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Dubinin and co-workers showed that a specific adsorbent adsorbs nearly equal volumes of similar compounds when their adsorption potentials are equal. They suggested a plot of volume adsorbed versus adsorption potential would produce a “characteristic curve,” applicable to that group of compounds for the specific adsorbent. Most people who use this type of isotherm equation have adopted W as the symbol for loading (volume adsorbed per unit mass or volume of adsorbent), but we will retain n*, and recognize that the units are specialized. From the measured data (moles or mass adsorbed), one calculates the volume adsorbed using Vm , the molar volume of the saturated liquid evaluated at the adsorption pressure, or evaluated at the normal boiling point, or another condition. Regardless, it should be consistent and clearly stated. It is easy, then, to extrapolate to other temperatures and other similar adsorbates for a given adsorbent. The main drawback is that the characteristic curve does not reduce to Henry's law at low coverage. The isotherms developed by Dubinin and co-workers employ a power to which the adsorption potential is raised that indicates the prevalent type of pores. The Dubinin-Radushkevich equation [(l) in Table 2] was intended for microporous adsorbents since the exponent is 2. The Dubinin-Astakhov equation (m) allows the exponent, n, to vary, but a reasonable lower limit is unity (for macroporous adsorbents). The Dubinin- Stoeckli equation (n) allows a distribution of pore sizes, which is a feature of many modern adsorbents. For this type of isotherm, n0 , represents the maximum loading, which correlates with pore volume among different adsorbents. The other isotherm parameters, k0 and $0 [no relation to the terms in eqs.(4) or (5)], represent the characteristic parameter of the adsorbent and an affinity coefficient of the compound of interest, respectively. The characteristic parameter, k0 , defines the shape of the n* vs. g curve. The affinity coefficient, $0 , adapts the compound of interest to the characteristic curve. It is a “fudge factor” that has been correlated to the ratio of molar volumes, parachors, or polarizabilities (via the Lorentz-Lorenz equation) of the compound of interest to that of a reference component (e.g., benzene or n-heptane). Those three methods are roughly equivalent in accuracy. The molar volume version is gi = gref Vi / Vref . The only controversy is whether to use the actual temperature to estimate volumes, or some other temperature such as the normal boiling point. Before leaving the topic of isotherms, it is fair to ask rhetorical questions. For example, given a set of data, what isotherm equation(s) might fit best? And what is the impact on calculations of fixed bed adsorption? Unfortunately, neither question can be answered fully. Some hint at the answer might be found in a specific example, however. (Before delving into the example, it should be stated that the complexity of fitting nonlinear isotherms is beyond the scope of this article. Many methods exist, but at ARI we use a specialized program that fits the equations illustrated here, plus many more, and plots the results because visual cues are usually better than numerical ones.) Thus, to illustrate the general principles of isotherm fitting, the most prevalent adsorbent/adsorbate pair in the world is fair: water vapor on silica gel, shown as symbols in Figure 4. In that same figure are the curves representing the best fits of the Langmuir, Freundlich, Redlich-Peterson and BDDT isotherms. 14PDF Image | ADSORBENT SELECTION
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