Particle number ration

In 1972-1973 Knox et al. [3, 4, 5] examined, in considerable detail, a number of different packing materials with particular reference to the effect of particle size on the reduced plate height of a column. The reduced plate height (h) and reduced velocity (v) were introduced by Giddings [6,7] in 1965 in an attempt to form a rational basis... 

The system is regarded as commensurate if there is a rational value for the ratio a=al2iT=L/N, where a denotes the mean space of the particles, 27t is the potential period, and L indicates the number of potential periods within the calculation window. By using different combinations of L and N, one may be able to alter the commensurability of the system, which will become incommensurate when a approaches an irrational value. The golden mean, a=( /5- l)/2 144/233, for example, is usually chosen to characterize a typical incommensurate case. 

In the second of their 1915 papers (Harkins and Wilson 1915b), Harkins and Wilson note from their study of the light elements (up to atomic number 27) that the main isotopic species had atomic masses which are integral multiples of 4. They concluded from this that, for those light nuclei, an important constituent must be the alpha particle just as it must be in the heavier radioactive nuclei which undergo alpha decay. In order to rationalize all the nuclei, including their nuclear charges, they... 

It was also observed, in 1973, that the fast reduction of Cu ions by solvated electrons in liquid ammonia did not yield the metal and that, instead, molecular hydrogen was evolved [11]. These results were explained by assigning to the quasi-atomic state of the nascent metal, specific thermodynamical properties distinct from those of the bulk metal, which is stable under the same conditions. This concept implied that, as soon as formed, atoms and small clusters of a metal, even a noble metal, may exhibit much stronger reducing properties than the bulk metal, and may be spontaneously corroded by the solvent with simultaneous hydrogen evolution. It also implied that for a given metal the thermodynamics depended on the particle nuclearity (number of atoms reduced per particle), and it therefore provided a rationalized interpretation of other previous data [7,9,10]. Furthermore, experiments on the photoionization of silver atoms in solution demonstrated that their ionization potential was much lower than that of the bulk metal [12]. Moreover, it was shown that the redox potential of isolated silver atoms in water must... 

In short, the same types of aerosol organic products have been identified both in model systems and in polluted urban ambient air and can generally be rationalized based on the oxidation of known constituents of air. The measured yields of organics in the particles can depend on the amount of particle phase available for uptake of the organic if it is semivolatile and partitions between the gas and condensed phases. This partitioning, and its dependence on the amount of condensed phase available, may be at least in part responsible for discrepancies in the yields of secondary organic aerosol reported in a number of studies. 

A small cluster or an atom of, for example, Pd (see Fig. 2) in an alloy has the same position of the d-band centroid as the full developed band of pure Pd this in contrast to the behavior of the small Pd particles. Such a difference can be rationalized if one assumes that due to the absence of a sufficient number of s-electrons in a small particle and due to their low mobility the screening is imperfect and thus the BEs are higher, whereas a hole on a Pd atom in a matrix of, say, silver can always be screened (extra-atomically) by s-electrons of the matrix. 

We can use these postulates to rationalize the various gas laws (Table 17.2). For Boyles Law, pressure decreases with increasing volume because the impact of gas particles against the walls of the container is spread out (diluted) over a greater area. For Charles s Law, volume increases with increasing temperatures because faster-moving particles demand more room. Similarly, for Avogadro s Law, volume increases with an increasing number of particles because more particles also demand more room. 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

It has been demonstrated [62] that nuclear synthesis can be rationalized in terms of continued a-particle (4He) addition, starting from the elementary units He (n = 2,3,4, 5), to yield the four modular series of nuclides shown in Figure 4.2. By assumption this process happens under cosmic conditions where all stable nuclides consist of protons and neutrons in the ratio Z/N = 1. The even mass number series, A = 4n and A = 4n + 2, result from the equilibrium chain reactions ... 

In addition, the mathematics of the undertaking, in terms of the number of combinations of possible structures, is against the rational approach. The number of variants of a small protein 80 amino acids in length far exceeds the estimated number of atomic particles in the universe. If an empirical approach works effectively, delivering more data and specific ligands in a shorter time, then one must consider the other approaches as irrational, or at least severely limited. 

FIGURE 12,20 Experimental void fraction of a two-component particle mixtures both having initial void fractions of 0.5. The numbers on the curves refer to the ration of the two particle sizes. Reprinted with permission from Furnas [57] copyright 1931 American Chemical Society. 

Thus, the wavelength-frequency relation (2.1) implies the Compton-effect formula (2.10). The best we can do is to describe the phenomena constituting the wave-particle duality. There is no widely accepted explanation in terms of everyday experience and common sense. Feynman referred to the experiment with two holes as the central mystery of quantum mechanics. It should be mentioned that a number of models have been proposed over the years to rationalize these quantum mysteries. Bohm proposed that there might exist hidden variables whieh would make the behavior of each photon deterministic, i.e., particle-like. Everett and Wheeler proposed the many worlds interpretation of quantum mechanics in which each random event causes the splitting of the entire universe into disconnected parallel universes in whieh eaeh possibility becomes the reality. 

A number of fundamental studies explore catalyst activity at an atomistic scale. DFT calculations can reveal how the rates of surface processes depend on the local electronic structure of surface atoms [233-238]. Monte Carlo simulations and mean-field approaches can incorporate this information in order to rationalize the effects of nanoparticle sizes and surface structure on the overall rates of current conversion [233, 239], Thereby the nontrivial dependence of reactivity on particle size could be explained. 

---