experimental isotherm by eq 25, its extrapolation beyond the experimentally determined domain of chemical potentials (η > -5.7) gives the plot which passes through a maximum at about η ) -3.5. At η > -3.5, the slope of isotherm in Figure 7 becomes negative which is an artifact (see below). To make the overall extrapolated isotherm more realistic, we added to the experimental points an additional point. That point corresponds to the last point in the domain of definition of the DR equation: θt(0) ) 1 (a completed monolayer at the saturated vapor pressure). This additional point was given the weight of 0.1 while the real experimental points were given the weights of 1. The extrapolated isotherm which used that additional point (extrapolation 1 in Figure 7) still fits the experimental points with a satisfactory accuracy of 0.65% and has qualitatively realistic behavior in the domain of definition of the DR equation (0 < P < Ps). The distribution densities φ(ξ) calculated by eq 26 from the extrapolated isotherms shown in Figure 7 are presented in Figure 8. Both of the graphs in Figure 8 have negative parts. The part of Figure 8 where φ(ξ) is negative has no physical meaning because a real distribution density is positive by definition. Comparison of Figures 7 and 8 shows that the negative values of φ(ξ) in Figure 8 arise at those values of ξ which are close to the values of η where the corresponding isotherm in Figure 7 has a negative slope. This is because the first derivative of the left-hand side of eq 22 θ′t(η) is always positive if φ(ξ) is positive. Therefore θ′t(η) can be negative only if φ(ξ) is negative at some values of ξ. Moreover, the values of ξ where φ(ξ)is negative should be close to the values of η where θ′t(η) is negative. This is due to the property of the solution of eq 22 which follows from the expansion of eq 24: The first, main term of this expansion (the so-called condensation approximation) shows that φ(ξ) = θ′t(ξ). The solid and the broken lines in Figure 8 are close to each other in the interval of experimental points in Figure 7 (ξ < -5.7) but deviate dramatically outside of that interval. Even in the interval -7 < ξ < -5.7, the two curves in Figure 8 deviate considerably. This is because the upper border of that interval is close to the negative part of the broken curve. The solid curve in Figure 8 is a more reliable representation of the true distribution density of adsorption sites than the broken curve. This fact also explains the reason why the solution of eq 23 which implicitly uses the fictitious adsorption sites can be more accurate for experimental isotherms than other exact solutions which do not use those sites, i.e., which consider only negative limits of integration in eq 22. Of course, the latter are more accurate for the theoretical isotherms of adsorption with θ′t(η) positive at all values of η in the infinite interval -∞ < η < ∞ (as, for example, eq 13). However, experimental isotherms of

Figure 7. Extrapolated isotherm of adsorption of pyridine on the silica surface at 100 °C. Dimensionless chemical potential is η ) µ/kBT eq 16. Extrapolation of experimental isotherm was performed by fitting it by eq 25 with n ) 3. Extrapolation 2, only experimental points were used for the fit. Extrapolation 1, an additional point (P ) Ps; θt ) 1) was added to the experimental points (see text).

Figure 8. Distribution density of adsorption sites corresponding to the extrapolated isotherms in Figure 7. Dimensionless free energy of a site is ξ in eq 17.

adsorption usually do not have such a broad domain of definition. When they are extended to the infinite interval of pressures 0 < P < ∞ which corresponds to -∞ < η < ∞, the negative slope of the isotherm such as that in Figure 7 can appear. A well-known example is the empirical DR equation which has the negative slope at P > Ps. From an experimental point of view, this is not a serious shortcoming of the DR equation because that oversaturated domain of pressures is either experimentally unreachable or uninteresting for the majority of applications. However the exact solution of eq 22 without fictitious adsorption sites, i.e., with more realistic limits of integration -∞ < ξ < 0 would give for the DR isotherm negative values of φ(ξ) for real adsorption sites. In contrast to this, eq 23 shifts those unrealistic, negative values of φ(ξ) into the domain of fictitious adsorption sites. The distribution density of sites φ(ξ) can be numerically integrated to obtain the free energy distribution of sites


