Beyond Ideal Isotherms

There are several causes for the ideal (Langmuir) equation to break down the most important are as follows  

The assumption of surface homogeneity is essential in the Langmuir equation otherwise a different value of K would apply to Equations 4.37 through 4.45 at various places, and eventually at every site, on the surface. For a few different values of K, the treatment of Section 4.4.3 can be applied. Attempts to deal with surface heterogeneity have been undertaken, and some of these will be detailed in Section 4.4.7. 

In many cases lateral interactions—that is, interactions between neighboring adsorbates—are present, causing those adsorbates that adsorb to anonanpty surface to hnd different conditions compared with those arriving to a clean surface (in the sense of absence of adsorbates other than the solvent). The interactions can be attractive, sometimes giving rise to S-type isotherms as discussed in Section 4.2.2, or repulsive, which can result in H-type isotherms, as the incoming adsorbates find it more difficult to adsorb as the surface becomes more covered. 

The adsorption process in condensed media is a complex one involving several different kinds of interactions solvent-solute, solvent-adsorbent, and solute-adsorbent. In adsorption form solution, as said before, what happens actually is an exchange process between the solute and the solvent a requirement for the Langmuir isotherm to remain valid is that the solvent-surface interaction is weaker than the solute-surface (Hiemenz and Rajagopalan 1997). Also, if the solution is not diluted, the solvent concentration may be comparable to that of the solute and not constant, thus its chemical potential should be included in Equation 4.44, and its activity should show up in the isotherm expression (Bartell and Sloan 1929). 

It is important to realize that from the adsorption equilibrium data (i.e., the experimental isotherm), generally it cannot be distinguished between the effects of surface heterogeneity and lateral interactions. Consider the adsorption equilibrium constant K it can be related to the Gibbs standard free energy of adsorption  

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The choice of the ideal interface in the Gibbs adsorption isotherm (3.52) for a two-component system is, in a certain view, arbitrary. It is, however, convenient. There are two reasons First, on the right side there are physically measurable quantities (a, 7, T), which are related in a simple way to the interfacial excess. Any other choice of the interface would lead to a more complicated expression. Second, the choice of the interface is intuitively evident, at least for ci > C2. One should, however, keep in mind that different spatial distributions of the solute can lead to the same T. Figure 3.6 shows two examples of the same interfacial excess concentration In the first case the distribution of molecules 2 stretches out beyond the interface, but the concentration is nowhere increased. In the second case, the concentration of the molecules 2 is actually increased. 

At higher loadings (beyond the Henry s law region) the equilibrium isotherms for microporous adsorbents are generally of Type I form in Brunauer s classification [2]. Several different models have been suggested to represent such isotherms, the simplest being the ideal Langmuir expression [3] ... 

Though the conditions of the experiment doubtlessly fall beyond those theoretically presumed in deriving the Langmuir formula (monomolecular layer, ideal gas), a considerable part of the isotherm conforms with this equation. 

A thermodynamically correct approach, the Ideal Absorbed Solution (IAS) theory, uses the Langmuir isotherm as the basic single conponent isotherm for the adsorption of mixtures. Since the details for these additional isotherms are beyond the scope of this book, readers who need to use more conplex isotherms are referred to Do (1998), Ruthven 119841 Valenzuela and Myers (1989), and Yang 11987 20031... 

Figure2.10 summarizes our results (solid lines— Eqs.(2.82) and (2.85) dashed line—Eq.(2.84)) and compares them to data (crosses) taken from the literature (here http //www.usatoday.com/weather/wstdatmo.htm Source Aerodynamics for Naval Aviators ). Notice that the temperature data are not direct measurements but rather data points computed from simple formulas describing the average temperature profile at different heights. Our calculation applies to the troposphere, i.e. to a maximum altitude of roughly 10000 m. Beyond the troposphere other processes determine the temperature of the atmosphere. We see that our result somewhat underestimates the actual temperature. The middle graph shows the pressure profile. We notice that the two theoretical models Eq.(2.84) (isothermal case ) and Eq.(2.85) (adiabatic case) bracket the true pressure profile. The bottom graph shows the compressibility factor, Z = PV/ nRT), versus h. The data points scatter because of scatter in the density values. Nevertheless the graph shows that our assumption of ideal gas behavior is very reasonable.

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