Isotherms experimental equilibrium isotherm

Adsorption of glutathione, which is an acidic peptide, on the crosslinked chitosan fiber (ChF) appeal technically feasible. The experimental equilibrium isotherm (q-C curve) for adsorption of glutathione on ChF was independent of the initial concentration of glutathione. But tite adsorbed amount of glutathione on ChF was effected by the pH value of the solution on tire q-pH curve, significantly. It appeared that the adsorption of glutathione was correlated by the Langmuir equation well. 

In this work, we investigate the possibility of using ChF for adsorption of acidic peptide. The experimental equilibrium isotherm and mechanism for adsorption of acidic peptide is presented and discussed. 

The experimental equilibrium isotherm q-C curve) for adsorption of glutathione on ChF was independent of the initial concentration of glutathione. 

Fig. 2 Experimental equilibrium isotherm for benzene in 13X zeolite at 458 and 513 K showing conformity with the SSTM isotherm (Eq. 11) with m < v/p = 5.0 and = 8.8 molecules/cage Torr at 458 K and 1.25 molecules/cage Torr at 513 K. From Ruthven [8]...

Fig. 3 Experimental equilibrium isotherm for hydrocarbons on NaX and NaY showing conformity with Eqs. 12-14. From Ruthven and Goddard [10]...

In the present example, using the cadmium chemical spill problem, the Langmuir equation provides a good correlation of the experimental equilibrium isotherm data. Equation (15.3) can be substituted into Eq. (15.2) and the mass of sorbent, M g peat, to bring the liquid phase concentration down to Qi, can be determined from Eq. (15.6). 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

The equilibrium isotherms were favorable type and the Langmuir equation represents our experimental data very well. 

The linear equilibrium isotherm adsorption relationship (Eq. 11) requires a constant rate of adsorption, and is most often not physically valid because the ability of clay solid particles to absorb pollutants decreases as the adsorbed amount of pollutant increases, contrary to expectations from the liner model. If the rate of adsorption decreases rapidly as the concentration in the pore fluid increases, the simple Freundlich type model (Eqs. 8 and 9) must be extended to properly portray the adsorption relationship. Few models can faithfully portray the adsorption relationship for multicomponent COM-pollutant systems where some of the components are adsorbed and others are desorbed. It is therefore necessary to perform initial tests with the natural system to choose the adsorption model specific to the problem at hand. From leaching-column experimental data, using field materials (soil solids and COMs solutions), and model calibration, the following general function can be successfully applied [155] ... 

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

Take a reversible cycle in a system where p, v, T, etc., change so slowly that the system passes through a series of small, equilibrium, isothermal steps before arriving back to its initial position. Let 6qa be the heat absorbed during one of these steps when the temperature is Ti. It is found experimentally that the sum of 6qJTa for all of the various steps taken during the cycle is equal to zero, i.e.. 

Diffusion-type models are two-parameter models, involving kt or Ds and La, while BDST models are one-parameter models, involving only 0, as gmax is an experimentally derived parameter. The determination of La requires the whole experimental equilibrium curve, and in case of sigmoidal or other non-Langmiur or Freundlich-type isotherms, these models are unusable. From this point of view, BDST models are more easily applied in adsorption operations, at least as a first approximation. 

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

The results of experimental studies of the sorption and diffusion of light hydrocarbons and some other simple nonpolar molecules in type-A zeolites are summarized and compared with reported data for similar molecules in H-chabazite. Henry s law constants and equilibrium isotherms for both zeolites are interpreted in terms of a simple theoretical model. Zeolitic diffusivitiesy measured over small differential concentration steps, show a pronounced increase with sorbate concentration. This effect can be accounted for by the nonlinearity of the isotherms and the intrinsic mobilities are essentially independent of concentration. Activation energies for diffusion, calculated from the temperature dependence of the intrinsic mobilitieSy show a clear correlation with critical diameter. For the simpler moleculeSy transition state theory gives a quantitative prediction of the experimental diffusivity. 

The theoretical lines in Figure 2 are calculated assuming constant values of D0 with the derivative d In p/d In c calculated from the best fitting theoretical equilibrium isotherm (Equation 8). The theoretical lines give an adequate representation of the experimental data suggesting that the concentration dependence of the diffusivity is caused by the nonlinearity of the relationship between sorbate activity and concentration as defined by the equilibrium isotherm. The diffusivity data for other hydrocarbons showed similar trends, and in no case was there evidence of a concentration-dependent mobility. Similar observations have been reported by Barrer and Davies for diffusion in H-chabazite (7). 

Figure 5 shows the experimental breakthrough curves obtained by Sheth (14) for saturation and regeneration of a 4A molecular sieve column with a feed stream containing a small concentration of ethylene in helium. The equilibrium isotherm for this system is highly nonlinear, and, as a result of this, the saturation and regeneration curves have quite different shapes. However, the theoretical curves calculated from the nonlinear analysis using the same values of the parameters bqB and D /rz2 for both... 

Experiments. Experimental adsorption isotherms were determined with a classical manometric apparatus in the equilibrium pressure range 1-760 torr and at temperatures 0°-80°C, thus ranging above and below the critical temperatures Tc for the adsorbates (see Table I). 

The amount of theoretical and experimental research focused on the interaction, equilibrium and dynamical properties of noble, simple and polyatomic gases within quasi-one-dimensional nanotubes is still limited [6-13]. Experimental adsorption isotherms have been reported for simple gases (Ar,N2) and alkanes (methane [11], ethane [12], propane-butane-pentane [13]) in monodisperse nanotubes of aluminophosphates. It is expected that similar experiment could be carried out soon in bundles of monodispersed carbon nanotubes. 

In the first step, in which the molecules of the fluid come in contact with the adsorbent, an equilibrium is established between the adsorbed fluid and the fluid remaining in the fluid phase. Figures 22-8 through 22-10 show several experimental equilibrium adsorption isotherms for a number of components adsorbed on various adsorbents. Consider Fig. 22-8, in which the concentration of adsorbed gas on the solid is plotted against the equilibrium partial pressure p° of the vapor or gas at constant temperature. At 40°C, for example, pure propane vapor at a pressure of 550 mm Hg is in equilibrium with an adsorbate concentration at point P of 0.04 lb adsorbed propane per pound of silica gel. Increasing the pressure of the propane will cause more propane to Be adsorbed, while decreasing the pressure of the system at P will cause propane to be desorbed from the carbon. 

Equations (1.8) and (1.9) represent the adsorption isotherm which is the relationship between the amount adsorbed by unit mass of solid and the equilibrium pressure (or relative pressure), at a known temperature. The experimental adsorption isotherm is usually presented in graphical form. 

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... 

Although conventional derivations of the conditions for chemical equilibrium sometimes are restricted to isothermal, isobaric processes (possibly because often dT = 0 and dp = 0 in experimental equilibrium determinations), the general equilibrium conditions do not depend on these assumptions, as will be seen from the following development. By substituting equation (1) into equation (2) and using equation (3) to eliminate dU from the resulting expression, we find that... 

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