Langmuir isotherm plots

Figure 7. Langmuir isotherm plot of ko/(T o — k) against the reciprocal of the phosphate concentration, where ko is the calcite growth rate constant in the absence of phosphate and k is the rate constant in the presence of phosphate ( ) ko = 0.824 (n)ko = 1.205 (O)ko = 0.790. Adapted from Ref. 43.

Fig. II The Langmuir isotherm. Plots of the fractional coverage 0 versus a) concentration and (b) potential or log C, for three values of the equilibrium constant K = I, 5 and 25.

Fig. 5 Electrochemical Langmuir isotherm. Plots of fractional coverage 9 versus...

Some examples of simple and complex Freundlich and Langmuir isotherm plots are shown in Fig. 10.12 from Domenico and Schwartz (1990). 

Figure 4. Langmuir isotherm plots for sorption of fresh and aged plutonium solutions data correspond to Figure 3...

Beside kinetic and thermodynamic studies, equilibrium studies were done in order to determine the influence of the initial concentration of the metals ions upon the efficiency of the adsorption process and in order to determine the maximum adsorption capacities of the phosphorylated chitin and chitosan materials in the removal process of various metal ions from aqueous solutions. The experimental equilibrium data were fitted to the Langmuir and Freundlich isotherms. In all the cases the experimental results showed a better fit to the Langmuir than to the Freundlich equation. The maximum adsorption capacities obtained from the Langmuir isotherm plot in the... 

In synthetic fibers the number of ionic groups or dye sites is relatively small, and may have been introduced dehberately to make the base polymer dyeable. The restrictions on dye absorption are therefore very great the dye molecule must find an available specific site from among the limited number of sites in the fiber. This situation follows a Langmuir isotherm, where the reciprocal of dye in fiber 1 /DF is direcdy proportional to the reciprocal of dye in the dyebath 1 /HT. A plot of 1/against 1/H therefore gives a straight line. 

Sigmoid, the characteristic S-shaped curves defined by functions such as the Langmuir isotherm and logistic function (when plotted on a logarithmic abscissal scale). 

A series of measurements of coverage against partial pressure can easily be tested for consistency with the Langmuir isotherm, by plotting 1/0 against 1/p, which should yield a straight line of slope 1/fC. 

It is often convenient to plot the Langmuir isotherm in a double logarithmic plot (Fig. 4.3). 

Plot of adsorption data in a double logarithmic plot. In a Langmuir isotherm the initial slope is unity. A Freundlich isotherm shows in a double log plot a slope of n < 1. Such a Freundlich isotherm is obtained if the adsorbent is heterogeneous (decreasing tendency for adsorption with increasing 6). (Modified from Morel, 1983)... 

Data on the adsorption of caprylic acid on a hydrophobic (mercury) surface in terms of a double logarithmic plot of Eq. (4.13) Panel a) compares the experimental values with a theoretical Langmuir isotherm, using the same values for the adsorption constant B for both curves. Panel b) shows that the adsorption process can be described by introducing the parameter a, which accounts for lateral interaction in the adsorption layer. Eq. (4.13) postulates a linear relation between the ordinate [= log [0/ 1 - 0)] - 2a 0 / (In 10)] and the abscissa (log c). If the correct value for a is inserted, a straight line results. For caprylic acid at pH 4, a value of a = 1.5 gives the best fit. 

The arrows show the isotherm evolution for continual addition of dissolved Me. The initial isotherm with the slope of 1 (in the double logaritmic plot) corresponds to a Langmuir isotherm (surface complex formation equilibrium). [Me]S0 = solubility concentration of Me for the stable metal oxide [Me]p = solubility concentration of Me for a metastable precursor (e.g., a hydrated Me oxide phase). 

Figure 2.2. The Langmuir isotherm 0 as a function of jc = kC. The experimental range of x and 6 is the shaded area 0 < x S and 0 0 S 1. The function 0(x) has been plotted here in the range -1 < x < 3 to stress the hyperbolic character of the curve. The center of the hyperbola is x = -1 and 0=1.

