Macroscopic treatment

Let us compare these results with the predictions of the theory formulated by Lampe etal. (24) in terms of a steady-state concentration of collision complexes. This is a classical macroscopic treatment insofar as it makes no assumptions about the collision dynamics, but its postulate of collision complexes implies that v8 = vp/2 for the system treated above. Thus, its predictions might be expected to coincide with those of the collision-complex model. Figure 3 shows that this is not so the points calculated from the steady-state theory (Ref. 25, Equation 10) coincide exactly with the curve for which v8 = vv. The reason for this is that the steady-state treatment assumes a constant time available for reaction irrespective oC the number of reactions occurring in any one reaction... 

A number of different techniques have been developed for studying nonhomogeneous radiolysis kinetics, and they can be broken down into two groups, deterministic and stochastic. The former used conventional macroscopic treatments of concentration, diffusion, and reaction to describe the chemistry of a typical cluster or track of reactants. In contrast, the latter approach considers the chemistry of simulated tracks of realistic clusters using probabilistic methods to model the kinetics. Each treatment has advantages and limitations, and at present, both treatments have a valuable role to play in modeling radiation chemistry. 

In the nineteenth century, when crystal morphology was systematized to fourteen types of unit cells, seven crystal systems and thirty-two crystal groups, the following two macroscopic treatments on the morphology of crystals emerged. 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

Though the original work is difficult to understand very good reviews about the van der Waals interaction between macroscopic bodies have appeared [114,120], In the macroscopic treatment the molecular polarizability and the ionization frequency are replaced by the static and frequency dependent dielectric permittivity. The Hamaker constant turns out to be the sum over many frequencies. The sum can be converted into an integral. For a material 1 interacting with material 2 across a medium 3, the non-retarded5 Hamaker constant is... 

Langbein209 has derived a corresponding expression for this so-called Void effect which is based on the Lifshitz macroscopic treatment of dispersion forces. 

Umklapp process In the interaction of a continuous wave (photon, electron, etc.) with the lattice, the quasi-momentum of the wave is conserved, modulo a vector in the reciprocal lattice. The introduction of these quanta of momentum leads to the Umklapp process. In many macroscopic treatments the matter is treated as a continuous medium and Umklapp processes are neglected. In our treatment, Umklapp processes are included in the coulombic interactions (calculation of the local field), but implicitly omitted in the retarded interactions, since we dropped the term (cua/c)2 in (1.64). 

The basic theory of dielectric relaxation behaviour, pioneered by Debye, begins with a macroscopic treatment of frequency dependence. This treatment rests on two essential premises exponential approach to equilibrium and the applicability of the superposition principle. In outline, the argument is as follows. 

If no forces are pushing particles in the direction of the flow, then what about the driving force for diffusion, i.e., the gradient of chemical potential (Section 4.2.1) The latter is only formally equivalent to a force in a macroscopic treatment it is a sort of pseudoforce like a centrifugal force. The chemical-potential gradient is not a true force that acts on the individual diffusing particles and from this point of view is quite unlike, for example, the Coulombic force, which acts on individual charges. 

To show the influence of various microscopic and structural factors on linear and non-linear effects in dense dielectrics, it is convenient to apply first a semi-macroscopic treatment of the theory, and then to proceed to its molecular-statistical interpretation, assuming appropriate microscopic models. The semi-macroscopic method was initially applied by Kirkwood and modified by Frohlich in the theory of linear dielectrics, and has beat successfully used in theories of non-linear tUelectrics. "... 

Molar Electric Polarization of Dense Media.— In Kirkwood s semi-macroscopic treatment of the linear proparties of isotropic dielectrics, one has the following relation between the relative electric permittivity Sr and the polarization P( ) induced in the medium ... 

It is appropriate to stress that the above argument is thermodynamical, and hence phenomenological. The transition from the film, with tension y to the droplet, with surface tension is taken as sharp, just as in the macroscopic treatment of contact angles. In other words, it is ignored that on a colloidal scale... 

However, Zettlemoyer (1968) has more recently shown that such a macroscopic treatment is inadequate to explain the nucleation of ice by well known nucleators, such as Agl. Conceptually, Zettlemoyer s results show that hydrophilic sites invariably present on hydrophobic surfaces are necessary for initiation of the crystalline phase. 

Surface creation and destruction—A rational basis for macroscopic treatment is essential for advanced applications in microelectronics, energy conversion and storage, electrocrystallization, and etching. These applications require improved precision, predictability, and freedom from trace impurities. Important topics include stability and evolution of surface texture and dendrites and the effect of electrochemical parameters on mechanical properties of the near-surface region. 

