Langmuir’s equation

The most widely used model of adsorption is Langmuir s equation for reversible molecular adsorption [163]. However, this is inappropriate for charged latex particles, because... 

This region of the isotherm is the Henry s law region the uptake is directly proportional to the pressure. At high pressures, KP 1, so that Langmuir s equation becomes IV =... 

One isotherm that is both easy to understand theoretically and widely applicable to experimental data is due to Langmuir and is known as the Langmuir isotherm. In Chapter 9, we see that the same function often describes the adsorption of gases at low pressures, with pressure substituted for concentration as the independent variable. We discuss the derivation of Langmuir s equation again in Chapter 9 specifically as it applies to gas adsorption. Now, however, adsorption from solution is our concern. In this section we consider only adsorption from dilute solutions. In Section 7.9c.4 adsorption over the full range of binary solution concentrations is also mentioned. 

From Langmuir s equation it is clear that it is possible to separate substances of the same vapor pressure but different molecular weights by molecular distillation. 

Diffusion Particles smaller than about 1 micron in diameter exhibit a Brownian motion which is sufficiently intense to produce diffusion. If a streamline containing these particles is sufficiently close to the collector, the particles may hit the collector and be removed. Contrary to the previous two mechanisms, the collection efficiency by diffusion increases with decreasing particle size or air velocity. The typical size of particles collected by this mechanism is less than about 0.5 jUm. The efficiency of collection by diffusion can be estimated by an equation analogous to Langmuir s equation, Eq. (8.34), as (Strauss, 1975) ... 

This disruption of lateral adhesion betw een the hydrocarbon chains was also found by Askew and Danielli. It is to be attributed to the hydrocarbon chains of the interfacial film sharing in the random translatory motions of the oil molecules, to which they are bound by van der Waals forces. In the case of a film which is liquid-expanded at the air-water surface this means that the term F0 in Langmuir s equation (6), p. 66, vanishes, i.e. that in the presence of the moving oil molecules the laterial adhesions of the hydrocarbon chains of the film molecules no longer prevent the films expanding indefinitely along the surface. 

Integration and rearrangement of trains give the time for a droplet of diameter d to evaporate to diameter d0 when the partial pressure in the surrounding medium is pt. This equation, sometimes known as Langmuir s equation, is... 

Calculations using Langmuir s equation show that the lifetime of very small volatile droplets in air is surprisingly short. But for large drops the lifetimes predicted by Langmuir s equation appear to be approximately correct, as indicated by experimental measurements. 

It is possible to extend the useful range of this equation to somewhat smaller drops. Fuchs (1959) initially pointed out that Langmuir s equation could not be correct for small particles having diameters approaching the mean free path of the gas since the equation predicted rates of molecular escape which exceeded the evaporation rate into a vacuum. To correct this difficulty, Fuchs considered the diffusion process to start a distance of approximately one mean free path from the droplet surface. 

Applying Fuchs modification to Langmuir s equation in the case of evaporation into a space where pt 0 gives... 

The effect of Fuchs modification on Eq. 15.15 is to increase the lifetime of very small drops fairly significantly and to even have some effect on drops having diameters of 10 gm or more. Figure 15.1 shows a plot of the ratio of Eq. 15.16 to Eq. 15.15 for similar conditions as a function of d, the droplet diameter. As an example, for 1-gm-diameter droplets, evaporation times as estimated by using the Fuchs modification are about 20 times greater than those estimated with Langmuir s equation alone. 

Zak (1936) found that the time for complete evaporation of a 1-mm diameter water droplet suspended from a thin glass fiber was 605 s at 20°C. Compare this value to one computed from Langmuir s equation, (a) assuming no temperature depression in the droplet and (6) considering a temperature depression (assume S = 0). 

Langmuir s equation indicates that a droplet will not evaporate when S = pip, a 1.0. But according to the Kelvin equation, curvature effects will cause small droplets to evaporate, even when S exceeds 1.0. How can this apparent contradiction be resolved One way is to replace the p, - pt term in Eq. 15.14 with an equivalent term from Kelvin s equation which takes curvature into account. Recalling Kelvin s equation... 

