Langmuir equation theory

ADSORPTION ON SOLID SURFACES 7.9a The Langmuir Equation Theory... 

Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n = 4 mol/kg, nl =. 3 mol/kg, Kipi = K2P2 = 1. Use the ideal adsorbed-solution theory to determine ni and n. Substituting the pure component Langmuir isotherm... 

The three isotherms discussed, BET, (H-J based on Gibbs equation) and Polanyi s potential theory involve fundamentally different approaches to the problem. All have been developed for gas-solid systems and none is satisfactory in all cases. Many workers have attempted to improve these and have succeeded for particular systems. Adsorption from gas mixtures may often be represented by a modified form of the single adsorbate equation. The Langmuir equation, for example, has been applied to a mixture of n" components 11). 

The Freundlich equation proved to be applicable to the adsorption of liquids with only limited ranges of concentration. It was replaced by the Langmuir equation (see later on) and others which had a theoretical basis in the kinetic theory of gases. It is clear that neither the Freundlich nor the Langmuir equation can describe isotherms of the shape shown in Figure 10.5. 

Rate equations for simple reversible reactions are often developed from mechanistic models on the assumption that the kinetics of elementary steps can be described in terms of rate constants and surface concentrations of intermediates. An application of the Langmuir adsorption theory for such development was described in the classic text by Hougen and Watson (/ ), and was used for constructing rate equations for a number of heterogeneous catalytic reactions. In their treatment it was assumed that one step would be rate-controlling for a unique mechanism with the other steps at equilibrium. 

This kinetic-theory-based view of the Langmuir result provides no new information, but it does draw attention to the common starting assumptions of the Langmuir derivation and the BET derivation (Section 9.5a). This kinetic derivation of the Langmuir equation is especially convenient for obtaining an isotherm for the adsorption of two gases. This is illustrated in Example 9.4. 

Langmufr s work on gas adsorption and insoluble monolayers prepared the way for more progress to be made in the interpretation of adsorption from solution data. In the light of the Langmuir theory, it seemed logical to suppose that the plateau of a solute isotherm represented monolayer completion and that the monolayer capacity could be derived by application of the Langmuir equation. 

Although the simple Langmuir equation is more applicable to some forms of chemisorption, the underlying theory is of great historical importance and has provided a starting point for the development of the BET treatment and of other more refined physisorption isotherm equations. It is therefore appropriate to consider briefly die mechanism of gas adsorption originally proposed by Langmuir (1916, 1918). 

By introducing a number of simplifying assumptions, Brunauer, Emmett and Teller (1938) were able to extend the Langmuir mechanism to multilayer adsorption and obtain an isotherm equation (the BET equation), which has Type II character. The original BET treatment involved an extension of the Langmuir kinetic theory of monomolecular adsorption to the formation of an infinite number of adsorbed layers. 

The Langmuir equation can also be derived kinetically. In fact, this was the route used by Langmuir himself in his famous seminal paper ). Langmuir had chemisorption in mind obviously the theory is also valid for this, provided the process is reversible. 

In dynamic equilibrium, the rate of desorption equals Nr" because r" Is the probability that an adsorbed molecule will desorb in one second. The rate of adsorption is determined by the available empty area a Ng- N), the pressure (i.e. by the number of molecules in the gas phase per unit volume) and by the rate at which they move. The result Is a p N N]/[2nmkT) molecules per second. The factor (27tmfcT) stems from the kinetic theory of gases and Is related to the collision frequency. Equating the two rates gives, after some rearrangements, the Langmuir equation with... 

Since 1916 when Langmuir published his fundamental paper on adsorption, the Theory of Activated Adsorption/Desorption Kinetics (TAAD) has, almost exclusively, been used for the interpretation of adsorption/desorption kinetics. However, contraiy to the success of Langmuir equation to represent the adsorption equilibria, a dramatic failure of TAAD was observed to represent by the Langmuir kinetic equation and its further modifications, [1-3] the monitored adsorption/desorption kinetics. 

Langmuir s theory for deriving an isotherm is a kinetic one, assuming the adsorption system is in dynamic equilibrium, where therate of adsorption is equal to that of desorption. The Langmuir isotherm, described by the following equation, is still the most useful for data correlation. 

The Mo adsorption process can be studied by using adsorption isotherms. Langmuir and Freundlich equations are the two major types of isotherms used to describe the Mo adsorption process. The Langmuir equation is based on the kinetic theory of gaseous adsorption onto solids, but is often used to model the adsorption of ions from solution (Ellis and Knezek, 1972). A common form of the Langmuir equation is... 

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