Halsey equation

instead of assuming Q is constant it is assumed that Q is u linear function of the fraction of surface occupied by adsorbate moles d, the Halsey equation develops [17], 

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On nonporous surfaces it has been shown that when W /W is plotted versus P/Po the data all approximately fit a common type II curve above a relative pressure of 0.3. This implies that when WJW = 3, for example, the adsorbed layer thickness t will be 10.62 A regardless of the adsorbent. The common curve is described closely by the Halsey equation which for nitrogen can be written as... 

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

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. 

Numerous scientist have calculated t-values as a function of p/p,).24,25,26,27 A general accepted formulation is the Halsey equation 28... 

Based on the above general principles, quite a number of models have been developed to estimate pore size distributions.29,30,31-32,33 They are based on different pore models (cylindrical, ink bottle, packed sphere,. ..). Even the so-called modelless calculation methods do need a pore model in the end to convert the results into an actual pore size distribution. Very often, the exact pore shape is not known, or the pores are very irregular, which makes the choice of the model rather arbitrary. The model of Barett, Joyner and Halenda34 (BJH model) is based on calculation methods for cylindrical pores. The method uses the desorption branch of the isotherm. The desorbed amount of gas is due either to the evaporation of the liquid core, or to the desorption of a multilayer. Both phenomena are related to the relative pressure, by means of the Kelvin and the Halsey equation. The exact computer algorithms35 are not discussed here. The calculations are rather tedious, but straightforward. 

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... 

Whilst the use of the Kelvin equation can be questioned in the case of smaller mesopores, this is not the case in the present case where, on the contrary, the pores are situated in the upper mesopore range. However, use of the BJH method implies the use of a t-curve. On commercial adsorption equipment, the software proposes the use of several equations to fit the t-curve. In the present case, the Harkins and Jura equation or Halsey equation is proposed. Unfortunately neither of these fit the original t-curve data of de Boer very well. 

The thickness of the adsorbed film can be estimated by using the BET isotherm. The appropriate results can be described closely by the Halsey equation, which for nitrogen is written as 49,59)... 

The pore size was calculated by BJH method with Halsey equation using the adsorption branch of the isotherm. 

In our studies of modified conventional mesoporous silicas [76,77] that had pore sizes greater than 10 nm, the BJH method was used, and calculations were performed on the basis of desorption data. These materials exhibited narrow HI hysteresis loops, which indicated facile pore connectivity and permitted the application of desorption data, which may be inappropriate for many other mesoporous silicas, as discussed above. The statistical fihn thickness curve used assumed the form of the Halsey equation t[nm] = 0.354[-5/ln(/i// o)] . ... 

Table 4.13 lists the best-known isotherm equations. The Langmuir equation can be applied in type I isotherm behavior. The Brunauer-Emmet-Tetter (BET) equation has been successfully applied to almost all kinds of materials, but especially to hydrophilic polymers for < 0.5. The Halsey equation is suitable for materials of types I, II, and III. The Henderson equation is less versatile than that of Halsey. For cereal and other field crops, the Chung... 

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