Critical molecular diameter

The primary requirement for an economic adsorption separation process is an adsorbent with sufficient selectivity, capacity, and life. Adsorption selectivity may depend either on a difference in adsorption equilibrium or, less commonly, on a difference in kinetics. Kinetic selectivity is generally possible only with microporous adsorbents such as zeolites or carbon molecular sieves. One can consider processes such as the separation of linear from branched hydrocarbons on a 5A zeolite sieve to be an extreme example of a kinetic separation. The critical molecular diameter of a branched or cyclic hydrocarbon is too large to allow penetration of the 5A zeolite crystal, whereas the linear species are just small enough to enter. The ratio of intracrystalline diffusivities is therefore effectively infinite, and a very clean separation is possible. 

Figure 6. Difiusivity of molecules of various critical molecular diameters.

Table 4-4. Effective pore diameter of different molecular sieves and corresponding adsorbate with smaller critical molecular diameter [4.11].

Critical molecular diameter The diameter of the smallest cylinder inside which the molecule will fit. 

In numerous applications of polymeric materials multilayers of films are used. This practice is found in microelectronic, aeronautical, and biomedical applications to name a few. Developing good adhesion between these layers requires interdiffusion of the molecules at the interfaces between the layers over size scales comparable to the molecular diameter (tens of nm). In addition, these interfaces are buried within the specimen. Aside from this practical aspect, interdififlision over short distances holds the key for critically evaluating current theories of polymer difllision. Theories of polymer interdiffusion predict specific shapes for the concentration profile of segments across the interface as a function of time. Interdiffiision studies on bilayered specimen comprised of a layer of polystyrene (PS) on a layer of perdeuterated (PS) d-PS, can be used as a model system that will capture the fundamental physics of the problem. Initially, the bilayer will have a sharp interface, which upon annealing will broaden with time. 

Hence, close to the critical point thermodynamic quantities at comparatively distant spatial locations become correlated. Especially in the case of liquid micro flows close to a phase transition, these considerations suggest that the correlation length and not the molecular diameter is the length scale determining the onset of deviations from macroscopic behavior. 

A. Smits and S. C. Bokhorst calculated the value of the constant b of J. D. van der Waals equation. This when taken to represent the size of the molecule becomes 0-00539, at the critical temp. the corresponding value calculated for the phosphorus atom in phosphine is 0-00124 so that there are 0-00539/0-00124=4-33 atoms in the molecule. This is taken to indicate a slight association of the P4-mols. J. J. van Laar calculated for b of J. D. van der Waals equation, 6=0-00140 for a, Va=0-066 and for the valency attraction A, VA—33. M. Trautz calculated 20xl0 8 cm. for the molecular diameter of phosphorus. 

The pair correlation function is a short range quantity in liquids, decaying to unity after a few molecular diameters, the correlation length However, in supercritical fluids g(r) has a much longer range and t, becomes considerably larger than the mean inter-molecular separation at the critical density. For instance, for carbon dioxide , = 5.5 nm at Tc compared to the mean intermolecular separation of 0.55 nm (Eckert, Knutson and Debenedetti 1996). 

While studying adsorption in mesopores using the molecular continuum model we have found [4,6,7] that there exist two critical diameters based on thermodynamic analysis of the adsorption, and two more when the mechanical stabihty of the meniscus is considered. These criticalities refer to the critical pore diameter below which there either exists a different mechanism of adsorption, or the adsorption is reversible. Here we provide a brief outline of these criticalities. The chemical potential of the fluid adsorbed in a cyfindrical pore of radius R can be expressed as [6,7] (r,R) = /jj (r,R) + (f>(R-r,R) = constant(/ ). After considering... 

The basic molecular characteristics of these species are reflected by the potential parameters given in Table I. Thus, the energy parameter c (proportional to the component critical temperature) increases from carbon dioxide to water, while the parameter a (molecular diameter) is the highest for acetone. 

There have been many analytic and numerical studies of the structure that solids induce in an adjacent fluid. Early studies focussed on layering in planes parallel to a flat solid surface. The sharp cutoff in fluid density at the wall induces density modulations with a period set by oscillations in the pair correlation function for the bulk fluid [169 173]. An initial fluid layer forms at the preferred wall fluid spacing. Additional fluid molecules tend to lie in a second layer, at the preferred fluid fluid spacing. This layer induces a third, and so on. The pair correlation function usually decays over a few molecular diameters, except near a critical point or in other special cases. Simulations of simple spherical fluids show on the order of 5 clear layers [174 176], while the number is typically reduced to 3 or less for chain or branched molecules that have several competing length scales [177 180]. 

When a fast chemical reaction or a rapid quench leads to the formation of a high density of condensable molecules, panicle formation may take place either by homogeneous nucleation, an activated process, or by molecular "coagulation a process in which nearly all collisions are successful. What determines which of these processes controls In principle, this problem can be analyzed by solving the GDE for the discrete distribution discussed in the previous section. An approximate criterion proposed by Ulrich (1971) for determining whether nucleation or coagulation is the dominant process is based on the critical particle diameter d that appears in the theory of homogeneous nucleation (Chapter 9)... 

Second, the correlation due to the intermolecular interactions is usually (except for perfect solids or near the critical point) of short range (normally, a few molecular diameters). For the present discussion, diameter could be defined as an average diameter of the species in the system. We do not need any more precise definition of this quantity here. All we need to assume is that the chosen molecular diameter is much smaller than the size of the macroscopic system. Hence, we assume that there exists a correlation distance Rc such that for R> Rc there are no correlations due to intermolecular interactions. The existence of such a correlation distance is supported both by experiments and by theoretical considerations (for more details see Appendix G). 

Routes II and III are identical in the sense that they use the same theoretical tools to achieve our goals. There is however one important conceptual difference. Clearly, molecular properties are microscopic properties. Additionally, all that has been learned about MDF has shown that in the liquid phase, and not too close to the critical point, molecular distribution functions have a local character in the sense that they depend upon and provide information on local behavior around a given molecule in the mixture. By local, we mean a few molecular diameters, many orders of magnitude smaller than the macroscopic, or global, dimensions of the thermodynamic system under consideration. We therefore rewrite, once again, route II in different words, but meaning the same as III, namely... 

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