Equilibrium isotherms equation

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). 

The corresponding equilibrium isotherm equations for these kinetic expressions are listed in the above table. 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

Adsorption — An important physico-chemical phenomenon used in treatment of hazardous wastes or in predicting the behavior of hazardous materials in natural systems is adsorption. Adsorption is the concentration or accumulation of substances at a surface or interface between media. Hazardous materials are often removed from water or air by adsorption onto activated carbon. Adsorption of organic hazardous materials onto soils or sediments is an important factor affecting their mobility in the environment. Adsorption may be predicted by use of a number of equations most commonly relating the concentration of a chemical at the surface or interface to the concentration in air or in solution, at equilibrium. These equations may be solved graphically using laboratory data to plot "isotherms." The most common application of adsorption is for the removal of organic compounds from water by activated carbon. 

The separation of bi-naphthol enantiomers can be performed using a Pirkle-type stationary phase, the 3,5-dinitrobenzoyl phenylglycine covalently bonded to silica gel. Eight columns (105 mm length) were packed with particle diameter of 25 0 fiva. The solvent is a 72 28 (v/v) heptane isopropanol mixture. The feed concentration is 2.9 g for each enantiomer. The adsorption equilibrium isotherms were determined by the Separex group and already reported in Equation (28) [33]. 

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

Wakeman found that the effect of increasing the equilibrium constant in the description isotherm equation is to increase the wash-ratio required. Is this confirmed by simulation and what is the explanation of this effect. 

In the mechanism illustrated by scheme B, significant inhibition is only realized after equilibrium is achieved. Hence the value of vs (in Equations 6.1 and 6.2) would not be expected to vary with inhibitor concentration, and should in fact be similar to the initial velocity value in the absence of inhibitor (i.e., v, = v0, where v0 is the steady state velocity in the absence of inhibitor). This invariance of v, with inhibitor concentration is a distinguishing feature of the mechanism summarized in scheme B (Morrison, 1982). The value of vs, on the other hand, should vary with inhibitor concentration according to a standard isotherm equation (Figure 6.5). Thus the IC50 (which is equivalent to Kfv) of a slow binding inhibitor that conforms to the mechanism of scheme B can be determined from a plot of vjv0 as a function of [/]. 

Poisoning is caused by chemisorption of compounds in the process stream these compounds block or modify active sites on the catalyst. The poison may cause changes in the surface morphology of the catalyst, either by surface reconstruction or surface relaxation, or may modify the bond between the metal catalyst and the support. The toxicity of a poison (P) depends upon the enthalpy of adsorption for the poison, and the free energy for the adsorption process, which controls the equilibrium constant for chemisorption of the poison (KP). The fraction of sites blocked by a reversibly adsorbed poison (0P) can be calculated using a Langmuir isotherm (equation 8.4-23a) ... 

And we see a further point the equilibrium constant K of a reaction is a direct function of A Gr according to the van t Hoff isotherm (Equation (4.55)). If the overall energy of reaction remains unaltered by the catalyst, then the position of equilibrium will also remain unaltered. 

Adsorption from liquids is less well understood than adsorption from gases. In principle the equations derived for gases ought to be applicable to liquid systems, except when capillary condensation is occurring. In practice, some offer an empirical fit of the equilibrium data. One of the most popular adsorption isotherm equations used for liquids was proposed by Freundlich 21-1 in 1926. Arising from a study of the adsorption of organic compounds from aqueous solutions on to charcoal, it was shown that the data could be correlated by an equation of the form ... 

Many solutions are available for equation 17.70 and its refinements. Three cases are considered to illustrate the range of solutions. Firstly, it is assumed that the bed operates isothermally and that equilibrium is maintained between adsorbate concentrations in the fluid and on the solid. Secondly, the non-equilibrium isothermal case is considered and, finally, the non-equilibrium non-isothermal case. 

