Equation Kiselev

This is useful since c can be estimated by means of the BET equation (see Section XVII-5). A number of more or less elaborate variants of the preceding treatment of lateral interaction have been proposed. Thus, Kiselev and co-workers, in their very extensive studies of physical adsorption, have proposed an equation of the form... 

Equation (3.73) is the basis of the method proposed by Kiselev for the evaluation of surface area from the Type IV isotherm. If perfect gas behaviour is assumed it becomes... 

The results of a comparison between values of n estimated by the DRK and BET methods present a con. used picture. In a number of investigations linear DRK plots have been obtained over restricted ranges of the isotherm, and in some cases reasonable agreement has been reported between the DRK and BET values. Kiselev and his co-workers have pointed out, however, that since the DR and the DRK equations do not reduce to Henry s Law n = const x p) as n - 0, they are not readily susceptible of statistical-thermodynamic treatment. Moreover, it is not easy to see how exactly the same form of equation can apply to two quite diverse processes involving entirely diiferent mechanisms. We are obliged to conclude that the significance of the DRK plot is obscure, and its validity for surface area estimation very doubtful. 

Contrary to the last two isotherms, which take into the account interactions between the neighboring molecities ortiy, the Kiselev model assumes the singlecomponent localized adsorption, with the specific lateral interactions among all the adsorbed molecules in the monolayer [4—6]. The equation of the Kiselev isotherm is given below ... 

Kiselev, using the above equation by graphical integration of the isotherm between the limits of saturation and hysteresis loop closure, was able to calculate surface areas for wide-pore samples in good agreement with BET measured areas. For micropores, the absence of hysteresis at the low-pressure end of the isotherm indicates that only adsorption and not condensation occurs, thereby rendering Kiselev s method inapplicable. 

Using the equilibrium equation developed by Hill (11) and adapted by Kiselev (12)... 

Equation 4 may be viewed as a three-constant extension pf the simpler equations, and the resulting improvement in agreement between the predictions and the experimental values may be attributed to the inclusion of extra arbitrary constants. If so, similar models, such as the Kiselev equation, which has the same number of constants, would be expected to provide the desired predictions. However, neither this nor other models did correlate the experimental data. Therefore, using models that include terms that describe the stipulated prevailing phenomena (heterogeneity, mobility, adsorbate interactions) provides a more realistic model of the actual mechanism and thus enables more accurate predictions. The application of experimental data to the proposed technique requires only slight increase in effort, which is negligible when computers are employed, despite the fact that the equations are more complex than the expressions derived from the simple models. 

The adsorption of D20 was first studied by Pimentel et al.44 and has been studied repeatedly since. Full references may be found in the work of Kiselev and Lygin.1 Using the D20 in the vapour phase gives superior results compared to liquid phase exchange. The exchange reaction follows equation (B). 

A number of different empirical equations have been proposed to allow for the deviations of physisorption isotherms from Henry s law. An approach which is analogous to that used in the treatment of imperfect gases and non-ideal solutions is to adopt a virial treatment. Kiselev and his co-workers (Avgul et al. 1973) favoured the form... 

The exponential form of the virial isotherm favoured by Kiselev and his coworkers (e.g. Avgul etal., 1973) was Equation (4.4), that is... 

Figure 1 shows the representation of the experimental isotherm (B. G. Aristov, V. Bosacek, A. V. Kiselev, Trans. Faraday Soc. 1967 63, 2057) of xenon adsorption on partly decationized zeolite LiX-1 (the composition of this zeolite is given on p. 185) with the aid of the virial equation in the exponential form with a different number of coefficients in the series i = 1 (Henry constant), i = 2 (second virial coefficient of adsorbate in the adsorbent molecular field), i = 3, and i = 4 (coefficients determined at fixed values of the first and the second coefficients which are found by the method indicated for the adsorption of ethane, see Figure 4 on p. 41). In this case, the isotherm has an inflection point. The figure shows the role of each of these four constants in the description of this isotherm (as was also shown on Figure 3a, p. 41, for the adsorption of ethane on the same zeolite sample). The first two of these constants—Henry constant (the first virial constant) and second virial coefficient of adsorbate-adsorbate interaction in the field of the adsorbent —have definite physical meanings. 

