Linearized Langmuir plot

The Langmuir plots in the linearized form, p/v = a + bP, are linear at the lowest pressures, over a wider range for the oxides, but reach maxima in both cases. 

For this reason, we sometimes construct Langmuir plots and find they are not particularly linear, particularly at the right-hand side, which represents higher concentrations. This deviation from... 

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 ... 

Typical adsorption isotherms that obey Eqs. (5) and (7) are shown in Figs. 2.1 and 2.2, respectively. It should be noted that a linear Langmuir plot can be obtained by plotting l/[nA]s against 1/P where the slope is 1/k and the intercept is l/[ng] as seen after rearrangement of Eq. (7). The adsorption isotherms are utilized primarily to determine the surface area of porous solids and the heats of adsorption. The isotherms yield the amount of gas adsorbed. By multiplying with the area occupied per molecule... 

Evaluate the Michaelis-Menten kinetic parameters by employing (a) the Langmuir plot, (b) the Lineweaver-Burk plot, (c) the Eadie-Hofstee plot, and (d) non-linear regression procedure. 

If the latter explanation is correct, it follows that the value of As as derived by either BET or Langmuir analysis (In fact, many microporous solids do not give linear BET plots although their Langmuir plots may be linear over an appreciable range of p/p°) cannot be accepted as the true surface area of a microporous adsorbent. On the other hand, if the slope of the isotherm is not too low at higher p/p° and provided that capillary condensation is absent, it should (in principle) be possible to assess the external surface area from the multilayer region. 

Fig. 16 The adsorbed amount of P-d-glucosidase per gram microgel Ta(js is plotted versus the concentration of free enzyme csoi in solution. The dashed line represents the fit of the experimental data by (2). The arrow marks the amount of entrapped enzyme used for kinetic investigation (620 mg P-D-glucosidase per gram microgel). The inset displays the data as a linear Langmuir plot. The adsorption was conducted at 4°C in 10 mM MOPS buffer solution (pH 7.2) and with a micro-gel concentration of 1 wt%. The immobilized enzymes do not prevent the microgel from shrinking at temperatures above the LCST [23]...

Figure 8. Langmuir plot for 2,4,5-T In accord with the Langmuir model for adsorption equilibrium, a linear trace is obtained for the isotherm for 2,4,5-T when (X) 1 is vlotted vs. (Ceq) 1. Intercept of plot is (Xm) 1, and the slope is (bXm) K...

Many examples of adherence of experimental data representing adsorption of ions at constant pH to Langmuir equation are reported in Tables 4.1 and 4.2. The apparent adsorption capacity (from the plateau level or from the slope of the linearized Langmuir plot, Eq. (5.5) depends on the pH. 

Therefore, a linear Langmuir plot is obtained by plotting 1 /o against 1 /P. Such a plot is shown for the adsorption of oxygen, carbon monoxide, and carbon dioxide on silica in Figure 3.18. 

In 1971 Bourne and coworkers in t-bultylbenzene cracking (already cited ( 56)) over silica-alumina determined the adsorption isotherm of benzene on the same material. A Langmuir plot was non-linear at one end they suggested that this portion of the isotherm indicated that site heterogeneity exists, of... 

FIGURE 2.6 Linear Langmuir plots for adsorption of iso-octane on different Saran charcoal [own work]. 

Goyal" ° and Goyal and Bansal" studied the adsorption isotherms of methylene blue on two samples of granulated activated carbons and two samples of activated carbon Fibers in the concentration range of 20 to 3000 ppm. The isotherms followed Langmuir isotherm equation. The linear Langmuir plots were used to calculate the monolayer capacity, which was further used to calculate surface areas covered by... 

When the linear Langmuir plots were extrapolated for the calculation of mono-layer capacities, the two linear plots crossed each other at higher equilibrium concentrations (Figure 7.4). This indicated that the adsorption capacity of the as-received sample was higher at equilibrium concentrations greater than 270 pg/1. Thus, these workers concluded that the adsorption of chloroform from aqueous solution was a function of the equilibrium concentration of the aqueous solution and the nature of the carbon surface. Nakamura et al. investigated the removal of chloroform from water using activated carbons and observed that the amount adsorbed did not depend upon the surface area or the pore volume of the carbon and the pH of the aqueous solution. The important factor in the adsorption of chloroform was the hydropho-bicity of the carbon surface. 

FIGURE 7.30 Linear Langmuir plots for as-received activated carbons. (After Goyal, M., Ph.D. thesis submitted to Panjab University, Chandigarh, India, 1997. With permission.)... 

Sample Identification BEKNj) Surface Area (mVg) Monolayer Capacity x (from Linear Langmuir Plots (mg/g) Surface Area Covered by Methylene Blue (mVg) % BET Area Covered... 

Figure 7.1 Adsorption isotherms for ammonia on charcoal (a) as plots of the volume of gas (per mass of solid), V, as a function of pressure (P) at constant temperature (T) and (b) in linearized Langmuir plots. Shaw (1992). Reproduced with permission from Elsevier...

The adsorption of the surfactant Aerosol OT onto Vulcan Rubber obeys the Langmuir equation [237] the plot of C/x versus C is linear. For C = 0.5 mmol/1, C/x is 100 mol/g, and the line goes essentially through the origin. Calculate the saturation adsorption in micromoles per gram. 

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. 

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... 

The plots are practically linear and show that the rates are constant. Consequently the order of reaction is zero at both initial pressures. If a Langmuir-Hinshelwood rate equation,... 

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. 

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