Langmuir specific surface area

Figure 2.11 Correlation between the hydrogen storage capacity and the apparent Langmuir specific surface area of different MOFs obtained from reference [86] and references [13, 88]. The...

Dye adsorption from solution may be used to estimate the surface area of a powdered solid. Suppose that if 3.0 g of a bone charcoal is equilibrated with 100 ml of initially 10 Af methylene blue, the final dye concentration is 0.3 x 10 Af, while if 6.0 g of bone charcoal had been used, the final concentration would have been 0.1 x Qr M. Assuming that the dye adsorption obeys the Langmuir equation, calculate the specific surface area of the bone charcoal in square meters per gram. Assume that the molecular area of methylene blue is 197 A. ... 

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. 

BET method. The most commonly used method for determining the specific surface area is the so-called BET method, which obtained its name from three Nobel prize winners Brunauer, Emmett and Teller (1938). It is a modification of the Langmuir theory, which, besides monolayer adsorption, also considers multilayer adsorption. The equation allows easy calculation of the surface area, commonly referred to as the BET surface area ( bet). From the isotherms also pore-radii and pore-volumes can be calculated (from classical equation for condensation in the pores). 

The Langmuir adsorption isotherm provides a simple mechanistic picture of the adsorption process and gives rise to a relatively simple mathematical expression. It can also be used to obtain a crude estimate of specific surface areas. More important, from the viewpoint of the chemical engineer, it serves as a point of departure for formulating rate expressions for heterogeneous catalytic reactions. 

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 estimation of the surface area of finely divided solid particles from solution adsorption studies is subject to many of the same considerations as in the case of gas adsorption, but with the added complication that larger molecules are involved whose surface orientation and pore penetrability may be uncertain. A first condition is that a definite adsorption model is obeyed, which in practice means that area determination data are valid within the simple Langmuir Equation 5.23 relation. The constant rate is found, for example, from a plot of the data, according to Equation 5.23, and the specific surface area then follows from Equations 5.21 and 5.22. The surface area of the adsorbent is generally found easily in the literature. 

Usually specific surface areas are determined from adsorption experiments. To illustrate this let us assume that adsorption of a specific sample is adequately described by the Langmuir Eq. (9.22). From fitting experimental results we obtain Tmon in units of mol/g. Then we assume a reasonable value for the cross-section area of a gas molecule a a, and obtain the specific surface from J2 = rmon ANA- In most practical applications the BET adsorption isotherm is used instead of the Langmuir Eq. (9.22) because it fits better. From a fit with the BET isotherm we get Tmon or nmon. Some cross-sectional areas for suitable gases in A2 are N2 16.2 02 14.1 Ar 13.8 n-C4Hi0 18.1. 

The most widespread method in determining the specific surface area of solid substrates is without doubt the Brunauer-Emmet-Teller (BET) method.3 It is based on a kinetic model of the adsorption process by Langmuir,7 in which the surface of the solid was regarded as an array of adsorption sites. A state of dynamic equilibrium... 

In general, adsorption isotherms obtained with this and other zeolitic substrates are of Brunauer s Type I, the simple hyperbolic form also known as the Langmuir isotherm. Consequently, the asymptotic limit of adsorption is used instead of the value of Vm normally derived from the BET evaluation of specific surface area. It is, of course, not possible to define exact monolayer or multilayer adsorption in these three-dimensional interconnected pore systems. 

In figure 1.12 we illustrate the different ways of plotting for the special case of a model Langmuir Isotherm. The "measuring points" are chosen such as to obey the Langmuir equation [1.1.31, with 0= VA (nion), where V(mon) is the volume adsorbed in a saturated monolayer. It is assumed that the specific surface area (via V(mon)) is not known, hence the data are given as V (in ml at standard p and T)as a function of pressure (in Pa or N m" ). The "original" plot is... 

Here, is the area of one site. In the Langmuir picture, the total area is N a. Any part of the surface between the sites are simply not "seen". This has to be considered when interpreting the notion of specific surface area and its determination from monolayer capacity. 

For type a curves the Intercept n° may be interpreted as n°, that is the real value of n° In a complete layer from which all 1 Is expelled (n = 0). This is the plateau value attained by the individual Isotherm of 2, Indicated by the dashed curve In fig. 2.23. When an assumption Is made about the molecular cross-section from n° the specific surface area can be obtained. In principle, this method Is not different from finding from, say the plateau In a Langmuir Isotherm, the only difference being that horizontal Langmuir plateaus (for adsorption from dilute solution) are replaced by linear upper parts, approaching zero at 1 (for excess adsorption from binary mixtures). 

