The Langmuir Equation

The Langmuir Equation. This is perhaps the most approximate but widely adopted equation  

Assuming the adsorbed molecule behaves as a three-dimensional harmonic oscillator and that the internal partition function (for electronic and nuclear energies) remains unchanged upon adsorption, A° may be expressed as  

Hence a plot of -RT In K is related to U through an axis shift RT In A . The accuracy of equation (69) depends on the system being studied but, as an example, for krypton at 77.5 K adsorbed on sodium chloride, the value of U may be in error by as much as 12% because of the neglect of the vibration-frequency-dependent terms in equation (68). 

Other methods of evaluating A have been suggested, de Boer and Hobson utilize the kinetic derivation to obtain a relationship  

The adsorption energy calculated by this method is the maximum adsorption energy (i.e. tZ + aED- 

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Although still used the Langmuir equation is only of limited value since in practice surfaces are energetic inhomogeneous and interactions between adsorbed species often occur. 

Assume that an aqueous solute adsorbs at the mercury-water interface according to the Langmuir equation x/xm = bc/( + be), where Xm is the maximum possible amount and x/x = 0.5 at C = 0.3Af. Neglecting activity coefficient effects, estimate the value of the mercury-solution interfacial tension when C is Q.IM. The limiting molecular area of the solute is 20 A per molecule. The temperature is 25°C. 

Assume that a salt, MX (1 1 type), adsorbs at the mercury-water interface according to the Langmuir equation ... 

If the Langmuir equation is obeyed, combination of Eqs. XI-4 and XI-13 gives the von SzysTkowski equation [24,25]... 

The Langmuir equation (Eq. XI-4) applies to many systems where adsorption occurs from dilute solution, but some interesting cases of sigmoid isotherms have been reported [54-56]. In several of these studies [54,55] the isotherms... 

Adsorption at a charged surface where both electrostatic and speciflc chemical forces are involved has been discussed to some extent in connection with various other topics. These examples are drawn together here for a brief review along with some more speciflc additional material. The Stem equation, Eq. V-19, may be put in a form more analogous to the Langmuir equation, Eq. XI-4 ... 

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. ... 

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. 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

This difference looks large enough to be diagnostic of the state of the adsorbed film. However, to be consistent with the kinetic derivation of the Langmuir equation, it was necessary to suppose that the site acted as a potential box and, furthermore, that a weak adsorption bond of ifi corresponding to 1 /tq was present. With these provisions we obtain... 

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. 

The basic assumption is that the Langmuir equation applies to each layer, with the added postulate that for the first layer the heat of adsorption Q may have some special value, whereas for all succeeding layers, it is equal to Qu, the heat of condensation of the liquid adsorbate. A furfter assumption is that evaporation and condensation can occur only from or on exposed surfaces. As illustrated in Fig. XVII-9, the picture is one of portions of uncovered surface 5o, of surface covered by a single layer 5, by a double-layer 52. and so on.f The condition for equilibrium is taken to be that the amount of each type of surface reaches a steady-state value with respect to the next-deeper one. Thus for 5o... 

Although the preceding derivation is the easier to follow, the BET equation also may be derived from statistical mechanics by a procedure similar to that described in the case of the Langmuir equation [41,42]. 

Equation XVII-70 bears a strong resemblance to the Langmuir equation (see Ref. 4)—to the point that it is doubtful whether the two could always be distinguished experimentally. An equivalent form obtained by Volmer [53] worked well for data on the adsorption of various organic vapors on mercury [54] (see Problem XVII-40). 

The rate of physical adsorption may be determined by the gas kinetic surface collision frequency as modified by the variation of sticking probability with surface coverage—as in the kinetic derivation of the Langmuir equation (Section XVII-3A)—and should then be very large unless the gas pressure is small. Alternatively, the rate may be governed by boundary layer diffusion, a slower process in general. Such aspects are mentioned in Ref. 146. 

Derive the general form of the Langmuir equation (Eq. XVII-11). 

The separate adsorption isotherms for gases A and B on a certain solid obey the Langmuir equation, and it may be assumed that the mixed or competitive adsorption obeys the corresponding form of the equation. 

Gas A, by itself, adsorbs to a of 0.02 at P = 200 mm Hg, and gas B, by itself, adsorbs tod = 0.02 at P = 20 mm Hg Tisll K in both cases, (a) Calculate the difference between (2a and (2b> the two heats of adsorption. Explain briefly any assumptions or approximations made, ib) Calculate the value for 6 when the solid, at 77 K, is equilibrated with a mixture of A and B such that the final pressures are 200 mm Hg each, (c) Explain whether the answer in b would be raised, lowered, or affected in an unpredictable way if all of the preceding data were the same but the surface was known to be heterogeneous. The local isotherm function can still be assumed to be the Langmuir equation. 

As a simple model of a heterogeneous surface, assume that 20% of it consists of sites of Q= 2.5 kcal/mol 45% of sites Q = 3.5 kcal/mol and the remainder, of sites of Q= 4.5 kcal/mol. Calculate Q(P, T) for nitrogen at 77 K and at 90 K, assuming the adsorption to follow the Langmuir equation with bo given by Eq. XVII-15. Calculate qsi for several 6 values and compare the result with the assumed integral distribution hinction. 

Because of the relatively strong adsorption bond supposed to be present in chemisorption, the fundamental adsorption model has been that of Langmuir (as opposed to that of a two-dimensional nonideal gas). The Langmuir model is therefore basic to the present discussion, but for economy in presentation, the reader is referred to Section XVII-3 as prerequisite material. However, the Langmuir equation (Eq. XVlI-5) as such,... 

Since in chemisorption systems it is reasonable to suppose that the strong adsorbent-adsorbate interaction is associated with specific adsorption sites, a situation that may arise is that the adsorbate molecule occupies or blocks the occupancy of a second adjacent site. This means that each molecule effectively requires two adjacent sites. An analysis [106] suggests that in terms of the kinetic derivation of the Langmuir equation, the rate of adsorption should now be... 

Equation (2.6) is the familiar Langmuir equation for the case when adsorption is confined to a monolayer. In practice B is an empirical constant and cannot be evaluated from the relationship in Equation (2.7). The question as to how well the Langmuir equation reproduces experimental isotherms will be dealt with in Chapter 4. 

Any interpretation of the Type I isotherm must account for the fact that the uptake does not increase continuously as in the Type II isotherm, but comes to a limiting value manifested in the plateau BC (Fig. 4.1). According to the earlier, classical view, this limit exists because the pores are so narrow that they cannot accommodate more than a single molecular layer on their walls the plateau thus corresponds to the completion of the monolayer. The shape of the isotherm was explained in terms of the Langmuir model, even though this had initially been set up for an open surface, i.e. a non-porous solid. The Type I isotherm was therefore assumed to conform to the Langmuir equation already referred to, viz. 

The Langmuir equation is based on the assumption that the heat of adsorption does not vary with the coverage 6 it is interesting that in the systems just quoted, the heat of adsorption varies with the amount adsorbed in the case of CO, but is virtually constant in the case of Ar. 

Conformity to the Langmuir equation, where it occurs, does not constitute proof of the correctness of the mechanism, since both B (or c) and n are disposable constants the quantities v, and , (cf. Equation... 

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... 

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