Ordering, homogeneous

Reactions involving collisions between two molecular species such as H2 and I2, or between two HI molecules are called bimolecular or second-order homogeneous reactions, because they involve the collision between two molecular species, and they are homogeneous since they occur in a single gas phase. The rates of these reactions are dependent on the product of the partial pressure of each reactant, as discussed above, and for the formation of HI, and the decomposition of HI,... 

An nth-order homogeneous liquid phase reaction is carried out in a batch tank reactor. 

The application of the time-independent Schrodinger equation to a system of chemical interest requires the solution of a linear second-order homogeneous differential equation of the general form... 

One of die most important second-order, homogeneous differential equations is that of Hennite It arises in the quantum mechanical treatment of the harmonic oscillator. Schrfidinger s equation for the harmonic oscillator leads to the differential equation... 

Like the entropy expression the fundamental relation as a function of U is also a first-order homogeneous function, such that for constant A,... 

A summary of the kinetic parameters is given in Table 11. With the exception of 3,3,3-trifluoropropyltrifluorosilane the decompositions are first-order homogeneous processes. The 2,2-difluoro compounds decompose by a four-centre transition state ... 

In the CE mechanism (Scheme 2.2), a first-order (or pseudo-first-order) homogeneous reaction precedes the electron transfer step. In the case where the initial electron transfer is fast enough not to interfere kinetically, the electrochemical response is a function of two parameters the first-order (or pseudo-first-order) equilibrium constant, K, and a dimensionless kinetic... 

The preceding approach applies to all linear systems that is, those involving mechanisms in which only first-order or pseudo-first-order homogeneous reactions are coupled with the heterogeneous electron transfer steps. As seen, for example, in Section 2.2.5, it also applies to higher-order systems, involving second-order reactions, when they obey pure kinetic conditions (i.e., when the kinetic dimensionless parameters are large). If this is not the case, nonlinear partial derivative equations of the type... 

The thermal isomerizations of a number of alkyl substituted cyclobutenes have been investigated in the gas phase. In all cases the reactions have been shown to be first order homogeneous reactions with no radical-chain component. The results obtained are shown in Table 6. 

Figure 6.21 Computed plot of the ratio of peak currents, /p(back//p(focwani), against log (kx) for an EC reaction, where t is the time-scale of the CV and k is the rate constant of the first-order homogeneous reaction. Notice how the plot has a similar shape to that shown in Figure 6.20, but the jc-axis is offset by the amount log k.

The electrode reaction (2.46) is followed by a first-order homogeneous chemical reaction (2.47), in which the product of the electrode reaction O is converted to a final electroinactive product Y. By analogy with the CE mechanism, the chemical step can proceed as ... 

This reaction scheme involves the second-order homogeneous disproportionation of NO2. It was stated that the rate of the latter process is independent of the electrode material, the supporting electrolyte, the presence of oxygen, and the pH of the solution. 

Laminar Natural Convection to a Vertical Plate with First-Order Homogeneous Reaction... 

Each uniform phase U a)(X a. ..,Xc+2) satisfies a mathematical scaling property (Euler first-order homogeneity )... 

The incorporation of high amount of niobium results in the formation of an additional pore system and the achievement of an incoherent structure. A stirring with water (at RT, for 8 h) leads to the well ordered homogeneous material, confirmed by the adsorption/desorption isotherm and the pore size distribution (Figure 1 A). 

It is always convenient to use intensive thermodynamic variables for the formulation of changes in energetic state functions such as the Gibbs energy G. Since G is a first order homogeneous function in the extensive variables V, S, and rtk, it follows that [H. Schmalzried, A.D. Pelton (1973)]... 

Fig. 2.21. Dispersed plug-flow with first-order homogeneous reaction. ft = k)C steady state, illustrating boundary conditions for a closed vessel...

Other steps could be added. I is a molecule of isomer and it may revert to a normal molecule either by a first or second order homogeneous reaction or at the walls. The fluorescence efficiency would be given by the equation... 

As the Gibbs energy is a first-order homogenous function of the extensive variables A7 and n. the application of Euler s theorem yields... 

As discussed in previous chapters (see Sects. 3.4.1, 4.5.1 and 6.3.1), of the processes with first- or pseudo-first-order homogeneous reactions coupled to the charge transfer, the catalytic mechanism is the simplest to study. The solution corresponding to SWV corresponding to planar, spherical, and disc electrodes will be discussed. 

It is often assumed that hemicellulose hydrolysis is a first-order homogeneous reaction in hemicellulose, that is... 

The latest contribution to the theory of the EC processes in SECM was the modeling of the SG/TC situation by Martin and Unwin [86]. Both the tip and substrate chronoamperometric responses to the potential step applied to the substrate were calculated. From the tip current transient one can extract the value of the first-order homogeneous rate constant and (if necessary) determine the tip-substrate distance. However, according to the authors, this technique is unlikely to match the TG/SC mode with its high collection efficiency under steady-state conditions. 

Fig. 8.11. Diffusion layer titration curve at a double hydrodynamic electrode (second order homogeneous reaction), (a) /det begins to rise when excess of B that did not react homogeneously reaches the detector electrode and A// /det = G(l/ar). (b) Assuming fast kinetics this is where linearity commences. From here...

In this technique the upstream electrode (generator) is galvanostati-cally controlled to generate a species that reacts with species X in solution in a second-order homogeneous reaction, the detector electrode being used to quantify the fraction of the electrogenerated species that did not react ... 

Before moving on to real Rosenbrock methods, consider again (4.66). The left-hand side contains a term in fy if we are dealing with a system of odes, this is called the Jacobian of the system. It is often constant, evaluable in advance. It will be seen in Chap. 9 that unless the diffusion problem has nonlinear concentration terms (for example from higher-order homogeneous reactions), the Jacobian is constant. If not, it must be evaluated at every step. 

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