F(G;T) )


∫-∞G/k T dξ φ(ξ) B

(27)

Here again we run into the problem of extrapolation since the values of φ(ξ) in the region below ξ < -9.34 are not available in Figure 8. Extrapolation of ξ into this region can be carried out by the same method as extrapolation of an isotherm to zero pressure described in subsection 2.1. This extrapolation is more reliable than the extrapolation of the isotherm shown in Figure 7. The reason for that is that the F-equation and the DR-equation mentioned in subsection 2.1 which constitute the first terms in the exponent of eq 25 can fit very well isotherms of adsorption at low coverage but totally fail in the limit of very high coverage: The F-isotherm tends to infinity as the concentration in eq 8 raises and the DR-isotherm has a negative slope when reduced chemical potential eq 16 exceeds that of liquid. The entropy distributions of adsorption sites can be obtained from their free energy distribution by eq 33 from ref 15

〈S〉G ) -

(∂G ∂T )

(28)

F

and the energy distribution then follows from eq 19

∂ G 〈E〉G ) G + T〈S〉G ) - T ∂T T F 2

()

(29)

This method of obtaining the energy distribution resembles that for obtaining isosteric heats of adsorption from isotherms measured at different temperatures (as mentioned in ref 14). A free energy distribution of sites is just an isotherm of adsorption in the condensation approximation F(ξ) = θt(ξ) (insert the n ) 0 term of eq 24 into eq 27). Besides, to obtain the free energy distribution from an isotherm one has to substitute η in θt(η) for ξ. Making the inverse substitution in the right-hand side of eq 29 and using eqs 16 and 17, one can convert the right-hand side of eq 29 into that of eq 10 and -〈E〉G into qst. Thus, the plot of coverage vs (-qst) gives the energy distribution of adsorption sites in the condensation approximation. The D ) 0 plot in Figure 9 was obtained from the D ) 0 isotherm in Figure 4. That isotherm was obtained from an experimental EP at 100 °C (similar to that in Figure 1) in the ECP approximation by eq 5 and integration described in section 2.1. Another isotherm (not shown) was obtained by exactly the same method from another EP at 110 °C. Two free energy distributions of sites were obtained from these two isotherms by the method described above. (The distribution density corresponding to the first isotherm is shown as the solid line in Figure 8.) Then the D ) 0 energy distribution in Figure 9 was obtained from those two free energy distributions by eq 29. The D ) 0.4 plot in Figure 9 was obtained by the same method. The only difference was that now the derivatives of the isotherms were obtained from the same two EPs not by eq 5 but by eq 4 with D ) 0.4 cm2/s and τkin ) 0.08 min-1. (Their dimensionless counterparts D h and τjkin were calculated by eq 12.). Finally, the qst plot in Figure 9 was obtained from the same two isotherms using eq 10 and ascribing to qst the negative sign (see discussion of eq 29). The energy distribution obtained from the isosteric heat of adsorption (qst plot in Figure 9) corresponds to a less heterogeneous surface than those obtained from solutions of eq 22. The energy distribution of sites for a homogeneous surface is a step function (the distribution density for such a surface is the Dirac delta-function). The qst plot in Figure 9 is closer to a step function than D ) 0 or D ) 0.4 plots. As shown above, the

Figure 9. Energy site distributions for pyridine on the silica surface. Triangles (qst), the isosteric heat plot D ) 0.4 in Figure 5 (turned by 90o counterclockwise); D ) 0, obtained from the ECP isotherm; D ) 0.4, obtained from the isotherm corrected for D ) 0.4 cm2/s and τkin ) 0.008 min-1 (cf. Table 1).

