Modeling of Gas Adsorption Equilibrium over a Wide Range of Pressure: A Thermodynamic Approach Based on Equation of State

Journal of Colloid and Interface Science 250, 49–62 (2002) doi:10.1006/jcis.2002.8311, available online at http://www.idealibrary.com on

Modeling of Gas Adsorption Equilibrium over a Wide Range of Pressure: A Thermodynamic Approach Based on Equation of State E. A. Ustinov,∗,1 D. D. Do,∗,2 A. Herbst,† R. Staudt,† and P. Harting† ∗ Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia; and †Institute of Non-Classical Chemistry, University of Leipzig, Permoserstrasse 15, D-04303 Leipzig, Germany E-mail: [email protected] Received October 29, 2001; accepted February 21, 2002; published online April 29, 2002

son for this peculiarity is well established and associated with the pressure range where the density in the bulk phase increases with pressure at a rate that is faster than that in the adsorbed phase. The difference between the excess and absolute adsorption isotherms is usually very small in the case of subcritical adsorption due to the low bulk fluid density. However, as shown by Sircar (15), all methods used for obtaining isotherms (gravimetric, volumetric, piezometric, and total desorption methods, column breakthrough methods, closed-loop recycle methods, and isotope exchange methods) measure the Gibbsian excess rather than the absolute amount adsorbed. In order to evaluate such thermodynamic functions as the isosteric heat of adsorption it is necessary to know the absolute adsorption isotherm (11), which is difficult to determine accurately because it requires the pore volume of an adsorbent to be exactly predetermined. There are different ways for the determination of the pore volume (12). Usually it requires some supplementary experiments (such as low-temperature nitrogen adsorption) or additional assumptions of an adsorbed phase density. Recently (12) a new method for this aim that resorts to simultaneously two assumptions was proposed: the excess adsorption obeys the Ono–Kondo equation (9), while the absolute adsorption isotherms are governed by the Langmuir equation, with the former being used for the extrapolation of the excess isotherm to zero amount. This clearly points to the principal difficulties of the exact pore volume determination in the case of supercritical adsorption. The reliability of excess adsorption isotherms seems to be even more problematic because their determination implies knowing the skeleton density of an adsorbent. This also requires some additional measurements, which are usually carried out with the adsorption of helium. For a long time it has seemed self-evident that the adsorption of helium is negligibly small. It always led to an overestimation of the “helium density” and such abnormalities as negative excess amount adsorbed. Only recently (16) was it shown that helium does not adsorb at approximately 400◦ C and that the “helium density” determined at room temperature is about 15% higher than that determined at 400◦ C in the case of activated carbon. In any case the adsorbent volume inaccessible for helium molecules is not the same as that inaccessible for the

A thermodynamic approach based on the Bender equation of state is suggested for the analysis of supercritical gas adsorption on activated carbons at high pressure. The approach accounts for the equality of the chemical potential in the adsorbed phase and that in the corresponding bulk phase and the distribution of elements of the adsorption volume (EAV) over the potential energy for gas–solid interaction. This scheme is extended to subcritical fluid adsorption and takes into account the phase transition in EAV. The method is adapted to gravimetric measurements of mass excess adsorption and has been applied to the adsorption of argon, nitrogen, methane, ethane, carbon dioxide, and helium on activated carbon Norit R1 in the temperature range from 25 to 70◦ C. The distribution function of adsorption volume elements over potentials exhibits overlapping peaks and is consistently reproduced for different gases. It was found that the distribution function changes weakly with temperature, which was confirmed by its comparison with the distribution function obtained by the same method using nitrogen adsorption isotherm at 77 K. It was shown that parameters such as pore volume and skeleton density can be determined directly from adsorption measurements, while the conventional approach of helium expansion at room temperature can lead to erroneous results due to the adsorption of helium in small pores of activated carbon. The approach is a convenient tool for analysis and correlation of excess adsorption isotherms over a wide range of pressure and temperature. This approach can be readily extended to the analysis of multicomponent adsorption systems. C 2002 Elsevier Science (USA)

1. INTRODUCTION

Physical adsorption of supercritical gases at high pressures is known to have some interesting features compared to subcritical fluid adsorption, which is important in the design of many adsorption processes and presents a challenge for fundamental theoretical researches. In many experimental isotherms the maximum of the excess adsorption is observed (1–14). The rea-

1 On leave from Saint Petersburg State Technological Institute (Technical University), 26 Moskovsky Prospect, St Petersburg 198013, Russia. 2 To whom correspondence should be addressed.

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 C 2002 Elsevier Science (USA)

