The Langmuir Isotherm

The Langmuir isotherm describing the formation of an adsorbed intermediate by charge transfer can be written in the form 

The dependence of Cl on coverage can be obtained by rewriting the Langmuir isotherm in the form  

Clearly, Cl has its maximum value, given by Eq. (11.28), at 0 = 0.5. Setting the concentration in Eq. (11.28) equal to unity, we note that the potential at which Cl is a maximtmi can be regarded as the standard potential Bq for the adsorption process. It is given by 

The physical meaning of this choice is that the standard state of the system is chosen to be 0 = 0.5 and cj, = 1.0 M. 

It is interesting to evaluate the numerical value of Cl, max- Taking a value of qi — 0.23 mC cm for a monolayer of single-charged species, we find 

This is the most commonly used isotherm. This adsorption assumes that at higher concentration the rate of adsorption decreases because of lack of space on the adsorbent surface. Therefore, the adsorption is proportional to concentration as well as empty surface available and rate of desorption is proportional to the surface area occupied by adsorbate. The general form of this isotherm is given below 

This isotherm can also be expressed in terms of partial pressure of adsorbate in the gas phase P or relative pressure PjP where is the saturated vapor pressure. 

If we express the concentration of gas then this equation reduces to Equation 

The isotherm that is most easily understood theoretically and widely applicable to experimental data is, known as the Langmuir isotherm. This is also the simplest model of a nonlinear isotherm. It accounts well for the adsorption of single components on homogenous surfaces at low to moderate concentrations, or for the adsorption to isolated type-I or type-II sites [103, 110, 111]. The Langmuir adsorption isotherm equation is written  

Here a is the equilibrium or Henry constant at infinite dilution, a is also equal to the initial slope of the adsorption isotherm. The coefficient b is the equilibrium constant per unit of surface area, and hence this coefficient is related to the adsorption energy. C is the mobile phase concentration of the analyte in equilibrium with q, the concentration of the analyte in the stationary phase. The monolayer capacity, qs (qs = alb) is the upper limit of concentration in the stationary phase (sometimes called specific saturation capacity of the stationary phase). The Langmuir equation can also be written as  

The Langmuir isotherm model can be extended to multi-component systems [109], When several components are simultaneously present in a solution, the amount of each component adsorbed at equilibrium is smaller than if that component were alone [13] because the different components compete to be adsorbed on the stationary phase. The adsorption isotherm for the / th component in a multicomponent system is written  

Here n is the number of components in the system, coefficients a, and b, are the coefficients of the single-component Langmuir adsorption isotherm for component /. The coefficient bt is the ratio of the rate constants of adsorption and desorption, so it is a thermodynamic constant. The ratio ajb, is the column saturation capacity of component / [13], 

Because of its simplicity and wide utility, the Langmuir isotherm has found wide applicability in a number of useful situations. Like many such classic approaches, it has its fundamental weaknesses, but its utility generally outweighs its shortcomings. The Langmuir isotherm model is based on the assumptions that adsorption is restricted to monolayer coverage, that adsorption is localized (i.e., that specific adsorption sites exist and interactions are between the site and a specific molecule), and that the heat of adsorption is independent of the amount of material adsorbed. The Langmuir approach is based on a molecular kinetic model of the adsorption-desorption process in which the rate of adsorption (rate constant /ca) is assumed to be proportional to the partial pressure of the adsorbate (p) and the number of unoccupied adsorption sites (N - n), where N is the total number of adsorption sites on the surface and n is the number of occupied sites, and the rate of desorption (rate constant d) is proportional to n. 

At equilibrium, the rates of adsorption and desorption will be equal so that 

Ignoring entropy effects, the equilibrium constant for the process will be eq = kJkB, SO that 

The fraction of the adsorption sites occupied at a given time, q, is given by 

A useful characteristic of the Langmuir isotherm is that it can be rearranged to the linearized form 

Dividing Eq. (3.1) by So and recognizing that the fraction of surface covered, 0, equals S2/So, then we have the conventional Langmuir equation 

Typical Langmuir isotherms are shown in Fig. 3.2(a) for several gases on a microporous silica membrane [1]. It is interesting to note that at low pressures, Eq. (3.4) reduces to a linear isotherm 

One conventional way to test the experimental data for the Langmuir isotherm is to rearrange Eq. (3.4) in the following linear form 

A plot of p/n versus p gives a straight line (shown in Fig. 3.2(b)) with the slope of l/wo and the intercept of l/bwo from which ng and b can be determined. 

The rate constants k and k2 can be related to the concepts of adsorption time, which is the average time an adsorbed molecule spent on the surface, and the Langmuir constant b can then be expressed as 

Molecules are adsorbed at a fixed number of well-defined localized sites. 

There is no interaction between molecules adsorbed on neighboring sites. 

Considering the exchange of molecules between adsorbed and gaseous phases Rate of adsorption - 0) 

This expression shows the correct asymptotic behavior for monolayer adsorption since at saturation p- cc, q- q, and 0- l.O while at low sorbate concentrations Henry s law is approached  

Since adsorption is exothermic (A// negative) b should decrease with increasing temperature. 

