Dubinin adsorption isotherm equation

3 Determination of the Micropore Volume 6.7.3.1 Dubinin Adsorption Isotherm Equation 

The Dubinin adsorption isotherm equation is a good tool for the measurement of the micropore volume. This isotherm can be deduced with the help of Dubinin s theory of volume filling, and Polanyi s adsorption potential [11,26], The Dubinin adsorption isotherm equation has the following form [11] 

P0 is the vapor pressure of the adsorptive at the temperature, T, of the adsorption experiment P is the equilibrium adsorption pressure P is a parameter called the characteristic energy of adsorption 

It is possible, as well, to express the Dubinin adsorption isotherm equation in linear form 

FIGURE 6.11 Dubinin plot for the adsorption of NH3 at 300K in the sample HC. 

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Wood, G O., "Affinity Coefficients of the Polanyi/Dubinin Adsorption Isotherm Equations a Review with Compilations and Correlations," Carbon. 2001 39 343 356. 

Observing the adsorption isotherm equation (4.2-2), we note that if the characteristic energy is independent of temperature, plots of the fractional loading versus the adsorption potential for different temperatures will collapse into one curve, called the characteristic curve. This nice feature of the Dubinin equation makes it convenient in the description of data of different temperatures. 

The adsorption on a solid surface, the types of adsorption, the energetics of adsorption, the theories of adsorption, and the adsorption isotherm equations (e.g., the Langmuir equation, BET equation, Dubinin equation, Temkin equation, and the Freundlich equation) are the subject matter of Chapter 2. The validity of each adsorption isotherm equation to the adsorption data has been examined. The theory of capillary condensation, the adsorption-desorption hysteresis, and the Dubinin theory of volume fllhng of micropores (TVFM) for microporous activated carbons are also discussed in this chapter. 

Hutson, N.D., and Yang, R.T., Theoretical basis for the Dubinin-Radushkevitch (D-R) adsorption isotherm equation. Adsorption, 3(3), 189-196 (1997). 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

Adsorption isotherms in the micropore region may start off looking like one of the high BET c-value curves of Fig. XVII-10, but will then level off much like a Langmuir isotherm (Fig. XVII-3) as the pores fill and the surface area available for further adsorption greatly diminishes. The BET-type equation for adsorption limited to n layers (Eq. XVII-65) will sometimes fit this type of behavior. Currently, however, more use is made of the Dubinin-Raduschkevich or DR equation. Tliis is Eq. XVII-75, but now put in the form... 

Porous texture characterization of all the samples was performed by physical adsorption of N2 at 77K. and CO2 at 273K, using an automatic adsorption system (Autosorb-6, Quantachrome). The micropore volume, Vpp (N2), was determined by application of Dubinin-Radushkevich equation to the N2 adsorption isotherm at 77K up to P/Po< 0.1. The volume of narrow micropores, Vnpp (DR,C02>, (mean pore size lower than 0.7 nm) was calculated from CO2 adsorption at 273 K. 

Comparison Between the Cohen-Kisarov and Dubinin-Radushkevich Equations. In a plot of log q vs. e2 the experimental points for one adsorption isotherm on zeolite frequently do not give a straight line, which would verify the Dubinin-Radushkevich equation. In this case, two distinct lines of different slopes are found (4). 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

As described earlier, one of the first methods used to obtain PSD from the Dubinin equation is the so-called Dubinin-Stoeckli method [38-43], For strongly activated carbons with a heterogeneous collection of micropores, the overall adsorption isotherm is considered as a convolution of contributions from individual pore groups. Integrating the summation and assuming a normal Gaussian equation for the distribution of MPV with respect to the K parameter (Equation 4.19), Stoeckli obtained an equation useful to estimate the micro-PSD. 

This equation is a powerful tool for the description of the adsorption data in microporous material. In Figure 6.11, the Dubinin plot of the adsorption isotherm in the range 0.001 < P/P0 < 0.03, describing the adsorption of NH3 at 300 K in the natural clinoptilolite sample HC is shown (see Table 4.1) [25], The adsorption data reported in Figure 6.11 were determined volumetrically in a home-made Pyrex glass vacuum system, consisting of a sample holder, a dead volume, a dose volume, a U-tube manometer, and a thermostat [25,31], It is evident that, in the present case, the experimental data is accurately fitted by Equation 6.20. 

In Figure 6.12 [2,25], the plot of the linear form of the osmotic isotherm equation, with B = 0.5, using adsorption data of NH3 adsorbed at 300 K in an homoionic magnesium natural zeolite sample labeled CMT (see Table 4.1), is shown. The adsorption data reported in Figure 6.12 were determined volumetrically in a Pyrex glass vacuum system, previously described in the case of the Dubinin equation [25,31], With this plot, it is possible to calculate the maximum adsorption capacity of this zeolite, which is m = Na = 5.07mmol/g and b = UK = -0.92 (Torr)05. 

