Number of single events

The rate constants for isomerization steps are similar in the forward and the reverse directions. For convenience, we choose to parameterize the kinetic model in terms of the forward rate constants. In this respect, we use the concept of single-event rate coefficients developed by Froment (127). According to Froment, the rate constant for a particular step is obtained by multiplying a single-event rate coefficient by the number of single events, ne, possible for the reactant. The expression for ne is... 

Remark The number of single-events could also be calculated using the notion of statistical factors (Bishop and Laidler, 1965, 1969). The statistical factor of a reaction is equal to the number of products that a reaction can form if a distinction is made between the atoms of each reactant. In our example, two methyl groups can shift in one direction and only one in the other direction ... 

Number of single events in the elementary step propyl carbenlum ion + butene as a function of the assumed activated complex. From Park and Froment [2001],... 

An example of how the configuration of the activated complex determines the number of single events of the elementary steps is shown in Fig. 2.4.3.1-1. 

The Number of Single Event Rate Coefficients, Assuming that the rates of hydride shift and methyl shift do not depend upon the number of C-atoms in a chain, but only on the type of carbenium ions involved (secondary or tertiary), the following single event rate coefficients are retained for the isomerizations,... 

This is applicable to both LT and HT capacitors. But it is more important in HT banks, which are relatively much larger and are built of a number of single units eonnected in series-parallel. These may encounter much higher fault currents in the event of a severe internal fault, even in one unit and are thus rendered more vulnerable to such ruptures. This phenomenon is more applicable to units that are externally protected w here (he intensity of fault may be more severe, than internally protected units. 

Six isotopes of element 106 are now known (see Table 31.8) of which the most recent has a half-life in the range 10-30 s, encouraging the hope that some chemistry of this fugitive species might someday be revealed. This heaviest isotope was synthsised by the reaction Cm( Ne,4n) 106 and the present uncertainty in the half-life is due to the very few atoms which have so far been observed. Indeed, one of the fascinating aspects of work in this area is the development of philosophical and mathematical techniques to define and deal with the statistics of a small number of random events or even of a single event. 

Because of its importance in natural phenomena, e.g., radioactive decay and population dynamics, let us introduce the exponential distribution through an illustration, n atoms are assumed to decay over the time interval [0 — 0], each atom having the same probability of decaying at any time in this interval. In other words, the time at which an atom decays is uniformly distributed over [0 — 0]. Let us call N(t) the number of decay events between 0 and t. The probability that a single atom has not decayed at time t, is 1—f/0 (Figure 4.3). The probability that none of the n atoms has decayed at time t is... 

The time of data collection depends on the complexity of the (5-pulse response. For a single exponential decay phase fluorometry is more rapid. For complex 5-pulse responses, the time of data collection is about the same for the two techniques in pulse fluorometry, a large number of photon events is necessary, and in phase fluorometry, a large number of frequencies has to be selected. It should be emphasized that the short acquisition time for phase shift and modulation ratio measurements at a given frequency is a distinct advantage in several situations, especially for lifetime-imaging spectroscopy. 

Figure 24.3 Temporal analysis of responses measured simultaneously in five different antennal lobe neurons in the moth Manduca sexta. The matrices show patterns of neural synchrony evoked by either of two pheromone components or a binary mixture at two concentrations. The number of synchronous events was averaged over 20 trials and calculated for 500 ms from stimulus onset. The gray scale ranges from 0 to 3.8 coincident spikes per stimulus. The horizontal displays below the matrices show the averaged spiking rate in each single neuron (gray scale ranges from 0 to 5.5 spikes per stimulus period). Neural synchrony was influenced not only by the odor quality but also by both stimulus intensity and blend interactions (redrawn from Christensen eta ., 2000).

---