Langmuir model linear form

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

The number of gas molecules can be measured either directly with a balance (gravimetric method) or calculated from the pressure difference of the gas in a fixed volume upon adsorption (manometric method). The most frequently apphed method to derive the monolayer capacity is a method developed by Brunauer, Emmett, and Teller (BET) [1], Starting from the Langmuir equation (monolayer adsorption) they developed a multilayer adsorption model that allows the calculation of the specific surface area of a sohd. The BET equation is typically expressed in its linear form as... 

The mechanism of the formation of phosgene according to the reaction, CO (A) + C12(B) =4 C0C12(C), is to be checked with given data at 30.6 C (Potter Baron, CEP 47 473, 1951). Six Langmuir-Hinshelwood equations and a power law model are examined. The rate equations are analyzed in linearized forms. Those that have negative constants are not physically realistic. 

The suitability of Langmuir model fo describe the photoadsorption phenomenon has been further checked by using the linear form of Langmuir equation [Equation (A18)]. Figure 8 reports the values of the Cl,o/(Ct,o Cl,o)... 

Approximation of the linear form is not necessary for the Langmuir isotherm, and the first plot of the adsorption data will determine whether or not the model is applicable, and also will allow calculation of the adsorption coefficients. Usually a single model will not be satisfactory for a wide range of adsorbate concentrations but will only serve in narrow range of concentration. At low concentrations, C/C 1, the BET model reduces to a Langmuir model. 

An example of a model nonlinear in parameters is Eq. (7-166). Here it is not possible through any number of transformations to obtain a linear form in all the parameters k0, E, K o, Eaa, Km, Ea. Note that for some Langmuir-Hinshelwood rate expressions it is possible to linearize the model in parameters at isothermal conditions and obtain the kinetic constants for each temperature, followed by Arrhenius-type plots to obtain activation energies (see, e.g., Churchill, The Interpretation and Use of Piate Data The Rate Concept, McGraw-Hill, 1974). 

Interestingly, while the adsorption of polyelectrolytes onto BaS04 does not conform to the usual assumptions of the Langmuir adsorption isotherm model, the Langmuir model nevertheless provides a good fit to these experimental results as it does in many other cases of polymer adsorption. The Langmuir adsorption isotherm can be written in linear form as shown in Equation 1. 

Theoretical breakthrough curves for nonlinear systems may be calculated by numerical solution of the model equations using standard finite difference or collocation methods. Such solutions have been obtained by many authors and a brief summary is given in Table 8.4. In all cases plug flow was assumed and the equilibrium relationship was taken to be of cither Langmuir or Freundlich form. As linearity is approached ( ->1.0) the linearized rate models approach the Anzelius model (Table 8.1, model la) while the diffusion models approach the Rosen model (Table 8.1, model la). The conformity of the numerical solution to the exact analytic solution in the linear limit was confirmed by Garg and Ruthven. ... 

Free gel was added to definite concentrations of the MB solutions (2-20 mg/1) at room temperature and was noted for its adsorption. It is clear from Figure 13.3 that the dye adsorption increases sharply with an increase in the initial dye concentration. When Cq was reached at 5 ppm and 10 ppm, the was reached at 10.04 and 20.81 respectively, which were much higher than reported Qe values of other adsorbents (Table 13.3). Equilibrium adsorption isotherm is an important criterion to determine the mechanism of dye adsorption on hydrogel. The Langmuir and Freundlich models are widely used to examine the adsorption isotherms. Freundlich isotherm models are based on the assumption that the surface of the adsorbent is not homogeneous. The experimental data in Figure 13.3 was also analyzed with the Freundlich isotherm model, which describes a heterogeneous system with multilayer adsorption. The linear form of Freundlich isotherm equation... 

If you want to test whether your data follow the Langmuir model, there is a better functional form than Equation (27.12). Equation (27.12) requires data over a wide range of pressures to ensure that you reach saturation, and this can often be difficult to obtain. Instead, you can rearrange Equation (27.12) into a linearized form ... 

Each data point represents the average value from three samples. Error bars show the standard deviations. (B) Adsorption data fit with the linear form of the Langmuir model, eq 2. The slope and intercept values are used to calculate the adsorption constant. Reproduced with permission from ref II. Copyright 2005 J. Am. Chem. Soc. 

Many adsorption phenomena especially of surfactants, polymers, proteins and the chemical adsorption of gases on solids can be well represented by the Langmuir adsorption isotherm. This equation can be expressed in a suitable linear form and we can obtain the two parameters of the model, of which one is the concentration or volume at maximum (full) coverage or the so-called monolayer coverage. Knowledge of this monolayer coverage and of the specific surface area of the solid can help us estimate the surface area occupied by a molecule at the interface and thus the amount needed for stabilization. The specific solid surface area can be obtained from gas adsorption measurements on the same solid. 

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... 

If the data fit this model, a plot of 1/F0 versus 1/(S0) should be linear with a slope K/Vmax and intercept l/Vmax. It is analogous to that used in determining the constants in the Langmuir equation for adsorption on solid surfaces. Other forms that may be used to prepare linear plots are... 

Non-linearities arising from non-reactive interactions between adsorbed species will not be our main concern. In this section we return to variations of the Langmuir-Hinshelwood model, so the adsorption and desorption processes are not dependent on the surface coverage. We are now interested in establishing which properties of the chemical reaction step (12.2) may lead to multiplicity of stationary states. In particular we will investigate situations where the reaction step requires the involvement of additional vacant sites. Thus the reaction step can be represented in the general form... 

The solid and liquid film linear driving force models can be written under the same general form of a second order Langmuir kinetic model [1]. We can insert the Langmuir isotherm equation q = qsbC)/ l bC)) in the partial differential equation of tire solid fihn linear driving force model (Eq. 14.3)... 

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