The next model, which assumes single-component localized monolayer adsorption with specific lateral interactions among all the adsorbed molecules, is the Kiselev model. The final equation of this model is... 

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23 Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Dillon, H. E., and Penoncello, S. G., A Fundamental Equation for Calculation of the Thermodynamic Properties of Ethanol, Int. J. Thermophys., 25(2) 321-335,2004. The source for viscosity is Kiselev, S. B., Ely, J. E, Abdulagatov, I. M., and Huber, M. L., Generalized SAFT-DFT/DMT Model for the Thermodynamic, Interfacial, and Transport Properties of Associating Fluids Application for n-Alkmols, Ind. Eng. Chem. Res., 44 6916-6927, 2005. The source for thermal conductivity is unpublished, 2004 however, the fit uses functional form found in Marsh, K., Perkins, R., and Ramires, M.L.V, Measurement and Correlation of the Thermal Conductivity of Propane from 86 to 600 K at Pressures to 70 MPa, J. Chem. Eng. Data, 47(4) 932-940, 2002. 

Methods for determining sorption isotherms by gas chromatography have been published by various authorsQ,-, ). The methods used have been elution and frontal chromatography. The first combines sorption and desorption so that any hysteresis in the equilibrium transport from gas to stationary phase and back to the gas phase can produce corresponding errors. The Kiselev-Yashin equation, as shown in Figure 1,... 

When three dimensional particles were used instead of coatings, the diffusional and structural kinetics showed marked differences in the rates of mass to area in the concentration dependent regions of partition coefficient (lower temperature and mass sorbed). Thus, the fundamental assumption of the Kiselev Yashin equation that the mass/area ratio was equal for prepeak and peak regions was not met, and agreement with static data was fortuitous at best. 

A number of more sophisticated theories have been developed to describe the adsorption of solutes from solutions in the whole concentration range. Most significant is the work pubhshed by Everett [22], Kipling [23], Schay and Nagy [24], Larionov et al. [25], Kiselev and Chopina [26], Siskova et al. [27], Minka and Myers [28], and Rusanov [29]. Unfortunately, this problem is far from being solved and most equations used in this area are empirical and approximate. 

Kiselev, S.B. Ely, J.F. Lue, L. Elliott, J.R. Computer simulations and crossover equation-of-state of square-well fluids. Fluid Phase Eq. 2002, 200, 121-145. 

The method devised by Barrett, Joyner, and Halenda (BJH) [35] is one of the earhest methods developed to address the pore size distribution of mesoporous sohds. This method assumes that adsorption in mesoporous solid (cylindrical pore is assumed) follows two sequential processes — building up of adsorbed layer on the surface followed by a capillary condensation process. Karnaukhov and Kiselev [45] accounted for the curvature in the first process, but Bonnetain et al. [46] found that this improvement has httle influence on the determination of pore size distribution. The second process is described by either the Cohan equation (for adsorption branch) or the Kelvin equation (for desorption branch). 

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23 Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Dillon, H. E., and Penoncello, S. G., A Fundamental Equation for Calculation of the Thermodynamic Properties of Ethanol, Int. J. Thermophys., 25(2) 321-335,2004. The source for viscosity is Kiselev,... 

For the adsorption of hexane on graphite we may use the experimental results of Isirikyan and Kiselev [18], who found that on graphi-tized carbon black (Graphon) n-hexane occupies 54 sq. A. per molecule and has a heat of adsorption (with liquid hexane as standard state) of 5.0 kcal. per mole. The value of y for graphite which gives a heat of adsorption of 5.0 kcal. per mole is found by Equation 8 to be 108 dynes per cm. 