At this point, it is feasible to correlate the liquid-phase adsorption equilibrium single component data, with the help of isotherm equations developed for gas-phase adsorption, since, in principle, it is feasible to extend these isotherms to liquid-phase adsorption by the simple replacement of adsorbate pressure by concentration [92], These equations are the Langmuir, Freundlich, Sips, Toth, and Dubinin-Radushkevich equations [91-93], Nevertheless, the Langmuir and Freudlich equations are the most extensively applied to correlate liquid-phase adsorption data. [2,87],... 

We expected that the adsorption isotherms could be described with help of Dubinin s equation, but analysis of adsorption isotherms was shown that ones have a breaking of line in region of low pressure and this single-parameter equation isn t able to describe them exactly. 

The micropore volume is defined as the pore volume of the pores < 2 nm. Microporous volumes calculated from the application of the Dubinin-Radushkevich equation to the N2 adsorption isotherms at 77 K. The mean pore size of each sample obtained from N2 adsorption was determined by applying Dubinin-Radushkevich equation. The hydrogen sorption isotherms were measured with the High Speed Gas Sorption Analyser NOVA 1200 at 77 K in the pressure range 0-0.1 MPa. 

The influence of temperature can be seen on Figs. 8-9. The storage capability is increasing for lower temperatures. Figure 9 compares the behaviour of the adsorption isotherms at different temperature levels for two of the more promising samples steam activated Busofit-M8 and wood-based carbon WAC 3-00 . The shape of the isotherms in the two cases is dissimilar. The isotherms for the 77 and 153 K exhibit a classical type 1 isotherm shape indicating a microporous material. The isotherms at room temperature exhibit a much less pronounced curvature (more like type II isotherm). As is seen from plots (Fig. 9) experimental data fit the calculated adsorption values (Dubinin-Radushkevich equation) with an error sufficient for practical purposes. 

Figure 9. Hydrogen adsorption isotherms for active carbon fiber Busofit-M8 (a), wood-based cardon WAC 3-00 (b) and different temperatures (1 - 77, 2 -153, 3 - 193, 4 - 293 K) experimental data - points, calculated data (Dubinin-Radushkevich equation) - lines.

Stoeckli (1993) has pointed out that the Dubinin-Astakhov equation (Equation (4.45)) can be derived from Equation (4.52), but McEnaney (1988) and others (e.g. Jaroniec et al. 1997) have drawn attention to the difficulty in arriving at an unambiguous interpretation of the energy distribution function. Indeed, Stoeckli et al. (1998) have now pointed out that Equation (4.45) can be usefully applied to a number of adsorption isotherms on non-porous solids. A comprehensive review of the significance and application of Equation (4.52) is given by Rudzinski and Everett (1992). 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

Textural characterisation of the samples was carried out by measuring apparent density (mercury at 0.1 MPa), mercury porosimetry and N2 and CO2 adsorption isotherms, at -196 and 0 °C, respectively. The apparent surface areas of the samples were obtained by using the BET equation [5]. The micropore size analysis was performed by means of the t-plot and the Dubinin-Astakhov methods [6]. 

The Dubinin-Astakhov (D-A) [6] equation was applied to the N2 adsorption isotherms. The accessible pore width, L, was calculated fi om the expression proposed by Stoeckli and Ballerini [9]. 

Several methods have been proposed for the characterisation of the Micropore Size Distribution (MPSD) that take into account the energetic heterogeneity of solid surfaces [9,10]. The Dubinin-Radushkevich (DR) and Dubinin-Astakhov (DA) equations have been used to describe the adsorption process on structurally heterogeneous solids [11,12]. From these equations, the adsorption isotherm can be expressed as follows ... 

The pore volumes of the obtained hard carbons were measured using the molecular probe method [3]. Adsorption isotherms of the probe molecules were measured at 298 K using an adsorption apparatus (Bel Japan, Belsorp 28). The employed probe molecules were CO2, C2H6> n-C4H o and (-C4H10 (minimum molecular dimensions 0.33, 0.40, 0.43 and 0.50 nm, respectively). By applying the Dubinin-Astakhov equation (n=2) [1] to the measured isotherms, the limiting micropore volumes corresponding to the minimum size of the adsorbed molecules were determined. 

A similar technique is based on the theory of micropore volume filling. It states that the total microporous volume accessible to a given adsorbate can be obtained from the Dubinin-Radushkevich equation as a function of the temperature, relative pressure, and characteristic energy of adsorption. When this procedure is applied to a few linear or spherical molecules (as probes) of different but known sizes, the adsorption isotherms of these gases at the same temperature can be employed in combination with their... 

The main porous structure characteristics (Table 2) were determined on the basis of benzene vapor adsorption isotherms using McBain-Baker sorption balances at 20°C (293 K), i.e., the specific BET surface area (5bht) [39], the surface area of mesopores (5 ,e), and the parameters of the Dubinin-Radushkevich equation (the volumes of the micropores and supermicropores. Woi and W 2, and the characteristic energies of adsorption, E, and o ) 136,37). In addition, the total micropore volume (ZVT, ) and geometric micropore surface area (5J 1168] were calcu-... 

Micropores in the lignocellulosic wastes and resulting chars were analysed from CO2 adsorption isotherms by applying the Dubinin-Radushkevich (DR) equation (14). DR plots... 

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