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Adsorption of CO2 on Glass Fibers T. I. Bakaeva,† V. A. Bakaev,*,‡ and C. G. Pantano† Department of Materials Science and Engineering and Department of Chemistry, 152 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802 Received October 19, 1999. In Final Form: April 3, 2000 Isotherms of adsorption of CO2 on two different glass fibers are measured at 158 and 185 K by the volumetric method. Heats of adsorption are determined. Experimental data are analyzed in terms of the Langmuir model of independent adsorption sites. Distributions of adsorption sites in free energy (not in energy) are determined at two temperatures, and their temperature dependence is discussed. This dependence makes it possible to obtain the entropy and energy distribution of adsorption sites and the relation of entropy to energy for a site. The latter enables one to assess the reliability of the distributions.
1. Introduction This paper may be considered an extension of our previous work1,2 where a new method of studying surface atomic structures was put forth. The method is based on (i) computer simulation of a model surface, (ii) computer simulation of adsorption on the model surface, and (iii) comparison of experimental and simulated adsorption characteristics to validate the model. It is intended to provide new information on the atomic structures of surfaces which are difficult to study by conventional methods of the surface structure analysis. This refers especially to amorphous oxide surfaces such as silica and silicate glasses. In principle, the atomic structure of a silicate glass surface can be obtained from a computer simulation using model interaction potentials for the atomic species present. However, at present, such a simulation needs to be validated by some independent method especially for the use of multicomponent silicates. As explained in refs 1 and 2, experimental data that can be used to validate the simulations of glass surfaces are scarce. Thus, we have considered comparison of simulated and experimental characteristics of physical adsorption of gases as a test of the validity of the surface atomic structure. This is because the adsorption characteristics of some molecules are very sensitive to the details of the surface atomic structure. This method was applied to the surface of vitreous silica in ref 2 where the binding energies of water molecules on this surface were studied. It was shown that the structure of amorphous silica accepted in the literature (with a large concentration of dangling bonds due to nonbridging oxygen atoms) should contain strong hydrophilic sites. This does not concur well with the fact that the surface of pure silica is often hydrophobic. Subsequently, a computer simulation of the vitreous silica surface was performed with special attention to the simulation of annealing.3 This confirmed the hydrophobic character of pure annealed silica by showing that the surface is composed only of bridging oxygen atoms. It was also shown that the adsorption characteristics (binding † ‡
Department of Materials Science and Engineering. Department of Chemistry.
(1) Bakaev, V. A.; Steele, W. A.; Bakaeva, T. I.; Pantano, C. G. J. Chem. Phys. 1999, 111, 9813. (2) Bakaev, V. A.; Steele, W. A. J. Chem. Phys. 1999, 111, 9803. (3) Bakaev, V. A. Phys. Rev. B 1999, 60, 10723.
energies) of water are exceedingly sensitive to the peculiarities of the surface atomic structure.2 In further work, we chose the carbon dioxide molecule to probe the surface atomic structure of vitreous silica.1 The reason was that the experimental study of carbon dioxide adsorption on glass fibers proved to be easier than that of water. Fortunately, there is not much difference between the simulation of water and carbon dioxide adsorption on model glass surfaces, although carbon dioxide is probably not as sensitive as water in probing the surface structure. In the present study, we report further on the experimental adsorption of carbon dioxide, but in this case we increased the accuracy and widened the range of temperatures. The most accurate isotherms were obtained with an ethyl ether freezing temperature bath (156.8 K). The reason for that is explained in section 2. We analyze our results in terms of the Langmuir model of independent adsorption sites. This is the most popular model of adsorption on heterogeneous surfaces.4-6 The peculiarity of our analysis is that we consider the distribution of adsorption sites in free energy and not the more typical energy.4-6 As a result, our distribution depends on temperature. Two distributions of adsorption sites in free energy obtained from two isotherms at different temperatures make it possible to determine the dependence of average entropy (defined in section 3) on average energy. This dependence has some universal meaning (see section 4) and allows one to check the reliability of the site distribution. The distribution of adsorption sites in energy (minimum of the adsorption potential at a site) should not depend on temperature; however, one usually obtains a slight temperature dependence for this distribution. This problem was discussed in the literature (see ref 4, p 63, and ref 5, p 107). It was connected to the dependence of the entropy of a molecule adsorbed on a site to its energy (see ref 5, p 119) or more specifically the dependence of the vibration frequency of an adsorbed molecule on its energy (4) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (5) Rudzinski W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (6) Papirer, E.; Balard, H. In Adsorption on New and Modified Inorganic Sorbents; Dabrowski, A., Tertykh, V. A., Eds.; Studies in Surface Science and Catalysis, Vol. 99; Elsevier: Amsterdam, 1996.
