ARTICLE pubs.acs.org/IECR
Influence of Energetic Heterogeneity and Lateral Interactions between Adsorbed Molecules on the Kinetics of Gas Adsorption Krzysztof Nieszporek* and Tomasz Banach Department of Theoretical Chemistry, Maria Curie-Sklodowska University, Pl. M. Curie-Sklodowskiej 3, 20-031 Lublin, Poland ABSTRACT: The kinetics of pure gas adsorption on energetically heterogeneous solid surfaces has been studied by statistical rate theory. The theoretical investigations take into account the influence of lateral interactions between adsorbed molecules. The energetic heterogeneity of the adsorption system has been described by the rectangular energy distribution function. The obtained kinetic expression is based on the exact solution of the integral equation approach. The experimental systems of alkane adsorption on activated carbons were used for quantitative analysis. As a result, a good simultaneous description of equilibria and kinetics in these systems was obtained.
1. INTRODUCTION The modern methods of gas separation and purification utilize adsorption phenomena. For example, a very popular method of industrial gas separation and purification competitive with the cryogenic method is pressure swing adsorption (PSA). Separation of gases in the PSA method can be based on the equilibrium properties of the adsorption system or performed by utilizing the kinetics of adsorption/desorption. Thus, the time dependence of the adsorption process is at least as important as the features of these systems in equilibrium. The adsorption kinetics at the solid/gas and solid/solution interfaces has been studied extensively by using different theoretical models. These models include pseudo-first-order,1 pseudosecond-order,2 and multiexponential equations.3 Recently an increase of interest in the statistical rate theory (SRT)4,5 has been observed. SRT links the rate of adsorption/desorption process to the difference between the chemical potentials of bulk and adsorbed phases. Its most important advantage is the possibility to determine various kinetic expressions related to different theoretical models of adsorption equilibria. Most applications of SRT, which take into account the energetic heterogeneity of adsorption system, use equilibrium isotherms derived using the condensation approximation (CA) and are thus subject to the CA associated error. This is due to the fact that CA leads to substantial simplification in the integral equation (IE) approach6 to describe adsorption on energetically heterogeneous surfaces. As CA can lead to serious errors,7,8 it is interesting to verify SRT of gas adsorption for the case when the equilibrium theoretical isotherm is the exact solution of the IE approach. The general purpose of this work is to show a new SRT of gas adsorption which also takes into account the lateral interactions between the adsorbed molecules. The proposed theoretical model will be verified using experimental kinetic data previously published.
adsorption equilibria by using adsorption isotherms of the pure components.6 IE is based on the integral representation of the total surface coverage θt(p,T): Z θl ðε, p, TÞ χðεÞ dε ð1Þ θt ðp, TÞ ¼
2. THEORY
Received: June 23, 2010 Accepted: January 13, 2011 Revised: December 22, 2010 Published: February 07, 2011
2.1. Adsorption Equilibrium. The integral equation (IE) approach is one of the most frequently used to predict mixed-gas r 2011 American Chemical Society
Ωi
where θl(ε,p,T) is the fractional coverage of a certain class of adsorption sites characterized by the adsorption energy ε (local adsorption isotherm) and χ(ε) is the differential distribution of the number of the adsorption sites among various values of ε. Most applications of IE require a local model of adsorption. In this case the local isotherm is represented by the Langmuir equation. To take into account the interaction effects between the adsorbed molecules, we use its following generalization: ε þ ωθt KL p exp kT ð2Þ θl ¼ ε þ ωθt 1 þ KL p exp kT In expression 2, θt = Nt/M is the partial surface coverage, Nt is the total number of molecules