Adsorption physical limitations

During Stages II and III the average concentration of radicals within the particle determines the rate of polymerization. To solve for n, the fate of a given radical was balanced across the possible adsorption, desorption, and termination events. Initially a solution was provided for three physically limiting cases. Subsequentiy, n was solved for expHcitiy without limitation using a generating function to solve the Smith-Ewart recursion formula (29). This analysis for the case of very slow rates of radical desorption was improved on (30), and later radical readsorption was accounted for and the Smith-Ewart recursion formula solved via the method of continuous fractions (31). 

Type I isotherms are encountered when adsorption is limited to, at most, only a few molecular layers. This condition is encountered in chemisorption where the asymptotic approach to a limiting quantity indicates that all of the surface sites are occupied. In the case of physical adsorption, type I isotherms are encountered with microporous powders whose pore size does not exceed a few adsorbate molecular diameters. A gas molecule, when inside pores of these small dimensions, encounters the overlapping potential from the pore walls which enhances the quantity adsorbed at low relative pressures. At higher pressures the pores are filled by adsorbed or condensed adsorbate leading to the plateau, indicating little or no additional adsorption after the micropores have been filled. Physical adsorption that produces the type I isotherm indicates that the pores are microporous and that the exposed surface resides almost exclusively within the micropores, which once filled with adsorbate, leave little or no external surface for additional adsorption. 

That the rates of adsorption be sufficiently slow in order that they may be isolated by the range of frequencies available due to the built in physical limitations of the apparatus (1 cycle/minute to 1 cycle/6 hours). 

The exponential function. The mathematical forms chosen to this point were those considered possible for tails. We now consider the exponential distribution function that may be relevant to loopy adsorption, at least for high molecular weight polymers. The exponential distribution function must be artificially truncated at the physical limit of the steric layer. This leads to a normalized segment density distribution of the form... 

Physical adsorption in microporous solids shows type I isotherms because the micropores limit the adsorption to a few molecular layers. Using a kinetic approach, Langmuir described the type I isotherm, considering that adsorption was limited to a monolayer (32). This approach assumes that the adsorption energy is constant and is independent of the fraction of the surface occupied by the adsorbed molecules. 

Both concepts of masses of an adsorbate discussed so far - the Gibbs surface excess (m g) based on proposition (PI) and calculated by Eq. (1.27) and the absolute mass adsorbed (m ) based on proposition (P2) and calculated by either Eq. (1.34) or (1.35) - do have their physical limitations. Hence it is desirable to mention other possibilities to define and to measure masses of adsorbates in gas adsorption systems. 

The resummed series is just Langmuir s adsorption isotherm, which is indeed the correct physical limit of 6 given by Eq. (1.166) when the lateral interactions are turned off. TMs immediately suggests the form of the Fade approximant for general z and = l/u. For low fugacities... 

The rate of nitrogen sorption was studied at 10 to 10 Torr and 298 to 473 K on continuously renewed scandium films at a condensation rate of 4.8x 10 atomcm s The rate of adsorption is limited by the rate of physical adsorption [239]. 

As stated in the introduction to the previous chapter, adsorption is described phenomenologically in terms of an empirical adsorption function n = f(P, T) where n is the amount adsorbed. As a matter of experimental convenience, one usually determines the adsorption isotherm n = fr(P), in a detailed study, this is done for several temperatures. Figure XVII-1 displays some of the extensive data of Drain and Morrison [1]. It is fairly common in physical adsorption systems for the low-pressure data to suggest that a limiting adsorption is being reached, as in Fig. XVII-la, but for continued further adsorption to occur at pressures approaching the saturation or condensation pressure (which would be close to 1 atm for N2 at 75 K), as in Fig. XVII-Ih. 

As also noted in the preceding chapter, it is customary to divide adsorption into two broad classes, namely, physical adsorption and chemisorption. Physical adsorption equilibrium is very rapid in attainment (except when limited by mass transport rates in the gas phase or within a porous adsorbent) and is reversible, the adsorbate being removable without change by lowering the pressure (there may be hysteresis in the case of a porous solid). It is supposed that this type of adsorption occurs as a result of the same type of relatively nonspecific intermolecular forces that are responsible for the condensation of a vapor to a liquid, and in physical adsorption the heat of adsorption should be in the range of heats of condensation. Physical adsorption is usually important only for gases below their critical temperature, that is, for vapors. 

Adsorption isotherms are by no means all of the Langmuir type as to shape, and Brunauer [34] considered that there are five principal forms, as illustrated in Fig. XVII-7. TVpe I is the Langmuir type, roughly characterized by a monotonic approach to a limiting adsorption at presumably corresponds to a complete monolayer. Type II is very common in the case of physical adsorption... 

The Freundlich equation is defective as a model because of the physically unrealistic/((2) consequences of this are that Henry s law is not approached at low P, nor is a limiting adsorption reached at high P. These difficulties can be patched by supposing that... 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

Forces of Adsorption. Adsorption may be classified as chemisorption or physical adsorption, depending on the nature of the surface forces. In physical adsorption the forces are relatively weak, involving mainly van der Waals (induced dipole—induced dipole) interactions, supplemented in many cases by electrostatic contributions from field gradient—dipole or —quadmpole interactions. By contrast, in chemisorption there is significant electron transfer, equivalent to the formation of a chemical bond between the sorbate and the soHd surface. Such interactions are both stronger and more specific than the forces of physical adsorption and are obviously limited to monolayer coverage. The differences in the general features of physical and chemisorption systems (Table 1) can be understood on the basis of this difference in the nature of the surface forces. 

A sharp separation results in two high purity, high recovery product streams. No restrictions ate placed on the mole fractions of the components to be separated. A separation is considered to be sharp if the ratio of flow rates of a key component in the two products is >10. The separation methods that can potentially obtain a sharp separation in a single step ate physical absorption, molecular sieve adsorption, equiHbrium adsorption, and cryogenic distillation. Chemical absorption is often used to achieve sharp separations, but is generally limited to situations in which the components to be removed ate present in low concentrations. 

The elution of such gels is an example not of size exclusion but rather of hydrodynamic fractionation (HDF). However, it must be remembered that merely being able to physically fit an insoluble material through the column interstices is not the only criterion for whether the GPC/HDF analysis of an insoluble material will be successful. A well-designed HDF packing and eluant combination will often elute up to the estimated radius in Eq. (5), but adsorption can drastically limit this upper analysis radius. For example, work in our laboratory using an 8-mm-bead-diameter Polymer Laboratories aqueous GPC column for HDF found that that column could not elute 204 nM pSty particles, even though Eq. (5) estimates a critical radius of —1.5 jam. 

---