10.1021/la991376x CCC: $19.00 © 2000 American Chemical Society Published on Web 06/03/2000
Adsorption of CO2 on Glass Fibers
Langmuir, Vol. 16, No. 13, 2000 5713
(see ref 5, pp 108 and 326). However, it has been shown by computer simulation that on a model heterogeneous surface there is no functional dependence between entropies and energies of (argon) atoms on adsorption sites of a model oxide. This is because adsorption sites are characterized by a two-dimensional distribution in adsorption energies and entropies, but these are not independent and there is a correlation between the entropy and the energy of an adsorbed molecule (instead of strict functional dependence).7 In the frame of the Langmuir model of adsorption sites, the distribution of adsorption sites in free energy is a more natural characteristic than the distribution of adsorption sites in energy. This is because in this model the occupation of a site strictly depends on its free energy. (In what follows, it will be convenient to understand by energy, entropy, etc. of an adsorption site the corresponding characteristic of a molecule adsorbed on the site.) To obtain a distribution of adsorption sites in energy, one has either to assume that entropies of all sites are equal6 or to assume a dependence between the entropy and energy of a site as mentioned above. The problem of conversion from free energy to energy is also solved by assuming that the partition functions of a molecule are equal in adsorption and gas states (see, e.g., ref 5, pp 97 and 421) or by using Hobson’s approximate relation ( eq 116). In this paper, we do not make such assumptions. Instead we assume that there is a two-dimensional distribution of adsorption sites in energy and entropy and then reduce it to the distribution of adsorption sites in free energy (see section 3). 2. Experimental Section Adsorption isotherms were measured volumetrically using a fully automated ASAP 2000 (Micromeritics) which usually employs a liquid nitrogen bath to maintain the sample temperature. To measure CO2 isotherms, we used two additional liquid/ solid thermostat baths: (i) a mixture of liquid and solid 2-propanol and (ii) a mixture of liquid and solid ethyl ether. The temperature of the bath was measured by a platinum thermometer. The overall accuracy of the temperature measurement was 0.05 K, and the sensitivity to the temperature changes was better than 0.01 K. The temperatures (T) and saturated vapor pressures of solid CO2 (P0) for these baths were, respectively, (i) T ) 185.1 K, P0 ) 329.8 Torr and (ii) T ) 156.8 K, P0 ) 15.5 Torr. The basic equation for the volumetric measurement of adsorption is
∆N )
(PV + P1Vd) P2(V + Vd) RT RT
V + Vd RT
(2)
where δP is a random experimental error of pressure measurement (experimental errors of temperature and volume measurements are neglected). In our version of the ASAP 2000, pressure is measured by two Baratron gauges: one for the interval 0-10 Torr and another for the interval 10-1000 Torr. The error of the first Baratron may be estimated as 0.01 Torr. Then at T ) 156.8 K where measurements of pressure are performed with this, more sensitive Baratron, V + Vd ≈ 100 cm3, and δ∆N ≈ 0.2 µmol. (7) Bakaev, V. A. Surf. Sci. 1988, 198, 571.
Si
Al
Ca + Mg
Ba
A B
21.6 24.5
1.3 1.5
5.2 3.8
0.4
Zn
Na + K
F
O
13.3 9.8
1.1
0.2
57.3 60.2
This estimate defines the range for accurate measurements of adsorption on glass fibers with this equipment. Reliable data were obtained for two samples of glass fiber which we designate here A and B. Their compositions are presented in Table 1. The masses of the A and B samples were 2.3 and 1.5 g, respectively. Their specific surfaces as measured by Kr (BET method; molecular cross-section 0.21 nm2) were 2.7 m2/g for A and 2.8 m2/g for B. Thus, the total surface areas were 4.6 m2 for A and 4.2 m2 for B. The effective diameter of these glass fibers can be estimated (on the basis of their specific surfaces and density, 2.2 g/cm3) as 0.7 µm. Electron microscopy showed that in reality the samples consisted of fibers with various lengths and diameters from 0.2 to 2 µm. Figure 2 shows that the increments of adsorption for the isotherms were about 1 µmol/m2, which is about 4 µmol/sample. Thus, the above evaluation of the incremental error gives an accuracy of 5%. This is a conservative estimate, but it shows that under these same conditions, one cannot reliably measure isotherms of adsorption of glass fibers with diameters on the order of 10 µm or greater due to the low values of their surface areas. It also shows that we also cannot reliably measure isotherms on samples A and B at higher temperatures where pressure exceeds 10 Torr. Such pressures must be measured by the second Baratron with the larger value of δP (which increases δN in eq 2).
