Modified Kelvin equation

In general, therefore, there are three processes, prior to the kind of capillary condensation associated with the hysteresis loop of a Type IV isotherm, which may occur in a porous body containing micropores along with mesoporesia primary process taking place in very narrow micropores a secondary, cooperative process, taking place in wider micropores, succeeded by a tertiary process governed by a modified Kelvin equation. 

Figure I Relation between filling pressure and pore width predicted by the modified Kelvin equation (MK), the Horvath-Kawazoe method (HK), density Junctional theory (DFT), and molecidar simulation (points) for nitrogen adsorption in carbon slits at 77 K [8].

By combining the DFT analysis of the sorption isotherm with X-ray diffraction data on the pore spacing, the pore wall thicknesses of a set of MCM-class adsorbents were determined. These thicknesses were found to be consistent across the set of adsorbents, which lends further strong evidence to the validity of the DFT adsorption model. The Kelvin and modified Kelvin equations, by contrast, overestimate the condensation pressures of nitrogen in cylindrical oxide pores [21]. 

Such more realistic models of porous materials can also be used to rigorously test existing characterization methods. The model material is precisely characterized (we know the location of every atom in the material, hence the pore sizes, surface area and so on). By simulating adsorption of simple molecules in the model material and then inverting the isotherm, we can obtain a pore size distribution for any particular theory or method. Such a test for porous glasses is shown in Figure 8, where the exactly known (geometric) PSD is compared to that predicted by the Barrett-Joyner-Halenda (BJH) method, which is based on the modified Kelvin equation. 

Figure 8 PSDs for model porous silica glasses [10]. A, B, C, D are sample glasses prepared by Quench Molecular Dynamics, and differ in mean pore size and porosity. The solid curves are the exact geometric PSDs for the models the dashed lines are PSDs predicted by analyzing simulated nitrogen adsorption isotherms for these materials using the BJH method (a form of the modified Kelvin equation). The BJH method gives mean pore sizes that are too small by about I nm in each case.

Now we turn to the pore-filling process. We argue that this process is governed by the following equation, which is similar in form to the modified Kelvin equation ... 

The difference between the above equation and the modified Kelvin equation is the use of the pore pressure. Substituting the pore pressure of Eqn (11.19) into the above equation gives... 

For large pores (mesopores and macropores), the contribution of solid-fluid potential is negligible (the first term in the RHS) and the above equation is reduced to the modified Kelvin equation. On the other hand, for small pores of molecular dimension the overlapping of potentials exerted by the two opposite walls is such that the overlapped potential outweighs the surface tension effect... 

The modified Kelvin equation for this case of cylindrical pore is ... 

The other estimate, from the modified Kelvin equation, yields the diameter along with the pore size distribution. For the data analyzed here this distribution is assumed to be a normal distribution in x- A more detailed analysis does not seem justified by the number of data points in the transition zone. 

Obviously, if there is some microporosity present then unless it can be separated in the isotherm then the above answer may be far from correct. A better method of obtaining the mesoporosity is as follows using the modified Kelvin equation. The x method is used here, but in principle any well-calibrated standard curve should work. 

Nearly all of the analysis of mesoporosity starts with the Kelvin-Cohan [14] formulation. Foster [15] proposed the Kelvin equation for the effect of vapor pressure on capillary rise but did not anticipate its use for very small capillaries where the adsorbate thickness is a significant geometrical perturbation. Cohan formulation subtracts the adsorbate film thickness from the radius of the pore to yield the modified Kelvin equation... 

Here is the condensation pressure for a slit-shaped pore of width L, Pq is the saturation pressure of the adsorbate fluid, p is the fluid density, y is the surface tension of the sorbate, and/ and T are the ideal gas constant and absolute temperature, respectively. However, the Kelvin equation seemed to fail as pore size approached very low values because it does not take the thickness of the monolayer formed within the pore into account. Several models for the estimation of a statistical film thickness have been proposed (cf. Refs. 7-9) with the view of improving the PSD prediction. A commonly used method based on the modified Kelvin equation is the technique by Barrett et al. (the BJH method) [10,11]. A detailed description of such methods is available in the reviews by Gregg and Sing [1] and Webb and Orr [9]. 

Activated carbons AX21, CXV, AC610, D52 Nz (77 K) LO-6.0 Modified Kelvin equation 7... 

Let us first demonstrate the differenee between the classical Kelvin equation (35) and the modified Kelvin equation in either of the forms (37)-(40). A nearly ideal mixture of 50% methane and 50% butane may be used for demonstration, as shown in Fig. 14. The mixture is in... 

Another deficiency of the Kelvin equation is that it does not feel the boimdaries of stability of the capillary condensate. These boundaries may be found only on the basis of the numerical calculations described in Sections III.C and III.E.5. As it has been shown in these subsections, the boundaries for stability of the capillary condensate lie very close to the phase envelope. It may be safely assumed that the modified Kelvin equation is rather precise within these limits. 

Mesoporous materials (with pore widths between 2 and 50 nm) experience pore condensation at pressures below the corresponding saturation pressure of the bulk liquid. The volume pore size distribution is generally determined according to the Barret-Joyner-Halenda model. This model considers that condensation occurs in pores when a critical relative pressure is reached according to the modified Kelvin equation... 

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