adsorbed at pressure p and temperature T, M is the maximum capacity, KL is the Langmuir constant, and ω is the product of the interaction energy between two molecules adsorbed on the two nearest-neighbor adsorption sites and the number of the nearest-neighbor sites on a given lattice of sites. Expression 2 is the well-known Fowler-Guggenheim, Bragg-Williams, or Frumkin isotherm and is the extension of the Langmuir equation for the case when the interaction effects between the adsorbed molecules are described by the mean field approximation. To calculate the integral equation (1), many adsorption energy distribution (AED) functions χ can be used. One of the
3078
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1-1 !Nt =M 10 !Nt =M l K L A@K m - KLl A L KLm ω Nt exp ð6Þ kT M
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In eq 6 there appear only four best-fit parameters: KlL, Km L , ω, and M. However, it is still the exact solution of the integral equation for the case of energetically heterogeneous surfaces. 2.2. Adsorption Kinetics. The adsorption rate is one of the most important factors which determine the practical application of many adsorption systems in industry. Although a number of theoretical models of adsorption kinetics has been proposed, there is still a need for an exact description of this phenomenon. Thus in recent decades a growth of interest in new concepts concerning the theoretical description of kinetics of adsorption has been observed. The most commonly used equations of adsorption kinetics are the pseudo-first-order (PFO) and pseudo-second-order (PSO) equations. The former was developed by Lagergren.1 It has the following form: dNðtÞ ¼ k1 ðN ðeÞ NðtÞÞ dt
ð7Þ
where N(t) is the amount adsorbed at time t, N(e) is the amount adsorbed at equilibrium, and k1 is the pseudo-first-order rate constant. The linear form of the above expression is known just as the Lagergren equation. Recently Rudzinski and Plazinski10 have shown that the linear form of the PFO equation can be used to decide whether the rate of adsorption is determined by surface reaction or diffusion effects. Another simple and widely used expression is the pseudosecond-order (PSO) equation:2 dNðtÞ ¼ k2 ðN ðeÞ - NðtÞÞ2 dt
ð8Þ
where k2 is the rate constant of the pseudo-second-order adsorption. As was shown by Azizian,11 the kinetic equation being the foundation and justification of the Langmuir isotherm explains both PFO and PSO equations as the boundary cases of the Langmuir kinetic models. At the beginning of the 1980s Ward and Findlay proposed the statistical rate theory of interfacial transport (SRT).4,5 SRT is based on the assumption that the adsorption rate depends on the difference in the chemical potentials of the molecules in the bulk and solid phases. In the case of monolayer, localized adsorption, the fundamental SRT expression for the adsorption kinetics reads g s Dθt μ - μs μ - μg 0 ¼ Kgs exp - exp ð9Þ Dt kT kT where t is the time, μsi and μgi are the chemical potentials of adsorbate in the solid and bulk phases, respectively, and K0gs is the exchange rate between the gas phase and the solid surface when an isolated system has reached equilibrium. One of the most important advantages of SRT is the possibility to calculate the adsorption rate by using the pure gas adsorption equilibrium data. The influence of lateral interactions between the adsorbed molecules on the adsorption kinetics has been considered by Panczyk and Szabelski.12 However, they studied only the case of adsorption on an energetically homogeneous surface. Our present studies are more general: we take into account both interactions and the energetic nonideality of the adsorption system. It is important to notice that at the initial stage of adsorption, i.e., when the surface coverage is very small, 3079
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Figure 1. Influence of the energetic heterogeneity and interaction effects between the adsorbed molecules on the theoretically calculated kinetic isotherms by using eq 12. The values of the parameters are noted in the figures.
