First-order solution

A third method, or phenomenon, capable of generating a pseudo reaction order is exemplified by a first-order solution reaction of a substance in the presence of its solid phase. Then if the dissolution rate of the solid is greater than the reaction rate of the dissolved solute, the solute concentration is maintained constant by the solubility equilibrium and the first-order reaction becomes a pseudo-zero-order reaction. 

To correct for nonideal behavior of the 1-propanol and tetramethylammonium at these high concentrations, a further optimization of the mimicking solution was required. A second-order solution for CTAB was elaborated using the first-order solution as a starting point. This second-order solution was found to reproduce not only the micellar rate constants for the hydrolysis of the substituted 1-benzoyl-1,2,4-triazoles 6a-f but the hydrolysis rate constant for 5 and the Ex30 value with good accuracy as well. " ... 

It has also been shown that it is possible to apply this method of analysis to more complex reaction schemes (5,9,10). For example, consider the following consecutive, first-order, solution phase reaction mechanism ... 

Applying the perturbation method, the first-order solution is given by... 

In the low-frequency limit only coii is set to be nonzero while all the higher modes are taken at equilibrium (cox = 0). Thence, when constructing via Eq. (4.182), by adding and subtracting a term with 4 (0), one can present the first-order solution in the form... 

Let us apply these general first-order solutions to two specific reactions irreversible consecutive and simple reversible reactions. Consecutive reactions (A —> B —> C) do not have reverse reactions. This leads to ... 

The first-order solution is to implement more conditioning in the section of the pad that experiences the wafer center paths. This need has generated the concept of zonal conditioning in different CMP tools the conditioning disk is forced to spend time at a pad location in proportion to the length of the wafer cord that intersects that pad location. On the surface, this concept... 

For a first-order solution, assuming linear velocity and temperature profiles satisfying Eq. (11.12), we have... 

The first known solution was that of Schwarzschild that provides the origin of the notion of a black hole. Let us also mention the Kerr solution, plane gravitational waves, the general first order solution, the successive approximations of the two black hole problem hundreds of rigourous solutions are known today. 

Another beautiful result is the general first order solution of the Gross-mann equation. We will choose the velocity c as our unit of velocity and obtain... 

If q is a first-order solution reaction that competes with r. then the probability A that R-... 

Figure 4.6 displays this solution and the first-order solution with a rate constant chosen such that the initial rates are equal. Notice that although both solutions have the same qualitative features, the second-order reaction decays more slowly to zero than the first-order reaction. 

In a direct attack upon the radial migration problem in essentially its full generality, Cox and Brenner (C18) succeeded in obtaining a first-order solution of the Navier-Stokes and continuity equations for the motion of a rigid spherical particle immersed in a Poiseuille flow within a circular tube of finite radius. No couple acts on the particle, so it is free to rotate. It is presumed in the analysis that the sphere center moves parallel to the tube axis. The lateral force required to maintain the sphere at a fixed distance from the axis is computed and converted into an equivalent radial migration velocity by application of Stokes law to this sidewise motion. 

For the probing experiment, applying the perturbation method, the first-order solution of Eq. (4.21) is given by... 

We have selected a negative constant (or zero, since A = 0 is still a possibility) on physical grounds, since it is clear that the first order solution... 

This procedure can be then iterated by taking further derivative of Eq. (4.465) with respect to the density, solving the obtained equation until the second order correction over above first order solution (4.469),... 

Several post-HF models have been developed to give more accurate results based on better treatment of electron correlation. Mpller-Plesset (MP) models include an approximation of electron correlation by adding further terms to the HF approach. The first-order solution calculates ground and excited states separately without interacting as in HF methods but the addition of second- and higher-order terms introduces perturbation. The second-order calculation, or MP2, therefore explicitly includes electron-electron interactions through the effects of electron promotion. ... 

The boundary values of the first-order solution p[ z) are obtained by solving the first-order equation (27) directly on the interval 0 < z < zq, where Eq. (27) can be replaced by the equation with constant coefficients (78) with the added inhomogeneity z). The solution of this equation is... 

The first-order solution can easily be extended to second-order boundary conditions, following the same reasoning as in the previous section. We find... 

An approximate solution for boo is obtained by truncating the infinite set of equations in some way. Here, the Chapman-Cowling method (Chapman Cowling 1970) is employed. A first-order solution is obtained by retaining the smallest number of nonvanishing terms of equivalent order in the expansion of the quantities This means that for viscosity one retains only s = t = 0, so that the first-order result for boo is obtained trivially from equation (4.16). For bulk viscosity and thermal conductivity in a first-order approximation the terms s = 1, t = 0 and = 0, t = 1 are retained because aoo = coo = 0. In this case one must solve a set of two equations each containing two terms to derive first-order approximations for a o, aoi and coi and hence the transport coefficients. In a second-order approximation one must include terms in the expansion of up to = 1, r = 1 for the viscosity and s = 2, t = 0 s = 0, t = 2 and s = t = 1 for bulk viscosity and thermal conductivity. It follows that it is possible by means of equations (4.12) to (4.14) to relate the transport coefficients directly to the quantities It is now, however,... 

Going one step further, the combination of molecular diffusion with Taylor dispersion has also been treated in Brodkey (1967, p. 326). Turbulent eddies carry what has to be mbted from one part of the fluid to another, which accelerates the breakdown of blobs of pure A. At the same time, moleeular diffusion is enhanced hy the increase in surface area and the steep gradients of concentration that occur due to the action of the mrbulent eddies. Using a statistical approach, the enhancement of nuxing due to mrhulent dispersion can be described with a simple first-order solution ... 

A "first-order" solution is obtained by the complete neglect of the terms a /(gB+Y) H in (6) and (7). This will usually only correspond within experimental error to the exact solutionwhen a is very small (less than 30 G). An exact solution is achieved by a process of iteration in which successive approximations to and a are substituted into the square root terms of (6) and (7) and the resulting linear simultaneous equations solved. The first-... 

For the first-order solution of the Boltzmann equation using Chapman-Enskog method, we get the constitutive laws of the N-S equation as... 

---