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adsorbate molecules, which can be a source of further discrepancies. The theory of adsorption of supercritical fluids at high pressure is not complete. There are a number of fitting approaches that rely on semi-empirical equations such as the Langmuir equation (12), Langmuir–Freundlich equation (10, 13, 17), and Toth equation and that modified by Jensen and Seaton (18, 19) for the absolute adsorption isotherm. Application of the potential theory and the Dubinin–Astakhov (DA) equation to supercritical fluid adsorption has been considered in a number of papers (20–24). A number of theories are devoted to the application of the 2-D equation of state (EOS) to the adsorbed phase [14, 25–31]. All these approaches have only correlative ability. The parameters of an EOS cannot be predetermined from analysis of the bulk phase behavior and are used as just fitting parameters. Using such a variable as the saturation pressure above the critical temperature in potential theories hardly has a physical meaning, although it should be noted that the single-temperature invariant characteristic curve can be plotted in some cases with high accuracy (21). For instance, Zhou and Zhou (24) replaced the saturation pressure in the DA equation by a temperature-independent parameter corresponding to the point of intersection of a family of linearized isotherms for the case of hydrogen adsorption. This approach was further developed on the basis of analysis of high-pressure nitrogen (32) and methane (33) adsorption. In these papers an empirical equation for absolute adsorption was applied to the range of low pressures and then extended to the range of high pressures. By comparing the experimental excess adsorption with that calculated by the linearized equation the authors came to the conclusion that the adsorbed phase volume changes with the amount adsorbed and is less than the pore volume. This result was, however, obtained by extrapolating the linear part of an absolute isotherm to the region where the bulk phase and adsorption phase densities were compatible, and using the implicit assumption that the adsorbed phase density does not vary over the adsorbed phase volume. More rigorous theoretical investigations based on the Ono– Kondo approach for the lattice theory (9, 34, 35) also involve fitting parameters and do not consider the real pore structure of adsorbents, in particular, the pore size distribution (PSD). The most adequate description of high-pressure adsorption can be obtained by grand canonical Monte Carlo (GCMC) simulations (11, 19, 36–38). However, these methods are time-consuming and sometimes lead to poor predictions for pure component adsorption (11, 19). As a compromise approach the simplified local density (SLD) theory was proposed (39–43). The main idea in these works is the application of the equation of state for the corresponding bulk phase to an adsorbed phase, but the parameters of the EOS are modified using the local density and the mean field approximations. At the present the authors of this theory have not applied the approach to PSD determination, which probably can be done in the future. The only obstacle is apparently associated with a complexity of evaluating of the EOS parameters in confined pores. It is not a problem for the

van der Waals equation (39), but more adequate equations of state require either additional assumptions or such complicated derivations that do not justify the term “simplified” in the name of the theory. Our aim in this paper is to create an approach capable of reliably determining the excess and absolute isotherms from gravimetric measurements without using experiments on helium adsorption. The idea of the paper may be expressed briefly as follows. We use the same Bender equation of state (44, 45) for the bulk phase and the adsorbed phase and the condition of equality of the chemical potential in these phases to determine the distribution of the pore volume of an activated carbon over the free energy change µ. The latter includes the potential energy of the fluid–solid interaction u in an element of adsorption volume and the part µs associated with the change of free energy of the compressed fluid compared to the bulk phase at the same density and temperature due to the effect of confined pores. This type of distribution and its dependence on the temperature provide important information concerning the state of an adsorbed phase. The algorithm developed for determining this distribution includes the adsorbent density as one of parameters to be determined by least-squares fitting. 2. EXPERIMENTAL

High-pressure adsorption experiments were carried out using a magnetic suspended gravimetric system (Rubotherm Pr¨azisionsmesstechnik GmbH, Bochum, Germany), cf. Fig. 1. The balance can be operated up to 50 MPa and has the advantage of noncontact weighing of samples (51, 52). The unit consists of a conventional high-pressure stainless steel sample cell, which was connected to a gas reservoir via an air-driven gas booster (Haskel, USA). The gases used were ultra-high-purity grade supplied by Air Liquid. A known amount of active carbon (∼4 g), which was outgassed at 423 K, was placed into the sample pan of the balance. Before the measurements the system was evacuated at 2.3 × 10−3 Torr. Equilibrium weights were achieved in 20–30 min. The measurements of excess adsorption isotherms of the pure gases on the adsorbents were carried out at four temperatures from 25 to 70◦ C over a wide pressure range from 0 to 50 MPa. These measured adsorption isotherms are typical excess isotherms with a distinct maximum. 3. MODEL

We consider that elements of the adsorption volume (EAV) of an adsorbent are distributed over the adsorption potential. Such a distribution is not a universal function for a given adsorbent because repulsive and attractive forces exerted by pore walls contribute differently to the adsorption potential, depending on the dimensions of the adsorbate molecules and pore. However, following the Polanyi idea that the potential of a gas–solid interaction due to dispersive forces is the product of the potential

51

MODELING OF GAS ADSORPTION EQUILIBRIUM

FIG. 1.

Experimental setup for gravimetric measurements of adsorption equilibria (51).

of a reference gas and a similarity coefficient, we may expect that the distribution function of a given adsorbent does not differ essentially for different species. If an equation of state for an adsorbate in confined pores could be found, it would be possible to determine the adsorbed phase density as a function of the bulk phase pressure and temperature and gas–solid interaction potential using the equality of chemical potentials of equilibrium phases. It is known that the potential energy is an additive component of the chemical potential. On the other hand, the chemical potential is proportional to the logarithm of fugacity. In the hypothetical case when the behavior of the bulk phase and adsorption phase obey the same equation of state one can write the equation RT ln f (a) + u = RT ln f.

[1]

Here f (a) and f are the fugacities of a gas in a pore and in the bulk phase, respectively; u is the overall potential energy of the gas–solid interaction; T is the temperature; and R is the universal gas constant. This equation is valid for subcritical fluids as well. In this case the fugacity f will be very close to the bulk phase pressure p, and the fugacity f (a) in the adsorbed phase will be limited above by the saturation fugacity f s . Once the fugacity in an element of the adsorption volume achieves the saturation pressure, the vapor–liquid transition occurs. Consequently, in the case of subcritical fluid adsorption each EAV can be nearly empty or completely filled, with the boundary between such

pores corresponding to the condition RT ln( p/ ps ) = u.

