Dubinin equation

Show that 5config = 0 for adsorption obeying the Dubinin equation (Eq. XVn-75). 

Assuming the form of the Dubinin equation to be eorrect, or more generally that the ratio T/T, is eonstant along an isostere then H can be expressed as a multiple of the latent heat L of the refrigerant at the system pressure ... 

The specific surface area of the fresh and used catalysts was measured by nitrogen adsorption method (Sorptometer 1900, Carlo Erba Instruments). The catalysts were outgassed at 473 K prior to the measurements and the Dubinin equation was used to calculate the specific surface area. The acidity of investigated samples was measured by infrared spectroscopy (ATI Mattson FTIR) by using pyridine (>99.5%, a.r.) as a probe molecule for qualitative and quantitative determination of both Bronstcd and Lewis acid sites (further denoted as BAS and LAS). The amounts of BAS and LAS were calculated from the intensities of corresponding spectral bands by using the molar extinction coefficients reported by Emeis (23). Full details of the acidity measurements are provided elsewhere (22). 

The following natural precursors have been selected for KOH activation coal (C), coal semi-coke (CS), pitch semi-coke (PS) and pitch mesophase (PM). An industrial activated carbon (AC) was also used. Activation was performed at 800°C in KOH with 4 1 (C KOH) weight ratio, for 5 hours, followed by a careful washing of the samples with 10% HC1 and distilled water. The activation process supplied highly microporous carbons with BET specific surface areas from 1900 to 3150 m2/g. The BET surface area together with the micro and the total pore volume of the KOH-activated carbons are presented in Table 1. The mean micropore width calculated from the Dubinin equation is designed as LD. 

The surface area of the catalysts was measured by nitrogen physisorption (Sorptometer 1900, Carlo Erba). The fresh and regenerated samples were outgassed at 150°C and the spent samples at 100°C for 3 hours. The specific surface area was calculated with the Dubinin equation. 

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. 

Stoeckli HF, Kraehenbuehl F, Ballerini L, and Debernardini S. Recent developments in the Dubinin equation. Carbon, 1989 27(1) 125-128. 

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. 

The energetic heterogeneity associated with micropores of microporous materials can be characterised by means of the adsorption potential distribution, and the distributions (X(A)), related to the Dubinin equations ... 

The volume of micropores can be obtained using the Dubinin equation [78],... 

Classical methods, like DS and HK, show shortcomings in the determination of MSD due to the assumptions involved in their formulation of the adsorption process Dubinin equation does not show linearity in the Dubinin plot for single slit pores and Horvath-Kawazoe equation assumes that at a given pressure a pore is either completely filled or completely empty, which is contrary to the behavior observed in computer simulations Resulting MSD are shifted respect to those obtained by Monte Carlo simulations, by amounts that vary with the actual distribution, and too small micropores are predicted... 

The empirical equations dealt with so far, Freundlich, Sips, Toth, Unilan and Keller et al., are applicable to supercritical as well as subcritical vapors. In this section we present briefly a semi-empirical equation which was developed originally by Dubinin and his co-workers for sub critical vapors in microporous solids, where the adsorption process follows a pore filling mechanism. More details about the Dubinin equation will be discussed in Chapter 4. 

The Dubinin equation (Dubinin, 1966, 1967, 1972, 1975) has its history in the development of theory for adsorption in activated carbon. Activated carbon has a very complex structure (see Chapter 1 for some details), with pores ranging from macropores of order of greater than 1000 A to micropores of order of 10 A. It is this micropore network where most of the adsorption capacity resides. Because of the pore dimension comparable to the dimension of adsorbate molecule, the adsorption mechanism in micropore is completely different from that on a surface of a large pore, where adsorption occurs by a layering process. In micropores, the mechanism is due to micropore filling because of the adsorption force field encompassing the... 

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. 

Here P and P2 can be calculated from the vapour pressure information at critical condition and at the normal boiling point at which Pq = 1 atm (Dubinin, 1975). Another approach is to evaluate the pseudo-vapour pressure at any temperature above T. by using the following Dubinin equation ... 

Modified Dubinin Equations for Inhomogeneous Microporous Solids... 

When the Dubinin equation is used as the local isotherm in eq. (6.3-1) or eq (6.3-2), there are a vast literature on the use of this approach. Like other local isotherm equations, Dubinin equations can be used with many forms of the energy distribution, and for each combination an isotherm equation can be obtained. In Chapter 4, we have discussed the use of Dubinin equations as the local isotherm to describe pure component adsorption in heterogeneous microporous carbon. Readers should refer to Chapter 4 for further exposition of this application. 

This equation suggests that the heat of adsorption is infinite at zero loading and approaches the heat of liquefaction when the solid is saturated. Similar behaviour is observed with the Dubinin equations (Chapter 4). 

B B structural parameter in Dubinin equation (mole/Joule) ... 

Eo EO characteristic energy used in Dubinin equation Joule/mole... 

Modified Dubinin equations for inhomogeneous microporous solids 172... 

The more important adsorption isotherms are the Langmuir, the Freundlich, the Temkin, the Brunauer-Emmett-TeUer (BET), and the Dubinin equations. The first three isotherm equations are very important for chemisorption, although the Langmuir and Ereundlich isotherms are eqnaUy important for physisorption. The BET eqnation and Dubinin equations are most important for analysis of the physical adsorption of gases and vapors on porons carbons. 

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