Horvath-Kawazoe method calculation

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. 

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

Two kinetic (CMS-Kl, CMS-K2) and one equilibrium (CMS-R) carbon molecular sieves, used originally for separation of gaseous mixtures, were investigated. The adsorption Nj isotherms at 77 K, in static conditions where obtained. In the case of the two first sieves the adsorption was so low that the calculation of parameters characterizing the texture was impossible. The volume of nitrogen adsorbed on the sieve CMS-R is remarkable From obtained results parameters characterizing micropore structure according to Dubinin -Radushkevich equation and Horvath - Kawazoe method were determined. 

Fig. 3. MPSDs calculated from the Horvath-Kawazoe method.

The methods differ in the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (lAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6], Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe methods are the most well known [7,8]. 

Figure 3.8. Differential pore volume of the activated carbon (AC) NORIT Rl EXTRA in the micropore region, calculated from the AI given in Fig. 3.7B by the Horvath-Kawazoe-method [3.29].

Table 2 Physical Properties for Pore Size Distnbution Calculation by Horvath-Kawazoe Method(I983)...

Fig. 2. 18. Effective pore size distributions of carbon molecular sieves calculated by Horvath-Kawazoe method.

A number of models have been developed for the analysis of the adsorption data, including the most common Langmuir [49] and BET (Brunauer, Emmet, and Teller) [50] equations, and others such as t-plot [51], H-K (Horvath-Kawazoe) [52], and BJH (Barrett, Joyner, and Halenda) [53] methods. The BET model is often the method of choice, and is usually used for the measurement of total surface areas. In contrast, t-plots and the BJH method are best employed to calculate total micropore and mesopore volume, respectively [46], A combination of isothermal adsorption measurements can provide a fairly complete picture of the pore size distribution in sohd catalysts. Mary surface area analyzers and software based on this methodology are commercially available nowadays. 

Recently, the Horvath-Kawazoe (HK) method for slit-like pores [40] and its later modifications for cylindrical pores, such as the Saito-Foley (SF) method [41] have been applied in calculations of the mesopore size distributions. These methods are based on the condensation approximation (CA), that is on the assumption that as pressure is increased, the pores of a given size are completely empty until the condensation pressure corresponding to their size is reached and they become completely filled with the adsorbate. This is a poor approximation even in the micropore range [42], and is even worse for mesoporous solids, since it attributes adsorption on the pore surface to the presence of non-existent pores smaller than those actually present (see Fig. 2a) [43]. It is easy to verify that the area under the HK PSD peak corresponding to actually existing pores does not provide their correct volume, so the HK-based PSD is not only excessively broad, but also provides underestimated volume of the actual pores. This is a fundamental problem with the HK-based methods. An additional problem is that the HK method for slit-like pores provides better estimates of the pore size of MCM-41 with cylindrical pores than the SF method for cylindrical pores. This shows the lack of consistency [32,43]. Since the HK-based methods use CA, one can replace the HK or SF relations between the pore size and pore filling pressure by the properly calibrated ones, which would lead to dramatic improvement of accuracy of the pore size determination [43] (see Fig. 2a). However, this will not eliminate the problem of artificial tailing of PSDs, since the latter results from the very nature of HK-based methods. 

In order to determine the PSD of the micropores, Horvath-Kawazoe (H-K) method has been generally used. In 1983, Horvath and Kawazoe" developed a model for calculating the effective PSD of slit-shaped pores in molecular-sieve carbon from the adsorption isotherms. It is assumed that the micropores are either full or empty according to whether the adsorption pressure of the gas is greater or less than the characteristic value for particular micropore size. In H-K model, it is also assumed that the adsorbed phase thermodynamically behaves as a two-dimensional ideal gas. 

The treatment with a flow containing SO2+H2O+O2 gives an amount of 400 mg of sulphuric acid. Heating up this sample, sulphuric acid is removed from the surface by reduction that leads to carbon consumption. This mild gasification can produce either an opening of the microporosity to mesoporosity and/or the creation of new microporosity. This can be followed by the increase of pore volume calculated by Horvath-Kawazoe (HK, for micropores) and Barret-Joyner-Halenda (BJH, for mesopores) methods. 

Specific surface area was calculated from the Brunauer-Emmett-Teller (BET) equation for N2 adsorption at 77 K (Micromeritics, ASAP 2010) [10], The t-method of de Boer was used to determine the micropore volume [11]. The pore size distribution curves of micropores were obtained by the Horvath-Kawazoe (H-K) method [12]. 

G. Horvath and K. Kawazoe, Method for the calculation of effective pore size distribution is molecular sieve carbon, J. Chem. Eng. Jpn. 16 (1983) pp. 470-475. 

Nitrogen isotherms were measured by using an ASAP (Micromeritics) at 77K. Prior to each analysis, the samples were outgassed at S73K for 10 - 12 h to obtain a residual pressure of less than 10 torr. The amount on nitrogen adsorbed was used to calculate specific surface area, and the micro pore volumes determined from the BET equation [14] and t-plot method [15], respectively. Also, the Horvath-Kawazoe model [16] was applied to the experimental nitrogen isotherms for pore size distribution. 

The calculation methods for pore distribution in the microporous domain are still the subject of numerous disputes with various opposing schools of thought , particularly with regard to the nature of the adsorbed phase in micropores. In fact, the adsorbate-adsorbent interactions in these types of solids are such that the adsorbate no longer has the properties of the liquid phase, particularly in terms of density, rendering the capillary condensation theory and Kelvin s equation inadequate. The micropore domain (0.1 to several nm) corresponds to molecular sizes and is thus especially important for current preoccupations (zeolites, new specialised aluminas). Unfortunately, current routine techniques are insufficient to cover this domain both in terms of the accuracy of measurement (very low pressure and temperature gas-solid isotherms) and their geometrical interpretation (insufficiency of semi-empirical models such as BET, BJH, Horvath-Kawazoe, Dubinin Radushkevich. etc.). 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

The principal drawback of the DFT method is that it is computationally intensive relative to the classical adsorption models, although it is still much less compute-intensive than full Monte Carlo molecular simulation. A semianalytic adsorption model that retains computational efficiency while accounting for gas-solid potential interactions in micropores was originally proposed by Horvath and Kawazoe [12], In the Horvath-Kawazoe or HK method, a pore filling correlation is obtained by calculating the mean heat of adsorption (/> required to transfer an adsorbate molecule from the gas phase to the condensed phase in a slit pore of width // ... 

Semiempirical models such as the Horvath-Kawazoe (HK) method [19] and the Dubinin model [20] and their derivatives These models generally make specihc assumptions regarding the shape of the pores and/or the distribution function that describes the pore sizes within the adsorbent. To varying degrees, the semiempirical methods incorporate adsorbate-adsorbent interaction energies into the calculation of the theoretical isotherm. 

In 1983 Horvath and Kawazoe [143] proposed a method to derive analytical equations for the average potential in a micropore of a given geometry, which in fact relate the adsorption potential with the pore size x. These equations are used to express the amount adsorbed in micropores as a function of the pore width and subsequently to calculate the micropore volume distributioa Thus, the Horvath-Kawazoe (HK) procedure is a logical extension of the metliod based on the Kelvin equation to the micropore range, and can be considered as an extension of the condensation approximation method to the region of fine pores [4]. Further improvements and modifications of this method are reported elsewhere [144, 153-157]. 

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