coverage vs negative isosteric heat curve is the condensation approximation for the energy distribution of sites. One can see from eq 22 and the definition of ξ and η in eq 14 that the condensation approximation becomes an accurate solution of eq 22 in the limit of T ) 0. Thus, one can obtain the true energy distribution by extrapolating the isosteric heat to the absolute zero of temperature (see the Introduction). At a very low temperature, adsorbed molecules occupy only the adsorption sites of lowest energy. When temperature increases, the energy fluctuations distribute adsorbed molecules over sites of higher energy. At a very high temperature, the adsorbed molecules are almost homogeneously distributed over adsorption sites as is the case on a homogeneous surface at much lower temperatures. Thus, when one obtains the energy distribution of sites directly from an isosteric heat plot at a relatively high temperature, that energy distribution corresponds to a more homogeneous surface than the surface is in reality. At a relatively high temperature, a more accurate distributions of adsorption sites can be obtained from adsorption isotherms as described above than directly from an isosteric heat plot. These are D ) 0 and D ) 0.4 plots in Figure 9. They differ from one another not by the way the energy distribution was obtained from an isotherm but by the way the isotherm was obtained from IGC. Those distributions reveal much stronger adsorption sites for pyridine on the silica surface than the sites shown by the qst plot in Figure 9. Before discussing those energy distributions we consider in detail what they really mean. As mentioned in the Introduction, an adsorption site in the Langmuir model is a potential well which can accommodate only one adsorbed molecule. A reasonable and temperature-independent characteristic of these sites on a heterogeneous surface is their distribution with respect to the minimal potential energy of the well, i.e., min in eq 14. However, as seen from eq 13, adsorption on such a site is determined not by min but by ξ. The latter is the dimensionless free energy of a molecule adsorbed on the site, its dimensional counterpart being G in eqs 17 and 19. Thus, one can determine from one isotherm only the free energy distribution of adsorption sites but not their energy distribution. The distribution density corresponding to the free energy distribution (usually called the free energy distribution in the adsorption literature) gives the number of adsorption sites whose free energy is very close to


Bakaev et al.

G. There are many sites with different values of E and S which have about the same value of G. These are the sites which are located along the straight line G ) E - TS (cf. Figure 1 in ref 15) on the E-S plane. One can calculate the average value of energy 〈E〉G among the sites having about the same value of G which is the left-hand side of eq 29. The value of 〈E〉G certainly depends on G. Since F(G) eq 27 is a monotonously increasing function, one can invert it and find the functions F vs 〈E〉G. The latter function is the function which we call here the energy distribution of adsorption sites. Its meaning is considered in more detail for a particular case of a bi-dimensional normal distribution of sites on the E-S plane.13 If all of the sites have the same values of S ) S0, then integration along the strip neighboring the straight line G ) E - TS on the E-S plane reduces to integration along the intersection of the former and the straight line S ) S0. In this case 〈E〉G ) G - TS0, and one can obtain the energy distribution of sites from their free energy distribution F(G) by subtraction of TS0 from G. Approximation of constant S is widely used in the theory of adsorption on heterogeneous surfaces. Usually it takes the form of constant K in eq 20. The methods of determining the value of K (which are in essence equivalent to finding S0) are reviewed in ref 18. Another approximation which was widely discussed in the literature but not that frequently used is the assumption that S strictly depends on E: S ) S(E).1,10 In this case, 〈E〉G again corresponds to some energy that can be found as a root of the equation: G ) E - TS(E). If the dependence of S on E is linear, the root is unique and one can again obtain the energy distribution from the free energy distribution simply by changing the G value for the root of the above equation. Finally, when S and E are independent variables, one cannot expect any direct connection between 〈E〉G and some specific energy, rather 〈E〉G is some average energy. In particular, when temperature is very high (T f ∞), the straight line G ) E - TS coincides with the E axis (because its slope tends to infinity and its interception with the S axis tends to zero (cf. Figure 1 in ref 15)). In this case, 〈E〉G is just the mean energy of adsorbed molecules over all adsorption sites, and from the adsorption point of view, the surface may be considered as homogeneous. The real situation is somewhat intermediate between those described in the two previous paragraphs. The values of S and E are not totally independent and not strictly dependent on one another, but there is a correlation between them. This correlation was reproduced in computer simulation for argon on a model amorphous oxide and discussed for other cases.25 In particular if one assumes, a bi-dimensional normal distribution of adsorption sites, one obtains the dependence of 〈S〉G and 〈E〉G on the parameters of that distribution as well as the linear dependence between 〈S〉G and 〈E〉G.13 The experimental dependence between 〈S〉G and 〈E〉G obtained from isotherms of adsorption of pyridine and benzene on the silica surface using eqs 28 and 29 is presented in Figure 11. It is really linear. The straight lines in Figure 11 obey the equation