3. Model of Adsorption Sites In this paper, experimental results are analyzed in terms of the model of independent adsorption sites.4-6 This model describes an experimental adsorption a(µ,T) at a given chemical potential µ and temperature T as an average over independent adsorption sites:
a(µ,T) )
∫µ′µ′
max
min
θ(µ,µ′,T) n(µ′) dµ′; n(µ′) g 0
θ(µ,µ′,T) ) 1/(1 + exp[(µ′ - µ)/kT])
(3) (4)
Chemical potential is taken as that of the perfect gas of linear molecules (CO2 in this case). It depends on the gas pressure P and molecular parameters8
µ ) kT ln
(kTP Λ Λ ); 3
2
I
Λ)
h ; (2πmkT)1/2 ΛI )
(1)
where V is a calibrated volume and Vd is the so-called dead volume, P1 and P2 are two neighboring points on the isotherm (R is the gas constant), ∆N is the adsorption increment (per sample) corresponding to these pressures, and P is the gas pressure in the calibrated volume. The error in ∆N can be evaluated by the following relation:
δ∆N ) 2δP
Table 1. Bulk Composition of Glasses (atom %) sample
h (5) 2π(IkT)1/2
Here h is the Planck constant and m and I are the mass and moment of inertia of a molecule. θ(µ,µ′,T) in eqs 3 and 4 is due to Langmuir and represents coverage of one adsorption site. The states of these sites (occupied vs empty) are assumed to be independent of each other, and each site can adsorb only one molecule; these are standard assumptions of the Langmuir model.4,5 In eq 4, µ′ is the free energy of a molecule on an adsorption site; that is
µ′ ) -kT ln
∑i exp(-i/kT)
(6)
where {i} is the energy spectrum of an adsorbed molecule.8 The object of this analysis is to obtain the distribution (density) of adsorption sites n(µ′). It is a solution to the integral equation (eq 3). Finding such a solution is known (8) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Addison-Wesley Series in Advanced Physics; Addison-Wesley: Reading, MA, 1958.
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Bakaeva et al.
as a so-called inverse problem.9 To solve this problem, we discretize the µ-axis
µ′j ) µ′min + h(j - 1); j ) 1, ..., M
(7)
µi ) µmin + h(i - i1); i ) i1, i1 + 1, ..., iN
(8)
(µi and µ′j are the same mesh points which differ only by their indexes) and replace the integral in eq 3 by a quadrature M
ai )
Aijnj; ∑ j)1
i ) i1, ..., iN
(9)
where ai ≡ a(µi,T), nj ≡ n(µ′j), and Aij ≡ θ(µi,µ′j,T) ∆µ′j. An extended trapezoidal rule with constant step h9 was used in the present case so that ∆µ′j ) h if j * 1, M and ∆µ′1 ) ∆µ′M ) h/2. Here
Figure 1. Calculation of the adsorption site density n(µ′) ([µmol/ m2]/[kJ/mol]) in free energy for sample B at 158.6 K: dotted curve, condensation approximation; broken curve, regularization with λ′ ) 0.034; solid curve, regularization with λ′ ) 0.11.
