Langmuir isotherm functional form

One may choose 6(Q,P,T) such that the integral equation can be inverted to give f Q) from the observed isotherm. Hobson [150] chose a local isotherm function that was essentially a stylized van der Waals form with a linear low-pressure region followed by a vertical step tod = 1. Sips [151] showed that Eq. XVII-127 could be converted to a standard transform if the Langmuir adsorption model was used. One writes... 

The form of the functions f depends on Che particular isotherm used for example the Langmuir isotherm gives the familiar relation... 

At a given temperature adsorption isotherms measure the number of adsorbed molecules as a function of pressure for the fluid that is in contact with the zeolite. The simplest form is the Langmuir isotherm which treats the zeolite as a collection of equivalent adsorption sites in the absence of adsorbate-adsorbate... 

When the functional form of the correlation is suggested by theory, there is a great deal more confidence that the correlation can be extrapolated into regions of P that have no experimental data, and can be used for other families of compounds other than the training set S. Examples of theory-suggested functional forms include the van der Waals equation of state for gases, the Langmuir isotherm for adsorption and catalysis, and the Clausius-Clapeyron equation for the vapor pressure of liquids. 

Let us consider some frequently used nonlinear functions. The Langmuir isotherm and the Michaelis-Menten kinetics are of the form... 

For example, the Langmuir adsorption isotherm specifically describes adsorption of a single gas-phase component on an otherwise bare surface. When more than one species is present or when chemical reactions occur, the functional form of the Langmuir adsorption isotherm is no longer applicable. Thus, although such simple functional expressions are very useful, they are not generally extensible to describe arbitrarily complex surface reaction mechanisms. 

The diffusion of the electroactive ions is both physical and due to electron transfer reactions.45 The occurrence of either or both mechanisms is a function of the electroactive species present. It has been observed that the detailed electrochemical behaviour of the electroactive species often deviates from the ideal thin film behaviour. For example, for an ideal nemstian reaction under Langmuir isotherm conditions there should be no splitting between the anodic and cathodic peaks in the cyclic voltammogram further, for a one-electron charge at 25 °C the width at half peak height should be 90.6 mV.4 In practice a difference between anodic and cathodic potentials may be finite even at slow scan rates. This arises from kinetic effects of phase formation and of interconversion between different forms of the polymer-confined electroactive molecules with different standard potentials.46... 

Pshezhetskii et al. (17) have recently expressed the view that in the equation for the rate of dehydrogenation, containing the function p in the form of the Langmuir isotherm, z is the ratio of the rate constants of some partial reactions. 

Figure 5 shows the adsorbent loading, Q, as a function of C°, based on data at various pHs. The result suggests that the neutral form of berberine was adsorbed on XAD-7. This result was consistent with the result that adsorption onto XAD-7 was due to hydrophobic interaction. In Fig. 5, the adsorption isotherm for berberine was of the Langmuir isotherm type. This result suggested that berberine might be adsorbed onto XAD-7 form a monolayer. 

Reinmuth [465] arrived at a solution for the Langmuir isotherm in the form of a series, involving the beta function. Levich et al. [362] had an approximate solution for a general adsorption isotherm. 

The more desirable approach is to determine f(Q) from an assumed 0(P,T,Q) and the experimental adsorption isotherm. Sips (16) showed that Equation 1 could be treated by a Stieltjes transform, so that in principle an explicit function could be written for f(Q), provided the experimental isotherm function, 0, could be expressed in analytical form. Subsequently, Honig and coworkers (10, 11, 12) investigated this approach further. The difficulty is that only for certain types of assumed functions 0 and 0 is the approach practical. As a consequence the procedure has been first to restrict the choice of 0 to the Langmuir equation, and second to assume certain simple functions for 0 such as the Freundlich and Temkin isotherm equations. The system is thus forced into an arbitrary mold and again it is not certain how much reliance should be placed on the site energy distributions obtained. 

Langmuir isotherm (Figure 2,6) shows that 50% occupancy occurs at P = 1 lb, and a reasonable first approximation is therefore to consider the curve as a step function in which 6 jumps from zero to unity at P = 1 lb. This amounts to assuming that, for a given pressure, sites of b values greater than 1 IP are completely filled, and the rest are completely empty. In effect, then, a plot of the observed isotherm in the form of 0 w. 1/P (as in Figures 1,6 and 2,a) becomes the first approximation -to the F vs. b function. 

Disregarding for a moment the electrochemical aspect of this isotherm, we note that 0 is proportional to logC, (as opposed to the Langmuir isotherm, where it is proportional to a linear function of the concentration.) A simitar "logarithmic isotherm" was developed by Temkin. His derivation is much more complex, but in the final analysis it is based on the same physical assumptions. It has, therefore, become common to refer to Eq. 141 as the Temkin isotherm, although Temkin has never used it in this form. It is this approximate form of the Frumkin isotherm which is applied to electrode kinetics, as we shall see below. 

Different methods have been proposed to solve Equation (9). The first ascribes a given analytical form to the distribution function y(e) such as a gaussian function [24,25], a gamma function [26] or a combination of two Langmuir isotherms discrete distributions [27] and try to reach the best fitting of the experimental isotherm, using a limited set of parameters. 

The form of the equation that describes the degree of occupancy of receptor sites as a function of analyte concentration is the same, whether the receptor is bound to a surface (Langmuir isotherm) or in solution (reversible reaction)... 

Functional Form and Limiting Behavior. The fnnctional form of the Langmuir isotherm can be rationalized by calcnlating the partition function of N indistinguishable molecules adsorbed on a solid surface, where the internal degrees of... 

The following relations between fractional surface coverage and gas pressure are useful for correlating experimental data on chemisorption. In most cases, there is theoretical justification for the functional form of the isotherm based on rates of adsorption and desorption. Langmuir (1918) and Sips (1948, 1950) proposed the following relation between a and pa -... 

In nonlinear adsorption systems where parallel diffusion mechanisms hold, the mass balance equation given in eq. (9.2-lb) is still valid. The difference is in the functional relationship between the concentrations of the two phases, that is the local adsorption isotherm. In general, this relationship can take any form that can describe well equilibrium data. Adsorption isotherm such as Langmuir, Unilan, Toth, Sips can be used. In this section we present the mathematical model for a general isotherm and then perform simulations with a Langmuir isotherm as it is adequate to show the effect of isotherm nonlinearity on the dynamics behaviour. The adsorption isotherm takes the following functional form ... 

The diffusivity matrix is a function of the concentrations of all species. Eq. (10.5-9) is the constitutive flux equation. The explicit functional form of the diffusivity matrix in terms of concentration depends on the choice of the adsorption isotherm. What we shall do in the next section is to show this form for the case of the extended Langmuir isotherm. 

The computation of the non-dimensional governing equations is carried out after we specify the functional form for the multicomponent isotherm. We shall do it here with the extended Langmuir isotherm (eq. 10.5-10). 

The statistical derivation shows that the Freundlich isotherm is expected to be valid at low surface coverages in fact, the isotherm successfully predicts that 0a —> 0 when Pa O but fails to predict 9a- -I when Pa — oo. The Freundlich isotherm can handle multicomponent adsorption to some extent, and in some cases, the Langmuir isotherm can be reduced to the power function form of the Freundlich isotherm. 

Equation (18) is the well-known Langmuir isotherm, apphcable and measurable for hquid free surfaces. It is evident that any measured and calculated explicit form of the function l/p(a )— according to Eq. (15)—yields the corresponding explicit excess isotherm equation. 

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

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