[2]

In this equation one can recognize the expression of the Polanyi idea of the temperature-invariant characteristic curve, which was used further in Dubinin’s theory of volume filling of micropores (46, 47). In this context it should be pointed out that Eq. [1] is more general and rigorous (in particular, it accounts for the compressibility of a fluid) and, hence, there are no principal obstacles in applying Eq. [1] to analysis of supercritical adsorption. However, in this approach we take into account that the behavior of an adsorbed gas in the general case does not obey the same equation of state valid for the bulk phase. Nevertheless Eq. [1] will remain correct, if the overall potential energy u is considered as the sum of the potential energy of the gas– solid interaction and the change in the chemical potential µs caused by differences in the state of the adsorbed phase compared to that of the bulk phase (for example, due to the decrease of the coordination number). Therefore Eq. [1] can be rewritten as RT ln f (a) + u + µs = RT ln f.

[3]

At very low densities fugacity becomes equal to pressure. In the general case, fugacity is a function of density and

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temperature, according to the equation (48)  p     1 1 f = p exp − dp , ρ RT p

In this paper we use the Bender equation of state (44, 45): [4]

+ (G + Hρ 2 )ρ 2 exp(−a20 ρ 2 )].

0

where p and ρ are the pressure and molar density, respectively. Combining Eqs. [3] and [4] one can write  µ + p

dp = 0. ρ

[5]

Here µ = u + µs , p is the pressure in the bulk phase, and  is the pressure in the EAV of the adsorbed phase, where the potential energy equals u. Given µ,  can be obtained as the root of Eq. [5]; that is, at a given temperature and pressure in the bulk phase  is a function of the potential energy. On the other hand,  is a function of density according to an equation of state for the adsorbed phase. Consequently, upon solving Eq. [5], one can find the local density of an adsorbed gas in the EAV. Finally what needs to be done is the integration of the local density over the adsorption space while accounting for a distribution function of the adsorption volume over the potential energy. Thus if we define ϕ(µ) as the volume distribution with respect to the change in chemical potential µ such that ϕ(µ) dµ is the volume of elements having the change of chemical potential to fall between µ and µ + dµ, then the total pore volume is given by ∞ W0 =

ϕ(µ) d(µ)

[6a]

−∞

and the mass excess is ∞

=



ρ (a) − ρ ϕ(µ) d(µ).

[6b]

−∞

Once the mass excess is known from Eq. [6b], the absolute amount adsorbed can be calculated from the following equation if the adsorbed phase volume is known: a = + ρW0 .

[6c]

For porous media, such as activated carbon dealt with in this paper, the adsorbed phase volume is taken to be the void volume of the pores. If the potential energy u is negative (which is usually the case due to attractive forces) and µs is small, the density in the adsorbed phase ρ (a) will be greater than the density ρ in the bulk phase because from Eq. [3] f (a) f

  −u − µs = exp > 1. RT

Hence the excess adsorption is positive.

p = ρT [R + Bρ + Cρ 2 + Dρ 3 + Eρ 4 + Fρ 5 [7]

Parameters B, C, . . . , H are functions of temperature, with the total number of constants for each substance equal to 20. Despite the form of this equation being rather complicated, it is, however, very convenient because it makes it possible to express fugacity analytically as a function of density. At a given pressure in the bulk phase solving the Bender equation [7] gives the bulk phase density and, consequently, the fugacity f from Eq. [4]. On the other hand, knowing the potential energy u and µs in an element of the adsorption volume, one can calculate the fugacity f (a) of this element by Eq. [3]. Once the fugacity of that element is determined, the density in the EAV with the chemical potential change µ can be easily evaluated numerically by any iteration technique. Hence if the distribution function ϕ with respect to µ is known, one can calculate the excess amount adsorbed. Inversely, if the adsorption isotherm data are available, the distribution function can be readily obtained. This procedure applies to supercritical as well as subcritical fluids. In the latter case it is necessary to compare the fugacity of the adsorbed phase with that of the vapor–liquid coexistence f s . If the fugacity exceeds f s , the condensation will occur at the liquid-like density. The fugasity f s , the saturation pressure, and the corresponding density can be determined from the same Bender equation [7]. Summarizing, we may express the algorithm of calculating the amount adsorbed as follows. 1. For a given pressure and temperature it is necessary to determine the value of fugacity for the bulk phase using the equation of state. This step may not be necessary for subcritical fluids because the fugacity would be very close to the bulk phase pressure. However, in the case of high-pressure adsorption of light gases, it is essential. 2. For the set of values of potential µ in the adsorbed phase, we may find the set of values of fugacity, each of which relates to a definite element of the adsorption volume. 3. Given the set of fugacities one can determine the set of densities distributed over the adsorption volume. Each time it is necessary to check whether the fugacity exceeds that for the saturation point. If so, the adsorbate would be liquid-like; otherwise it would be gas-like. 4. Superposition density dependence on the potential energy and distribution function previously determined yields a value of the amount adsorbed. 3.1. Adapting the Method to the Special Case of Experimental Measurements The peculiarity of the device used for obtaining experimental data allows us to carry out measurements of weight change of a sample minus the Archimedean forces acting on the sample and

MODELING OF GAS ADSORPTION EQUILIBRIUM

sample holder. The balance of forces can be written as  g = M m

∞

4. RESULTS AND DISCUSSION



Argon on Norit R1

ρa ϕdµ − m(W0 + 1/ρs )ρ − V0 ρ  . [8]

−∞

Here g is the weight change due to adsorption, which is measured by the device; M is the molecular mass; m is the mass of a sample; and ρs is the picnometric density of an adsorbent (inaccessible volume), which is usually less than the actual density of graphite. We will further use the picnometric density term in order to distinguish it from the helium density and the skeleton (true) density. The inverse value is the volume of graphite material of an adsorbent, so the sum of the pore volume and the inverse of density is the specific total volume of the sample. V0 is the volume of the sample holder. Experimental dependence of g on the pressure p in the bulk phase can be used in the analysis of matching between data and theory. The functional to be minimized is given by the equation =



{[m i i − (m i /ρs + V0 )ρi ] − gi /Mi }2 .

