Langmuir equation linear form

Either Equation 2 or Equation 3 may be used to linearize the data that conform to the Langmuir equation. The form chosen usually depends on the range and spread of the data and on the particular range of data to be emphasized. 

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... 

Brunauer, Emmett, and Teller extended the Langmuir theory to multimolecular layer adsorption [8]. They related the condensation rate of gas molecules onto an adsorbed layer and the evaporation rate from that layer for an infinite number of layers. The linear form of the relationship is called the BET equation ... 

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... 

Use the linear form of the Langmuir equation to evaluate (n/w)sat and K for the adsorption of pentane on carbon black from the higher pressure values in the following data. Use the ratio p/p0 rather than p only to normalize pressures relative to the equilibrium vapor pressure of pentane at different temperatures. (All pressures are in torr.)... 

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. 

By equivalent mathematical transformations, the Langmuir equation can be expressed in linear forms. From Equation 1.61, for example,... 

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)... 

Figure 4 shows the form of Langmuir adsorption isotherm. The equation can be rearranged to yield a linearized form of y = mx + b ... 

A linear form of the Langmuir equation can be obtained if in equation 10.4 is replaced by v/vm, where v is the maximum volume of gas adsorbed per unit weight of adsorbent (representing a monolayer of adsorbed molecules, with all S surface sites occupied), and v is the volume of gas (measured at STP ) adsorbed per unit weight of adsorbent at a pressure, P. Equation 10.4 then becomes... 

It is common in soil chemistry for adsorption of organics from solution to be of more interest than adsorption from vapor. This usually does not involve multilayer adsorption because of the tendency of water molecules to compete effectively for adsorption sites, at least if physical adsorption is the main process. In such cases, the Langmuir equation may be more accurate in describing adsorption results than the BET equation. The Langmuir equation is readily converted from the linear form used in gas... 

When A"Ce 1, the Langmuir isotherm reduces to a linear form analogous to Henry s Law. Sorbate-sorbent pairs that display a Type 1 isotherm can use the Langmuir equation. 

When adsorption follows the Langmuir equation, determination of the values of rm and a permits calculation of the area per adsorbed molecule at surface saturation and the free energy of adsorption at infinite dilution. To determine whether adsorption is following the Langmuir equation and to permit calculation of the values of rm and a, the equation is usually transformed into linear form by inverting it. Thus,... 

Large values of C show a large difference in heat of adsorption between the first and subsequent layers. When C becomes large, the isotherm increasingly resembles the Langmuir adsorption isotherm. For volumetric adsorption measurements, Equation (625) can be rearranged in a linear form... 

The parameters K and qm are treated as empirical constants, obtained by fitting the nonlinear equation directly to experimental adsorption data or by first linearizing the Langmuir equation to the form... 

Adsorption equilibrium data of SO2, NH3 and CO2 were analyzed by the linearized forms of the Langmuir and Freundlich equations (see Eqns. 2 and 4). The water vapour isotherm was analyzed by the linearized forms of the Dubinin-Astakhov equation (6), because of the above mentioned condensation phenomena. 

The interpretation of data on adsorption in terms of the Langmuir isotherm is most easily accomplished using the procedure previously described for reaction rate data, that is examination of the results first through a linear form of the isotherm equation. Rearrange equation (3-10) to... 

The units of x and xj are typically cm (STP)/g, and C is a nondimensional constant. As in the case of the rearrangement of the Langmuir isotherm, equation (3-14), this is a pseudo-linear form of equation which, given experimental data on P versus x, can be used to extract the parameters Xj and C from the slope and intercept in the... 

The condition of local mass equilibrium can exist even when adsorption and chemical reaction are taking place (Whitaker, 1999, Problem 1-3). When local mass equilibrium is not valid, one must propose an interfacial flux constitutive equation. The classic linear form is given by (Langmuir, 1916, 1917)... 

Specific to the metal, the adsorption isotherm will show an initial rapid increase of uptake with pressure, followed by a linear region of low slope. The monolayer capacity is often reported (when anything at all is said) as the intercept at zero pressure obtained by extrapolation of the linear part. Alternatively, and perhaps better, the results are plotted according to the one of the linearised forms of the Langmuir equation from which the monolayer volume can be obtained. 

---