leads to the following expression for the rate of adsorption:
Table 1. Main Textural Characteristics of Adsorbents Used21,22
( )! εm εl þ Δεθt exp - exp kT kT Δεθt exp -1 kT 3 Δεθt 7 exp -1 7 1 kT 7 ( ) ! - 0 7 Kp 5 ωθt εl þ Δεθt εm exp - exp exp kT kT kT 2
particle bulk density
total porosity
BET surface
adsorbent
[kg/m3]
[pores/m3 particle]
area [m2/g]
Norit AC
595.6
0.74
1052
Ajax AC
660.0
0.71
1200
molecular interactions have no significant influence on the rate of adsorption. In this case the adsorption rate is controlled by the energy of adsorption and/or by diffusion effects. The role of interactions between the adsorbed molecules increases with the growing surface coverage. Most applications of the SRT approach concerning the adsorption rate are based on the adsorption isotherms determined by using IE with the condensation approximation.12-15 For example, by using the Tiemkin isotherm (which corresponds to the uniform AED function (3)), the SRT approach leads to the Elovich equation.16 Nevertheless it is well-known that CA sometimes can lead to serious errors.7,8 For that purpose, it is interesting to determine the expressions describing the adsorption rate free from such errors. To our knowledge it is possible in the case of uniform energy distribution (2). Therefore, let us use the SRT approach with isotherm 4. In such a case the chemical potential of the adsorbed phase can be expressed as follows: Δεθt exp -1 kT s μ ¼ kT ln ωθt εm εl þ Δεθt exp - exp KL exp kT kT kT
ωθt 6 6 exp kT Dθt 0 6 ¼ Kgs 6K 0 p 6 Dt 4
ð11Þ where K 0 = KL exp{μg0 /kT}. The constant K0gs can be combined with K 0 . Following Azizian, 17 we can rewrite eq 11: ( )! m ωθt ε εl þ Δεθt exp exp - exp kT kT kT Dθt ¼ Ka p Δεθt Dt exp -1 kT Δεθt exp -1 Kd kT ( ) m ! p ωθt εl þ Δεθt ε exp - exp exp kT kT kT ð12Þ where the adsorption and desorption rate constants are defined as (
g
μ Ka ¼ Kgs KL exp 0 kT
ð10Þ
0
While assuming the ideal gas phase, the SRT equation (9) 3080
) ð13Þ
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Figure 2. Comparison of the experimental equilibrium isotherms for ethane/Norit activated carbon adsorption and propane/Norit activated carbon measured by Qiao and Hu21 with the theoretical isotherms calculated using eq 4. The values of the obtained best-fit parameters are collected in Table 1.
Table 2. Values of the Parameters Obtained by Applying eq 4 to the Experimental Equilibrium Isotherms of Ethane and Propane Adsorbed on Norit Activated Carbon21 adsorbate T [K] M [mmol/g] KlL [kPa-1] ω/k [T] Δε/k [T] Error1 C2H6
C3H8
303
9.44
2.32 10-6
3.887
-6
333
8.73
2.42 10
363
7.17
3.28 10-6
303
10.53
1.05 10-8
333
10.39
1.49 10-8
363
9.58
2.54 10-8
469.22
3651.67 17.592 13.206 17.391
634.27
6232.50 18.716 2.667
0
Kd ¼
Kgs (
g
μ KL exp 0 kT
)
ð14Þ
Rudzinski et al. showed also the possibility of calculating K 0gs by considering the physical conditions of the kinetic experiment. 18 Then 1 0 m ε þ ωθt C B 1 þ Kp exp C B kT 0 C B kT ( ) C ð15Þ B ln Kgs ¼ Kgs p 1 B l Δε ε þ ωθt C A @ 1 þ Kp exp kT Expression 15 has been obtained with the assumption that the gas pressure above the surface does not change much. In other words, in the nonequilibrium adsorption process the gas in the gaseous phase above the surface exceeds the amount of the adsorbed portion significantly. 18 Recently19,20 it has been shown that the SRT of gas adsorption gives more correct results when changes in pressure between the bulk vapor phase and the near-surface region are taken into account. It is especially important when the adsorption process is
