Adsorption linear isotherm explained

Cluster-assembled carbon film are very porous with a pore diameter peaked at 3-4 nm, as shown by adsorption/desorption isotherm analysis [32]. The gases used in our experiments have a molecular or atomic size much smaller than the average pore size, so it is reasonable to assume that they equally diffuse in the mesoporous film network and interact with the same amount of linear carbon structures. This explains the similar Rq value observed for the three gases. The time required to reach the asymptotic value of is affected by the momentum transferred from the molecule to the film network during collisions, hence by the mass of the gas molecule. We note that H2 does not seem to chemically interact with sp structures. 

Fig 7 4), which corresponds to the Langmuir equilibrium with separation factor rl in the Freundlich isotherm system In these cases, a constant profile of mass transfer zone is established while adsorption proceeds in a column (Fig 7 5) This is in contrast to linear isotherm systems where mass transfer zone continues to spread with increase of traveling time The reason for the formation of a constant pattern is explained as follows The speed of movement of the point on the mass transfer zone whose concentration is C, F(C), can be related to the equilibrium through the basic equation... 

The application of dispersion forces with surface polarization to account for a potential function in first-layer adsorption such as that described by Equation 7a, in which correlations with other adsorbate molecules result only in repulsion over the entire first-layer filling, seems more difficult to justify. Thus, b is apparently negative for the entire region below point B in most typical Type II isotherms, at least those in which point B appears at x < 0.05. If molecules adsorbed on the first layer of a Type II isotherm were to cause only repulsion, one would expect them to adsorb always in a pattern such as to remain as far apart as possible. But then when they are all a long distance from each other, as they would be near zero coverage, a repulsion term of sufficiently long range to account for a linear bO relationship is difficult to explain. Perhaps it may be possible to explain this situation in a Model 4 type adsorption process—i.e., in a model in which both the adsorbate and the adsorbent suffer polarization upon adsorption. 

A mathematical difficulty arises in the evaluation of the integral in the region below the first experimental point (p=px). First, the exact form of the adsorption isotherm is unknown in this range and secondly the ratio n/p is indeterminate as p tends to zero. A possible solution is to assume a linear variation of n versus p (i.e. Henry s law, as explained in Chapter 4) and then... 

An exaggerated emphasis on heats of chemisorption has probably been harmful in the proper understanding of the role of chemisorption in surface phenomena. Thus the marked nonuniformity of all surfaces with respect to heats of chemisorption has led to rather elaborate treatments where models of surface heterogeneity (statistical distribution of energy sites) or, less successfully, specific forces of interaction between adsorbed species have been invoked to explain the non-Langmuirian adsorption isotherms. For instance the Frumkin isotherm can be obtained with a linear variation of heats of adsorption with coverage, and the Freundlich isotherm is attributed to an exponential variation of heats of adsorption. 

Ferguson and Anderson (31) investigated the adsorption of arsenate and arsenite on Tron and aluminum hydroxides and found that for both adsorbents the adsorption of arsenate followed a Langmuir isotherm while the sorption of arsenite varied linearly with concentration. In the concentration range used in the experiment described above, they found that arsenate was much more strongly adsorbed than arsenite. They did not explain the rather surprising difference in behavior between arsenate and arsenite however, their results give support to the results found in this experiment. 

An experimental example of this kind Is shown in flg. 5.29, for a mixture of two dextran fractions (of equal mass fractions and u> ) adsorbing from water on to silver iodide ). The top figure gives F(Cp) adsorption isotherms at three different values of as explained in sec. 5.3d this parameter is the ratio V/A between the solution volume V and the surface area A in the system. The isotherms consist of three linear sections in the initial steep rise the surface is unsaturated emd all molecules can find a place on the surface, in the linearly increasing branch the small molecules are gradually displaced by big ones, and in the pseudo-plateau only the long cheiins are found on the surface. 

It would appear that, at fluid/fluid interfaces, proteins give adsorption isotherms for which interfacial pressure is a linear function of the logarithm of the bulk concentration over appreciable ranges as has been found for simpler compounds. What has not been satisfactorily explained is the reason for the very low values of A. Joos has inter-... 

If, because of the small sample size used and the numerical values of the parameters, the isotherm for a two-site surface behaves linearly, a certain degree of peak tailing can still be explained by assuming that the kinetics of adsorption-... 

This isotherm was proposed to explain the linear dependence of the heat of adsorption on surface coverage (88). In the following derivation, however, induced surface heterogeneity will be implied rather than intrinsic heterogeneity, originally considered by Temkin. This isotherm is applicable for intermediate total surface coverage (0.2 < 0j < 0.8). 