The method of Neimark [7, 8, 33, 34] for the determination of the surface fractal dimension of a microporous solid is based on the adsorption isotherm equation that was developed by Kiselev [35]. This equation relates the surface area of pores filled by the adsorbate S(x) to the amount of adsorbed molecules N(x) at a given relative pressure x = pf po. ... 

A more phenomenological approach to describe crossover critical phenomena in simple fluids has been developed by Kiselev and coworkers [76-79]. This approach starts from the asymptotic power-law expansion including the leading correction-to-scaling terms which is then multiplied by an empirical crossover functions so that the equation becomes analytic far away from the critical point. A comparison of this approach with the crossover theory based on a Landau expansion has been discussed in earlier publications [13, 78]. One principal difference is that in the application of the results of the RG theory to the Landau expansion the leading correction to asymptotic scaling law is incorporated in the crossover function and recovered upon expanding the crossover function [18]. 

Kiselev, S.B. (1998) Cubic crossover equation of state, Fluid Phase Equilibria 147, 7-23. 

Krasnov, M. L., Kiselev, A. I. Makarenko, G. I. (1978). Problems on ordinary differential equations. Vysshaya Shkola, Moscow (in Russian). 

The method differs from that of Cranston and Inkley also in that instead of the Kelvin equation, the equation of Kiselev is used ... 

Associated Adsorbate ModeV° This model, which leads to the Berezin-Kiselev equation, accounts for lateral interactions within the adsorbed phase by proposing the formation of associates in the monolayer. The equation may be written as ... 

The representation of pVTx properties of mixtures by using the cubic EOS is still a subject of active research. Kiselev (1998), Kiselev and Friend (1999), and Kiselev and Ely (2003) developed a cubic crossover equation of state for fluids and fluid mixtures, which incorporates the scaling laws asymptotically close to the critical point and is transformed into the original classical cubic equation of state far away from the critical point. Anderko (2000) and Wei and Sadus (2000) reported comprehensive review of the cubic and generalized van der Waals equations of state and their applicability for modeling of the properties of multicomponent mixtures. 

This crossover equation of state (CREOS) (2.61)-(2.64) has been applied for dilute aqueous NaCl solutions (Belyakov et al, 1997), aqueous toluene (Kiselev et ai, 2002) and n-hexane (Abdulagatov et al., 2005) mixtures, and H2O + NH3 (Kiselev and Rainwater, 1997) solution near flie critical point of pure water and supercritical conditions. The values of the parameters were found from fit of equation... 

According to the classification of Scott and Konynenburg (1970, the binary systems of Type I, have only one critical locus between both critical points of the pure components and do not have the inuniscibility phenomena. For this type binary aqueous solutions, the functions Tc(x),pc(x), and Pc(x) were represented as simple polynomial forms (see Equations (2.65)-(2.67) of x and (1 - x) (Kiselev and Rainwater, 1997, 1998 Kiselev et al, 1998). Water -I- toluene system corresponds to a Type 111 mixture (Scott and Konynenburg, 1970), in which there is a three-phase immis-cibility region L1-L2-V with two critical endpoints (Li = V-L2 and Li = L2-V) where the VLE critical locus, originated in the critical point of pme more-volatile component (toluene) and the LLE critical locus, started in the critical point of less-volatile component (water), are terminated. 

In Figure 2.13, the partial molar volumes of n-hexane and toluene derived from thepVTx measurements (Abdulagatov et al, 2001, 2005 Rabezkii et al, 2001 Degrange, 1998) and the values calculated with semiempirical equation developed by Majer et al (1999) and crossover model (Kiselev et al, 2002 and Abdulagatov et al, 2005) are shown as a function of pure solvent (water) density along the various near-critical and supercritical isotherms. 

Figure 2.14 show the density dependencies of the partial molar volumes at infinite dilution for HjO + NaCl solutions along the supercritical isotherms calculated with the crossover model (Belyakov et al., 1997 and Kiselev and Rainwater, 1997) and the semiempirical equation developed by Sedlbauer et al (1998). 

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