Virial isotherm equation

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

Vii ial equation of state in two dimensions, 931 Virial isotherm, 936 Visible radiation, 797 Volcanoes, in electrocatalysis, 1284 Volmcr, Max, 1048,1474 Volmer. Weber, electrodeposition. 1303. 1306 Volta, 1423, 1455 Volta potential difference, 822 Voltammetry. 1432 1434 cyclic, 1422 1423 diffusion control reactions, 1426 electron transfer reaction, 1424... 

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 ... 

The virial isotherm equation, which can represent experimental isotherm contours well, gives Henry s law at low pressures and provides a basis for obtaining the fundamental constants of sorption equilibria. A further step is to employ statistical and quantum mechanical procedures to calculate equilibrium constants and standard energies and entropies for comparison with those measured. In this direction moderate success has already been achieved in other systems, such as the gas hydrates 25, 26) and several gas-zeolite systems 14, 17, 18, 27). In the present work AS6 for krypton has been interpreted in terms of statistical thermodynamic models. 

These coefficients appear in the virial isotherm Equation 4. Even for osmotically ideal solutions the coefficients A do not vanish in Equation 4. One has in this case... 

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

Concentration dependances for n = 1 or 2 are given for isotherms based on equations of state or sorbed ffuid in Figure 12b. Where n = 1, for larger values of 6 the activity correction is dominant for both Volmer s and van der Waal s isotherms, and Da rises rapidly with increasing concentration. On the other hand, for the virial isotherm with n = 1 and the present choice of coefficients A, the reverse is true. The curvature of the isotherm requires that A/ be positive values of Ai> are usually smaller than Ai and negative. 

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) ... 

Combined with the Gibbs isotherm (Eq. 3.6) it gives the virial isotherm equation... 

In equation (10.8.42) which is known as the virial isotherm, the surface excess of adsorbed ions Fa is expressed in terms of their charge density Qad using the charge on one mole of ions Zj F. 

The values of these coefficients may be deduced directly by matching experimental isotherm data. Equation (3.105) thus provides a useful semiempirical correlation in which the parameters have a simple and well-defined physical significance. In that sense Eq. (3.105) is comparable with the virial isotherm [Eq. (3.43)]. In general the coefficients are temperature dependent but we have found that for many nonpolar sorbates the temperature dependence is modest. For such systems an approximate correlation of equilibrium data over a wide range of temperatures may therefore be obtained with a single set of constant coefficients, considering the Henry constant as the only temperature-dependent parameter (see Figure 4.9). 

While (a In S/d In c)e is usually <1 as > 0, it is of interest to note that R can be >n when attractive interactions (or equivalent solvent structure changes in the interphase) arise (i.e., for positive r in the equations of Table 2). For the case of the Hg-solution interphase, the virial or the Frumkin isotherms are of special interest, as these represent the effects of interactions in the adlayer. Some calculated values for (aln /alnc) for the virial isotherm are shown in Table 3. 

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... 

Let us compare the theoretical isotherm of Equation 3-1 with the tc-T relationship represented as the standard expansion with the second a and third 3 virial coefficients as in... 

Nevertheless, surfactant sorption isotherms on natural surfaces (sediments and biota) are generally non-linear, even at very low concentrations. Their behaviour may be explained by a Freundlich isotherm, which is adequate for anionic [3,8,14,20,30], cationic [7] and non-ionic surfactants [2,4,15,17] sorbed onto solids with heterogeneous surfaces. Recently, the virial-electrostatic isotherm has been proposed to explain anionic surfactant sorption this is of special interest since it can be interpreted on a mechanistic basis [20]. The virial equation is similar to a linear isotherm with an exponential factor, i.e. with a correction for the deviation caused by the heterogeneity of the surface or the energy of sorption. 

Fig. 5.4.7. Sorption isotherms of C12-LAS on different sediments. The lines were calculated with the virial equation (from Westall et al. [20]).

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. 

Numerous representations have been used to describe the isotherms in Figure 5.5. Some representations, such as the Van der Waals equation, are semi-empirical, with the form suggested by theoretical considerations, whereas others, like the virial equation, are simply empirical power series expansions. Whatever the description, a good measure of the deviation from ideality is given by the value of the compressibility factor, Z= PV /iRT), which equals 1 for an ideal gas. 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

Figure 22. Excess internal energy, Eex/N, and virial pressure, PP/p, calculated with the ODS integral equation versus the reduced densities p = pa3, along the isotherms T = 297.6, 350 and 420 K (from bottom to top), by using the two-body potential alone (dotted lines) and the two- plus three-body potentials (solid lines). The experimental data (open circles) are those of Michels et al. [115], Taken from Ref. [129].

Derive an equation for the work of mechanically reversible, isothermal compression of 1 mol of a gas from an initial pressure P, to a final pressure P2 when the equation of state is the virial expansion [Eq. (3.10)] truncated to... 

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... 

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. 

If the isotherm curvature is not too great, the simple virial plot of In (it/p) versus it provides a useful means of obtaining H. Although the linear range of these plots is generally confined to very low values of n, the evaluation of kH is achieved by extrapolation and the application of Equation (4.6) (Cole el al., 1974 Carrott and Sing, 1989). 

It is generally agreed that a virial form of isotherm equation is of greater theoretical validity than the DA equation. As explained in Chapter 4, a virial equation has the advantage that since it is not based on any model it can be applied to isotherms on both non-porous and microporous adsorbents. Furthermore, unlike the DA equation, a virial expansion has the particular merit that as p — 0 it reduces to Henry s law. 

The two-constant versions of Equation (11.2) and other virial expansions can be applied to the low fractional filling section of isotherms on the faujasite zeolites, provided that the temperature is not too low. In this manner it is then possible to obtain the Henry s law constant, kH. 

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