very fast. To account for this effect, eq 12 should be solved with the following expression:20 Dp Dθt ¼ Rðp0 - pÞ - β ð16Þ Dt Dt where p0 is the bulk gas pressure and the parameters R and β are expressed as follows: rffiffiffiffiffiffiffiffiffi 1 kT SA kT ð17Þ R¼ β ¼ N0 h 2πm SG h where SG, SA, h, and N0 are the quantities characterizing the properties of the surface of the adsorbing phase: SG is the area of the interface between the near-surface region and the bulk gas phase, h is the distance between SG and the area of adsorbing surface SA, N0 is the number of adsorption centers per unit of surface area, and m is the mass of the adsorbing molecule. The theoretical basis of eq 16 can be found elsewhere.20 Although SRT is based on the assumption that the rate of surface reaction is the rate-determining step in the adsorption process, it can be used also when surface diffusion plays a significant role in adsorption kinetics. It is only necessary to use eq 16, which simulates the influence of diffusion on the adsorption rate very well.20 Summing up, we determined two new equations, eqs 4 and 12, which describe the equilibrium and kinetics of single-gas adsorption. During the theoretical analysis of the experimental equilibrium isotherms, the use of eq 4 leads to the following values of the best-fit parameters: M, εl/kT, εm/kT, ω/kT, and KL. These values are necessary in the theoretical analysis of kinetic isotherms by means of eq 12. Such a calculation technique reduces the number of best-fit parameters determined from the kinetic curves. The application of SRT, eq 12, requires only two best-fit values of the rate constants Ka and Kd. An attentive reader can notice that the adjustment of the theoretical adsorption isotherm (4) or (5) to the experimental data is not as easy as it seems to be. This is due to the fact that the parameters KL, εl/kT, and εm/kT appear in eq 4 as the product of KL exp{εl/kT} and KL exp{εm/kT}. Then there exist many 3081
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different values of these parameters which can lead to satisfactory adjustment of isotherm 4 to the experimental data. Such a case is frequently called a “correlation of the parameters”. The problem is to determine “true” values of KL, εl/kT, and εm/kT. It can be done by analyzing concurrently a few experimental isotherms of a given adsorption system. Another possible solution of that problem is the theoretical analysis of calorimetric effects accompanying gas adsorption. Unfortunately, it is not easy to find in the literature simultaneously measured equilibrium and kinetic isotherms as well as enthalpic effects accompanying them. Therefore, we suggest another technique of calculations. That is, the analysis of equilibrium isotherms by means of eq 5 gives inter alia the values m of KlL = KL exp{εl/kT} and Km L = KL exp{ε /kT}. Then, the kinetic single-gas adsorption isotherms can be adjusted by the following modification of eq 12: Dθt Kd ¼ Ka pKLl F - m Dt pKL F where
F ¼ exp
ωθt kT
exp
Δε Δεθt - exp kT kT Δεθt exp -1 kT
ð18Þ
ð19Þ
and Δε can be calculated as follows: Δε ¼ kT ln
KLm KLl
ð20Þ
We have often mentioned that the accuracy of calculations can be easily increased by analyzing a few equilibrium adsorption isotherms of a given adsorption system measured at different temperatures concurrently. In that case, we can adjust the best-fit parameters ω/k and Δε/k simultaneously to all equilibrium isotherms. This is due to the fact that these values are temperature independent and should be common for different isotherms of a given adsorption system. An attentive reader can notice that also the maximum adsorbed amount M should be temperature independent. From the theoretical point of view it is true because the presented calculations are based on the lattice model of adsorption. The number of adsorption centers should not be temperature dependent. In practice, the maximum adsorbed amount M determined from the pure gas adsorption equilibrium isotherms varies with temperature: the experimentally measured isotherms at different temperatures have different shapes and achieve the plateau at different surface coverages. With the increasing temperature, the plateaus of adsorption isotherms usually shift to the region of lower surface coverages.
3. RESULTS AND DISCUSSION Now we are going to analyze the applicability of derived equations (4) and (18). Although the theoretical isotherm (4) has already been reported in the literature, the new SRT expression (18) derived with the AED function (3) has not been investigated. For that purpose, it is reasonable to perform some parametric studies. Figure 1 shows the influence of the values of the best-fit parameters on the behavior of the theoretically calculated kinetic isotherms.
