The Corrected Kelvin Equation

Around 1967, Broekhoff and de Boer [16], following Derjaguin [17], pointed out that the supposition introduced in the derivation of the Kelvin equation, viz. the equality of the thermodynamic potential of the adsorbed multilayer to the thermodynamic potential of the liquified gas (see Eqn. 12.28), cannot be correct. This can be seen immediately from an inspection of the common t curve (Fig. 12.5) at each t value lower than, say, 2 run, the relative equilibrium pressure is lower than 1, the equilibrium pressure of the liquefied gas. 

The correct value of the thermodynamic potential of the adsorbed multilayer is a function if t and is given by the common t curve itself  

Introduction of Eqn. 12.35 into the derivation of the Kelvin equation, along the same lines of reasoning as following in Section 12.10, gives the corrected Kelvin equation  

This equation can be used to explain that stable adsorption on the inner wall of a capillary tube is possible up to a certain critical thickness. Capillary condensation starts from this critical thickness. The important aspect is explained further in ref. 16. 

An excellent material for checking the validity of the corrected Kelvin equation is chrysotile, Mg3(0H)4.Si20s, which consists of hollow needles, the pore volume distribution of which can be measured both by means of calibrated electron microscopy and nitrogen capillary condensation [18]. It appears that capillary 

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Pore radii and pore volume distributions can be calculated on the basis of the classical Kelvin equation which can be adapted to various pore shapes. For t materials the corrected Kelvin equation according to Broekhoff and de Boer leads to better quantitative results. The Broekhoff-de Boer theory also explains why stable adsorption on the inner walls of pores is possible up to a certain critical thickness of the adsorbed layer, without giving rise to immediate capillary condensation of the gas. 

It is of interest to compare the values of pore diameter obtained by molecular simulation and by the use of the corrected Kelvin equation. By comparing the nitrogen isotherm in Figure 12.6 with molecular simulation model isotherms, Maddox et al. (1997) have arrived at pore diameters of 4.1-4.3 nm. As indicated in Table 12.4, the corrected Kelvin diameters are 3.3-4.3 nm. The corresponding surface areas are 631 and 655 m2 g 1. In view of the assumptions in the model and the shortcomings of the Kelvin and BET equations, this level of agreement must be considered to be encouraging. 

As expected, the initial part of the tts-plot for sample B in Figure 2 is similar to that for A, but in this case there is a third stage of pore filling. This is due to capillary condensation in mesopores, which according to the corrected Kelvin equation would have an effective pore width in the range 3 - 7 nm. The results of the micropore structure analysis for sample B are summarised in Table 2 and the mesopore size analysis by the BJH method in Table 3 (assuming a slit-shaped mesopore configuration). 

Equation (13.30) leads to the corrected Kelvin equations for the adsorption and desorption branch ... 

This equation can be used to explain that stable adsorption on the inner wall of a capillary tube is possible up to a certain critical thickness. Capillary condensation starts from this critical thickness. It appears that capillary condensation in the cylindrical pores with radii between 2 and 7 run is well described by the corrected Kelvin equation. The deviation from the ideal behavior can amount to up to +20%. Nevertheless, it is difficult to estimate the limits of the applicability of the Kelvin equation. It seems likely that distortions of the meniscus in small pores may occur and that the local pore geometry may have a marked influence [1,15]... 

The HK micropore volume distribution for a slit-like microporous structure can be obtained by multiplying the adsorption potential distribution [see Eq. (24) and Fig. 10] by Eq. (77). For cylindrical and spherical micropore geometries another expressions for the derivative dAldx should be used [160]. An illustration of the HK pore volume distributions is shown in Fig. 12 for the WV-A900, BAX 1500 and NP5 active carbons. Similarly, the mesopore volume distribution can be calculated from the multilayer and capillary condensation range of the adsorption isotherm. In this case, the corrected Kelvin equation should be used to calculate the derivative dAldx. 

If one could ignore the influence of this potential the uncorrected Kelvin equation would lead to a critical radius of curvature, R. The corrected Kelvin equation renders a value instead of R, for a spherical geometry, as well as another value R In place of R / 2 for a cylindrical geometry, both as functions of the relative pressure. Table 1 presents some comparative values for the condensation of nitrogen at 77 K. 

At the junction of the adsorbed film and the liquid meniscus the chemical potential of the adsorbate must be the resultant of the joint action of the wall and the curvature of the meniscus. As Derjaguin pointed out, the conventional treatment involves the tacit assumption that the curvature falls jumpwise from 2/r to zero at the junction, whereas the change must actually be a continuous one. Derjaguin put forward a corrected Kelvin equation to take this state of affairs into account but it contains a term which is difficult to evaluate numerically, and has aroused little practical interest. 

The classical Kelvin equation assumes that the surface tension can be defined and that the gas phase is ideal. This is accurate for mesopores, but fails if appUed to pores of narrow width. Stronger sohd-fluid attractive forces enhance adsorption in narrow pores. Simulation studies [86] suggest that the lower limit of pore sizes determined from classical thermodynamic analysis methods hes at about 15 nm. Correction of the Kelvin equation does lower this border to about 2 run, but finally also the texture of the fluid becomes so pronounced, that the concept of a smooth hquid-vapor interface cannot accurately be applied. Therefore, analysis based on the Kelvin equation is not applicable for micropores and different theories have to be applied for the different ranges of pore sizes. 