The microscopic behaviour between the ions in dilute systems results from multipolar interaction. On the other hand in the rate equations which are used for measurement of macroscopic data such as quantum efficiencies of fluorescence, the multipole questions are absent. The macroscopic treatment of energy transfer was performed recently independently by Fong and Diestler (5) and Grant ( >) who conclude that the concentration... 

Velazquez-Marcano et. al. (42) found in their study of college general chemistry that both a particulate animation and a macroscopic demonstration of the phenomena were needed for the maximum effect when students were asked to predict the outcome of fluid experiments at the macroscopic scale. Both the particulate and the macroscopic treatments were needed however, the order of the visualizations did not matter for significantly better scores. There was no gender effect found in this study. The authors called for the use of multiple types of visualizations in instruction. 

Since AGsqIv is expressed on a per-mole basis, the right sides of these equations should be multiplied by the Avogadro constant iV. ) Although energy is both a molecular and a macroscopic property, both entropy and free energy are macroscopic but not molecular properties. The dielectric continuum model uses the macroscopic property of the solvent, and the macroscopic treatment of the solvent allows us to find AG°oi ei, which is a contribution to a macroscopic property. The dielectric continuum treatment of solvation is a combined quantum-mechanical and statistical-mechanical treatment. 

The point of this exercise is to show that eq. (4.r2 whose basis is entirely molecular, produces the same final result for entropy as the classical macroscopic treatment. 

The overall biosynthetic equation, Equ. 2.8, is the net equation resulting from hundreds of metabolic reactions in the living cell. The various cycles and chains in the metabolic network, including the ATP system and other energyhandling systems that do not result in the output of new cells or products, do not contribute to the net reaction. Hence, detailed knowledge of these cycles is unnecessary in a macroscopic treatment using the net stoichometric equations. Bioprocess analysis, therefore, is made less complex with no significant loss of information. 

A macroscopic treatment of solute-solvent electrostatic interactions is more justified, mainly because of the long range nature of electrostatic forces. As soon as one has been able to define the surface which separates the molecule from the continuum, one is able to use classical electrostatics to analyze the polarization of the solvent AAp, and of the averaged electrostatic potential in the cavity, which permits us to define the perturbation that the solute undergoes under the influence of the solvent. 

As we have seen, the macroscopic treatment of diffusion using Pick s first and second laws makes no distinction about whether the diffusion process occurs in a solid, liquid, or gaseous medium. In general, these macroscopic laws apply fairly well to all three phases. The differences between solid-, liquid-, and gas-phase diffusion mostly show up in the magnitude of the diffusion coefficient Z),. This parameter quantifies the relative ease with which atoms or molecules can be transported via diffusion in a material. Because diffusion occurs by a series of discrete random movements, for example, as a species jumps from lattice site to lattice site in a solid, or veers from one collision event to another collision event in a liquid or gas, both the speed (y,) at which a species moves and the average distance traveled during each movement (A) are embedded in the diffusion coefficient. In a gas or liquid, this dependence is often expressed as... 

I agree with Prof. B. Pullman that in a number of other studies there have been various choices of the value of the dielectric constant, which introduces a great deal of arbitrariness. It might be instructive to have a discussion devoted to this question. My opinion is that in any calculations which take into account all operative interactions in a system explicitly the dielectric constant should be taken as unity. Incidentally, in a recent article (S. Nir /. Chem. Phys. 61,2316 (1974)) it was shown that the Lifshitz macroscopic treatment of dispersion interactions between dielectric bodies in a fluid, in which macroscopic values of the dielectric constant have been employed, reduces to a microscopic treatment based on the London formula, which considers e=l in the interaction between any two molecules. 

It is interesting to compare this "microscopic treatment of radiative processes with the "macroscopic" treatment of classical physics. In the... 

It is interesting to compare this microscopic treatment of radiative processes with the macroscopic treatment of classical physics. In the latter, the atom is considered as an exponentially damped harmonic oscillator with an amplitude (Fig. 4.4)... 

In macroscopic treatments of growth, the main physical quantity is the height of the surface h(r, t), which depends on the position on the surface r and evolves with... 

Dispersion forces are universal because they attract all molecules together, regardless of their specific chemical nature. The potential energy of dispersion attraction between two isolated molecules decays with the sixth power of the separation distance. Based on the so-called Hamaker theory (i.e., the method of pair-wise summation of intermolecular forces) or the more modern Lifshitz macroscopic treatment of strictly additive London forces, it is possible to develop the so-called Lifshitz-Van der Waals expression for the macroscopic interactions between macroscopic-in-size objects (i.e., macrobodies) [19, 21], Such an expression strongly depends on the shapes of the interacting macrobodies as well as on the separation distance (non-retarded or retarded interaction). For two portions of the same phase of infinite extent bounded by parallel flat surfaces, at a distance h apart, the potential energy of macroscopic attraction is ... 

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