The kinetic curves theoretically calculated by using eq 12 show, that with the growing interaction energy ω, the rate of adsorption increases. However, at the initial time of adsorption, the interaction energy ω does not influence significantly the rate of adsorption. The increase of the energetic heterogeneity of adsorbent is reflected by a stronger increase of θt drawn vs time. In other words, the increase of the force field acting on the adsorbed molecule (created by the increase of adsorption energy or the presence of other molecules) causes the increase of adsorption rate under the same bulk conditions. The true verification of the expressions presented in section 2 will be their application to describe the real adsorption system. As eqs 4 and 18 are based on the exact solution of integral equation (1), they are applicable to the surfaces of arbitrary adsorption energy heterogeneity. It seems that eqs 4 and 18 can be used to analyze not only the gas adsorption on truly heterogeneous adsorbents as activated carbons but also the adsorption of gases on materials which are nearly energetically homogeneous. The practical calculations of the rate of adsorption by SRT, eq 18, and by taking into account the change of the pressure of gas phase near the surface region, eq 16, seem to be difficult. The problem is to solve the set of two differential equations with four best-fit parameters which are chosen by a numerical procedure to improve the agreement between the theoretically calculated and experimentally measured adsorption rate curves. We have coped with it by using the Mathematica 7.0 software. By assuming physical limits of the adjusted best-fit parameters, calculations have been performed by applying the random walk Monte Carlo procedure with 106 steps. At each step, the values of the best-fit parameters occurring in eqs 16 and 18 were generated randomly to find the best correlation quality of experimental data. Each calculation was repeated a few times with a gradual decrease of the search limits of each parameter. To present the quality of correlations of equilibrium and kinetic isotherms, we used two error functions. Obviously, this fit is better when the value of an error function is smaller. First is the well-known and frequently used residual sum of squares: X ðexp - theorÞ2 Error1 ¼ ð21Þ n
where n is the number of experimental points. As during the numerical calculations we show the comparison of the quality of fit of the SRT equation (18) and the Lagergren equation (7) fit to the experimental data, we also should employ the sum of squared differences metric weighted by the number of degrees of freedom in the fit. This is because these models have different numbers of free parameters. In that case we used the following error function: Error2 ¼
X ðexp - theorÞ2 n-f n
ð22Þ
where f is the number of the best-fit parameters in a given model of calculations. We showed previously that some information necessary for theoretical predictions of kinetic curves is determined by using the equilibrium adsorption isotherms (they are M, KL, ω/k, and Δε/k). Then, the numbers of the best-fit parameters in the models used are as follows: • Lagergren equation (7): f = 1 (N(e)) • SRT equation (18) and eq 16: f = 4 (Ka, Kd, R, β) It can be seen that, comparing the quality of different theoretical models fit to the experimental data, the influence of 3082
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Table 3. Values of the SRT Best-Fit Parameters Obtained When Fitting the Measured Kinetic Isotherms Presented in Figures 3-5 (Norit AC) T [K]
adsorbate C2H6
303
C3H8
Ka
gas phase component [%]
4.23 10
2
R
Kd -5
2.97 10
5.42 10
31.12
Error2 -3
1.74 10-3
-2
6.96 10
1.11 10 1.88 10-4
3.47 10 2.92 10-11
8.17 10 3.37 10-6
43.51 59.76
1.16 10 1.51 10-2
2.90 10-3 3.78 10-3
303
2
2.89 10-5
2.01 10-7
3.34 10-1
1069.67
2.56 10-3
6.40 10-4
-3
3.26 10
-5
-7
-3
1.23 10-3
1.47 10
-7
-2
2.65 10-3
4.92 10
303
-5
3.80 10
10
-6
Error1
5 10 5
-16
β -5
303 303 303
-4
-7
2.05 10
-1
3.75 10
1051.50 1387.35
4.92 10 1.06 10
Table 4. Lagergren Parameters of Ethane and Propane on Norit Activated Carbon21 adsorbate
T [K]
gas phase component [%]
k1
C2H6
303
2
3.550 10-3
C3H8
-3
C3H8
CO2
0.4918
3.06 10-2
4.37 10-3
-2
Error2
5
5.070 10
0.9477
2.94 10
4.20 10-3
303
10
6.624 10-3
1.4139
2.50 10-2
3.57 10-3
303
2
2.064 10-3
1.6769
1.07 10-2
1.53 10-3
303 303
5 10
2.905 10-3 4.018 10-3
2.1852 2.5759
2.10 10-2 2.38 10-2
3.00 10-3 3.40 10-3
adsorbate T [K] M [mmol/g] KlL [kPa-1] ω/k [T] Δε/k [T] Error1