S ) S0 +

E Ti

(30)

where S0 ) 0.1379 kJ/(mol K) and Ti ) 393.8 K for pyridine and S0 ) 0.1217 kJ/(mol K) and Ti ) 343.3 K for benzene. Equation 30 was used in our previous publications to convert the free energy distribution of sites into the energy distribution.11,13 Finally, one can see from Figures 9 and 10 that the interaction of benzene with the silica surface is in general weaker than that

Figure 10. Energy site distributions for benzene on the silica surface. Triangles (qst), the isosteric heat plot D ) 0.4 in Figure 6 (turned by 90o); D ) 0, obtained from the ECP isotherm; D ) 0.4, obtained from the isotherm corrected for D ) 0.4 cm2/s and τkin ) 0.015 min-1 (cf. Table 1).

Figure 11. Entropy/energy relationships. Diamonds, pyridine; triangles, benzene; solid lines, least-square fits of corresponding points.

of pyridine. This is immediately obvious from the comparison of their EPs in Figure 1. This should be expected from the higher polarity of a pyridine molecule in comparison to that of benzene. Besides, pyridine is a stronger organic base than benzene. The energy distribution of pyridine in Figure 9 reveals about 1% of very strong adsorption sites on the silica surface. The magnitude of the energy of those sites (250 kJ/mol) is 6.7 times larger than the heat of liquefaction of pyridine at 100 °C. Since the energy of physical adsorption is generally close to the heat of liquefaction and the heats of liquefaction of pyridine and benzene are close to one another, the strongest sites on Figure 9 might indicate some specific interaction of a pyridine molecule with the fumed silica surface. The fumed silica is an oxide consisting of nanoparticles (size of a particle is about 10 nm). Its bulk atomic structure is amorphous. It has been shown by computer simulation of argon on a generic model of an amorphous oxide that such a model amorphous surface has an energy distribution of adsorption sites which resembles that experimentally observed by Drain and Morrison on TiO2 (rutile).25 However, the bulk atomic structure of rutile is not amorphous but crystalline. In fact, the adsorbent studied by Drain and Morrison was a powder consisting of needlelike rutile nanoparticles 40 nm long and 10 nm wide with

