First-order point processes

An important question concerning energy trapping is whether its kinetics are limited substantially by (a) exciton diffusion from the antenna to RCs or (b) electron transfer reactions which occur within the RC itself. The former is known as the diffusion limited model while the latter is trap limited. For many years PSII was considered to be diffusion limited, due mainly to the extensive kinetic modelling studies of Butler and coworkers [232,233] in which this hypothesis was assumed. More recently this point of view has been strongly contested by Holzwarth and coworkers [230,234,235] who have convincingly analyzed the main open RC PSII fluorescence decay components (200-300 ps, 500-600 ps for PSII with outer plus inner antenna) in terms of exciton dynamics within a system of first order rate processes. A similar analysis has also been presented to explain the two PSII photovoltage rise components (300 ps, 500 ps)... 

Surface water estimated t/2 = 9.9-32 d in surface waters at various locations in case of a first order reduction process t/2 = 3-30 d in rivers, t,/2 = 30-300 d in lakes and ground waters (Zoeteman et al. 1980) t,/2 = 25 d in spring at 8-16°C, 14 d in summer at 20-22°C and 12 d in winter at 3-7°C when volatilization dominates, and t/2 = 12.1 d and 12.0 d for experiments with and without HgCl2 as poison respectively in September 9-15 in marine mesocosm (Wakeham et al. 1983) t,/2 = 4320-8640 h, based on aerobic river die-away test data (Mudder 1981 quoted, Howard et al. 1991) and saltwater sample grab data (Jensen Rosenberg 1975 quoted, Howard et al. 1991) calculated t/2 = 10 d and 32 d concentration reduction between sampling points on the Rhine River and a lake in the Rhine basin, respectively (Zoeteman et al. 1980 quoted, Howard 1990) t,/2(aerobic) = 180 d, t/2(anaerobic) = 98 d in natural waters (Capel Larson 1995). 

Point-Slope Methods. Euler s method follows directly from the initial condition as a starting point and the differential equation as the slope (Fig. 3). Consider the simple model of a single differential Eq. (13) with one first-order rate process ... 

This PBE is written in a general form and contains the terms representing accumulation, real-space advection, phase-space advection, phase-space diffusion, and second-, first-, and zeroth-order point processes. (See Chapter 5 for more details on these processes.) Let us... 

Another approach to obtain a parameter that describes the dissolution rate is to use statistical moments to determine the mean dissolution time (MDT) (von Hattingberg 1984). This method has the advantage of being applicable to all types of dissolution profiles, and it does not require fitting to any model. The only prerequisite is that data points are available close to the final plateau level. The MDT can be interpreted as the most likely time for a molecule to be dissolved from a solid dosage form. In the case of zero- and first-order dissolution processes, the MDT corresponds to the time when 50 and 63.2 percent have been released, respectively. The MDT is determined from ... 

Fig. 12.12 The non-isothermal thermogiavimetry runs in air for polypropylene with Mg(OH)2, the rate of heating 5 °C/min. The numbers denote the initial concentration of Mg(OH>2 in wt%. Points represent the examples of the fit of the non-isothermal thennogravimetry course by Eq. 12.2 from Sect. 12.3.1 for the sum of the three first-order decomposition processes (/ = 3). Arrows A point to the steep slope of the experimental run...

Figure 4.3b is a schematic representation of the behavior of S and V in the vicinity of T . Although both the crystal and liquid phases have the same value of G at T , this is not the case for S and V (or for the enthalpy H). Since these latter variables can be written as first derivatives of G and show discontinuities at the transition point, the fusion process is called a first-order transition. Vaporization and other familiar phase transitions are also first-order transitions. The behavior of V at Tg in Fig. 4.1 shows that the glass transition is not a first-order transition. One of the objectives of this chapter is to gain a better understanding of what else it might be. We shall return to this in Sec. 4.8. 

Step 4 of the thermal treatment process (see Fig. 2) involves desorption, pyrolysis, and char formation. Much Hterature exists on the pyrolysis of coal (qv) and on different pyrolysis models for coal. These models are useful starting points for describing pyrolysis in kilns. For example, the devolatilization of coal is frequently modeled as competing chemical reactions (24). Another approach for modeling devolatilization uses a set of independent, first-order parallel reactions represented by a Gaussian distribution of activation energies (25). 

Figure 13.6 shows the influence of temperature on specific volume (reciprocal specific gravity). The exaet form of the eurve is somewhat dependent on the crystallinity and the rate of temperature change. A small transition is observed at about 19°C and a first order transition (melting) at about 327°C. Above this temperature the material does not exhibit true flow but is rubbery. A melt viseosity of 10 -10 poises has been measured at about 350°C. A slow rate of decomposition may be detected at the melting point and this increases with a further inerease in temperature. Processing temperatures, exeept possibly in the case of extrusion, are, however, rarely above 380°C. 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

As pointed out in Chapter 11, radioactive decay is a first-order process. This means that the following equations, discussed on pages 294-295, apply ... 

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