Dimensional Consideration

We will now consider the more general collision problem in which the nuclear potential energy for a given electronic state of the interacting system is a function V(x, X2) of two nuclear coordinates x and X2 Such is the case of the colinear three-atomic reaction 

In the initial state (before the collision) x X2, so that x-j describes the relative motion of the incoming atom A and X2 the vibration of molecule BC. In the final state (after the collision) X2 x- hence X2 describes the relative motion of the outgoing atom C, and x the vibration of molecule AB. Therefore, the asymptotic solutions of the stationary-state Schrodinger equation 

Two corresponding equations for the final state are obtained by replacing the index 1 by 2 and vice versa, the suitable reduced masses being 

The solutions p (x ) of (103.11) may be approximately expressed by the BWK-wave functions (68.11) which turn into plane waves at large x-j-values (x-j X2) where the potential V-j (x) and momentum p-j become constant. The solutions (104.11) represent os- 

If the system is initially in a given quantum state n, it may be reflected in the same and in any other initial state m, or transmitted in any final state n. The current conservation requires that the total current density in reaction direction be the same in reactants and products regions hence, 

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We start with the reaction of abstraction of a hydrogen atom by a CH3 radical from molecules of different matrices (see, e.g., Le Roy et al. [1980], Pacey [1979]). These systems were the first to display the need to go beyond the one-dimensional consideration. The experimental data are presented in table 2 together with the barrier heights and widths calculated so as to fit the theoretical dependence (2.1) with a symmetric gaussian barrier. 

Using the definition of Pep and Nr and the nominal values from Table I for the parameters of interest, the conditions under which any of these dimensional considerations apply can be... 

From dimensional considerations, the drag coefficient is a function of the Reynolds number for the flow relative to the particle, the exponent, nm, and the so-called Bingham number Bi which is proportional to the ratio of the yield stress to the viscous stress attributable to the settling of the sphere. Thus ... 

Three-Dimensional Considerations Discrimination of common protein folds application of protein structure to sequence/structure comparisons, 266, 575 three-dimensional profiles for measuring comparability of amino acid sequence with three-dimensional structure, 266, 598 SSAP sequential structure alignment program for protein structure comparison, 266, 617 understanding protein structure using scop for fold interpretation, 266,... 

An expression for the length x0 of the laminar path can be obtained from dimensional considerations by noting that the state of turbulence near the wall can be characterized by a, p, and rj. Dimensional analysis leads to... 

In contrast to the case of a solid boundary, dimensional considerations do not provide an expression for t in the case of a liquid boundary. In order to obtain explicit expressions for t, two extreme cases can be considered. In one, t is assumed to be independent of [Pg.82]

In the case of the mass transfer from a bubble of radius R and velocity U, A should be a function of the latter two quantities and dimensional considerations provide the equation... 

Denoting, for the simplicity of writing, the multiple average mass transfer coefficient also by k, dimensional considerations lead to... 

In contrast to Section IV,M, where the turbulent diffusivity was employed to derive an expression for the mass transfer coefficient, in this section expression (401), which is based on a physical model, constitutes the starting point. Concerning the renewal frequency s, the following dimensional considerations can lead to useful expressions. The state of turbulence near the interface can be characterized by a characteristic velocity ua = (gSf)i/2, the dynamic viscosity rj, the surface tension a, and the density p. Therefore... 

The total kinetic energy U contributing to molecular dislocations in the given direction A — C (one dimensional consideration) may be supplied by more than one thermal motion simultaneously (e.g. oscillations and hindered rotations). In all cases where the total energy U is given by Ut + U2 = U the probability for U will be calculated by multiplication and integration over all cases satisfying the condition U2 = U-Uu... 

Since both the governing differential equations and all boundary conditions are homogeneous in the concentrations, can depend upon C", Coo, and C, only in some dimensionless combination. The only other variables are D and t, so that dimensional considerations require that... 

Feind s results (F2) have been given already by Eq. (104). Partly from dimensional considerations, Brotz (B21) showed that, in the turbulent region,... 

The probability of the atom ionization per unit of time is in proportion with the probability of electron transfer through the barrier created by the potential (5). Most probable is tunneling along the direction of the field so, to a first approximation, one-dimensional rather than three-dimensional considerations can be used (see Fig. 3)... 

The main role in tunneling through the barrier belongs to the electron velocity component which is perpendicular to the barrier. Thus, to reveal the peculiarities of the current-voltage characteristic of the transition it is possible to confine oneself to a first approximation to a one-dimensional consideration. The left-to-right current of electrons (Fig. 11) is evidently equal to... 

In this formula, vibration frequency in a separate potential well and p is the momentum within the region where classic motion is forbidden. The physical meaning of this formula is simple. The splitting energy is determined chiefly by the coordinate region between the potential wells and, consequently, is proportional to exp( - j p dx). The magnitude of the preexponential factor can be further determined (with an accuracy of up to n) on the basis of the dimensional consideration. 

