Cranston-Inkley method

This suffices to show the difference between this corrected modelless method and the Cranston-Inkley method. For detailed description and examples of calculations reference must be made to the original paper. 

Cranston and Inkley Method. Cranston and Inkley (39) used the known thick-. ness, /, of the Him of nitrogen on the inner walls of the pores, along with the diameter of pores filled by nitrogen according to the Kelvin equation, to develop a procedure for calculating the volume and size of pores from the desorption or adsorption isotherm. Use is made of the portion of the isotherm for p/p above 0.3 where at least a monomolecular layer of nitrogen is adsorbed. 

A further discussion of the Cranston and Inkley method and some useful graphs have been given by Hougen (159). However, it is not easy to translate a set of equations into actual calculations, which is why the above steps have been given in detail. 

Figure 6. Comparison between methods, (a) Isotherms. The circles are the experimental data the line is the theoretical fit (b) Pore size distributions. The plain hne is the molecular method PSD, the circles are from the Micropore method and the squares are from the Cranston and Inkley method.

The desorption isotherm approach is the second generally accepted method for determining the distribution of pore sizes. In principle either a desorption or adsorption isotherm would suffice but, in practice, the desorption isotherm is much more widely used when hysteresis effects are observed. The basis of this approach is the fact that capillary condensation occurs in narrow pores at pressures less than the saturation vapor pressure of the adsorbate. The smaller the radius of the capillary, the greater is the lowering of the vapor pressure. Hence, in very small pores, vapor will condense to liquid at pressures considerably below the normal vapor pressure. Mathematical details of the analysis have been presented by Cranston and Inkley (16) and need not concern us here. 

The ideas of Wheeler that condensation and evaporation occur within a center core during adsorption and desorption and that an adsorbed film is present on the pore wall has led to the proposal of various methods for pore size analysis. In addition to the methods of Pierce and the BJH technique, other schemes have been proposed, including those by Shull, Oulton, Roberts, Innes, and Cranston and Inkley. These ideas are all based upon some assumption regarding the pore shape. 

Many different mathematical procedures based on the Kelvin equation have been proposed for the calculation of the meso-PSD from nitrogen adsorption isotherms. The most popular method was proposed by Barrett et al. [49] (known as BJH method), but others like Cranston and Inkley [50], Dollimore and Heal [51], and Robert [52] methods are also currently used. 

Over the period 1945-1970 many different mathematical procedures were proposed for the derivation of the pore size distribution from nitrogen adsorption isotherms. It is appropriate to refer to these computational methods as classical since they were all based on the application of the Kelvin equation for the estimation of pore size. Amongst the methods which remain in current use were those proposed by Barrett, Joyner and Halenda (1951), apparently still the most popular Cranston and Inkley... 

The last section has shown the basic concepts of capillary condensation and how they can be utilized in the determination of pore size distribution (PSD). In this section, we address a number of practical approaches for PSD determination. One of the early approaches is that of Wheeler and Schull and this will be presented first. A more practical approach is that of Cranston and Inkley, and this will be discussed next. Finally, the de Boer method is presented, which accounts for the effect of pore shape on the calculation of the statistical film thickness and the critical pore radius. 

Cranston and Inkley (1957) presented a refined method over that of Barrett et al. (1951). Instead of using the pore length distribution and giving an equation... 

This method, like the other methods, requires the knowledge of the statistical film thickness of the adsorbed layer on a flat surface. The experimentally determined thickness of the adsorbed layer for nitrogen on a flat surface was obtained by Cranston and Inkley as a function of reduced pressure as shown in Figure 3.10-1 as symbols and are tabulated in Table 3.10-1. 

The t-plot method of plotting the amount of nitrogen adsorbed versus the thick ness of the adsorbed nitrogen film on a flat, nonporous surface was proposed by Shull (38). Cranston and Inkley (39) assembled available data and published a com posite / plot which they and others have used widely in characterizing porous solids. 

Many workers contributed to the development of methods for calculating pore size distribution from the adsorption isotherm,- as described by Broekhoff and Linsen (156). The original approach and general equation developed by Barret, Joyner, and Halenda (157) was followed up by Pierce (158) and later by Cranston and Inkley. The subsequent evolution of the subject has been described in detail by Gregg and Sing (7). 

Cranston and Inkley concluded that for many silica gels the method should be applied in reverse, starting from p/po = 0.3 and following steps along the adsorption isotherm. 

The method differs from that of Cranston and Inkley also in that instead of the Kelvin equation, the equation of Kiselev is used ... 

The following analysis need not be interpreted in terms of physical quantities. Thus it yields an analytical form which one could use more easily with more traditional pore size analysis systems as well as x theory or DFT. Included in the traditional digital methods is the pore length method originated by Wheeler [7] and developed by Shull [8], the Barrett loyner and Halenda (BJH) [9] and the Cranston and Inkley [10]. It is, however, easier to visualize and it may be possible that once the parameters for a particular isotherm are obtained one could attach different meanings to them. Indeed, the x plot representation has been presented [11] as a method to empirically construct an analytical expression for the standard curves. 

Cranston and Inkley [23] derived a curve of t against x from published isotherms on 15 non-porous materials by dividing the volume of nitrogen adsorbed by the BET surface area. They state that their method may be applied either to the adsorption or desorption branch of the isotherm and that the indications were that the desorption branch should be used, a proposal which was at variance with current practice. They assumed cylindrical pores closed at one end but stated that this assumption was unnecessary. 

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