On Inverse Adsorption Chromatography. 2. a specific surface area of 85 m2/g.26 The adsorption isotherms of argon at 85 K on crystalline faces of rutile obtained by computer simulations deviate drastically from the experimental isotherm,8 whereas those simulated on the generic amorphous model of the amorphous surface mentioned above fit the experimental isotherm very well at the reasonable choice of the interatomic potential parameters.27 This suggested a hypothesis that the surfaces of highly dispersed solids consisting of crystalline particles of nanoscale sizes may be, in fact, amorphous in an atomically thin layer. This hypothesis is discussed in subsection 2.1 of ref 25. One of the arguments mentioned there is Figure 4 in ref 28 (this reference is misprinted in ref 25) where the isosteric heats of adsorption for nitrogen on diamond, amorphous carbon, and P-33 (graphitized carbon black) are shown. The first and the last samples are crystalline, but the middle one is amorphous. On one hand, isosteric heats of nitrogen on crystalline and amorphous carbon surfaces are qualitatively different which testifies to the high sensitivity of this adsorption characteristic to the atomic structure of the surface layer. On the other hand, isosteric heats of nitrogen on the diamond and amorphous carbon surfaces almost coincide which suggests that the surface of diamond dust is, in fact, amorphous. Of course, the hypothesis that the surfaces of very small crystallites (nanoparticles) are typically amorphous still needs additional substantiation. The IGC method considered above may prove to be very useful in this respect because it gives the energy distribution of the strongest adsorption sites where defects of the crystalline structure play the major role. This method can be a valuable technique for nanoparticle characterization. 4. Conclusion 1. The method of inverse adsorption chromatography allows one to obtain an important characteristic of a surface: the derivative of the adsorption isotherm with respect to the chemical potential (concentration, partial pressure) of the adsorbate. This measured characteristic can be integrated to obtain an isotherm of adsorption and isosteric heats of adsorption. These in turn allow one to obtain the energy distribution of adsorption sites on the surface with respect to a chosen probe molecule. However, to calculate these adsorption characteristics of a surface from an isotherm derivative, one has to know a constant of integration. In other words, one has to know the value of adsorption at some initial concentration in addition to the isotherm derivative. This fact did not receive special attention in the literature because in many cases the constant of integration can be set to zero or easily estimated by the method of thermodesorption (see section 2.1). However, this is not always the case. It is shown in this paper that in the case of adsorption of pyridine on the silica surface the estimate of the integration constant by the method of thermo-desorption gives qualitatively wrong isosteric heats of adsorption. Since isosteric heats of adsorption are closely related to the energy distribution of adsorption sites, the latter is also qualitatively wrong. This explains why the energy and entropy distributions of adsorption sites for butanol on the silicate glass surfaces obtained in our previous publications by IGC method displayed qualitatively wrong behavior for the strongest adsorption sites.11,13 The reason was the method of thermodesorption which we used for obtaining the integration constant. It is shown in this paper that one can obtain reasonable isosteric heats and the energy distributions of adsorption sites for pyridine on the silica surface if one evaluates the integration constant by extrapolation of the adsorption isotherm (and its derivative) to very low coverage

