Virial adsorption isotherm

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

Virial treatment provides a general method of analysing the low-coverage region of an adsorption isotherm and its application is not restricted to particular mechanisms or systems. If the structure of the adsorbent surface is well defined, virial treatment also provides a sound basis for the statistical mechanical interpretation of the adsorption data (Pierotti and Thomas, 1971 Steele, 1974). As indicated above, Kl in Equation (4.5) is directly related to kH and therefore, under favourable conditions, to the gas-solid interaction. 

The coefficient of the first term of the virial series in Equation I is the inverse Henry constant for the temperature considered. From the graph of the isotherm in Figure 3 of the survey paper, it follows that the linearity of the initial section of the adsorption isotherm of ethane on zeolite LiX in accordance with the calculated Henry constant is observed for adsorption values not exceeding 0.7 mmole/g. According to K. N. 

Fig. 2. Adsorption isotherms in all four model systems. F is the Gibbs excess adsoiption. The pressures corresponding to the three configurations shown in Figure 3 are marked with arrows. The pressure is plotted relative to the vapor pressure of the model fluid, as determined by independent Gibbs Ensemble Monte Carlo simulations. Chemical potentials were converted to pressures using a virial equation of state.

Whether the adsorption isotherm has been determined experimentally or theoretically from molecular simulation, the data points must be fitted with analytical equations for interpolation, extrapolation, and for the calculation of thermodynamic properties by numerical integration or differentiation. The adsorption isotherm for a pure gas is the relation between the specific amount adsorbed n (moles of gas per kilogram of solid) and P, the external pressure in the gas phase. For now, the discussion is restricted to adsorption of a pure gas mixtures will be discussed later. A typical set of adsorption isotherms is shown in Figure 1. Most supercritical isotherms, including these, may be fit accurately by a modified virial equation. ... 

Table 1 Constants of Eq. (1) for reference adsorption isotherms of gases in NaX zeolite at 293.15 K. Virial coefficients C, in units of kg mol ...

Analysis of Heterogeneity. The monolayer analysis consists of three elements an adsorption isotherm equation, a model for heterogeneous surfaces, and an algorithm such as CAEDMON, which uses the first two elements to extract the adsorptive energy distribution and the specific surface from isotherm data. Morrison and Ross developed a virial isotherm equation for a mobile film of adsorbed gas at submonolayer coverage (6) ... 

To obtain a manageable equation for the adsorption isotherm in this model, without loosing the generality of a continuous model, we make use of a virial expansion for the two-dimensional spreading pressure (f) of the adsorbed phase [24] ... 

It is clear that the calculation of gas—solid virial coefficients is very difficult, so that only the first few of them could be evaluated. This means that the model will be useful only at low values of the adsorbed phase density. But on the other hand, the most important effects of heterogeneity can be seen for the low-pressure part of the adsorption isotherm. 

Once the virial coefficients have been evaluated, the adsorption isotherm for low pressure is obtained through... 

Fluid phase interactions - Second virial coefficients Solution interactions - Partial molar volumes - Solubilities Surface interactions - Adsorption isotherms Diffusion coefficients Mass transfer coefficients Molecular masses... 

Data are given in [mmol/gAC]. The lines show the correlation ofthe data using a virial adsorption isotherm, cp. Chap. 10. 

The interpretation of impedance measurements, i. e. curves C = C(v) in adsorption equilibria, or capacitance versus time curves C = C(t) at constant frequency (v = const) for adsorption processes, is not easy and still a field for analytic and simulative investigations. Also the dielectric equation of state (EOS) (Cr = 8r(T, p, m ))of an adsorption system does not seem to have been investigated in a systematic way -except some virial like series expansions considered in [3.40]. Correspondingly interrelations between the dielectric EOS and the adsorption isotherm or the heat of adsorption are - though they must exist, cp. [6.16, 6.17] - unknown to the best knowledge of the authors. 

Theoretical values of the second to the fourth gas-siuface virial coefficients are available for some specific intermolecular potentials [5,17,30,33]. Moreover, a relationship between the gas-surface virial coefficients and the real 2D virial coefficients can be obtained. Thus, for the second virial coefficient and by using the Gibbs adsorption isotherm equation, Eq. (2), one finds [5,30,197]... 

Comparison between the experimental adsorption isotherms (adsorption pressure as a function of the amotmt of matter adsorbed) and the theoretical expression for the adsorption pressure as a function of the density of the monolayer, obtained from Steele s two-dimensional approximation, Eq (23). In this case, Henry s constant must be previously calculated from experimental data or through the gas-solid virial coefficient [210,211], Eq. (4). 

Despite the importance of mixtures containing steam as a component there is a shortage of thermodynamic data for such systems. At low densities the solubility of water in compressed gases has been used (J, 2 to obtain cross term second virial coefficients Bj2- At high densities the phase boundaries of several water + hydrocarbon systems have been determined (3,4). Data which would be of greatest value, pVT measurements, do not exist. Adsorption on the walls of a pVT apparatus causes such large errors that it has been a difficult task to determine the equation of state of pure steam, particularly at low densities. Flow calorimetric measurements, which are free from adsorption errors, offer an alternative route to thermodynamic information. Flow calorimetric measurements of the isothermal enthalpy-pressure coefficient pressure yield the quantity 4>c = B - TdB/dT where B is the second virial coefficient. From values of obtain values of B without recourse to pVT measurements. 

If the experimental data of adsorption are fitted into the virial isotherm, the equilibrium constant P" is given by the equation... 

In Section 6.8.5 we were able to derive one specific isotherm, the virial isotherm. However, can all the adsorption systems be described by this isotherm There is a difficulty. The isotherms, similarly to the equations of state in the gas phase, have restrictions that make them suitable for use only under certain conditions.57 For... 

Virial Isotherm Equation. No isotherm equation based on idealized physical models provides a generally valid description of experimental isotherms in gas-zeolite systems (19). Instead (6, 20, 21, 22) one may make the assumption that in any gas-sorbent mixture the sorbed fluid exerts a surface pressure (adsorption thermodynamics), a mean hydrostatic stress intensity, Ps (volume filling of micropores), or that there is an osmotic pressure, w (solution thermodynamics) and that these pressures are related to the appropriate concentrations, C, by equations of polynomial (virial) form, illustrated by Equation 3 for osmotic pressure ... 

Because the 77 K isotherms are very steep (i.e. high adsorption affinity), it is very difficult to undertake any form of virial analysis. The few detailed nitrogen isotherms so far determined (Reichert et al., 1991) at higher temperatures (i.e. 293-373 K) on Silicalite and HZSM-5 have indicated that Henry s law is obeyed at low fractional... 

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