It will be noted that the universal isotherm equation as written here has formal similarity to pressure explicit forms of Langmuir, Langmuir-Freundlich and LRC models. One key advantage of the universal form is that the heat of adsorption and the adsorption equilibrium are bound to be self-consistent. 

The equilibrium models of chromatography are given by the mass balance equation given in Equation 10.8 and a proper isotherm equation, q = f(C), should be used to relate the mobile phase and stationary phase concentrations. 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

There exists an equilibrium at the interface between a charged suspension and its solution consisting of a dynamic interchange of both cation and anion. If the suspension be positively charged the adsorption of the positive ion is greater than that of the negative at that particular concentration. On application of the Freundlich isotherm equation to each ion... 

On thermodynamic grounds pore-size distributions are measured using the desorption isotherm, (see equations 8.8 and 8.9). The exception to this are bottle-neck pores exhibiting type E hysteresis. In this case the equilibrium isotherm is that of adsorption because the unstable state is associated with the condition that the wide portion of the pore is unable to evaporate until the narrow neck empties. Regardless of which isotherm is used, however, the mathematical treatment remains the same. 

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 statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

At higher sorbate concentrations, such that an appreciable fraction of the cavities contain more than one occluded molecule, the equation for the equilibrium isotherm may be expressed as (21)... 

Figure 1. Equilibrium isotherms for sorption in 5A zeolite and H-chabadte data of Glessner and Myers (26), data of Barrer and Davies (7), O data of Derrah (3), X theoretical lines from Equation 8.

Tphe breakthrough curve for a fixed-bed adsorption column may be pre-dieted theoretically from the solution of the appropriate mass-transfer rate equation subject to the boundary conditions imposed by the differential fluid phase mass balance for an element of the column. For molecular sieve adsorbents this problem is complicated by the nonlinearity of the equilibrium isotherm which leads to nonlinearities both in the differential equations and in the boundary conditions. This paper summarizes the principal conclusions reached from a recent numerical solution of this problem (1). The approximations involved in the analysis are realistic for many practical systems, and the validity of the theory is confirmed by comparison with experiment. 

K denotes the equilibrium constant, C is in moles dm 3 of solution, and the coefficients A depend upon T but not on C. This isotherm equation approaches Henry s law as C becomes small enough and has already been shown adequate, for a suitable choice and number of coefficients A, to represent experimental isotherms well (4, 6, 20). One to three such coefficients were needed according to the rectangularity of the isotherms. It lends itself to thermodynamic analysis of gas-solid distribution equi-... 

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. 

Equation 3 was verified experimentally (3) over wide ranges of temperature and equilibrium pressure for the adsorption of various vapors on active carbons with different parameters for the microporous structure. For adsorption on zeolites, this equation fitted the experimental results well only in the range of high values of 0 (4, 5, 6, 7). Among other equations proposed for the characteristic curve (4, 5, 8, 9, 10) we chose to use the Cohen (4) and Kisarov s (10) equation, which starts from the following adsorption isotherm equation ... 

This rate equation must satisfy the boundary conditions imposed by the equilibrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to zero at equilibrium. 

The model involves four basic assumptions. First, it is postulated that an adsorbed phase of silicic acid exists on the particle surface, the equilibrium of which can be expressed by a Langmuir isotherm (Equation 1, symbols are defined in table of nomenclature). 

The equilibrium concentrations in the solid and liquid phases at the surface of the adsorbent particle are qsfc end Cs and afc, bk and 8k are isotherm constants for each solute corresponding to the single solute isotherm equation (4). 

It may seem from the above discussion that it is impossible to use even batch isotherm measurements to design HPLC (or SPE) separations. This is not so, however, at least when the HPLC separation occurs under near-equilibrium conditions. Nonlinear chromatographic peaks can be simulated [38] once the corresponding isotherms have been measured. In this case one does not need a physical interpretation of the isotherm equation s constants they can be regarded merely as interpolation factors. Separately measured isotherms of the two compounds are satisfactory in many cases because - as mentioned above - competition often has only minor influence on the separated peaks position and shape. 

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