(AT‚A + λBT‚B)‚n ) AT‚a
(15)
where µmin and µmax are minimal and maximal experimental values of µ in eq 3 and N is to be chosen. In eq 8, µi is determined in the interval
where A is the matrix {Aij} in eq 9, the subscript T indicates a transpose matrix, a and n are vectors {ai} and {nj} in eq 9, and B is the matrix of a regularizing operator:9 Bij ) -1 if i ) j or j ) i + 2; Bij ) 2 if j ) i + 1; else Bij ) 0 i ) 1, ..., M - 2, j ) 1, ..., M. This choice of B stabilizes the second derivative of the distribution n(µ′).9 The parameter λ in eq 15 may be scaled by the rule9
i1 e i e iN; iN ) i1 + N - 1
λ ) λ′ Tr(AT‚A)/Tr(BT‚B)
h ) (µmax - µmin)/(N - 1)
(10)
and µmin ) µ′i at i ) i1 so that
i1 ) 1 + (µmin - µ′min)/h
(11)
Now all the parameters in eqs 7-9 are determined except N and M. Consider a(µmax,T). Since a is a monotonic function of µ (as seen from eq 3), this is the maximal value of a. To obtain it from eq 3, one has to extend the integration over µ′ beyond µmax. But as the difference µ′ - µmax increases, θ(µ′,µmax,T) decreases as seen from eq 4. We fix upper limit of integration in eq 3 at the level
µ′max ) µmax + h INT(RkT/h + 0.5)
(12)
which gives µ′max - µmax ≈ RkT and θ(µ′max,µmax,T) ) 1/(1 + exp(R)). (Here R is just a constant that fixess together with T and hsthe limits of integration.) The lower limit of integration in eq 3 can be fixed at the level
µ′min ) µmin - h INT(RkT/h + 0.5)
(13)
which gives θ(µ′min,µmin,T) ) 1/(1 + exp(-R)). (In eqs 12 and 13, INT(x + 0.5) is the integer closest to x. This function makes i1 and M in eqs 11 and 14 integers.) Thus, θ in the integrand of eq 3 is big at the lower limit of integration, and one has to assume (and check post factum) that n(µ′min) is small. From eqs 7, 10, 12, and 13
M ) N + 2 INT(RkT/h + 0.5)
(14)
The problem now is that the number of unknowns M in eq 9 is larger than the number of linear algebraic equations. This reflects the well-known fact that the inverse problem is ill-posed.9 To obtain a unique solution, one solves a system of normal equations9 (9) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in FORTRAN; Cambridge University Press: Cambridge, U.K., 1992.
(16)
One chooses 0 e λ′ e 1. If λ ) 0, the matrix on the lefthand side of eq 15 is singular. If λ′ ) 1, one can obtain a unique solution to eq 15, but it can be too distorted.9 In fact, λ in eq 15 introduces some a priori information on the shape of n(µ′). Thus, it would be good to have some preliminary idea concerning the shape of this distribution for a reasonable choice of λ′ in eq 16. This information can be obtained from the evaluation of n(µ′) by the so-called condensation approximation.4,5 This approximation substitutes θ(µ,µ′,T) by its limit at T f 0. In this limit θ(x) is just a step function and
n(µ) ) ∂a(µ,T)/∂µ
(17)
An example of the above analysis is presented in Figure 1. As seen by the dotted curve, the experimental isotherm is measured between the limits: µmin ) -59.4 kJ/mol; µmax ) -53 kJ/mol. In eqs 12-14, R ) 2.2 and N ) 50. At the lower value of λ′ the broken curve has negative values, which contradict its definition (cf. eq 3). Thus, the value of λ′ was increased until n(µ′) was everywhere positive. This is the solid curve in Figure 1. The latter is a formally valid solution in the sense that its substitution into eq 3 as n(µ′) restores the experimental points a(µ,T) with accuracy less than a fraction of a percent. This does not mean, however, that the solid curve in Figure 1 is the real distribution we are looking for. The problem with it is the following. With our choice of R in eqs 12 and 13 we have in eq 3 θ(µmax,µ′max,T) ) 0.1 and θ(µmin,µ′min,T) ) 0.9. Thus, the ordinates of the solid curve with abscissas less than µmin ) -59.4 kJ/mol in Figure 1 should be small or at least decrease in the direction of smaller free energies. This is because they are an unreliable extrapolation of experimental data and are not suppressed by decreasing values of θ(µ,µ′,T) in contrast to the right end of the curve. The problem is that we were not able to measure our isotherms at lower values of pressures due to the limited range of our pressure gauge. It will be shown in the next section that we can partially obviate the problem by least-squares fitting the experimental isotherms and extrapolating them to lower pressures.
