Rate constant space dependent

J In each case, it is necessary to work above certain minimum values. When the trapping potential is less than 0.4 V, the measured rate constant is dependent upon the value of the potential and is too low. When the drift field is less than 0.35 V cm , calculated residence times in the resonance region disagree with those measured from linewidths. An important feature of the flat cell used in these measurements (1,27 cm spacing between drift electrodes) is that measured total ion currents are independent of magnetic field strength. 

A space-charge region is also formed, and the bands bent, when a potential is apphed to the electrode. As above, the band edges remain pinned at the electrode/solution interface, which arises because the potential drop between the bulk semiconductor and the solution is essentially entirely across the space-charge region rather than at the semiconductor interface. As a consequence, the intrinsic electron transfer rate constant is independent of applied potential. Nevertheless the current (and hence the effective rate constant) does depend on the apphed potential because the concentration of electrons (the majority carriers) at the electrode surface relative to its bulk concentration has a Boltzmann dependence on the energy difference between the band edge and the interior of the electrode. (The Fermi Dirac distribution reduces to a Boltzmann distribution when E > Fp-)... 

Haynes G R, Voth G A and Poliak E 1994 A theory for the activated barrier crossing rate constant in systems influenced by space and time dependent friction J. Chem. Phys. 101 7811... 

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

In Fig. 28, the abscissa kt is the product of the reaction rate constant and the reactor residence time, which is proportional to the reciprocal of the space velocity. The parameter k co is the product of the CO inhibition parameter and inlet concentration. Since k is approximately 5 at 600°F these three curves represent c = 1, 2, and 4%. The conversion for a first-order kinetics is independent of the inlet concentration, but the conversion for the kinetics of Eq. (48) is highly dependent on inlet concentration. As the space velocity increases, kt decreases in a reciprocal manner and the conversion for a first-order reaction gradually declines. For the kinetics of Eq. (48), the conversion is 100% at low space velocities, and does not vary as the space velocity is increased until a threshold is reached with precipitous conversion decline. The conversion for the same kinetics in a stirred tank reactor is shown in Fig. 29. For the kinetics of Eq. (48), multiple solutions may be encountered when the inlet concentration is sufficiently high. Given two reactors of the same volume, and given the same kinetics and inlet concentrations, the conversions are compared in Fig. 30. The piston flow reactor has an advantage over the stirred tank... 

A central problem in studying ion-molecule reactions is the dependence of the microscopic cross-section, a or the rate constant k upon the relative velocity of the ion and the molecule. Only from reliable, established data on this dependence can one choose among the various theoretical models advanced to account for the kinetics of these processes such as the polarization theory of Gioumousis and Stevenson (10) or the more recent phase-space treatment of Light (26). 

One of Perrin s students, the brilliant Rene Marcelin who perished in the First World War, set to work on the general problem, demonstrating that, in addition to the Arrhenius activation energy, the rate constant had to contain an activation entropy term. 76 In his thesis, defended in 1914, Marcelin developed a general theory of absolute reaction rates, describing activation-dependent phenomena by the movement of representative points in space. 

Discussion. Fixed bed cracking reactors as well as commercial moving bed reactors operate under steady state or pseudo-steady state conditions ( ). Observed selectivity (eg., ratio of yield of branched to n-paraffin) in a steady state catalytic reactor is independent of space velocity (1, 17). The selectivity depends on intrinsic rate constants and diffusivities of the reacting species which depend on temperature. Thus, the selectivity observations reported here are applicable to commercial FCC units operating at space velocities different from that employed in this study. 

In Eq. (14), C5- monotonically increases with x through the hexane concentration, and therefore is uniquely related to x. Equation (14) allows the selectivity rate constant matrix to be fit in a selectivity space with the observed C5- concentration as the independent variable and all C6+ compositions as the dependent variables. 

As shown above, the concentration of a compound that is transported in a river is affected by various mixing and elimination processes. Their relative importance for reducing the riverbome mass flow and the maximum concentration depends on the characteristics of the river as well as on the compound under consideration. This section gives a summary of the relevant rate constants by emphasizing the simplest descriptions. In order to compare their relative importance, all processes will be approximated by first-order rate constants, either in time (fc-rates, dimension T-1) or in space along the river (e-rates, dimension L 1). 

Figure 11. The error threshold of replication and mutation in genotype space. Asexually reproducing populations with sufficiently accurate replication and mutation, approach stationary mutant distributions which cover some region in sequence space. The condition of stationarity leads to a (genotypic) error threshold. In order to sustain a stable population the error rate has to be below an upper limit above which the population starts to drift randomly through sequence space. In case of selective neutrality, i.e. the case of equal replication rate constants, the superiority becomes unity, Om = 1, and then stationarity is bound to zero error rate, pmax = 0. Polynucleotide replication in nature is confined also by a lower physical limit which is the maximum accuracy which can be achieved with the given molecular machinery. As shown in the illustration, the fraction of mutants increases with increasing error rate. More mutants and hence more diversity in the population imply more variability in optimization. The choice of an optimal mutation rate depends on the environment. In constant environments populations with lower mutation rates do better, and hence they will approach the lower limit. In highly variable environments those populations which approach the error threshold as closely as possible have an advantage. This is observed for example with viruses, which have to cope with an immune system or other defence mechanisms of the host.

The rate depends on the number of reactant systems Nr(t), Eq. (5.22), as we have seen above through the dependence of p, so when we are going to determine the rate constant we need a relation between the number of system points in phase space and the number of reactant molecules in the reaction. Typically, the reactant molecules behave independently of each other, when the concentrations are small, and this is assumed in the following. That is, in the reactant phase space, the total Hamiltonian is... 

The form of the expressions in Eqs (5.98) and (5.114) is closely related to the classical expressions for the rate constant given in Section 5.1. The quantum mechanical trace becomes in classical statistical mechanics an integral over phase space [9] and the Heisenberg operators become the corresponding classical (time-dependent) functions of coordinates and momenta [8]. Thus, Eq. (5.78) is the classical version of Eq. (5.114). Furthermore, note that Eq. (5.98) is related to Eq. (5.49), i.e., the relevant classical (one-way) flux through Ro, at a given time, becomes S(R - Ro)(p/p)9(p/p), exactly as in Eq. (5.49). 

The first assumption, that phase space is populated statistically prior to reaction, implies that the ratio of activated complexes to reactants is obtained by the evaluation of the ratio between the respective volumes in phase space. If this assumption is not fulfilled, then the rate constant k(E, t) may depend on time and it will be different from rrkm(E). If, for example, the initial excitation is localized in the reaction coordinate, k(E,t) will be larger than A rrkm(A). However, when the initially prepared state has relaxed via IVR, the rate constant will coincide with the predictions of RRKM theory (provided the other assumptions of the theory are fulfilled). 

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