Dubinin

Where Pg is the pressure of vapor in equilibrium with the adsorbed film. The characteristic curve is now just PF In Pq/P versus x (or against [Pg.627]

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... 

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

Blinov L M, Dubinin N V, Mikhnev L V and Yudin S G Polar 1984 Langmuir-Blodgett films Thin Solid Films 120 161-70... 

The pore systems of solids are of many different kinds. The individual pores may vary greatly both in size and in shape within a given solid, and between one solid and another. A feature of especial interest for many purposes is the width w of the pores, e.g. the diameter of a cylindrical pore, or the distance between the sides of a slit-shaped pore. A convenient classification of pores according to their average width originally proposed by Dubinin and now officially adopted by the International Union of Pure and Applied Chemistry is summarized in Table 1.4. 

For other adsorptives the experimental evidence, though less plentiful than with nitrogen, supports the view that at a given temperature the lower closure point is never situated below a critical relative pressure which is characteristic of the adsorptive. Thus, for benzene at 298 K Dubinin noted a value of 017 on active carbons, and on active charcoals Everett and Whitton found 0-19 other values, at 298 K, are 0-20 on alumina xerogel, 0-20-0-22 on titania xerogel and 017-0-20 on ammonium silicomolybdate. Carbon tetrachloride at 298 K gives indication of a minimum closure point at 0-20-0-25 on a number of solids including... 

A different approach is followed by Kadlec and Dubinin who calculate the theoretical tensile strength from a 6-12 relation for molecular forces (cf. Section 1.3) as... 

These various considerations led Pierce, Wiley and Smith in 1949, and independently, Dubinin, to postulate that in very fine pores the mechanism of adsorption is pore filling rather than surface coverage. Thus the plateau of the Type 1 isotherm represents the filling up of the pores with adsorbate by a process similar to but not identical with capillary condensation, rather than a layer-by-layer building up of a film on the pore walls. 

Fig. 4.6 Plat of the net differential heat of adsorption q — qj against the relative adsorption n/no.j, where H(,.2 is the adsorption at p/p° = 0-2. (A) active carbon (B) carbon black. (After Dubinin.)...

According to Dubinin s ideas, the process involved is volume filling of the micropores rather than layer-by-layer adsorption on the pore walls. A second parameter is therefore the degree of filling of the micropores, defined by... 

On the assumption that the pore size distribution is Gaussian, Dubinin and Radushkevich arrived at the expression... 

Fig. 4.17 Plot of log,o(n/(mmol g ) against logfo (p7p) for the adsorption of benzene at 20°C on a series of progressively activated carbons prepared from sucrose. (Courtesy Dubinin.)...

In the attempt to extend the scope of the DR treatment, Dubinin and Astakhov have put forward a more general equation,... 

The original DR equation is thus a special case of the Dubinin-Astakhov equation, with m = 2 parameter of Equation (4.18) for m = 2 is related to the structural constant B of the DR treatment through the simple expression... 

An example from Dubinin s paper illustrates the application of the new equation. For benzene at 293 K, the plot of Equation (4.18) with m = 2 gave a line concave to the log (p°/p) axis. A revised value of m was therefore required, and was obtained as follows. A provisional value of IVq (= 408 mm g ) was estimated from the plateau of the isotherm the relative pressure (p/p°), corresponding to IV/tV = 0-368 was then read off, and a first value of S calculated from... 

These procedures proposed by Dubinin and by Stoeckli arc, as yet, in the pioneer stage. Before they can be regarded as established as a means of evaluating pore size distribution, a wide-ranging study is needed, involving model micropore systems contained in a variety of chemical substances. The relationship between the structural constant B and the actual dimensions of the micropores, together with their distribution, would have to be demonstrated. The micropore volume would need to be evaluated independently from the known structure of the solid, or by the nonane pre-adsorption method, or with the aid of a range of molecular probes. 

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