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Langmuir, Vol. 16, No. 13, 2000 5715
The above method of numerical solution of eq 3 is not principly new. It is very close to that described in ref 10. One of the motivations for publication of ref 10 was the scarcity of subroutines for regularization methods at that time (see at the end of their section Regularization Method). So they developed the program INTEG for solution of adsorption integral equations similar to eq 3. However, ref 10 does not contain a detailed description of INTEG (however, INTEG can be placed at the disposal of interested persons10). Since that time the situation changed. Now one has easily accessible standard subroutines for solution of problems similar to that described above.9 Thus, eqs 7-16 are, in fact, just detailed instructions on how to apply the methods of ref 9 to our particular problem. There are other methods of solution to that problem. They are described in refs 4, 5, and 10 (see also ref 11 and references therein). Now we introduce some new elements in the analysis. First, as seen from eqs 3 or 17, the distribution of adsorption sites in free energy µ′ depends on temperature. An adsorption site may be thought of as a minimum of the adsorption potential at the surface. The minimal energy of a molecule adsorbed on this site does not depend on temperature, which is also true for other mechanical properties such as vibrational frequencies, etc. Thus, the distribution of adsorption sites in these mechanical properties is multidimensional (two sites may differ not only in their minimal energies but also in the second derivatives of their energies, which determine frequencies of vibrations of adsorbed molecules, etc.) and does not depend on temperature. However, according to eq 4, the adsorption properties of a site depend only on its free energy µ′ (not to be confused with chemical potantial µ), which is a function of mechanical properties mentioned above (as represented by the energy spectrum {i} in eq 6) and temperature. Since µ′ is a free energy, its dependence on temperature is explicitly expressed as8
µ′ ) ′ - Ts′
(18)
where ′ and s′ are the average energy and entropy of a molecule on a site. The latter is sometimes called thermal entropy. The dependence of ′ and s′ on temperature is much weaker than that of µ′ so that we can consider them as temperature independent in the narrow temperature interval considered below (see the discussion of Figure 8). Consider now the two-dimensional distribution of adsorption sites in energy and thermal entropy n2(s′,′). As far as we assume s′ and ′ to be independent of temperature, so is n2(s′,′). This function gives a more detailed description of surface heterogeneity than n(µ′). The latter (now its dependence on T is explicitly shown) can be obtained from n2(s′,′):
n(µ′,T) )
∫ds′∫d′ n2(s′,′) δ(µ′ + Ts′ - ′)
(19)
Here δ(x) is the Dirac δ-function that selects the sites on the line of eq 18. The cumulative distribution corresponding to n(µ′,T) is
N(µ′,T) ) )
∫-∞µ n(µ′,T) dµ′ ∫ds′∫d′ n2(s′,′) H(µ′ + Ts′ - ′)
(20)
(10) Von Szombathely, M.; Bra¨uer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17. (11) Puziy, A. M.; Matynia, T.; Gawdzik, B.; Poddubnaya, O. I. Langmuir 1999, 15, 6016.
where H(x) is a unit step function (an integral of the δ-function). Now differentiate N(µ′,T) to obtain
(∂µ′ ∂T )
)-
N
∂N ∂N ∂N / )/n(µ′) ∂T ∂µ′ ∂T
(21)
From eq 20 one finds
∂N ) ∂T
∫ds′∫d′ n2(s′,′) s′δ(µ′ + Ts′ - ′)
(22)
Thus
(∂µ′ ∂T )
) -〈s′〉
(23)
N
where 〈s′〉 is (cf. eqs 19, 21, and 22) the thermal entropy averaged over all the adsorption sites with the free energy µ′. It follows from eq 18 that
〈′〉 ) µ′ + T〈s′〉 ) -T2
∂ µ′ ∂T T
()
N
(24)
where 〈′〉 is the average (in the same sense as 〈s′〉) energy of a site. According to their physical sense, 〈s′〉 and 〈′〉 depend upon µ′, but from eqs 23 and 24 it follows that they are also functions of N:
〈s′〉 ) s′(N); 〈′〉 ) ′(N)
(25)
Now one may exclude N from these equations to obtain
〈s′〉 ) f(〈′〉)
(26)
This equation is known in various fields of physical chemistry (see ref 7 and the discussion below). It follows from eqs 17 and 20 that
N(µ′,T) ) a(µ′,T)
(27)
which means that in the condensation approximation the cumulative distribution function of adsorption sites in free energy is just an experimental isotherm of adsorption (adsorption vs chemical potential). On the left-hand side of this equation µ′ is the free energy of a molecule on a site, but on its right-hand side µ′ is in fact the chemical potential of the perfect gas µ, eq 5. Thus, eq 24 is closely connected to that determining the isosteric heat of adsorption. Substitute µ for µ′ in eq 24 and use eq 5 (rescaled per mole) to obtain
〈′〉 ) -qst + 7/2RT
(28)
where qst is the isosteric heat of adsorption13
ln P (∂ ∂T )
qst ) RT2
a
(29)
4. Results and Discussion The basic isotherms of adsorption are presented in Figure 2. These are the isotherms measured at the lowest temperature in this experiment. As explained in section 2, the experimental errors are the smallest for these isotherms. The curves in Figure 2 are cubic spline interpolation of the experimental adsorption points (these points can be seen in Figure 4), and the symbols are desorption points. These lie slightly higher than the adsorption isotherm, which might be an indication of some slight chemisorption that is known to take place for carbon dioxide on some oxides although it was not observed on
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Bakaeva et al. Table 2. Coefficients of Eq 31 no.
sample
T (K)
a0
a1 × 104
a2 × 109
1 2 3 4 5 6
A A A B B B
156.8 156.8 184.4 156.8 156.8 185.2
-9.2130 -7.7654 5.9968 -6.6946 -5.2631 6.7539
-6.7057 -6.1545 -0.53449 -5.6173 -5.0016 -0.19799
-9.4390 -8.9256 -2.7463 -8.3094 -7.6477 -2.3874
Figure 2. Isotherms of adsorption (µmol/m2) at 158.6 K: solid line, sample A; broken line, sample B; symbols, desorption points.
