Statistical thickness

This equation describes the additional amount of gas adsorbed into the pores due to capillary action. In this case, V is the molar volume of the gas, y its surface tension, R the gas constant, T absolute temperature and r the Kelvin radius. The distribution in the sizes of micropores may be detenninated using the Horvath-Kawazoe method [19]. If the sample has both micropores and mesopores, then the J-plot calculation may be used [20]. The J-plot is obtained by plotting the volume adsorbed against the statistical thickness of adsorbate. This thickness is derived from the surface area of a non-porous sample, and the volume of the liquified gas. 

If the utmost rigor were used, it would be correct to modify the area contributed by previously emptied pores since their statistical thickness diminishes with each successive decrement. However, this procedure would be cumbersome and of questionable value in view of the many other assumptions which have been made. Nevertheless, the BJH method attempts to make this modification by introducing an average inner core based on its variation with each decrement of relative pressure. 

In the second decrement the liquid volume desorbed iVuq)2 i st be corrected for the decrease in the adsorbed film depth remaining on the walls of previously emptied pores. By assuming the pores are cylindrical, the core volume V )2 can be calculated from the decrease in statistical thickness t, as... 

Figure 8.6 Statistical thickness versus relative pressure from the Halsey equation.

When measuring mesopores the statistical thickness t is used as a correction to allow for desorption from the adsorbed film. However, the value of t is much more critical when measuring micropores because in the MP method, t is the actual measure of the pore size. 

The usual method of measuring the statistical thickness is to divide the liquid volume of nitrogen adsorbed, by the BET surface area... 

The assumption usually made is that the ratio Fu /Sbet has the same value at a given relative pressure independent of the solid. A plot therefore of t versus P/Pq should give the same curve for any non-porous solid (see Fig. 8.6). In fact, plots of the number of adsorbed layers versus P/Pq show some discrepancies which for the analysis of large pores is not significant. Therefore, the Halsey equation can be used for the statistical thickness in that application. However, for micropore analysis, a statistical thickness must be taken from a t versus P/Pq curve that has approximately the same BET C value as the test sample. The unavailability of t versus P/Pq plots on numerous surfaces with various C values would make the MP method of passing interest were it not for the fact that t can be calculated from equation (8.36). This implies that surface area can be accurately measured on microporous samples. Brunauer points out that in most instances the BET equation does correctly measure the micropore surface area. 

External surface area m2/g ASTM D 5816 STSA (Statistical Thickness Surface Area) calculated from the nitrogen adsorption isotherm... 

A convenient method is provided by the t-plot of Lippens and De Boer.37 It consists of plotting the volume of gas adsorbed vs t, the statistical thickness of the adsorbed film, t is a function of p/p0, as measured in the standard isotherm, according to the Halsey equation or one of its modifications.38,39,40,41... 

The left-hand side abscissa gives the statistical thickness of the adsorbed multilayer, f, in nm. The t values on the abscissa are calculated as follows. 

Figure 1.28. Statistical thickness of an adsorbed water film on a number of different non-porous adsorbents. The temperature of the measurements is indicated. Pretreatment 4h in vacuo at 30°C.

BJH and Dollimore-Heal methods are based on the assumption that the statistical thickness of the adsorbed layer is independent of the surface curvature and assume that the meniscus between vapor and condensed phase is hemispherical. This kind of meniscus is met in the case of a cylindrical pore during desorption. The hypothesis of constant thickness, independently of surface curvature, is justified for large mesopores, but generally leads to underestimation of pore size [7]. 

For the partial pressures indicated on the ordinate we have chosen the symbol p /po in order to indicate that we are dealing here with equilibrium pressures above approximately flat surfaces. This will be seen to be important in the discussion of capillary condensation in Section 13.10. The left-hand side abscissa gives the statistical thickness of the adsorbed multilayer, t, in nm. 

The isotherms presented in Fig. 16 are shown in de Boer s representation [V = f(t)]i where t is the statistical thickness of the adsorptive layer [51,52]. If a non-porous adsorbent of plain surface is considered, this representation gives a straight line starting from... 

The N2 adsorption-desorption isotherms were collected on an ASAP 2010 analyser (Micromeritics). Prior to analysis, the samples were degassed (palumina-based materials) during 5 h. The contributions of microporosity to the overall surface area were estimated fi om a t-plot (Harkin-Jura) analysis of the adsorption curve (0.3 nm < t < 0.5 nm, with t being the statistical thickness). The pore size distribution was calculated from the desorption branch of the isotherm using the B JH model. 

If we let t to represent the statistical thickness of the adsorbed layer (which is a function of pressure), then the effective pore radius available for condensation is related to the true pore radius as follows ... 

It is known that the BET statistical film thickness of a practical porous solid is larger than the experimental thickness for flat surfaces in the high pressure region (Schull, 1948). Figure 3.10-1 shows a plot of the statistical thickness t calculated from eq. (3.10-9) with C= 100. 

Also plotted in the same figure is the statistical thickness calculated from equation (3.9-25). The two curves deviate significantly in the high pressure region. The circle symbols on this figure are experimental data obtained on many non-porous solids (Cranston and Inkley, 1957). We see that eq. (3.9-25) agrees very well with the experimental data and it is then a better choice of equation for the calculation of the statistical film thickness. 

If the adsorption occurs in a free surface (no restriction on the number of layers which can be built up on top of the surface), the statistical thickness of the adsorbed layer is a function of the reduced pressure as seen in Figure 3.10-1 or Table 3.10-1, that is... 

For pores of cylindrical or slit shape, the behaviour of the calculated statistical thickness does not follow that of a flat surface as the pore shape can influence the statistical film thickness. This is explained as follows. For cylindrical pores, the solid will take up more sorbates than a free surface, that is... 

The equilibrium data of the amount adsorbed versus the reduced pressure are plotted as the amount adsorbed versus the statistical thickness by using the data of nonporous material in step 1. 

The next step is to plot the amount adsorbed V (cc/g) versus the statistical thickness t as shown in Figure 3.11-2. 

FIG. 15 Volume of nitrogen gas adsorbed versus statistical thickness. C-0. C-IO, C-20, and C-30 are collagen fibers alkali-treated for 0, 10. 20, and 30 days, respectively. = amount adsorbed converted into liquid t = adsorbed thickness. (From Ref. 4.)... 

From ASTM international NSA, nitrogen surfaee area, STSA, statistical thickness surface area, dibutylphthalate absorption. 

Ae is the excess of the evaporation heat due to the interference of the layering on the opposite wall of pores (determined as a varied parameter using local isotherm approximation [LIA]) tip, / p) is the statistical thickness of adsorbed layer flni is the BET monolayer capacity... 

The statistical thickness t of the adsorbed layer was calculated by Duchet equation [15] which takes into account the adsorbate-adsorbent interaction of pure zirconia samples. 

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