[9]

i

Here i is the number of an experimental point. In such a way, it is possible to derive the necessary parameters of the model. The subscript i at the sample mass and the molecular mass denotes that in the general case we can simultaneously process a family of isotherms, which includes measurements at different temperatures for different gases and different samples of the same adsorbent. The picnometric density ρs was considered as one of the fitting parameters. We accounted for the slight variance in ρs for different species, which was necessary due to the difference in molecular size of the adsorbates. One of our aims in this analysis was the determination of the distribution function ϕ. Undoubtedly, the simplest way is finding ϕ in the form of either a symmetrical or a nonsymmetrical function in terms of the change of chemical potential µ, which usually involves three fitting parameters. Even though these parameters are sufficient for an accurate description of the experimental data, this does not mean it circumvents the socalled ill-posed problem, where the derived distribution function is very sensitive to experimental errors. This problem is nonexistent if we know a priori the form of the distribution function. Otherwise, the fitting of the theory, using almost any reasonable distribution function, against the experimental data could give rise to an acceptable agreement. By this reason we replace the mass excess Eq. [6b] with the following equation written in terms of a sum involving the volume of the element of adsorption, which is a function of µ:

=





Vk ρ (a) − ρ .

53

Argon has an apparent advantage to being chosen as a standard gas because its molecule is spherical and does not have dipole and quadrupole moments. By this reason argon is commonly used for determining pore size distribution. Figure 2 shows the experimental data of argon at 25, 40, 55, and 70◦ C obtained by the device described under Experimental. At low pressures the increase of pressure in the bulk phase leads to an increase in the amount adsorbed. The dependence of density on pressure is a convex curve, and a further increase of the pressure decreases the difference between the adsorbed phase and bulk phase densities ρ (a) − ρ. In this situation the Archimedian force acting on the sample and sample holder dominates, resulting in a decrease of the measured “weight” into the negative range. Solid lines in the figure are the results of calculations after the distribution function had been obtained by simultaneously fitting Eq. [6] against all argon isotherm data. We note that the correlation ability of the approach is excellent. Figure 3 presents the distribution functions of EAV over potential change µ. The distributions as presented by lines are obtained separately for each adsorption isotherm, while the distribution with symbol A is derived from the processing of all argon experimental data simultaneously. One clear feature manifested from these distributions is the asymmetry with respect to change of the chemical potential. Another feature noted is that some of the EAV correspond to positive potentials resulting mostly from repulsive forces. It is interesting to note that in the region of low potential change the distribution curve A differs from those calculated separately for each temperature. It is one of the problems associated with ill-posed problems. It is difficult to say which result is more reliable. On the one hand, simultaneous treatment of the isotherms’ family based on the assumption

[10]

k

The vector V was found by a nonnegative regression with regularization described by Tikhonov (49, 50).

FIG. 2. Adsorption of argon on activated carbon Norit R1. Measured change of sample weight with pressure. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

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FIG. 3. Distribution function of EAV over potentials for the system Ar–Norit R1. Temperature (◦ C): (solid line) 25; (medium dashed line) 40; (short dashed line) 55; (dash-dot line) 70. (Points) common distribution function determined for all isotherms simultaneously.

of the same distribution function reduces the total number of fitting parameters, hence increasing the reliability of the results. However, on the other hand, very small systematical deviations between different runs (which are unavoidable) may cause artificial humps and troughs. Nevertheless, one can state that there are at least two overlapping peaks on the distribution function in the range of approximately −5.2 and −1.4 kJ mol−1 , which may be attributed to micropore and mesopore structure, respectively. The area under this curve is equal to the total volume of pores W0 that in this case is 0.56 cm3 g−1 . The excess adsorption for different temperatures is plotted in Fig. 4. As is seen all isotherms have maximal points in the range of 15–20 MPa. It should be noted that experimental points involve a fitting parameter, which is the picnometric density ρs . As mentioned above, this parameter is obtained from the

FIG. 4. Excess adsorption of Ar on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

FIG. 5. Absolute amount adsorbed as a function of pressure. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

same system of adsorbent–adsorbate rather than from the helium adsorption data by a reason discussed below. The value of ρs obtained by a least-squares fitting was 1.866 g cm−3 , which is quite reasonable. Solid lines in Fig. 4 are plotted from the previously determined distribution function shown in Fig. 3. One can see that the agreement between calculated curves and experimental points is excellent. Reconstruction of isotherms in terms of absolute loading is presented in Fig. 5. The form of these isotherms is quite common, being not significantly convex (as argon is a supercritical gas) and indicating a decrease in loading with temperature. Nitrogen on Norit R1 In the case of adsorption of nitrogen we had a series of isotherms at 25, 40, 55, and 70◦ C (supercritical) and one isotherm in the subcritical region at 77 K, which is especially interesting for comparison. The low-temperature isotherm obtained by analyzer ASAP 2000 is shown in Fig. 6. The adsorption data measured for high temperature and pressure are presented in Fig. 7. Distribution functions obtained separately for 25, 40, 55, and 70◦ C are plotted in Fig. 8. One can see from the figure that all curves are quite close to each other and reflect the same feature of the adsorbent as in the case of argon adsorption. There are again at least two peaks presumably corresponding to micropores and mesopores. A slight shift of these curves toward lower values of the potential with temperature can be explained as follows. The potential µ contains the chemical potential change µs induced by deviations between the state of the adsorbed phase in confined pores compared to the state in the bulk phase. This change in the chemical potential mostly associated with the deficit of intermolecular interactions in the adsorbed phase is positive. An increase in temperature suppresses the adsorbed phase density, lowering the difference of this phase from the bulk phase. Apparently by this reason the contribution of change in the potential decreases with temperature, shifting the

MODELING OF GAS ADSORPTION EQUILIBRIUM

FIG. 6.