Figure 4. Least-squares fits of the experimental isotherms (adsorption in µmol/m2, chemical potential in kJ/mol): symbols, experimental points; lines, least-squares fit. Curves marked by crosses and squares correspond to coefficients given on lines 2 and 3 of Table 2, respectively.
Isosteric heats of adsorption (µmol/m2):
Figure 3. crosses and solid line, sample A; open squares and solid line, sample B; error bars, computer simulation (see the text) on pure silica; filled square, another experiment (see the text) on porous silica gel.
pure silicas (see ref 12 and references therein). Thus, the experimental errors for the adsorption isotherms (only adsorption points are used in the analysis given below) are less than the difference between the desorption points and the curves in Figure 2. Another indication of the accuracy of the adsorption isotherms at 158 K can be revealed in Figure 3, which presents isosteric heats of adsorption calculated by the following equation:
qst(a) ) RT1T2[ln(P1/P2)/(T1 - T2)]
(30)
always seeks to occupy the strongest vacant adsorption site.) In comparison to the glass fiber curves, Figure 3 also includes data corresponding to the surface of dehydroxylated silica (characteristic of a more homogeneous surface). This curve was obtained by computer simulation where the method of simulation and the parameters of CO2/silica interaction were chosen to provide agreement between simulated and experimental isotherms of adsorption.1 Finally, the filled square in Figure 3 is the experimental result obtained long ago for a silica gel.14 At that time the methods of adsorbent characterization were not well developed. The result is referred to in a later paper concerned with the heats of adsorption of CO2 on various adsorbents other than silica.15 This point is in semiquantitative agreement with our results since adsorption in porous silica gel is usually higher than that on a nonporous silica.13 The adsorption branches of all the isotherms were fitted to the equation
This is the common approximation of eq 29 that is widely used in adsorption measurements.13 Here P1 and P2 are the equilibrium pressures corresponding to the adsorption a at temperatures T1 and T2, respectively. The solid curves are the interpolations of experimental points obtained from one isotherm at 158 K and the isotherm at 185 K. The symbols closest to these curves are obtained by another isotherm (on the same sample) at 158 K and the same isotherm at 185 K. Thus, whatever the accuracy of the isotherm at 185 K the deviation of the symbols from the solid curves in Figure 3 is a measure of the accuracy of the lowest temperature isotherm. This accuracy is high everywhere except at the lowest adsorption points. These points in Figure 3 are unreliable because the pressure gauges could not reliably measure the associated equilibrium pressures (cf. section 2). The decrease of isosteric heats with an increase of coverage (in Figure 3) is characteristic of a heterogeneous surface composed of adsorption sites of different strengths. (Roughly speaking, this is because an adsorbed molecule
where µ is the chemical potential in eq 5 and the leastsquares fit coefficients a0, a1, and a2 are presented in Table 2. The first and the second as well as the fourth and the fifth lines of the table correspond to two independent measurements on the same sample at the same temperature. In Figure 4, the fitted isotherms of eq 31 are compared with experimental points. Now we use eq 31 to obtain distribution functions. This can be done in several ways (see ref 6 and references therein). We use here the methods described in section 3. First, a smoothed set of 50 points (N ) 50 in eq 14) corresponding to line 5 in Table 2 were calculated in the interval [-60.5, -44.1] of chemical potentials (kJ/mol). This interval is extended by 20% to the left with respect to eight original experimental points in the interval [-50.4, -44.1]. In eqs 12-14 R ) 4 and in eq 16 λ′ ) 0.01. The distribution function calculated with these parameters is
(12) Lemcoff, N. O.; Sing, K. S. W. J. Colloid Interface Sci. 1977, 61, 227. (13) Gregg S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982; p 17.