Particle Surface Area Determination Methods From the standard definitions of particle surface area, it can be seen that various determination methods are used for surface area measurement, such as adsorption (including Langmuir s equation for monolayer adsorption and the BET equation for multilayer adsorption), particle size distribution, and permeability methods. The different methods are rarely in agreement because the value obtained depends upon the procedures used and also on the assumptions made in the theory relating the surface area to the phenomena measured. The most common methods used for measuring particle surface area are described below. 

Christiansen s formula cannot be used because two adsorbed species react with one another. If the rate is controlled by the surface reaction, reactants and products are at adsorption quasi-equilibrium, that is, for each the adsorption and desorption rates are practically equal. With Langmuir s equations for these rates ... 

However, the pressure is given in dynes/cm which is inconvenient. Equation 7-2 revises Langmuir s equation to have the pressure in torr and the amount distilled in grams ... 

Under potential flow Ej, can be expressed by Sutherland s equation, by Langmuir s equation for a potential flow. Note that Langmuir obtained the equation from numerical calculations of the differential equations for the particle trajectory. In this equation the particle is considered as a point mass, i.e. the particle dimension is absent in Eq. (10.10). This means there is no direct influence of the particle dimension on the trajectory. However, the particle mass, the drag coefficient, and the Stokes number depend on the particle size, only later Langmuir s equation was derived for a finite particle size. This result cannot be used in... 

The data in Fig. 11.9 correspond to results given in Fig. 3 by Hewitt et al. (1994) where the curves are calculated by means Schulze superposition model. The solid line is calculated by Hewitt et al. using Schulze s superposition and for a retarded bubble surface of 2 mm diameter. If the bubble surface is free of surfactant Sutherland s Eq. (10.11) yields E , in Schulze s approximation and Langmuir s equation can be used to estimate the effect by inertia (Section... 

According to Langmuir s equation with ps as the surface oxygen concentration calculated as mentioned above from the main stream values, we have... 

This relationship is expressed in Langmuir s equation for the rate of evaporation [142] ... 

Points plotted as reciprocal isotherms in accordance with Langmuir s equation ... 

Starting from Langmuir s equation, we proceed now to derive the equation of Bunauer, Emmet, and Teller. It is interesting to note that in the paper by Bmnauer,... 

Therefore, it seemed that an explanation of multilayer adsorption was quite possible. Furthermore, no new constants were needed in order to explain the experimental data. The two relevant factors are the attraction that the molecules experience in the first layer (which had already appeared in Langmuir s equation) and the interaction between the various layers of adsorbed substance. This latter constant could be derived from the vapor pressure at each temperature. 

When A = 1, this expression reduces to Langmuir s equation for monomolecular adsorption. If 0 < A < 1, there is a finite maximum hygroscopic moisture content. While many textile fibers approach such a moisture content asymptotically at high relative humidities, some man-made fibers such as nylon and viscose appear to have well-defined maximum hygroscopic moisture contents [9]. In many cases, the coefficient k is greater than 1. Jaafar and Michalowski [8] interpret this behavior as the thermal effect of adsorption being equal to the heat of condensation only after a multimolecular layer has been formed. 

Further discrepancies may result from effects such as those mentioned on p. 64 and embodied in Langmuir s equation (37). Besides, in this chapter we have gone no further than to give a brief outline of the thermodynamic theory. More detailed accounts and extensions to more special cases can be found elsewhere In particular we mention Guggenheim s extension to mixtures in which the solvent molecules also occupy more than one lattice point. 

Considering the fraction of sites occupied 0a as the ratio of the concentration of the adsorbate on the solid phase (mg g-i) and the concentration of adsorbate in solid phase in saturation (mg g-i) also known as maximum adsorption capacity, we have Equations 5 and 6, the latter being known as the Langmuir s Equation. 

The Langmuir s equation can be linearized, yielding experimental values of the equilibrium constant Ka and maximum adsorption capacity according to Equation 8. 

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