Second-order equations particles

The special form of second-order equation in which the right-hand side is a function only of the dependent variable also turns up in the theory of diffusion and reaction in a slablike particle. Corresponding to equations (123-125) for the sphere, we would have, thanks to the reduction described in Chapter 2 and the example of a first-order nonisothermal reaction given by Eq. (129),... 

The increase in the rate of recombination upon illumination in the CC14 absorption band is due to the CCl photoionization with an electron being transferred to a continuous spectrum and subsequently captured by an arbitrary MP+ particle. As seen from Fig. 16, in this case the process kinetics (curve 1) is described by the second-order equation... 

Stone et al. (S29) developed by a mathematical analysis the functional relationship between the rate of extraction of silica from pure quartz in sodium hydroxide solution and time, temperature, sodium hydroxide concentration, and particle size. With the use of response surface methodology, a comprehensive picture of this dissolution process was obtained from a few well-chosen experiments. The fractional extraction of silica can be expressed by a second-order equation. The effect of quartz particle size and temperature are predicted to be about equal and greater than the influence of sodium hydroxide concentration and reaction time. The reaction rate is controlled by the surface area of the quartz. An increase in sodium hydroxide concentration increases the activation energy for the reactions and is found to be independent of quartz size. 

This example illustrates the complexity of even elementary second order equations when practical boundary conditions are applied. Fig. 2.2 illustrates the reactor composition profiles predicted for plug and dispersive flow models. Under quite realizable operating conditions, the effect of backmbcing (diffusion) is readily seen. Diffusion tends to reduce the effective number of stages for a packed column. Its effect can be reduced by using smaller particle sizes, since Klinkenberg and Sjenitzer (1956) have shown that the effective diffusion coefficient (Dg) varies as Vgdp, where dp is particle size. This also implies that Peclet number is practically independent of velocity Pe L/dp). 

It is assumed here that the particle surface is molecularly smooth and that the gap h R infinity as the integration limit has the meaning of distances comparable to h. The particles may not necessarily be spherical in the general case, instead of n, one can introduce a dimensionless coefficient dependent on the main curvatnre radii of both particles, k = k(R[, R2, R , R ) At the same time, there is a limitation associated with the nse of the Pythagoras theorem both surfaces must conform to a second-order equation, that is, must be quadratic. 

The singles and triples make their first appearance in the second-order equations (14.3.20) and are modified by higher-order corrections to the amplitudes. The quadruples do not enter to second order since the commutator of 4> and is a three-electron operator - see the discussion of excitation ranks and commutators in Section 13.2.8. In general, the nth-order excitations enter first to order n — 1 since the particle ranks of the commutators in the equations of order n — 1 are at most n. The only exception to this rule are the singles, which - because of the Brillouin theorem - enter the equations to second order. These results are summarized in Table 14.2. 

In this section we consider the classical equations of motion of particles in cases where the highest-frequency oscillations are nearly harmonic The positions y t) = j/i (t) evolve according to the second-order system of differential equations... 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

The main conclusion which can be drawn from the results presented above is that dimerization of particles in a Lennard-Jones fluid leads to a stronger depletion of the proflles close to the wall, compared to a nonassociating fluid. On the basis of the calculations performed so far, it is difficult to conclude whether the second-order theory provides a correct description of the drying transition. An unequivocal solution of this problem would require massive calculations, including computer simulations. Also, it would be necessary to obtain an accurate equation of state for the bulk fluid. These problems are the subject of our studies at present. 

Although the Klein-Gordon equation is of second order in the time derivative, for a positive energy particle the knowledge of at some given time is sufficient to determine the subsequent evolution of the particle since 8ldt is then given by Eq. (9-85). Alternatively Eq. (9-85) can be adopted as the equation of motion for a free spin zero particle of mass m. We shall do so here. 

Pyrolysis of more complex molecules proceeds via production of free radicals. Then formula (4.5) fails, because reactions of creation and recombination of radicals in these systems are irreversible. Therefore, the steady-state concentration of active particles in these systems depends on conditions of pyrolysis, determining the first or the second order of recombination of active particles, and is governed by the following equations [8]... 

A single classical particle is described by a second-order differential equation... 

The diffusion of particles can be described by the second-order differential equation (Pick s second law) ... 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

Solution First we evaluate kr, using Equation (32). It is convenient to use cgs units for this calculation therefore we write kr = 4 (1.38 10-16) (293)/(3 )(0.010) = 0.54 10-11 cm3 s 1. Recall that the coefficient of viscosity has units (mass length-1 time-1), so the cgs unit, the poise, is the same as (g cm -1 s -1). As a second-order rate constant, kr has units (concentration -1 time -1), so we recognize that the value calculated for kr gives this quantity per particle, or kr = 0.54 10-11 cm3 particle-1 s-1. Note that multiplication by Avogadro s number of particles per mole and dividing by 103 cm3 per liter gives kr = 3.25 109 liter mole-1 s-1 for the more familiar diffusion-controlled rate constant. 

Second-Order Moment. The linearity of y U /2L vs. l/t/B2 is shown in Figure 4. From the slope of the straight line, the axial dispersion coefficient D can be calculated. With the assumption that kR = Z)Ab, Da and Di can be calculated from the second and third terms in the bracket of the right-hand side of Equation 6 by varying the particle size. The results are given in Table II. As expected, both inter- and intracrystalline diffusion coefficients increase with temperature. The values obtained for Di in Na mordenite are somewhat smaller than those obtained by Satterfield and Frabetti (7) and Satterfield and Margetts (8) which were obtained at a lower temperature. However, Frabetti reported that diffusion co-... 

Since the collision radius for two particles of equal size is two times the particle radius, the effective volume swept out will be four times that given by Eq. 9-33. Since both particles are diffusing, the effective diffusion constant will be twice that used in obtaining Eq. 9-28. Thus, the effective volume swept out by the particle in a second will be eight times that given by Eq. 9-33. The volume swept out by one mole of particles is equal to /cD (recall that the second-order rate constant has dimensions of liter mol-1 s 1). Thus, when converted to a moles per liter basis and multiplied by 8, Eq. 9-33 should (and does) become identical with the Smolu-chowski equation (Eq. 9-30). 

According to this kinetic model the collision efficiency factor p can be evaluated from experimentally determined coagulation rate constants (Equation 2) when the transport parameters, KBT, rj are known (Equation 3). It has been shown recently that more complex rate laws, similarly corresponding to second order reactions, can be derived for the coagulation rate of polydisperse suspensions. When used to describe only the effects in the total number of particles of a heterodisperse suspension, Equations 2 and 3 are valid approximations (4). 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

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