(14) Magnus, A.; Ka¨lberer, W. Z. Anorg. Allg. Chem. 1927, 164, 357. (15) Avgul’, N. N.; Belyakova, L. D.; Vorob’eva, L. D.; Kiselev, A. V.; Muttik, G. G.; Chistozvonova, O. S.; Chicherina, N. Yu. Kolloidn. Zh. 1974, 36, 928; Colloid J. USSR (Engl. Transl.) 1974, 36, 841.
ln a ) a0 + a1µ + a2µ2
(31)
Adsorption of CO2 on Glass Fibers
Figure 5. Adsorption site density ([µmol/m2]/[kJ/mol]) in free energy for sample B (Table 2, no. 5): dotted line, condensation approximation.
Figure 6. Marginal distribution in energy (number of sites in µmol/m2).
presenten in Figure 5 as a solid curve. Substitution of this function (n(µ′)) in eq 3) reproduces its left-hand side with the average accuracy 0.05%, the maximal deviation being 0.04%. We will conditionally call the distribution function obtained from the eq 15 the exact distribution which may be compared with the approximate distribution based on the condensation approximation (CA). Comparison of the exact distributions in Figures 1 and 5 shows that oscillations in Figure 1 are an artifact due to the fact that the experimental isotherm was truncated at a too high value of µmin (pressure). Extrapolation of experimental isotherms to much lower pressures (chemical potentials) changes the character of the exact distribution and makes it qualitatively similar to the CA distribution. Still, the difference between the exact and CA distribution in Figure 5 is considerable especially at higher values of the chemical potential. At lower values of the chemical potential that difference is smaller, but it does not mean that this part of the exact distribution is reliable: It is obtained by a reliable method, but the data which this distribution is based upon are an extrapolation of the original experimental data to lower pressures, and this is usually unreliable. Now, we use eq 20 to obtain cumulative distributions from the solid curve in Figure 5 and the similar curve at higher temperature corresponding to line 6 in Table 2 as well as eqs 23 and 24 to obtain 〈′〉 and 〈s′〉. The former is presented in Figure 6. This is 〈′〉 vs N. The inverse of itsN vs 〈′〉sis the cumulative distribution of adsorption sites in 〈′〉. This is the characteristic closest to the distribution of adsorption sites in energy usually discussed in the literature.4-6 The distribution (cumulative or density) of adsorption sites in energy is usually obtained on the assumption that in eq 18 s′ is the same for all adsorption sites.6 We do not make such an assumption here, and as a result we obtain Figure 6. Here the ordinate is not the energy of a site but 〈′〉sthe average energy over all the sites with the same value of free energy µ′.
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Figure 7. Marginal distribution in entropy (number of sites in µmol/m2).
Figure 8. Entropy/energy relation.
Another peculiarity of Figure 6 is that it has a minimum. This is probably an artifact: the decreasing part of the curve to the left of the minimum was obtained from extrapolated values of isotherms at coverages smaller than those measured experimentally. It gives an unrealistic double-valued distribution function of sites in energy. Thus, only the part of the curve to the right of the minimum is reliable. The thermal entropy (〈s′〉) vs number of sites (N) relation in Figure 7 also has a spurious minimum. Only the part of the curve to the right of that minimum is reliable. The inverse of this curve is the cumulative distribution of sites in the average thermal entropy, the averaging being the same as explained above for energy. Figure 8 is obtained from the plots in Figures 6 and 7 in the interval of the number of sites 2 < N < 8. This is a relation between some average entropy and the average energy of an adsorption site. Similar relations between entropy and energy have been observed in various fields of physical chemistry (see ref 7 and references therein).
S ) S0 + E/Ti
(32)
Here S0 and Ti (isokinetic temperature) are constants and S and E are entropy and energy (enthalpy). In heterogeneous catalysis, eq 32 refers to the compensation effect, isokinetic dependence, or θ rule, S and E being the changes of entropy and enthalpy of activation that accompany the modification of a catalyst. If S and E are the standard entropy and enthalpy of solution, eq 32 is known as the Barclay-Butler rule; it also holds for changes of entropy and enthalpy of some chemical reactions when the conditions or the structure of the reagents is changed in a systematic way. Finally, eq 32 also holds for adsorption on heterogeneous surfaces (see ref 7 and references therein). In particular, it has been shown that this equation with Ti ) 740 K describes the correlation between s′ and ′ for adsorption of argon on different sites of a model oxide surface.7 The part of the plot to the right of the minimum in Figure 8 has a slope corresponding to Ti ≈ 330 K. This is
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and take them as adsorption sites. Such an approach was used for argon on a model oxide surfaces7 and for water on amorphous silica.2 It makes it possible to obtain directly the distribution of adsorption sites in energy or in entropy.7 5. Conclusion
Figure 9. Comparison of density distribution for two samples of glass fibers (adsorption site density ([µmol/m2]/[kJ/mol]) in free energy): dotted line, Table 2, no. 1; solid line, Table 2, no. 5.