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Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

The form of the functions f depends on Che particular isotherm used for example the Langmuir isotherm gives the familiar relation... 

In order to test the Langmuir isotherm against experimental data. Equation (4.1) may be rewritten in the form... 

For adsorbed hydrocarbons, the adsorption—desorption process can be thought of as a reaction and the adsorption isotherm as a description of the reaction at equihbtium. For the Langmuir isotherm,... 

The classical isotherm for a homogeneous flat surface, and most popular of all nonhnear isotherms, is the Langmuir isotherm... 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972) AIChE J., 18, 761 (1972)]... 

The Langmuir isotherm, Eq. (16-13), corresponds to the constant separation rac tor isotherm with... 

For noncoustaut diffusivity, a numerical solution of the conseiwa-tion equations is generally required. In molecular sieve zeohtes, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is ... 

In LC, at very low concentrations of moderator in the mobile phase, the solvent distributes itself between the two phases in much the same way as the solute. However, as the dilution is not infinite, the adsorption isotherm is not linear and takes the form of the Langmuir isotherm. 

Equation (3) is one form of the Langmuir isotherm and it should be noted that, when (Cm) tends to zero,... 

Volume overload results from too large a volume of sample being placed on the column, and this effect will be discussed later. It will be seen that volume overload does not, in itself, produce asymmetric peaks unless accompanied by mass overload, but it does broaden the peak. Mass overload, however, frequently results in a nonlinear adsorption isotherm. However, the isotherm is quite different from the Langmuir isotherm and is caused by an entirely different phenomenon. 

Brunauer further developed the Langmuir isotherm expression to include multilayer adsorption ... 

Inserted in Eq. (3), this gives the Langmuir isotherm P = Zi ,exp( u ABr)= - exp(-KoABr,, from which we get the isosteric heat of adsorption... 

Two limiting cases of the Langmuir isotherm are of interest. When 0 is very small, as when the pressure (or concentration) is low, or the constant a is small, then equation 20.18 reduces to... 

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

The assumptions made to derive the Langmuir isotherm (Eq. 2.7) are well known Energetic equivalence of all adsorption sites, and no lateral (attractive or repulsive) interactions between the adsorbate molecules on the surface. This is equivalent to a constant, coverage independent, heat (-AH) of adsorption. 

For associative (not dissociative) coadsorption of A and B on a catalyst surface the Langmuir isotherm takes the form ... 

There are several reasons for deviations from the LHHW kinetics Surface heterogeneity, surface reconstruction, adsorbate island formation and, most important, lateral coadsorbate interactions.18,19 All these factors lead to significant deviations from the fundamental assumption of the Langmuir isotherm, i.e. constancy of AHa (and AHB) with varying coverage. 

Despite the already discussed oversimplifications built into the Langmuir isotherm and in the resulting LHHW kinetics, it is useful and instructive at this point to examine how a promoter can affect the catalytic kinetics described by the LHHW expressions (2.11) to (2.14). 

The larger the value of kj, the stronger is the adsorption of j on the catalyst-electrode surface. More generally the Langmuir isotherm (6.23) can be written as ... 

It should be noted that within the context of the Langmuir isotherm (energetically equivalent adsorption sites, no lateral interactions) Eq. (6.28), which relates two surface properties, i.e. aj and 0j, remains valid even when the surface activity of Sj, aj, is different from the gaseous activity, pJ5 i.e. when Pj(g) Pj(ad). 

We start by noting that the Langmuir isotherm approach does not take into account the electrostatic interaction between the dipole of the adsorbate and the field of the double layer. This interaction however is quite important as already shown in section 4.5.9.2. In order to account explicitly for this interaction one can write the adsorption equilibrium (Eq. 6.24) in the form ... 

The above equation reduces to the one used to derive the Langmuir isotherm (Eq. 6.26) when Xj=0 or A0=O. 

Consequently, one can also write the Langmuir isotherm as 0a = KaPa9 (a convenient form to use in solving a kinetic scheme if the fraction of unoccupied sites is not yet known). 

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. 

In essence, we have used the Langmuir isotherms for the adsorbing and desorbing species. By substituting the coverages into the rate expression for the ratedetermining step we obtain... 

For each step in quasi-equilibrium we can either start from the differential equations as before or immediately use the Langmuir isotherm ... 

If we assume equilibrium between adsorption and desorption we find the Langmuir isotherm. 

Figure 5.19 shows an idealized form of the adsorption isotherm for physisorption on a nonporous or macroporous solid. At low pressures the surface is only partially occupied by the gas, until at higher pressures (point B on the curve) the monolayer is filled and the isotherm reaches a plateau. This part of the isotherm, from zero pressures to the point B, is equivalent to the Langmuir isotherm. At higher pressures a second layer starts to form, followed by unrestricted multilayer formation, which is in fact equivalent to condensation, i.e. formation of a liquid layer. In the jargon of physisorption (approved by lUPAC) this is a Type II adsorption isotherm. If a system contains predominantly micropores, i.e. a zeolite or an ultrahigh surface area carbon (>1000 m g ), multilayer formation is limited by the size of the pores. 

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