A numerical solution, based on a linear adsorption isotherm, was performed by Ziller et al. (1985) using a finite difference technique. An example of the numeric results is given in Fig. 4.16. The dramatic drop of the surface concentration at x=L/K is explained by the boundary condition, which assumed a 10-fold expansion of the interface. Due to the complexity of the problem, it seems impossible to derive an analytical solution without further serious simplifications. 

Figure 6. The adsorption isotherms of Cd and PO4 plotted on a double logarithmic scale. The adsorption of Cd on goethite is linear at low Cd loading (low pH and concentration), as shown by the slope of the line (n=l), in contrast to the adsorption at high loading at pH=9 (n=0.45). The isotherm of PO4 on goethite (pH=4, 0.01 M) is extremely non linear (n<0.1). The (non) linearity can be explained on the basis of electrostatics (see text). Data are taken from Venema et al.(1996b) and Geelhoed et al.(1997).

Explain the meaning of each term in the development of the solute movement equations and use this theory for both linear and nonlinear isotherms to predict the oudet concentration and temperature profiles for a variety of different operations including elution chromatography, adsorption with thermal regeneration, PSA, SMB, and ion exchange... 

Crosslinked styrene (St)/maleic anhydride (MA) eopolymers have been synthesized, hydrolyzed with dicarbojgrlie aeid, and converted to bear dihydrojg hosphino functionalities. On the dihydro3QT>hosphino-funetionalized styrene-methyl acrylate (20% divinylbenzene) copolymer, the adsorption of Pb showed a linear relationship with the concentrations and fitted the Langmuir isotherm. The kineties of Pb adsorption on this dihydrojg hosphino-functionalized eopolymer were studied. The metal-ion adsorption kinetics of this copolymer appeared to show particle diffusion. They explained that the moving boundary advanced from the surface of the molecule towards the center. ... 

Numerous attempts have been made to derive mathematical expressions to fit the adsorption isotherms, but no single adsorption isotherm equation has been found to explain all the adsorption data. The common feature of these isotherms, however, is that all isotherms tend to be linear at low pressures and correspondingly low adsorption values i.e the amount adsorbed x is proportional to pressure p. This low-pressure region of the adsorption isotherm is sometimes referred to as the Henrys law region, because it is in consonant with the Henrys law, according to which the solubility of a gas in a liquid is proportional to its pressure. 

The point at low P/Po values where the almost linear middle section of the isotherm begins is normally, it is explained, where monolayer coverage is complete and multilayer adsorption begins. Typically a large uptake occms close to saturation pressures due to substantial vapor condensation on external smfaces... 

In some instances the enthalpy curves (e.g. for toluene + hexane mixtures) were linear indicating a constant value of A hf in most, however, a break was detected and interpreted as a jump in A h at a certain surface coverage. What is not explained is how A 2 can exhibit a break when the analysis is based on the assumption that Aw 2 is directly proportional to x 2. Nor is it clear in the case, for example, of the acetone + nitrobenzene system how a smooth curve of x" against x l can give rise to a sharp break in AG". Attempts to relate various breaks and discontinuities in the thermodynamic curves to the presence of two types of adsorption site cannot have any validity. Without access to the original data it is difficult to assess this work, which contains some potentially useful experimental results. In any case it is important to remember, as stressed by the author that the thermodynamic quantities of adsorption are calculated from individual isotherms, which depend strongly on the method of their determination . 

Such deviations occur when distributions of adsorption site energies do not fit a Gaussian-type (or related distribution function). Then, the obtained experimental isotherm will not be linearized by the conventional Langmuir, BET and DR adsorption equations. If the continuity of the distribution curve is disturbed in some way (e.g. by selective oxidation to widen some parts of the porosity during an activation process) then deviations will occur from the model equations. Elaborations of equations to obtain a better fit are mathematical devices to correct for deviations to the distribution curves but do little to explain the causes. 

In the case where the adsorption isotherm is linear and the process is reversible, analytical solutions exist, as described in Section 7.2 above (Van Geneuchten, 1981). Physically this means that, although the polymer effluent is retarded at the frontal part, the material will appear again later after the rear of the tracer slug has appeared. The analysis of this case is very simple and is dealt with as explained above. 

The linear section of isotherms as those shown in Fig. 21 defines the boundary of supercritical adsorption. As argued earlier, both absolute adsorption and the volume of the adsorbed phase are constant for the scope of linearity therefore, the density of the adsorbed phase must also be constant. The constant equals the gas-phase density pj, when the linear section touches the abscissa, and it becomes less than pj, upon the further inerease in pressure. Because the gas phase is at the top of the adsorbed phase, a thermodynamically unstable system has thus been established. The lower end of the linear section marks the boundary of adsorption, beyond which any recorded isotherm cannot be explained by adsorption, but rather by compression due to the density difference. 

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