Paths, defined

As one raises the temperature of the system along a particular path, one may define a heat capacity C = D p th/dT. (The tenn heat capacity is almost as unfortunate a name as the obsolescent heat content for// alas, no alternative exists.) However several such paths define state functions, e.g. equation (A2.1.28) and equation (A2.1.29). Thus we can define the heat capacity at constant volume Cy and the heat capacity at constant pressure as... 

A particle entering the cyclone finds a poiat where its velocity with respect to the fluid is equal to the radial velocity, and hence is at rest with respect to radial movement to or from the wall. Because the radial velocity changes with radius, the particle spirals down the cone section, foUowiag a path defined... 

For a molecule at RTP this is of the order of a few hundred molecular diameters. In our ideal gas there is a distribution of velocities of the molecules about a mean value c. The mean free path defines a length scale in gases. As the density of the gas is increased and the mean free path approaches the molecular dimensions, a short-range molecular order develops and the material condenses to a liquid. The diffusional length scale is now much shorter range as a molecule encounters its... 

Throughout this section we use the notation introduced in Eq. (7) in which the phase label is absorbed into an extended macrostate label CC. The path defined by the set of macrostates f may (and sometimes will) extend from one phase to another. 

For the reaction Ha + Ha- Hs + H,270 VBCI calculations were performed on all the points considered. Five attitudes of approach of H2 and Hi and three attitudes of retreat of and H were considered, while the geometries of H2, Hi, and H+ were fixed at their respective calculated equilibrium values. The steepest approach potential is found to be one in which the H+ axis bisects the Ha axis. The steepest retreat potential is one in which the H atom leaves the triangular H, ion on the same axis on which it entered. With this general reaction path defined, the calculations were repeated with the optimization of the H-H bond distance. A minimum was found at H-H and H-H + distances of 1.66 a.u. and 4.11 a.u., respectively, while the distance from the leading H atom of the H+ ion to the bisector of the Ha molecule is 1.44 a.u. The binding energy is found to be 3.6 kJ mol-1. 

To effectively express concentrations and fluxes for this example, we introduce the shorthand notations LJ,tJ,n,n,I],Z],ll ,C,LJ,-U,rt, n,I, U, fl, and C, where each of these symbols represents a product of three first-order or pseudo-first-order rate constants. These symbols (called directional diagrams) represent the product of rate constants along the path defined by the diagram. The symbol U represents the product +2 +3 +4. Where substrate or product concentrations participate in one of the state transitions for a diagram, the pseudo-first-order rate constant (which incorporates the steady state reactant concentration) is used. Therefore the diagram U represents the product A 4/C 3A 21I,... 

It has been demonstrated that the network of bond paths defined by the unique axis of the [3, —1] critical points coincides with the network formed by linking together those pairs of atoms which are presumed to be bonded chemically. 

If we move on the activation surface A, the energy will remain Qq. If we move in the direction from which the path defining the activated state comes, or if we move in the direction of this path, the energy will decrease, because the activation point is the highest one of this path. In all directions perpendicular to the direction of steepest descent, however, El will increase. Otherwise, the highest point of an adjoining... 

A is the mean molecular free path defined as the average distance the molecule travels between successive collisions The value of A can be calculated using Equation 19.25 ... 

The same method based on the bond contribution w for the Wiener index calculation was proposed by Pisanski-Zerovnik [Pisanski and Zerovnik, 1994] together with a more general edge weight iv based on paths, defined as ... 

Ultimately, the intrinsic reaction path defines a unique way in which the structure of a system is allowed to evolve. Thus, in a molecular-dynamics calculation the system is no longer allowed to change structure completely freely, but has to change along the intrinsic reaction path. This leads to an extra constraint on the possible structural degrees of freedom that has to be included in a practical molecular-dynamics calculation. This was done recently by Michalak and Ziegler130 who showed how the approach could be combined with the Car-Parrinello method that we discussed in Section 2.3. 

Each monomer is constrained to stay fairly close to the primitive path, but fluctuations driven by the thermal energy kT are allowed. Strand excursions in the quadratic potential are not likely to have free energies much more than kT above the minimum. Strand excursions that have free energy kT above the minimum at the primitive path define the width of the confining tube, called the tube diameter a (Fig. 7.10). In the classical affine -and phantom network models, the amplitude of the fluctuations of a... 

Minami proposed a diffusion path model, in which a path is defined by connectivities of sites with different potential depths. At least three distinct pairs of interactions are present in AgI-Ag2Mo04 glasses, which characterize such potential wells. The Ag" ion migrates in a minimum energy path defined by the connectivities of the wells. 

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