Fractional Time Methods

The functional dependence of the half-life on reactant concentration varies with the reactant order. From the integrated rate equations we obtain these results  

These relationships offer a means for the determination of order by studying the dependence of on initial concentration. Although this method may not seem to offer an advantage over other procedures, it can provide additional evidence that 

Wilkinson has generalized the fractional time method in the following way. For rate equation dcldt = —kc , the integrated equation, for n 1, is 

Define the fraction reacted, p, by p = 1 — c/cq. Combination with Eq. (2-40) gives 

The left side is expanded in a binomial series, which is truncated after the quadratic term. Combination leads to 

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Use cf Integrated Equations 24 The Isolation and Pseudo-Order Techniques Initial Rate Method 28 Fractional Time Methods 29... 

One approach that allows increased chromatographic flow rates without loss of resolution entails the use of microparticulate stationary-phase media of very narrow diameter. This effectively reduces the time required for molecules to diffuse in and out of the porous particles. Any reduction in particle diameter dramatically increases the pressure required to maintain a given flow rate. Such high flow rates may be achieved by utilizing high-pressure liquid chromatographic systems. By employing such methods, sample fractionation times may be reduced from hours to minutes. 

Fractional lifetime method. The half-life, tm, of a reactant is the time required for its concentration to decrease to one-half its initial value. Measurement of tll2 can be used to determine kinetics parameters, although, in general, any fractional life, r, can be... 

Fractional time stepping is widely used in reacting-flow simulations (Boris and Oran 2000) in order to isolate terms in the transport equations so that they can be treated with the most efficient numerical methods. For non-premixed reactions, the fractional-time-stepping approach will yield acceptable accuracy if A t r . Note that since the exact solution to the mixing step is known (see (6.248)), the stiff ODE solver is only needed for (6.249), which, because it can be solved independently for each notional particle, is uncoupled. This fact can be exploited to treat the chemical source term efficiently using chemical lookup tables. 

The intra-cell processes are common to all PDF codes, and are treated the same in both Eulerian and Lagrangian PDF codes.8 On the other hand, inter-cell processes are treated differently in Eulerian PDF codes due to the discrete representation of space in terms of x . In PDF codes, fractional time stepping is employed to account for each process separately. Methods for treating chemical reactions and mixing are described in Section 6.9. Thus we will focus here on the treatment of inter-cell processes in Eulerian PDF codes. 

Fractional Life Method The half-life method can be extended to any fractional life method in which the concentration of reactant drops to any fractional value F = C /Cao in time The derivation is a direct extension of the half-life method giving... 

Tantalum was discovered by the Swedish chemist Anders Ekeberg in 1802, although for a long time after his discovery many chemists believed tantalum and niobium were the same element. In 1866, Marignac developed a fractional crystallization method for separation of tantalum from niobium. Ekeberg named the element in honor of Tantalus, who was Niobe s father in Greek mythology. 

Fractional Change Method From the equations of half life tor reactions of various orders except first order reaction, time required to complete a definite fraction of the reaction is inversely proportional to af- where n is the order of reaction and a is initial concentration. 

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

It was first intended to remove styrene homopolymer and unreacted cellulose (in the form of triacetate) by alternate extractions with benzene and a 1 1 methylene chloride-methanol mixture, but this was not successful. Therefore, a fractional precipitation method was adopted. The acetylated apparent graft copolymer was dissolved in a methylene chloride-methanol mixture (80 20 by volume). Methanol was very slowly added to the solution to precipitate the styrene homopolymer and the true graft copolymer. Dissolution in the methylene chloride-methanol mixture and precipitation with methanol were repeated four times. The final solution contained 45.0-46.3% methanol. 

Fractional-life methods. If a reaction is known to be first order and at constant fluid density, its apparent rate coefficient can be found very quickly. For batch and differential recycle reactors, the relationship between the rate coefficient and the time ty required for all but a fraction y of the reactant to be consumed is... 

With variable-time methods, standards should be chosen that have concentrations near those of the samples. Variable-time techniques are most suited to measurements where the signal is nonlinear with concentration and to the measurement of concentrations of catalysts such as enzymes, where A[I] represents a constant fraction of a reaction (the catalyst being regenerated). 

The pressure-based method was introduced by Harlow and Welch [67] and Chorin [30] for the calculation of unsteady incompressible viscous flows (parabolic equations). In Chorines fractional step method, an incomplete form of the momentum equations is integrated at each time step to 3ueld an approximate velocity field, which will in general not be divergence free, then a correction is applied to that velocity field to produce a divergence free velocity field. The correction to the velocity field is an orthogonal projection in the sense that it projects the initial velocity field into the divergence free... 

Fractional step methods have become quite popular. There are many variations of them, due to a vast choice of approaches to time and space discretizations, but they are all based on the principles described above. To... 

In this method the time required for a given fractional decrease in starting concentration (frequently the half-time of the reaction) is measured as a function of initial reactant concentrations. A log-log plot of fractional lives vs. initial concentration will give the reaction order (Gardiner, 1969). Often fractional lives methods will be used as a convenient way of analyzing kinetic data. Boyd et al. (1947) and Kressman and Kitchener (1949) used a fractional-lives type approach to analyze their ion-exchange kinetic data. A reaction half-time approach to analyzing soil kinetic data was discussed by Seyfried et al. (1989). 

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