First-order equations particles

For N particles in a system there are 2N of these first-order equations. For given initial conditions the state of the system is uniquely specified by the solutions of these equations. In a conservative system F is a function of q. If q and p are known at time t0, the changes in q and p can therefore be determined at all future times by the integration of (12) and (13). The states of a particle may then be traced in the coordinate system defined by p(t) and q(t), called a phase space. An example of such a phase space for one-dimensional motion is shown in figure 3. 

Water loss is identified [13] as being diffusion controlled across a dehydration layer of significant thickness that advances into the reactant particles (see Figure 8.1.). The Ca(OH)2 structure is maintained during the greater part of reaction [14]. The rate of this reaction is more deceleratory than the requirements of the contracting volume equation and data satisfactorily fit the first-order equation. The anhydrous residue later recrystallises to CaO but the principal water elimination zone and the phase transformation interface are separated and may advance independently. 

The loss rate of an atmospheric molecule resulting from the uptake by liquid or solid particles is generally expressed by a pseudo-first-order equation... 

If the sinplified pseudo first order equation for the conversion rate of glucose to fructose, equation (6), is assumed to be sufficiently accurate, the conversion rate of particles in which whole cells are immobilized is given by ... 

The BRE shows that only one kinetic equation can exist for various geometries of the biological mass. In the past, attempts were often made to formally utilize zero-order equations with microbial particles (floes) and to formally use first-order equations with microbial films, and this discrepancy was the starting point of Atkinson s work. 

In Equation (5,14), (77j ) is found by interpolating existing nodal values at the old time step and then transforming the found value to the convccted coordinate system. Calculation of the componenrs of 7 " and (/7y ) depends on the evaluation of first-order derivahves of the transformed coordinates (e.g, as seen in Equation (5.9). This gives the measure of deformation experienced by the fluid between time steps of n and + 1. Using the I line-independent local coordinates of a fluid particle (, ri) we have... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Decompositions may be exothermic or endothermic. Solids that decompose without melting upon heating are mostly such that can give rise to gaseous products. When a gas is made, the rate can be affected by the diffusional resistance of the product zone. Particle size is a factor. Aging of a solid can result in crystallization of the surface that has been found to affect the rate of reaction. Annealing reduces strains and slows any decomposition rates. The decompositions of some fine powders follow a first-order law. In other cases, the decomposed fraction x is in accordance with the Avrami-Erofeyev equation (cited by Galwey, Chemistry of Solids, Chapman Hall, 1967)... 

In a thin flat platelet, the mass transfer process is symmetrical about the centre-plane, and it is necessary to consider only one half of the particle. Furthermore, again from considerations of symmetry, the concentration gradient, and mass transfer rate, at the centre-plane will be zero. The governing equation for the steady-state process involving a first-order reaction is obtained by substituting De for D in equation 10.172 ... 

Solve the above equation for a first-order reaction under steady-state conditions, and obtain an expression for the mass transfer rate per unit area at the surface of a catalyst particle which is in the form of a thin platelet of thickness 2L. 

Radioactive decay provides splendid examples of first-order sequences of this type. The naturally occurring sequence beginning with and ending with ° Pb has 14 consecutive reactions that generate a or /I particles as by-products. The half-lives in Table 2.1—and the corresponding first-order rate constants, see Equation (1.27)—differ by 21 orders of magnitude. 

Water soluble impurities and their effect can be easily included in equation (1-4), through which they are going to directly affect the particle nucleation rate, f(t). If one assumes a first order reaction of an active radical with a water soluble impurity (WSI) to give a stable non-reactive intermediate, then one simply has to add another term in the denominator of equation (1-4), of the form kwsr[WSI](t)-kv, and to account for the concentration of WSI with a differential equation as follows ... 

A phase space is established for a typical particle, whose coordinates specify the location of the particle as well as its quality. Then, ordinary differential equations describe how these phase coordinates evolve in time. In other words, the state of a particle in a processing system is specified by the values of a number of phase coordinates z. The only requirement on z is that they describe the state of the particle fully enough to permit one to write a set of first order ode s of the form ... 

The parameter y reflects the sensitivity of the chemical reaction rate to temperature variations. The parameter represents the ratio of the maximum temperature difference that can exist within the particle (equation 12.3.99) to the external surface temperature. For isothermal pellets, / may be regarded as zero (keff = oo). Weisz and Hicks (61) have summarized their numerical solutions for first-order irreversible... 

In the following section, we only consider the integration of the equation of linear motion Eq. (20) the procedure for the equation of rotational motion, Eq. (21), will be completely analogous. Mathematically, Eq. (20) represents an initial-value ordinary differential equation. The evolution of particle positions and velocities can be traced by using any kind of method for ordinary differential equations. The simplest method is the first-order integrating scheme, which calculates the values at a time t + 5t from the initial values at time t (which are indicated by the superscript 0 ) via ... 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

Equation 8.5-11 applies to a first-order surface reaction for a particle of flat-plate geometry with one face permeable. In the next two sections, the effects of shape and reaction order on p are described. A general form independent of kinetics and of shape is given in Section 8.5.4.5. The units of are such that is dimensionless. For catalytic reactions, the rate constant may be expressed per unit mass of catalyst (k )m. To convert to kA for use in equation 8.5-11 or other equations for d>, kA)m is multiplied by pp, the particle density. 

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