2.2 times smaller than Ti mentioned above, which may be explained by the different meanings of S and E in eq 32 in both cases. In ref 7, S and E correspond to s′ and ′ of a single site, and here they are 〈s′〉 and 〈′〉saverage values of s′ and ′ over a group of sites with the same value of free energy µ′. The meaning of eq 32 for adsorption on heterogeneous surfaces is that the larger the energy of the adsorption site, the smaller (on average) the mobility of an adsorbed atom, which diminishes s′.7 Other interpretations of eq 32 are also possible.7 Aside from the interpretation of the entropy-energy relation, eq 32 is a useful criterion for validating the adsorption site analysis of an isotherm developed above. It shows, for example, that adsorption sites with energies to the left of the minimum in Figure 8 are unreliable. These are the strongest adsorption sites obtained from extrapolation of experimental isotherms described above. The absolute values of energies and entropies in Figure 8 are 16-22 kJ/mol and 0.16-0.17 kJ/(mol‚K) correspondingly. They refer to the interval of temperatures from 157 to 185 K (cf. Table 2). The heat capacity of adsorbed molecules can be estimated as 3R (Dulong and Petit law). Thus, the variations of the thermal energy and entropy of a molecule in this temperature interval (171 ( 15) are 0.4 kJ/mol and 2 × 10-3 kJ/(mol‚K) correspondingly. Thus, s′ and ′ are constant in this temperature interval with an accuracy of about 1% as has been assumed above (cf. eq 19). Finally, the distributions n(µ′) of adsorption sites in free energy at 157 K for samples A and B of glass fibers are compared in Figure 9. The dotted line refers to sample A. The surface of this glass fiber has more adsorption sites of high energies than that of sample B. This is in accord with the fact that the isotherms and heats of adsorption in Figures 2 and 3 lie higher for sample A than for sample B. We are not in the position now to discuss the source of this difference. In line with our general approach described in section 1, we have to simulate the atomic structure of these multicomponent glass surfaces and then simulate adsorption of carbon dioxide on those surfaces (as has been done for the surface of amorphous silica in ref 1). One can also determine the positions of minima for an adsorption potential at those model surfaces
We have measured experimental isotherms of adsorption of carbon dioxide on glass fibers. The main problem here is that the specific surfaces of many kinds of glass fibers can be exceedingly small. In this work, we took relatively high surface area fibers and used conventional volumetric techniques of adsorption measurement. The accuracy of an isotherm mesurement is limited by the errors in pressure measurements. It is shown that it is high enough at lower temperature (158 K) to reliably determine the difference in adsorption characteristics of two different glass fibers. The adsorption isotherms are analyzed in terms of the model of independent adsorption sites. The peculiarity of our analysis is that we determine the distribution of adsorption sites not in energy (as usual) but in free energy. This distribution depends on the temperature, and from this dependence one can determine the entropy and energy of adsorption sites (averaged as explained above) and the entropy vs energy relation for sites. The latter makes it possible to evaluate the reliability of the distribution. In particular, it shows that the high (in absolute value) energy part of our distribution is unreliable. It was obtained by a standard solution (linear regularization) to the so-called inverse problem.9 It is shown that this method requires an experimental isotherm to be measured to much lower equilibrium pressures (coverages) than those accessible with our equipment. We dealt with this problem by extrapolating experimental isotherms to lower coverages. This enabled us to obtain qualitatively reasonable distributions of adsorption sites in free energy, but only part of those distributions passed the entropy/energy test. On one hand, this shows that one has to develop another experimental method to study adsorption characteristics of glass fibers and other low surface area materials (this work is underway now). On the other hand, this shows how the method developed in this paper can be used to asses the reliability of the energy distribution obtained from adsorption experimental data. The results described in this paper are in support of a more general goal that includes computer simulation of the surfaces of multicomponent glasses and adsorption on those surfaces (in the form of fiber). The simulated adsorption characteristics (isotherms, site distribution functions, etc.) on model glass fibers will be compared to experimental ones such as those obtained in this work to assess the validity of computer simulation and to determine the nature of glass surfaces. Acknowledgment. We thank W. A. Steele for helpful discussion. This material is based upon work supported by the National Science Foundation under Grant DMR 9803884. LA991376X