The locations of the maximal slope of 0 and are clearly discernible in the maxima in the plots of the slopes of the BI. Thus, in Fig. 3.4, the maximum slope of 0 is always at the point logjQ X. = 0, but as q increases the location of the maximum slope of 0 moves leftward. Note that 0 is similar in shape to the Langmuir isotherm, but the form of 0/ is quite different. As can be seen from the lower panel of Fig. 3.4, in this particular case there are two points at which the curve... 

A typical N2 adsorption measurement versus relative pressure over a solid that has both micropores and mesopores first involves essentially a mono-layer coverage of the surface up to point B shown in isotherm IV (lUPAC classification) in Figure 13.1. Up to and near point B the isotherm is similar to a Langmuir isotherm for which equilibrium is established between molecules adsorbing from the gas phase onto the bare surface and molecules desorbing from the adsorbed layer. The volume of adsorbed N2 that covers a monolayer volume, hence the surface area of N2 can then be determined from the slope of the linearized Langmuir plot when P/V is plotted against P ... 

If a straight hne is obtained when is plotted against x/m at low concentrations, the Langmuir equation is applicable. However, due to the restrictive assumptions, Langmuir isotherms usually are of miuor importance in heterogeneous media... 

The adsorption of various aliphatic alcohols from benzene solutions onto silicic acid surfaces has been studied.t The experimental isotherms have an appearance consistent with the Langmuir isotherm. Both the initial slopes of an n/w versus c plot and the saturation value of n/w decrease in the order methanol > ethanol > propanol > butanol. Discuss this order in terms of the molecular structure of the alcohols and the physical significence of the initial slope and the saturation intercept. Which of these two quantities would you expect to be most sensitive to the structure of the adsorbed alcohol molecules Explain. 

EXAMPLE 9.3 Analysis of Adsorption Data Using the Langmuir Isotherm. Slope and intercept values for the linearized plots of the data in Figure 9.7 are as follows ... 

In a Langmuir isotherm, the reciprocal of dye in fiber 1 /OF, is directly proportional to the reciprocal of dye in the dyebath /DF,. A plot of I/O, against I/O, therefore gives a straight line. Langmuir isotherms are typically found with ionic synthetic fibers and ionic dyes. e.g.. dyeing polyacrylonitrile with modified basic dyes, and on hydrophilic fibers in situations when the number of sites becomes very low. This may arise when the internal pH is such that only a small number of sites ionize. 

The acidic sites on iron oxides are believed to be FeOH sites (32), much like the well-known SiOH sites on silica. Heats of adsorption on iron oxide of bases of known Cg and Eg, having appreciably different ratios of Cg to Eg ("hardness" or "softness"), allow estimation of the and for the acidic sites of iron oxide. Our initial studies were done by measuring adsorption isotherms at two or more temperatures (Figure 7) and from the temperature coefficient of the equilibrium constant K the enthalpy of adsorption was calculated. In Figure 7 the adsorption data is plotted as a Langmuir isotherm ... 

For a gas-solid system following the Langmuir isotherm, how would you plot 0(P) so that a fit to a straight line would provide b T)l From Eq. (77), how would you plot b(T) so that a fit to a straight line would provide C/0 ... 

Figure 4.5 Relationship between the surface coverage of 20H-AQ and its bulk concentration, where the continuous line represents the best fit to the Langmuir isotherm. The inset illustrates data plotted according to the linearized Langmuir isotherm. The supporting electrolyte is 1.0 M HCIO4. Reprinted with permission from R.J. Forster, Anal. Chem., 68,3143 (1996). Copyright (1996) American Chemical Society...

Most of the assumptions in deriving the Langmuir isotherm are not accurate. It works best for cases of chemisorption where only one layer is formed. A straightforward check on the validity of the Langmuir isotherm is to plot the inverse of the surface coverage, 0, versus the inverse of the equilibrium pressure ... 

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