C2H6
Error1
303
Table 5. Values of the Parameters Obtained by Applying eq 4 to the Experimental Equilibrium Isotherms of Ethane and Propane Adsorbed on the Ajax Activated Carbon22
CH4
N(e)
258
8.04
7.95 10-6
0.5548
273
7.93
6.67 10-6
3.7281
283
7.82
7.00 10-6
303
7.38
283 303
10.5851 10.3435
371.55
2395.07
-6
7.29 10
1.97 10-4 1.50 10-4
1.5900 3.1108
47.45
24.6611 2038.74 30.5918 31.8950
8.62054
1.46 10-4
303
8.5864
333
8.53183
1.07 10-4 133.38 7.68 10-5
3073.66 13.1548 3.0917
258
7.55582
1.07 10-3
273
7.27558
303
6.97362
7.38 10-4 197.94 4.45 10-4
333
9.71165
1.22 10-4
283
26.6350
2.4493 1201.15
6.7088 10.8992
the number of best-fit parameters f on the Error2 value decreases with the increasing number of experimental points. The experimental data used to verify the presented new expression should include the single-gas equilibrium and kinetic isotherms. The data reported by Qiao and Hu21,22 satisfy the above requirements. In a few papers the authors studied the adsorption of simple gases on various activated carbons (ACs). The brief characteristics of adsorbent porosity are listed in Table 1. We start our numerical studies by using the experimental data concerning the adsorption of ethane and propane on the Norit activated carbon.21 All details related to the experimental procedures can be found in ref 21. As the single-gas adsorption isotherms were measured at different temperatures, this makes it possible to determine correctly the temperature-independent values of ω/k and Δε/k because they are common for all isotherms of a given adsorption system. It is particularly important in the case of the adsorption isotherms reported by Qiao and Hu21 which were determined in low-range adsorbate pressures, thus making the correct theoretical description of the
experimental adsorption isotherms complicated (e.g., correct determination of the maximum capacity). The results of adjustment of eq 4 to the single-gas isotherms reported by Qiao and Hu21 are shown in Figure 2 and Table 2. A very good agreement between the theoretically calculated and the experimentally measured equilibrium isotherms is found. Looking at the values of ω/k collected in Table 2, it can be seen that the interaction effects between the adsorbed molecules play an important role in adsorption of C2H6 and C3H8 on Norit AC. The values of ω/kT determined for ethane are in the range from about 1.3 to 1.5, whereas those determined for propane range from 1.7 to 2.1. Also the heterogeneity parameter Δε/k determined for propane is about twice as large as that for ethane. It is worth noting here that energetic heterogeneity is a property of the entire adsorption system, not only of the adsorbent or adsorbate. The heterogeneity parameters Δε/k collected in Tables 2 and 5 show that the heterogeneous character of an adsorption system increases with the increasing number of repeating units in an alkane molecule. Such regularity can be confirmed by analyzing the calorimetric effects accompanying gas adsorption. This is due to the fact that the heat of adsorption is more sensitive to the nature of the adsorption system than the corresponding equilibrium adsorption isotherms. More precisely, an increase in the heat of adsorption with gas loading is characteristic of homogeneous adsorbents with constant gas-solid interaction energies. The increase is due to cooperative interactions between the adsorbed molecules. A decrease in the heat of adsorption with gas loading is characteristic of highly heterogeneous adsorbents with a wide distribution of gas-solid interaction energies. A constant heat of adsorption with gas loading indicates a balance between the strength of cooperative gas-gas interactions and the degree of heterogeneity of gas-solid interactions. The measured kinetic isotherms were analyzed by using the equation system composed of eqs 16 and 18. It has the form 8 Dθt Kd > > ¼ Ka pKLl F - m < Dt pKL F ð23Þ Dp Dθ > > : ¼ Rðp0 - pÞ - β t Dt Dt 3083
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Figure 3. Comparison of the experimental kinetic isotherms for ethane/Norit activated carbon adsorption and propane/Norit activated carbon adsorption measured by Qiao and Hu21 (points) with the theoretical isotherms (solid lines) calculated using eq 23. Dashed lines are the rate of adsorption calculated using the Lagergren equation (7). The values of the obtained best-fit parameters are collected in Tables 2-4. The bulk phase concentration is denoted in the figure.