FIGURE 8.5 Relative vapor pressure vs. inverse of pore radius of n-hexadecane in nanoconfinement is plotted on a semi-log scale. Straight line represents the classical Kelvin equation, dashed line represents corrected Kelvin equation, solid symbols represent experimental data (Fisher, L. R. and Israelachvili, J. N. J. Colloid Interface Sci. 80 528, 1981), and oscillating line represents data (Adapted from Chen, Y. et al., J. Colloid Interface Sci. 300 45, 2006. With permission.)... 

It is assumed that the surface tension and the molar volume are independent of the radius of curvature associated with the porosity. Corrections or adjustments can be made to the above Kelvin equation because in an adsorption process, at the stage when mesopore filling occurs, already the walls of the pore contain adsorbed material such that the effective diameter of the pore is reduced and /"p = tk + (where t is the thickness of the adsorbed layer). Another correction is where the pore shape is not cylindrical but is perhaps slit-shaped when the meniscus is hemi-cylindrical and the effective pore width Wp = + 2t. However,... 

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. 

Different asymptotic forms of the Kelvin equation, which are equivalent within the second-order corrections, may be proposed. In order to improve the accuracy of Eq. (38) and to make it closer to the original Kelvin equation (35), an asymptotic relation %— 1 may be used. For... 

In principle, the use of a suitable adsorptive should also make it possible, as Kamaukhov has pointed out, to reduce the magnitude of the f-correction, which is always a source of some uncertainty. From the Kelvin equation... 

You get 1 point for the prediction that the reaction is spontaneous. The setup (plugging into the equation) is worth 1 point if you remember to change the temperature to kelvin and convert joules to kilojoules. An additional 1 point comes from the answer. If you got the wrong value in either part (a) or (b), but used it correctly, you will still get the point for the answer. The free-energy equation is part of the material supplied in the exam booklet. Subtract one point if all your answers do not have the correct number of significant figures. 

When you use this equation, be sure that all your units match. For example, if your Cp has units of J/(g-K), don t expect to calculate heat flow in kilocalories. A common source of error in solving specific heat problems is the need to use the correct temperature units be sure to pay attention to whether your temperature is in kelvins or degrees Celsius. 

Sbet - BET specific surface area V, - single-point total pore volume w - pore width at the maximum of the pore size distribution calculated using the BJH method with the corrected form of the Kelvin equation [34]. 

Specific surface areas of the materials under study were calculated using the BET method [22, 23]. Their pore size distributions were evaluated from adsorption branches of nitrogen isotherms using the BJH method [24] with the corrected form of the Kelvin equation for capillary condensation in cylindrical pores [25, 26]. In addition, adsorption energy distributions (AED) were evaluated from submonolayer parts of nitrogen adsorption isotherms using the algorithm reported in Ref. [27],... 

Figure 1. (a) Experimental relations between the capillary condensation pressure and the pore diameter (hollow circles) and between the capillary evaporation pressure and the pore diameter (filled circles) for nitrogen adsorption at 77 K. The dashed line corresponds to the Kelvin equation with the statistical film thickness correction. The solid line corresponds to Eq. 2 derived using the KJS approach, (b) Relation between pore diameters calculated on the basis of Eq. 1 and the KJS-calibrated BJH algorithm using nitrogen adsorption data at 77 K. 

It should be recognized that the Kelvin equation provides the core radius instead of the actual pore radius. The core radius represents the radius of the inner free space in the pore, which is not yet filled with the adsorbate. Therefore, in order to obtain the actual pore radius, the Kelvin equation needs to be corrected for the actual thickness of adsorbed molecule layers fad. The radius of a cylindrical pore, rv is given by... 

The state of a pure liquid without any viscoelastic film coverage is designated as I when e = 0 and /< = 0. Al this point, the frequency a>0 agrees with Eq. 10 while the damping coefficient a is given by the Stokes equation [64] with a correction similar to the one used for the Kelvin equation ... 

The correct answer is (A). Ideal gas equation. Temperature is in Celsius, so it will need to be converted to Kelvin ... 

If the Type IV isotherm has a short plateau, which is followed by an upward swing, the amount adsorbed at the plateau can be regarded as the capacity of the particular range of mesopores. A useful assessment of the upper limit of mesopore size can then be obtained with the aid of the Kelvin equation corrected for multilayer adsorption. 

Many early attempts were made to correct the Kelvin equation (see Brunauer, 1945). As already indicated, when the Kelvin equation is applied to capillary condensation it is normally assumed that the reduction in chemical potential is entirely dependent on the curvature of the meniscus. This assumption implies a sharp discontinuity between the state of the adsorbed layer and the condensate. However, as Detjaguin first pointed out (1957), the transition is more likely to be a gradual one. This problem was also discussed by Everett and Haynes (1973). 

As already indicated, by applying the Kelvin equation (assuming hemispherical meniscus formation) and correcting for the adsorbed layer thickness, we are able to calculate the ranges of apparent pore width recorded in Table 12.5. The values of mean pore diameter, w, are obtained from the volume/surface ratio, i.e. by applying the principle of hydraulic radius (see Chapter 7) and assuming the pores to be non-intersecting cylindrical capillaries and that the BET area is confined to the pore walls. 

The close agreement between the corresponding values of wp and w in Table 12.5 is probably deceptive. It must be re-emphasized that it is unlikely that the Kelvin equation provides a reliable basis for the calculation of pore widths of around six molecular diameters. Also, as pointed out in Chapter 7, the application of the standard statistical multilayer thickness correction may be an oversimplification. For these reasons, the values of pore widths in Tables 12.4 and 12.5 should be regarded as apparent rather than real pore sizes. 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

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