Nitrogen isotherm adsorption on Norit R1 at 77 K.

55

FIG. 8. Distribution function for the system N2 –Norit R1. Temperature (◦ C): (solid line) 25; (medium dashed line) 40; (short dashed line) 55; (dash–dot line) 70.

distribution function to the left. More detailed investigations, however, should be done in the future on the basis of more precise measurements. Analysis of the low-temperature isotherm was identical to that carried out for the other isotherms, but, of course, in this case it was not necessary to determine the picnometric density ρs . Phase transition in pores was taken into account, with all calculations being built in the scheme based on the Bender equation of state. Figure 9 shows the distribution determined from the analysis of the low-temperature adsorption isotherm (line with points) and the distribution by simultaneous treatment of four isotherms under supercritical conditions. It is interesting to note that the distribution functions obtained are very close to each other despite the difference in temperature exceeding 220◦ . In both cases the main peculiarity of these distribution functions is reproduced showing two overlapping peaks, which correspond

to the same values of the gas–solid interaction potential. As we discussed in the last paragraph, a decrease of temperature slightly shifts the distribution function to the right. However, it is likely that further temperature drops in the subcritical range do not lead to any significant change in the coordination number of the adsorbed phase confined in separate pores because the fluid density in the most pores has reached its maximum value. Hence the chemical potential change associated with the difference in the state of the adsorbate confined in a pore and compressed gas at the same pressure remains nearly constant, which probably explains the temperature invariance. The total adsorption volume W0 found from the lowtemperature isotherm is 0.605 cm3 g−1 while the high-pressure nitrogen adsorption gives a quite close value of 0.582 cm3 g−1 .

FIG. 7. Measurements of weight change with pressure in the case of N2 adsorption on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

FIG. 9. Distribution function for the system N2 –Norit R1 obtained at supercritical and subcritical conditions. (Line with points) adsorption at 77 K. (Solid line) result of simultaneous treatment of isotherms at 25, 40, 55, and 70◦ C.

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FIG. 10. Excess adsorption of N2 on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

Excess nitrogen adsorption at four temperatures from 25 to 70◦ C is shown in Fig. 10. As in the case of argon adsorption one can see that the agreement between experimental points and calculated curves is excellent. Methane on Norit R1 We have obtained three isotherms at 25, 40, and 55◦ C. Experimental data of methane adsorption are shown in Fig. 11. Solid lines are calculated from the single distribution function, previously determined by simultaneous treatment of isotherms at all temperatures. A high degree of agreement between experimental points and calculated curves implicitly confirms the state that the temperature invariance is a reasonable assumption, although it is not so evident from Fig. 12, where we plotted three distri-

FIG. 12. Distribution function for the system CH4 –Norit R1. Temperature (◦ C): (solid line) 25; (dashed line) 40; (dash-dotted line) 55. (Points) result of simultaneous treatment of isotherms at 25, 40, and 55◦ C.

bution functions calculated by processing isotherms at different temperatures separately. The resulting curve obtained by simultaneous treatment of all isotherms is plotted as the curve with points. This discrepancy can be explained by an apparent lack of information in the range of relatively low pressure, where micropores filling predominantly occurs. Indeed, it is seen from Fig. 11 that there are only a few experimental points on each isotherm in the range from zero coverage to the maximal points of these curves. That is why the accuracy of determining the distribution function at low potential u is not high. Nevertheless, from our viewpoint, the situation is essentially improved by simultaneous treatment of all isotherms, which allows us to obtain more reliable results, although the existence of the third peak with maximum at −12 kJ mol−1 is rather questionable. Anyway, one can state that the other two overlapping peaks do exist as in the previous cases. Excess adsorption isotherms for methane are plotted in Fig. 13. It is interesting to note that these isotherms have a point of intersection at approximately 20 MPa. At higher pressure an increase in temperature even leads to a small increase in the excess adsorption. Absolute adsorption isotherms are plotted in Fig. 14. These isotherms are sharper than those for argon adsorption, indicating stronger fluid–solid interaction, but loading at high pressure is less than that for argon. Ethane on Norit R1

FIG. 11. Measurements of sample weight change with pressure in the case of methane adsorption on Norit R1. Temperature (◦ C): () 25; () 40; () 55. Solid lines are correlated by the approach.

At room temperature ethane is very close to the boundary between the subcritical and supercritical regions. The critical temperature of ethane (calculated by the Bender equation) is 305.42 K. Consequently, at 25◦ C ethane behaves as a subcritical fluid, with the saturation pressure of 4.2 MPa. At 40, 55, and 70◦ C ethane is a supercritical gas. The results of direct measurements of ethane adsorption at these temperatures are

MODELING OF GAS ADSORPTION EQUILIBRIUM

FIG. 13. Excess methane adsorption on Norit R1. Temperature (◦ C): () 25; () 40; () 55. Solid lines are correlated by the approach.

shown in Fig. 15, where black points denote data under subcritical conditions (25◦ C). The last point measured at 25◦ C corresponds to 3.99 MPa, which is close to the saturation pressure of 4.2 MPa. The solid lines are the results of calculation from the distribution function obtained by processing all isotherms simultaneously, which is shown in Fig. 16. It is seen that this curve essentially differs from those obtained for argon, nitrogen, and methane. It is difficult to say definitely what factor this difference could be attributed to. It may be associated with a lack of information at relatively low pressure, where ethane has strong interaction with the micropores. Nevertheless, as in the previous cases there are again two overlapping peaks. The correspondence between experimental points and calculated excess isotherms is shown in Fig. 17. One can see that the agreement is excellent, although the form of these curves

FIG. 14. Absolute adsorption of methane on Norit R1. Notations are the same as described in the legend to Fig. 12.