Specific surface areas from the Langmuir equation (0.01 < p/p° < 0.05, interval of relative pressure). 

The most relevant characteristic of porous materials is the disposal of a high effective surface/volume relationship, usually expressed in terms of their specific surface area (area per mass unit), which can be determined from nitrogen adsorption/desorption data. Different methods are available for determining the specific surface area (Brunauer-Emmett-Teller, Langmuir, and Kaganer), micropore volume (f-plot, ttj, and Dubinin-Astakhov), and mesopore diameter (Barrett-Joyner-Halenda Leroux et al., 2006). Table 1.1 summarizes the values of specific surface area for selected porous materials. 

Kruk, M., Jaroniec, M., and Gadkaree, K. (1999). Determination of the specific surface area and the pore size of microporous carbons from adsorption potential distributions. Langmuir, 15 1442-8. 

Once the isotherm is obtained, a number of calculation models can be applied to different regions of the adsorption isotherm to evaluate the specific surface area (i.e., BET, Dubinin, Langmuir, and the like) or the micro- and mesopore volume and size distributions (i.e., Barett-Joyner-Halenda, Dubinin-Radushkevich, Horvath and Kawazoe, Saito and Foley, and the like). 

The catalysts were characterized by using various techniques. X-ray diffraction (XRD) patterns were recorded on a Siemens D 500 diffractometer using CuKa radiation. The specific surface areas of the solids were determined by using the BET method on a Micromeritics ASAP 2000 analyser. Acid and basic sites were quantified from the retention isotherms for two different titrants (cyclohexylamine and phenol, of p/Ta 10.6 and 9.9, and L ,ax 226 and 271.6 nm, respectively) dissolved in cyclohexane. By using the Langmuir equation, the amount of titrant adsorbed in monolayer form, Xm, was obtained as a measure of the concentration of acid and basic sites [11]. Also, acid properties were assessed by temperature-programmed desorption of two probe molecules, that is, pyridine (pKa= 5.25) and cyclohexylamine. The composition of the catalysts was determined by energy dispersive X-ray analysis (EDAX) on a Jeol JSM-5400 instrument equipped with a Link ISI analyser and a Pentafet detector (Oxford). 

If the Langmuir equation describes methylene blue adsorption, calculate the specific surface area of the clay in square meters per gram. Assume that each adsorbate molecule occupies 65 on the surface. 

When the specific surface area of the solid adsorbent as is not known, n] may be plotted against C and the Langmuir equation takes the form... 

Adsorption from solutions is often used as a relatively simple method for the determination of the specific surface area of adsorbents [15]. The decrease in the surfactant concentration, Ac, is determined after a solution of known volume, V, has been equilibrated over an adsorbent of known weight, m, and the adsorption isotherm, T (c), is established (note that the adsorption in this case is expressed in moles per gram). The limiting adsorption, T max, can be obtained from the Langmuir equation, e.g. as c/T = a/T max + c/T max (see Chapter II, 2). Using independently acquired data for the area occupied by one surfactant molecule, on a similar solid substrate, one can obtain the specific surface area (in m2/g) ... 

The pore distribution functions in relation to their radii show that the materials studied can be classified as mesoporous adsorbents with the additional presence of micropores. The exception is the sample A2 in which a contribution from micropores on the surface is not observed. From the sorptometry data (Figure 13) the specific surface area (5), total pore volume (V) and pore diameter (Z> k) were calculated. These quantities were determined from the three Brunauer-Emmett-Teller (BET), Barrett-Joyner-Holenda (BJH) and Langmuir theories. 

Inasmuch as the Langmuir equation does not allow for nonuniform surfaces, interactions between neighboring adsorbed species, or multilayer adsorption, a variety of theoretical approaches that attempt to take one or more of these factors into account have been pursued by different investigators. The best known alternative is the BET isotherm, which derives its name from the initials of the three people responsible for its formulation Brunauer, Enunett, and Teller (8). It takes multilayer adsorption into account and is the basis of standard methods for determining specific surface areas of heterogeneous catalysts. The extended form of the BET equation (4) can be used to derive relations for all five types of isotherms as special cases. 

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