Figure 5. Comparison of the experimental kinetic isotherms for ethane/Norit activated carbon adsorption and propane/Norit activated carbon adsorption measured by Qiao and Hu21 (points) with the theoretical isotherms (solid lines) calculated by means of eq 23. Dashed lines are the rate of adsorption calculated using the Lagergren equation (7). The values of the obtained best-fit parameters are collected in Tables 2-4. The bulk phase concentration is denoted in the figure.
and should be solved with the following boundary conditions: θt(t = 0) = 0 and p(t = 0) = p0. Qiao and Hu21 determined kinetic isotherms as “fractional uptake” versus time. The fractional uptake is defined as the uptake at any time divided by its value at final equilibrium (t f
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Figure 4. Comparison of the experimental kinetic isotherms for ethane/Norit activated carbon adsorption and propane/Norit activated carbon adsorption measured by Qiao and Hu21 (points) with the theoretical isotherms (solid lines) calculated using eq 23. Dashed lines are the rate of adsorption calculated using the Lagergren equation (7). The values of the obtained best-fit parameters are collected in Tables 24. The bulk phase concentration is denoted in the figure.
¥). In order to compare the results of calculations with the experimental data, the values of ∂θ/∂t should be multiplied by M/N(e), where N(e) can be calculated by using eq 4 for p = p0. During calculations we assumed the values of the best-fit parameters collected in Table 2 and adjusted the next four values: Ka, Kd, R, and β. For comparison we decided to calculate also the rate of adsorption by applying the Lagergren equation (7). This is due to the fact that the pseudo-first-order equation is one of the most commonly used in adsorption kinetics. The equilibrium adsorbed amount N(e) was calculated by using eq 4 for given bulk phase concentrations with the best-fit parameters collected in Table 2. The results of these calculations are shown in Tables 3 and 4 and Figures 3-5. As the kinetic experimental data reported by Qiao and Hu21 have been measured at 303 K, we used the equilibrium best-fit parameters collected in Table 2 determined from the equilibrium isotherms measured at the same temperature. The values of Error1 and Error2 functions collected in Tables 3 and 4 show that, in comparison to the Lagergren equation (7), the SRT equation (23) is in better agreement with the experimental data. With the growing bulk phase concentration, the agreement between the theoretically calculated uptake curves and the experimental data diminishes. This may be caused by an underestimation of the values of best-fit parameters determined from the equilibrium isotherms. Figures 3-5 show that the isotherms were not measured at a sufficiently wide range of adsorbate pressures: the experimental adsorption isotherms did not reach a plateau. As the other example, we present here an analysis of the experimental data published by Qiao and Hu.22 This paper reports adsorption of CH4, C2H6, C3H8, and CO2 on the Ajax activated carbon. All details related to the experimental procedures are similar to the previously analyzed data and can be found 3084
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Figure 6. Comparison of the experimental equilibrium isotherms for ethane/Ajax activated carbon adsorption and propane/Ajax activated carbon measured by Qiao and Hu22 with the theoretical isotherms calculated using eq 4. The values of the obtained best-fit parameters are collected in Table 4.