57

FIG. 15. Measurements of sample weight change with pressure in the case of ethane adsorption on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

is more complex than that for argon, nitrogen, and methane. At 50 MPa the excess amount adsorbed is only 25–30% of that at the maximal points of these isotherms. What is more interesting is that in the pressure range above 5 MPa (especially in the region from 5 to 20 MPa) the excess amount adsorbed essentially increases with temperature. Carbon Dioxide on Norit R1 Adsorption data for carbon dioxide are available at 25, 40, 55, and 70◦ C. Carbon dioxide has the critical temperature 304.08 K (also calculated by the Bender equation). Hence the situation with carbon dioxide adsorption is very close to that for ethane adsorption. At 25◦ C the saturation pressure is 6.44 MPa. At 40, 55, and 70◦ C carbon dioxide behaves as a supercritical gas. The results of direct measurements by the device are presented

FIG. 16.

Distribution function for the system C2 H4 –Norit R1.

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FIG. 17. Excess ethane adsorption on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

in Fig. 18. The method of treatment was the same as in the previous cases. The distribution functions obtained for different temperatures (excluding 25◦ C) are plotted in Fig. 19. As seen in this case the distribution function is rather complex. It should be emphasized that it is not necessary for these curves to be similar, as the potential u includes the change of the chemical potential due to structural differences between the adsorbed phase and the bulk phase. Excess adsorption isotherms obtained for four temperatures from the single-distribution function are shown in Fig. 20. One can see the same peculiarity of these isotherms as in the case of ethane adsorption. In the pressure range above 10 MPa the excess amount adsorbed grows with temperature.

FIG. 18. Initial experimental data for the system CO2 –Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

FIG. 19. Distribution function for the system CO2 –Norit R1. Temperature (◦ C): (solid line) 40; (dashed line) 55; (dash-dotted line) 70. (Points) result of simultaneous treatment of isotherms at 40, 55, and 70◦ C.

4.1. Problem of Affinity of Distribution Functions We have shown that there exists a temperature invariance in the distribution function of volume in terms of the change in chemical potential, which allows us to predict an isotherm if the isotherm of the same component at other temperature is known. It is also interesting (and practically reasonable) to check whether there is a similarity between distribution functions determined from the adsorption of different species on the same adsorbent. To this end we have treated all isotherms (excepting ethane, which behaves rather differently from the other species) simultaneously, considering that there is a common distribution function in the coordinates ϕ − βµ, where β is the affinity coefficient defined for each substance. As a standard gas we accepted argon, whose molecules are the simplest ones. However, the picnometric density was assumed to be different for all

FIG. 20. Excess adsorption of carbon dioxide on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70. Solid lines are correlated by the approach.

MODELING OF GAS ADSORPTION EQUILIBRIUM

FIG. 21. Resulting distribution function obtained by simultaneous treatment of isotherms for Ar, N2 , CH4 , and CO2 .

cases. The resulting distribution function is presented in Fig. 21. Having this single curve one can recalculate the excess amount adsorbed for all species. Figure 22 shows isotherms for argon, nitrogen, methane, and carbon dioxide at 40◦ C. It is quite convincing that the correlative ability of this approach is very high. For comparison there is plotted the helium adsorption isotherm at 40◦ C. As is seen the adsorption of helium cannot be considered as negligibly small. This is worth taking into consideration because the methods of determining the inaccessible volume of adsorbents by helium could lead to significant mistakes. We will return to this problem below. 4.2. Pressure and Density in Adsorbed Phase The Bender equation of state makes it possible to calculate the pressure and density if the chemical potential (or fugacity) is

FIG. 22. Predicted excess isotherms adsorption for Ar (), N2 (), CH4 (), CO2 (), and He (×) at 40◦ C.

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FIG. 23. Dependence of ethane density on potential inside pore volume of activated carbon Norit R1 at different pressures in the bulk phase. The bulk phase pressure is (from left to the right) 0.005, 0.05, 0.5, 5, and 50 MPa. Temperature is 40◦ C.

known. We consider the distribution of the pressure and density over the potential energy for the case of ethane adsorption. At a definite pressure in the bulk phase the adsorbed phase density varies over the adsorption volume. In those elements of the adsorption volume where the gas–solid interaction potential is greater, the density and pressure will be higher. Figure 23 presents the change in ethane density with the potential in EAV along constant values of the bulk phase pressure at 40◦ C. It is seen from the figure that in each curve there is a region where density drops with increasing potential very sharply. This situation is very close to the case of phase transition, which is not a complete surprise because 40◦ C is only 7.7◦ higher than the critical temperature. Each curve, from left to right, corresponds to the bulk phase pressure 10 times greater than that for the previous curve. Increasing pressure in the bulk phase shifts these curves to the right, creating impression of successive filling of EAV. However, the main peculiarity of high-pressure supercritical or close to critical gas adsorption is significant adsorbed phase compressibility. Indeed, the liquid phase density of ethane at 25◦ C is 10.4 mmol cm−3 , while the density at 40◦ C and 50 MPa in an element of adsorption volume where potential is equal to −16 kJ mol−1 exceeds 20 mmol cm−3 . Such great compression is a result of extremely high pressure induced by gas–solid interactions in the adsorbed phase. The dependence of the adsorbed phase pressure  on potential u at different pressure p values in the bulk phase is shown in Fig. 24 for the case of ethane adsorption. As is seen, even if the bulk phase pressure is as small as 5 kPa, the pressure  achieves 3 MPa in the element of the adsorption volume where potential equals −16 kJ mol−1 . If the bulk phase pressure is increased up to 50 MPa, the pressure  in the same EAV approaches 350 MPa. It is feasible that such high pressure could induce a deformation