J. Phys. Chem. C, Vol. 111, No. 20, 2007 7485 (see section 2.1). The method of extrapolation improves the adsorption properties obtained from IGC even for benzene on the silica surface despite the fact that benzene interacts with that surface much weaker than pyridine, the integration constant for benzene is very small, and the improvement occurs only in the region of very small coverage. 2. The currently used method of obtaining the adsorption isotherm derivative from inverse adsorption chromatography, in particular from IGC, is the so-called ECP method. This method totally ignores the role of kinetics in adsorption and the longitudinal diffusion in the chromatographic process. These factors are approximately taken into account in the modified ECP method described in the accompanying (theoretical) paper I (see ref 29). To use the modified ECP method one has to know the numerical value of the longitudinal diffusion coefficient D and the kinetics of adsorption coefficient τkin. It is shown in this paper that the former can be obtained from the elution profile (peak) of methane (section 2), and its numerical value is not critical for adsorption characteristics and the latter can be chosen in such a way that a calculated isotherm do not depend on the flow rate of the carrier gas (see discussion of Figure 4). Taking account of these factors makes considerable difference for the isosteric heats of adsorption and the energy distribution of adsorption sites. 3. A peculiarity of inverse adsorption chromatography (IGC in particular) is that it is well adapted for determining adsorption characteristics in the region of very small coverage. It is a convenient method of determining the Henry constants for a homogeneous surface or determining the energy distribution of the strongest adsorption sites on a heterogeneous surface. An extrapolation of an adsorption isotherm to the region of small (cf. first paragraph of this section) and large (cf. section 3.2) coverage plays an important role for this method. Such an extrapolation can be successful if one can describe an isotherm using an empirical equation. This is the case for adsorption on a wide class of heterogeneous surfaces such as amorphous oxide surfaces (e.g., the silica surface) as well as for nonporous surfaces of many relatively high surface area oxides (e.g., composed of nano scale particles). There are many other surfaces, for example microporous active carbons, for which the applicability of the method presented in this paper would require additional validation. 4. The energy distribution of adsorption sites for pyridine on the silica surface reveals a small number (about 1%) of sites whose adsorption energy is much higher than the conventional energy of physical adsorption. This raises the obvious question about their identity and local chemical structure. Acknowledgment. The authors gratefully acknowledge the Center for Glass Research for their partial support of this work. References and Notes (1) Honig, J. M. Ann. N.Y. Acad. Sci. 1954, 58 (6), 741. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (3) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (4) Cerofolini, G. F.; Rudzinski, W. in Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces (Studies in surface science and catalysis Vol. 104); Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997. (5) Papirer, E; Ballard, H. in Adsorption on new and modified inorganic sorbents(Studies in surface science and catalysis Vol. 99); Dabrowski, A., Tertykh, V. A., Eds.; Elsevier: Amsterdam, 1996. (6) Bakaev, V. A.; Chelnokova, O. V. Surf. Sci. 1989, 215, 521. (7) Heer, C. V. J. Chem. Phys. 1971, 55, 4066. (8) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 1372. (9) Drain, L. E.; Morrison, J. A. Trans. Faraday Soc. 1952, 48, 316.

7486 J. Phys. Chem. C, Vol. 111, No. 20, 2007 (10) Drain, L. E.; Morrison, J. A. Trans. Faraday Soc. 1953, 49, 654. (11) Bakaeva, T. I.; Pantano, C. G.; Loope, C. E.; Bakaev, V. A. J. Phys. Chem. B. 2000, 104, 8518. (12) Bakaev, V. A.; Bakaeva, T. I.; Pantano, C. G. J. Chromatogr. A 2002, 969, 153. (13) Bakaev, V. A.; Bakaeva, T. I.; Pantano, C. G. J. Phys. Chem. B 2002, 106, 12231. (14) Bakaeva, T. I.; Bakaev, V. A.; Pantano, C. G. Langmuir 2000, 16, 5712. (15) Bakaev, V. A. Surf. Sci. 2004, 564, 108. (16) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics (translated from the Russian); Butterworth Heinemann: Oxford, U.K., 1987. (17) Guiochon, G.; Guillemin, C. L. QuantitatiVe gas chromatography: for laboratory analyses and on-line process control; Elsevier: Amsterdam, 1988. (18) Jaroniec, M. Surf. Sci. 1975, 50, 553.

Bakaev et al. (19) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in FORTRAN. The Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1992. (20) Kiselev, A. V.; Yashin, Ya. I. Gas-Adsorption Chromatography (translated from the Russian); Plenum Press: New York, 1969. (21) Isirikyan, A. A.; Kiselev, A. V. Zhur. Fiz. Khim. 1958, 32, 679. (22) Yaws, C. L. Handbook of Vapor Pressure; Gulf Publ. Comp.: Houston, 1994. (23) Temkin, M.; Levich, V. Zhur. Fiz. Khim. 1946, 20, 1441. (24) Landman, U.; Montroll, E. W. J. Chem. Phys. 1976, 64, 1762. (25) Bakaev, V. A. Surf. Sci. 1988, 198, 571. (26) Morrison, J. A.; Los, J. M.; Drain, L. E. Trans. Faraday Soc. 1951, 47, 1023. (27) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 1379. (28) Graham, D. J. Phys. Chem. 1960, 64, 1089. (29) Bakaev, V. A. J. Phys. Chem. C. 2007, 111, 7463-7472.

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