It is sufficient for us, however, to note that the boundary conditions do not contain quantities with the dimension of length, and the dependence of the width of the reaction zone on the rate constant and on the coefficient of diffusion are uniquely determined from dimensional considerations ... 

It is easy to verify that dimensional considerations establish only three relations between the four exponents (since all the quantities are expressed in a dimensional system with only three basic units mass, length, and time). 

The piston s velocity is completely determined when the pressure at the piston is given however, to find u(t) from a given function n(t) requires solution of the gasdynamic equations. In any case, from dimensional considerations it is easy to establish that E — jp3/2Tp 1/2, where 6 is a dimensionless number depending in some way on the dimensionless... 

In order to construct the expression for the equilibrium number of nuclei in a unit volume (the dimension of b(x)dx is cm-3, the dimension of b(x), when x is defined as the radius, is cm-2), we must multiply the exponent exp(- /fcT), where is determined by (17), by a quantity of dimension cm-2. Exact evaluation of a pre-exponential factor is presently an unsolved problem of statistical mechanics. Erom dimensional considerations we may propose d 2 or x 2, where d is the linear size of a molecule of liquid and x is the radius of a bubble. In the present problem of evaluating the critical (i.e., minimum) value of the equilibrium concentration, we are dealing with a region where the factor in the exponent is large and exact evaluation of the pre-exponential factor is not actually necessary. 

Results pertinent to the theory of critical diameter are contained for the most part in earlier works by English authors. Despite his erroneous assumptions, Holm obtained the correct relation between the critical diameter and the flame velocity (1.4.6). The remarkable work by Daniell on the theory of flame propagation contains an analysis of the influence of heat losses. The losses enter directly into the equation describing the temperature distribution in the flame zone. A solution exists only for heat losses which do not exceed a certain limit, and under critical conditions (at the limit of propagation), the flame velocity drops to a certain fraction (40-50%) of the theoretical flame velocity. Daniell was also the first to indicate definitely that the flame velocity cannot be constructed from thermal quantities alone and by dimensional considerations must be proportional to the square root of the reaction rate. 

In accordance with the general expression which follows from dimensional considerations... 

The flame velocity should not enter into the governing criteria since it must itself be determined by the chemical reaction rate, the thermal conductivity and other properties of the mixture. From dimensional considerations it follows that the flame velocity in a tube is... 

Given the condition NO/[NO] = 0 at t/r = 0 we must find the limit of NO/[NO] at t/r —> oo. It follows from the form of the equation that at given fi and /2 the quantity NO/[NO] depends only on the product fcm[NO]r. This statement coincides with the content of the similarity theory introduced in the preceding section on dimensional considerations. We see that the validity of the theory depends on the existence of the functions fx and /2, which must be the same for different explosions. But the rate coefficient and the equilibrium quantity depend on the temperature. Hence the form and the very existence of the functions f1 and /2 depend on the law of cooling. On the other hand, it is evident that the law of cooling must be formulated in such a manner that under the given conditions the cooling rate will depend only on the temperature of the gas, but not on the heat of activation of the reaction or the equilibrium quantity of nitric oxide. It can be shown that both conditions are satisfied only by the law ... 

The rate factors rr interrelate just two wells, the well 1 and the well r. For just one barrier separating the two wells, the classical path under the barrier does not split into branches. The classical path in the restricted area is approximately separable (see Section 5 later). Therefore, in JT systems, after the weight factors mr(r) are established, the multidimensional tunneling problem reduces to a one-dimensional consideration. 

A relation between the mean end-to-end distance of the entire chain (R2) and the mean end-to-end distance of a subchain b can be found from simple speculation. This relation includes temperature T, mean distance b between the nearest along chain particles, excluded volume parameter v and the number of particles on the chain N. When dimensional considerations are taken into account, the relation can be written in the form... 

Free energy of the system in volume V, due to general relation (1.35), depends on the parameters n, T, V, N, b and parameters of interaction, whereby the arbitrary quantity N cannot influence the free energy of the system. So, after dimensional considerations has been taken into account, one has to write free energy for unit of volume... 

Hunt [37] developed self-similar solutions to the von Smoluchowski equation based on dimensional considerations. Three assumptions that were required are as follows. 

Pettinelli deduced from dimensional considerations that ... 

PettineUi,2 from dimensional considerations, deduced the equation (g =density, M=mol. wt.) ... 

To make this approach more quantitative, actual data and dimensional considerations are needed. Values for A S can be found in general textbooks on physical chemistiyl or derived from and tabulations. Their average... 

This definition arises from dimensional considerations. Introducing the factor 2 is customary, but not essential. Its reciprocal is... 

A crucial parameter-free test of the theory is provided by its application to micelle formation from ionic surfactants in dilute solution [47]. There, if we accept that the Poisson-Boltzmann equation provides a sufficiently reasonable description of electrostatic interactions, the surface free energy of an aggregate of radius R and aggregation number N can be calculated horn the electrostatic free energy analytically. The whole surface free energy can be decomposed into two terms, one electrostatic, and another due to short-range molecular interactions that, from dimensional considerations, must be proportional to area per surfactant molecule, i.e. 

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