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FIG. 24. Dependence of adsorbed phase pressure on potential inside pore volume in the case of ethane adsorption. Notations are the same as described in the legend to Fig. 22.

of the adsorbent, which must be taken into account for further development of this approach. 4.3. Helium Adsorption Even though helium adsorption is rather weak, it does not mean that this adsorption can be neglected. As easily seen from Eq. [6], if the adsorption of helium is a linear function of the bulk phase density, we will never distinguish the contribution of the Archimedian force acting on the sample and, consequently, we will not be able to determine the picnometric density of the sample correctly. It does not matter that the dependence of the measured weight g on pressure would be linear or close to linear. Such dependencies are plotted in Fig. 25 at different temperatures. This figure shows that all curves are very close to

FIG. 25. Dependence of weight change on the bulk phase density for helium adsorption on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70.

FIG. 26. Excess helium adsorption on Norit R1. Temperature (◦ C): () 25; () 40; () 55; () 70.

straight lines, which are nearly coinciding. If excess adsorption were zero, the value ρs calculated from these lines for 25, 40, 55, and 70◦ C would be 2.163, 2.125, 2.119, and 2.098 g cm−3 , respectively. One can see from this sequence that the picnometric density determined in such a way decreases with temperature, pointing out the decrease of helium adsorption. The more detailed analysis shows that these values are indeed overestimated. Very small adsorption of helium does not allow us to find the distribution function, as done for the other gases. To overcome this difficulty we used the following method. Assuming that the distribution function for the case of helium adsorption remains the same as in the cases of the other gases (excepting ethane), only the affinity coefficient and inaccessible volume of the adsorbent may be considered as fitting parameters. The accuracy of the experimental measurements is quite enough to determine these parameters are reliable. In such a way we obtained the values for the picnometric density of 1.964, 1.939, 1.940, and 1.928 g cm−3 for temperatures of 25, 40, 55, and 70◦ C, respectively. The affinity coefficient was found to be 0.0702. Reconstructed excess isotherms for helium are presented in Fig. 26. What is interesting is that this isotherm is very close to that obtained by Malbrunot et al. (16) for the activated carbon, having a specific area of 1030 m2 g−1 . In particular, the reported amount adsorbed at 56.5 MPa and 25◦ C is 0.684 mol kg−1 , while in our calculations the helium amount adsorbed at 50.3 MPa and 25◦ C is 0.676 mol kg−1 . Main parameters determined for the gases investigated are summarized in Table 1. The values of the picnometric density obtained by the different gases are quite close to each other and seem reasonable, although these values cannot be the same due to the difference in molecular size. The smaller the molecule, the closer to the graphite surface inside a pore this molecule is able to approach and therefore the less would be the inaccessible volume.

MODELING OF GAS ADSORPTION EQUILIBRIUM

TABLE 1 Parameters for High-Pressure Adsorption on Activated Carbon Norit R1

β ρs , g cm−3 W0 , cm3 g−1

Ar

N2

CH4

C2 H6

CO2

He

1 1.866 0.561

1.106 1.938 0.582

1.442 1.962 0.516

— 1.919 0.527

1.335 1.874 0.535

0.0702 1.943 0.547

Total pore volume W0 is only a little less than that determined by low-temperature nitrogen adsorption (0.605 cm3 g−1 ). Relatively high dispersion of this value determined by the different gases is a result of very small contributions of EAV, where potential is close to 0, to the amount of the excess adsorption, which explains the significant sensitivity of the pore volume to experimental errors. 5. CONCLUSION

A new approach for analyzing adsorption of supercritical gases at high pressure is suggested. The approach is based on the application of the Bender equation of state to the adsorbed phase, accounting for the equality of the chemical potential in the adsorbed phase and in the bulk phase. It is shown that activated carbon can be characterized by the distribution of elements of its adsorption volume over the potential energy of the gas–solid interaction. This distribution function reflects the pore size distribution but is free from additional assumptions concerning activated carbon topology (slit shape pores, for example) and the equation of fluid–solid interactions. The distribution function proved to be temperature invariant with good accuracy, which is confirmed by the comparison of the distribution function obtained from nitrogen adsorption at 77 K and that for high-pressure nitrogen adsorption in the range from 25 to 70◦ C. It is also shown that in the first approximation the distribution function obtained for the same adsorbent by different gases is similar, giving the possibility of prediction isotherms by the standard one. The method allows us to determine total pore volume and the volume of activated carbon inaccessible to an adsorbing gas. In our viewpoint this thermodynamically consistent approach may be considered as a reasonable compromise between simplified and theoretically vague concepts and those more accurate theories that are rather complicated and involve superfluous assumptions. The suggested approach can be further extended to multicomponent adsorption, which is now being developed. ACKNOWLEDGMENT Support from the Australian Research Council is gratefully acknowledged.

REFERENCES 1. Menon, P. G., Chem. Rev. 68, 277 (1968).