Table 6. Values of the SRT Best-Fit Parameters Obtained When Fitting the Measured Kinetic Isotherms Presented in Figures 7-9 (Ajax AC) adsorbate C2H6
T [K] 303 303 303
C3H8
gas phase component [%] 5 10 20
Ka
R
Kd -5
7.82 10
-4
1.66 10
-4
2.94 10
-9
6.22 10
-9
1.01 10
-10
6.27 10
-5
-5
β
1.77 10
-5
3.01 10
-6
1.38 10
-10 -1
Error2
58.93
1.04 10
-2
73.04
2.01 10
-2
6.70 10-3
94.75
3.01 10
-2
1.51 10-2
-3
8.96 10 5.41 10-3
2.99 10-3 2.71 10-3
5.58 10-4
2.79 10-4
303 303
5 10
4.46 10 7.68 10-5
5.20 10 2.41 10-6
3.78 10 6.38 10-1
2254.86
303
20
1.07 10-4
1.17 10-6
2.21 10-1
1496.66
in ref 22. In the case of methane and ethane the authors report both kinetic and equilibrium isotherms of single- and mixed-gas adsorption systems. The adsorption data of methane and carbon dioxide relate only to the equilibrium single- and mixed-gas adsorption and the mixed-gas kinetic isotherms. Then we cannot calculate the kinetic single-gas isotherms of these gases.
Error1
3357.59
3.47 10-3
Nevertheless, to show applicability of eq 4, we adjusted also the equilibrium adsorption isotherms of CH4 and CO2. The results of the correlation of equilibrium adsorption isotherms of CH4, C2H6, C3H8, and CO2 are shown in Table 5 and Figure 6. It can be seen that isotherm 4 in all cases describes the experimental data very well. Since all adsorption isotherms 3085
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Table 7. Lagergren Parameters of Ethane and Propane on Ajax Activated Carbon22 adsorbate
T [K]
gas phase component [%]
k1
C2H6
303
5
4.064 10-3
303 303 C3H8
303 303 303
10 20 5 10 20
-3
5.331 10
-3
6.482 10
-3
1.772 10
-3
3.036 10
-3
4.498 10
N(e)
Error1
0.7608
1.73 10-2
2.88 10-3
-3
1.35 10-3
-3
1.92 10-3
-3
1.24 10-3
-3
1.14 10-3
-2
3.68 10-3
1.2735 1.9703 2.3774 2.9622 3.5605
Error2
8.07 10 9.60 10 7.46 10 5.71 10 1.84 10
Figure 7. Comparison of the experimental kinetic isotherms for ethane/Ajax activated carbon adsorption and propane/Ajax activated carbon adsorption measured by Qiao and Hu22 (points) with the theoretical isotherms (solid lines) calculated using eq 23. Dashed lines are the rate of adsorption calculated using the Lagergren equation (7). The values of the obtained best-fit parameters are collected in Tables 5-7. The bulk phase concentration is denoted in the figure.
Figure 8. Comparison of the experimental kinetic isotherms for ethane/Ajax activated carbon adsorption and propane/Ajax activated carbon adsorption measured by Qiao and Hu22 (points) with the theoretical isotherms (solid lines) calculated using eq 23. Dashed lines are the rate of adsorption calculated using the Lagergren equation (7). The values of the obtained best-fit parameters are collected in Tables 5-7. The bulk phase concentration is denoted in the figure.
were determined at a few temperatures, we can minimize the effect of correlations between the best-fit parameters in eq 4. Similarly to the adsorption on Norit AC, we can state that in the case of alkane adsorption the energetic heterogeneity increases with the growing number of CH2 repeating units. In comparison to the Norit AC adsorption data, the interactions between the adsorbed molecules are visibly smaller. The values of ω/kT are, except for methane adsorption, always less than 1. While studying the kinetics of mixed-gas adsorption, Qiao and Hu21,22 concluded that propane is more strongly adsorbed than ethane. It confirms the best-fit values collected in Tables 2 and 5. l By calculating Km L = KL exp{Δε/kT}, it can be seen that the value m of KL drastically increases with the growing number of CH2 repeating units in the chain of adsorbed species. While assuming that the values of the Langmuir constant KL are similar for the m studied alkanes, Km L = KLexp{ε /kT} is directly connected with the maximum energy of adsorption. Then, free energy of adsorption becomes more favorable in the series: methane < ethane
