Dubinin-Polanyi isotherms

Obviously to get the entire isotherm Eq. (180) must be subtracted from Eq. (179) and the results doubly integrated form x= — co (which is PIP=Q) to whatever x is of interest. (A similar equation is given in Chapter 5 in the discussion of the Freundlich and Dubinin-Polanyi isotherms. There the match to the second derivatives was used as being a more sensitive test.) This yields a rather messy but quite useable and easily calculated 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]... 

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

The virial isotherm [Eq. (3.43) has also been widely used to correlate experimental equilibrium data. " With two virial coefficients in addition to the Henry constant it is possible to correlate the equilibrium data for many systems with high accuracy. However, little detailed information concerning the nature of the adsorbed phase can be deduced from the virial coefficients so the method is essentially one of empirical data correlation. As such it is less convenient than the Dubinin-Polanyi approach since more parameters, each one in principle dependent on temperature, are required. However the virial isotherm reduces naturally to Henry s law in the low-concentration limit, and this is a significant advantage over the Dubinin-Polanyi approach, the validity of which becomes questionable at low concentrations. 

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

Gradients of Dubinin-Polanyi plots forC02/273K isotherms X 10 ... 

The relationship to the Freundlich isotherms is important for two reasons. First the question as to whether the % theory can predict isotherms such as the Freundlich (of which = 1 is a special case), Dubinin-Astakov, Dubinin-Radushkevich and Toth isotherms All but the Toth isotherm will be referred to as the Dubinin-Polanyi (DP) isotherm. Second, the reason for the observation that in most cases P appears to approach 0 as approaches 0. Even though there are cases where P approaches a finite value, thus disproving the universal application of Henry s law , this is not convincing without an explanation as to why it is observed in many cases. 

The comparison to isotherms, when there is a distribution, comes back to the Henry s law question. Why is it that sometimes one observes the Freundlich isotherm and thus at least the appearance that the pressure and adsorbate amount simultaneously approach zero. As demonstrated in Ch ter 4 a log-normal distribution in yields the Dubinin-Polanyi (DP) set of isotherms of which the Freundlich isotherm is a subset. The Toth isotherm is similar but mathematically not in this class. The question becomes, are the generated isotherms, and not just the energy distributions, similar. 

Correlation of the experimental isotherms by a generalized one is encouraged by the possibility that if the generalized isotherm for a given adsorbate is known, the adsorption values at any temperature and pressure can be predicted. Such a representative generalized isotherm is the Dubinin-Polanyi characteristic curve. According to the potential theory, the plot of adsorbed volume versus adsorption potential is temperature independent and, hence, characteristic. The adsorbed volume is determined by [55]... 

Semiernpirical Isotherm Models. Some of these models have been shown to have some thermodynamic inconsistencies and should be used with due care. They include models based on the Polanyi adsorption potential (Dubinin-Radushkevich, Dubinin-Astakhov, Radke-Prausnitz, Toth, UNI LAN. and BET). 

Many other equilibrium relationships have been applied to model sorption. For example, the Langmuir (Eq.44) and the Polanyi-Dubinin (Eq.45) isotherms have been widely applied to adsorption in zeolites [9] ... 

Dubinin was the pioneer of the concept of micropore filling. His approach was based on the early potential theory of Polanyi, in which the physisorption isotherm data were expressed in the form of a temperature-invariant characteristic curve . 

Figure 6. Sulfur isotherm for Westvaco carbon at J000 F, Polanyi-Dubinin model...

The free energy change associated with the desorption of a molecule (or mole). Employed by Polanyi and Dubinin and coworkers to fit isotherm data. See Equation (14.7). 

The potential theory of adsorption was introduced by Polanyi in 1914. Dubinin [48,49] and Stoeckli et al. [50] improved the theory and termed it the theory of volume filling of micropores (TVFM). This theory has been widely used in correlating the effect of temperature on the adsorption isotherms of pure gases. The modern formulationof TVFM is the Dubinin-Astakhov (DA) equation, which is expressed as... 

Based on the Polanyi potential theory, different approaches to describe the adsorption behavior of a purely microporous material (isotherm type I, Figure 21.25) have been undertaken by Dubinin and Stockli in collaboration with different other scientists. The simplest relationship that can be considered the base for all other variants is the Dubinin-Radushkevich equation [58] ... 

The Polanyi potential theory successfully represents the temperature dependence of adsorption. It is also the only theory that gives quantitative description of physical adsorption on strongly heterogeneous surfaces, such as those of active carbons and oxide gels. However, the significance of the theory had been limited for a long time because it did not provide an analytical expression for the adsorption isotherm. This problem was solved by Dubinin and coworkers. ... 

The DR isotherm remained known to a relatively small group of researchers until Hobson called attention to its applicability to non porous surfaces. In a letter to the Journal of Chemical Physics, Hobson, not without wonder, observed that the appearance of the Dubinin-Radushkevich equation in the present context (submonolayer adsorption of nitrogen on Pyrex) is surprising for two reasons. First, it is a particular equation within the Polanyi potential theory, which is a theory of condensation and might not be expected to apply to physical adsorption at very low coverage. Second, most of the adsorbents to which the Dubinin-Radushkevich equation have been applied have been porous, whereas our conclusion suggests that Pyrex is non-porous for nitrogen. Thus, unless and until a basic derivation for this equation is provided, it can only be considered as a useful empirical relation . 

The DR isotherm, initially proposed by Dubinin and Radushkevich for the description of adsorption on porous adsorbents, was found by Hobson to describe adsorption in submonolayer range on non-porous surfaces too. This empirical discovery was a challenge to theoreticians since the DR isotherm was expressed in terms of the Polanyi potential, which is expected not to apply to adsorption in the submonolayer range. 

The adsorption and desorption isotherms of water vapor are drawn at 25°C for dealuminated HY zeolites upon framework Si/Al ratio. The isotherms are compared to that of the parent NaY zeolite. The isotherm changes in shape from the type I to the type IV with an hysteresis loop changing from the type H4 to the type H2, as increases the Si/Al ratio. The POLANYI-DUBININ theory is used to determine the micropore volume accessible to water. It decreases with increasing Si/Al ratios, down to zero at a Si/Al ratio of 35. Such a result is accounted by the adsorption on the hydrophilic centers which are the cations (H ) associated with the structural aluminium ions, each cation being coordinated by 8H2O. 

Adsorption of hydrocarbons or nitrogen reflects the same trend as did water adsorption, with respect to the biporous character of dealuminated samples (structural micropores and secondary mesopores) (refs. 2, 4-6). The non polar character of n-hexane leads to use such a molecule for determining the micropore volume by the POLANYI-DUBININ theory. The study of the pore size distributions from adsorption isotherms and mercury porosimetric measurements characterizes two pore diameters of the secondary pore network, the one between 3 nm and 4 nm and the second to about 20 nm (refs. 7, 8). 

The POLANYI-DUBININ adsorption potential theory is used to characterize the micropore network of zeolites (ref. 10). An isotherm at a given temperature T (expressed in volume adsorbed per activated zeolite mass unit, W, as a function of the relative pressure p/Pq) is treated in the DUBININ-RADUSHKEVICH model (ref. 11) (denoted D-R) in the linear form log W = f[(Tlog Po/p) ] ... 

A. Elm Chaouri, M.H. Simonot-Grange, Hydration thermodynamic functions of hectorite and montronite from experimental Isotherms and the Polanyi-Dubinin theory, 9th International Clay Conference, Strasbourg, France, August 28 September 1, 1989. 

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