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2. Ozawa, S., Kusumi, S., and Ogino, Y., J. Colloid Interface Sci. 56, 83 (1976). 3. Specovius, J., and Findenegg, G. H., Ber. Bunsenges Phys. Chem. 82, 174 (1978). 4. Wakasugi, Y., Ozawa, S., and Ogino, Y., J. Colloid Interface Sci. 79, 399 (1981). 5. Blumel, S., Foster, F., and Findenegg, G., J. Chem. Soc. Faraday Trans. II 78, 1753 (1982). 6. Findenegg, G., Korner, B., Fischer, J., and Bohn, M., Ger. Chem. Eng. 6, 80 (1983). 7. Malbrunot, P., Vidal, D., Vermesse, J., Chahine, R., and Bose, T., Langmuir 8, 577 (1992). 8. Jiang, S., Zollweg, J., and Gubbins, K., J. Phys. Chem. 98, 5709 (1994). 9. Benard, P., and Chahine, R., Langmuir 13, 808 (1997). 10. Zhou, L., and Zhou, Y., Ind. Eng. Chem. Res. 35, 4166 (1996). 11. Salem, M. M. K., Braeuer, P., v. Szombathely, M., Heuchel, M., Harting, P., Quitzsch, K., and Jaronec, M., Langmuir 14, 3376 (1998). 12. Murata, K., and Kaneko, K., Chem. Phys. Lett. 321, 342 (2000). 13. Zhou, L., Zhou, Y., Li, M., Chen, P., and Wang, Y., Langmuir 16, 5955 (2000). 14. Humayun, R., and Tomashko, D. L., AIChE J. 46, 2065 (2000). 15. Sircar, S., Ind. End. Chem. Res. 38, 3670 (1999). 16. Malbrunot, P., Vidal, D., Vermesse, J., Chahine, R., and Bose, T., Langmuir 13, 539 (1997). 17. Kikic, I., Alessi, P., Cortesi, A., Macnaughton, S. J., Foster, N. R., Spika, B., Fluid. Phase Equilibria 117, 304 (1996). 18. Jensen, C. R. C., and Seaton, N. A., Langmuir 12, 2866 (1996). 19. Heuchel, M., Davies, G., Buss, E., and Seaton, N., Langmuir 15, 8695 (1999). 20. Agarwal, R. K., and Schwarz, J. A., Carbon 26, 873 (1988). 21. Amankwah, K. A. G., and Schwarz, J. A., Carbon 33, 1313 (1995). 22. Kaneko, K., Shimizu, K., and Suzuki, T., J. Chem. Phys. 97, 8705 (1992). 23. Kaneko, K., and Murata, K., Adsorption 3, 197 (1997). 24. De Boer, J. H., “The Dynamical Character of Adsorption.” Oxford Univ. Press, London, 1953. 25. Zhou, L., and Zhou, Y., Chem. Eng. Sci. 53, 2531 (1998). 26. Ross, S., and Olivier, J. P., “On Physical Adsorption.” Interscience, New York, 1964. 27. Hoori, S. E., and Prauznitz, J. M., Chem. Eng. Sci. 22, 1025 (1967). 28. Payne, H. K., Sturdevant, G. A., and Lealand, T. W., Ind. End. Chem. Fundam. 7, 363 (1968). 29. Friederich, R. O., and Mullins, J. C., Ind. End. Chem. Fundam. 11, 439 (1972). 30. Zhou, C., Hall, F., Gasem, K., and Robinson, R., Ind. Eng. Chem. Res. 33, 1280 (1994). 31. Zheng, Y., and Gu, T., J. Colloid Interface Sci. 206, 457 (1998). 32. Zhou, L., Zhou, Y., Bai, Sh., Lu, Ch., and Yang, B., J. Colloid Interface Sci. 239, 33 (2001). 33. Zhou, L., and Zhang, J., Langmuir 17, 5503 (2001). 34. Aranovich, G., and Donohue, M., J. Colloid Interface Sci. 180, 537 (1996). 35. Aranovich, G., and Donohue, M., J. Colloid Interface Sci. 194, 392 (1997). 36. Jiang, S., Zollweg, J., and Gubbins, K., J. Phys. Chem. 98, 5709 (1994). 37. Gusev, V., O’Brien, J., and Seaton, N., Langmuir 13, 2815 (1997). 38. Suzuki, T., Kobori, R., and Kaneko, K., Carbon 38, 630 (2000). 39. Rangarajan, B., Lira, C. T., and Subramanian, R., AIChE J. 41, 838 (1995). 40. Subramanian, R., Pyada, H., and Lira, C. T., Ind. Eng. Chem. Res. 34, 3830 (1995). 41. Subramanian, R., and Lira, C. T., in “Fundamentals of Adsorption” (M. D. LeVan, Ed.), p. 873. Kluwer Academic, Norwell, MA, 1996.

62

USTINOV ET AL.

42. Chen, J. H., Wong, D. S. H., Tan, C. S., Subramanian, R., Lira, C. T., and Orth, M., Ind. Eng. Chem. Res. 36, 2808 (1997). 43. Soul, A. D., Smith, C. A., Yang, X., and Lira, C. T., Langmuir 17, 2950 (2001). 44. Polt, A., Platzer, B., and Maurer, G., Chem. Techn. 22, 216 (1992). 45. Platzer, B., and Maurer, G., Fluid Phase Equilibria 84, 79 (1993). 46. Dubinin, M. M., in “Progress in Surface and Membrane Science” (D. A. Cadenhead et al., Eds.), p. 1. Academic Press: New York, 1975.

47. Dubinin, M. M., Carbon 21, 359 (1983). 48. Sandler, S., “Chemical and Engineering Thermodynamics,” third ed., p. 280. Wiley, New York, 1999. 49. Tikhonov, A. N., Dokl. Akad. Nauk SSSR 39, 195 (1943). 50. Tikhonov, A. N., Dokl. Akad. Nauk SSSR 153, 49 (1963). 51. Beutekamp, S., “Adsorption von Gasgemische bis zu Dr¨ucken von 15 MPa,” Ph.D. thesis. Institut f. Nichtklassissche Chemie e. V. an der Universit¨at Leipzig, Universit¨at Leipzig, in preparation. 52. Herbst, A., and Harting, P., Adsorption, in press.