First order current

The morphological stability of initially smooth electrodeposits has been analyzed by several authors [48-56]. In a linear stability analysis, the current distribution on a low-amplitude sinusoidal surface is found as an expansion around the distribution on the flat surface. The first order current distribution is used to calculate the rate of amplification of the surface corrugation. A plot of amplification rate versus mode number or wavelength separates the regimes of stable and unstable fluctuation and... 

It may be emphasized here that such an elaboration is possible for any small amplitude perturbation technique. It is only necessary to explicitize either the first-order current or the first-order interfacial potential, corresponding to the type of perturbation, to be able to derive expressions for 7q, 1 and AEl. So, the treatment is also useful to estimate the error due to second-order non-linearity in the step methods. However, a separate measurement of the second-order effect can only be done with (sinusoidal) a.c. perturbation. In Table 5, the explicit expressions for SF pertaining to the four methods mentioned in Sect. 2.4.1 are given in such a way that the connection between them is clearly shown. 

An outline of this method can be found in [11] a comprehensive presentation of the theoretical model including the multiple scattering terms in the first order current is presently in preparation. 

The first order current involves a number of multiple scattering terms, described by multiplications of the Mik matrices. So far these terms remain unconsidered. However, we also obtain terms with Mik 2 in the first order expression, which lead to a modification of the result for finite bias voltage. These terms amount to a tunneling current ... 

Figure 14. Direction of the induced, first-order current density for the CO2 moiecuie, in the molecular plane, obtained with the CTOCD DZ method of Zanasi et ai. Note the outer diamagnetic circulation and the paramag netic vortical circulation near the center of symmetry. (Adapted and reprinted with permission from ref 337. Copyright 1995 American Institute of Physics.)...

Figure 15. Projected direction of the induced, first order, current-density field for benzene, in the molecular plane (d= 0) and in planes parallel with it at distances d= 0.13, 0.26, 0.39, and 0.52 bohr, obtained by the CTOCD DZ method. (Adapted and reprinted with permission from refs 4 and 338. Copyright 2000 Elsevier Science B.V.)...

The first-order current rate constant Kave... 

A con jugate gradicri I method differs from the steepest descent technique by using both the current gradient and the previous search direction to drive the rn in im i/ation. , A conjugate gradient method is a first order in in im i/er. 

Let us in fact consider the expectation value of the current operator in the no-particle state in<0 (a ) 0>ln. In order to obtain an insight into this quantity, we first treat the case of a very weak external field so that only effects to first order in the external field need be... 

The growth current is characterized by the coefficient lG. Figure 46 is a log-log plot of lG vs. NaCl concentration, which yields a linear relation with the slope of 2.02 lG is proportional to the second order of NaCl concentration. However, in Eq. (112), lG is apparently in proportion to the first order of NaCl concentration. This apparent discrepancy can be solved by assuming that the coefficient B is a function of the coverage 0, which depends on NaCl concentration as shown in Fig. 44. So, including the... 

FIGURE 6.9 Plots of the total current density vs. (or K ) for (1) first-order and (2) zeroth-order reactions. 

When only taking into account the concentration polarization in the pores (disregarding ohmic potential gradients), we must use an equation of the type (18.15). Solving this equation for a first-order reaction = nFhjtj leads to equations exactly like (18.18) for the distribution of the process inside the electrode, and like (18.20) for the total current. The rate of attenuation depends on the characteristic length of the diffusion process ... 

Mass-spectrometric research on silane decomposition kinetics has been performed for flowing [298, 302-306] and static discharges [197, 307]. In a dc discharge of silane it is found that the reaction rate for the depletion of silane is a linear function of the dc current in the discharge, which allows one to determine a first-order reaction mechanism in electron density and temperature [302, 304]. For an RF discharge, similar results are found [303, 305]. Also, the depletion and production rates were found to be temperature-dependent [306]. Further, the depletion of silane and the production of disilane and trisilane are found to depend on the dwell time in the reactor [298]. The increase of di- and trisilane concentration at short dwell times (<0.5 s) corresponds to the decrease of silane concentration. At long dwell times, the decomposition of di- and trisilane produces... 

Currently, benzene alkylation to produce ethylbenzene and cumene is routinely carried out using zeohtes. We performed a study comparing a zeohte Y embedded in TUD-1 to a commercial zeolite Y for ethylbenzene synthesis. Two different particle diameters (0.3 and 1.3 mm) were used for each catalyst. In Figure 41.7, the first-order rate constants were plotted versus particle diameter, which is analogous to a linear plot of effectiveness factor versus Thiele modulus. In this way, the rate constants were fitted for both catalysts. 

This value represents the upper limit of a first order reaction rate constant, k, which may be determined by the RHSE. This limit is approximately one order of magnitude smaller that of a rotating electrode. One way to extend the upper limit is to combine the RHSE with an AC electrochemical technique, such as the AC impedance and faradaic rectification metods. Since the AC current distribution is uniform on a RHSE, accurate kinetic data may be obtained for the fast electrochemical reactions with a RHSE. 

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