Zero-order approximation

The sum of the properties of the diatomic species X and Y2 is the same as twice the property of XY. This is the zero-order approximation to additivity rules. 

As a naive or zero-order approximation, we can simply ignore the V12 term and allow the simplified Hamiltonian to operate on the Is orbital of the H atom. The result is... 

The proliferation of acidity functions is a consequence of the activity coefficient cancellation assumption. According to Eq. (8-89), a plot of log(cB/cBH+) against Hq should be linear with unit slope. Such plots are usually linear (for bases of closely related structure), but the slopes often differ from unity. - This behavior is an indication that the cancellation assumption (also called the zero-order approximation) is not valid, and several groups have devised alternatives. We will use the symbolism of Cox and Yates. ... 

Let us compare the expressions for pXbh+ under ideal (Eq. 8-95), nonideal zero-order approximation (Eq. 8-96), and nonideal first-order approximation (Eq. 8-97) conditions ... 

A Perturbation Theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second order correction for the energy greatly simplifies because of the special property of the zero order solution. It is pointed out that the development of the higher order approximation involves only calculations based on a definite one-body problem. 

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

From Eqs. (1-76), the zero-order approximations to the pressure tensor and heat flow vector are ... 

The pressure profiles obtained from isothermal runs at five different temperatures using this method are shown in Figure 2. It can be observed that in general, the pressure rise is fairly linear for most of the duration of the experiments so that a zero-order approximation may be used to fit the data. This linearity was found to hold even after 5 days for the 175 °C isotherm, reaching a pressure level of approximately 300 psia (this was the longest duration of all the experiments). In the case of the 225 °C isotherm, the pressure accumulation finally levels off at approximately 1100 psia after one day. 

Rate dependence on added base (formate) concentration. zero order approximately first order approximately first order approximately first order... 

Stein, M., van Lenthe, E., Baerends, E. J. and Luhitz, W. (2001a) G- and A-tensor calculations in the zero-order approximation for relativistic effects of Ni-complexes as models for the active center of [NiFej-hydrogenase./. Rhys. Chem., A 105, 416-25. 

The original model regarding surface intermediates is a system of ordinary differential equations. It corresponds to the detailed mechanism under an assumption that the surface diffusion factor can be neglected. Physico-chemical status of the QSSA is based on the presence of the small parameter, i.e. the total amount of the surface active sites is small in comparison with the total amount of gas molecules. Mathematically, the QSSA is a zero-order approximation of the original (singularly perturbed) system of differential equations by the system of the algebraic equations (see in detail Yablonskii et al., 1991). Then, in our analysis... 

The simplest zero order approximation for the steady state gives... 

The zero-order approximation of the steady state depends only on the sign of inequality between kji fcjj- 21 12 almost all concentration... 

Exactly as it was in the previous example, the zero-order approximation of the steady state depends only on the sign of inequality between k and 54. If k ks4 then almost all concentration in the steady state is accumulated inside A. After restoring the cycle A3 Ai A2 A5 As A3 we find that in the steady state almost all concentration is accumulated in A (the component at the beginning of the limiting step of this cycle, A(, A3). The eigenvector for zero... 

Let us assume again that functions a are non-degenerate, in which case they are automatically orthogonal. If one of 4>a is a reasonable approximation of the exact wave function xp for the state investigated, then this zero-order approximation is called the model function ... 

The purpose of PT is to find a scheme for generating a sequence of successive improvements to this zero-order approximation. Let us define... 

As was mentioned earlier, model function xpo corresponds to the zero-order approximation and the remaining part, Qxp, can be considered as a correction . If xpo is generated in an independent particle model (e.g., Hartree-Fock), then Qxp is often referred to as the correlation function [75, 76]. 

Zero-order energy of the central field approximation, described by the central symmetric part of the potential, does not contain interaction of the momenta. Therefore, in zero-order approximation all states of a given configuration differing from each other by quantum numbers m, m, i.e. by different orientation of orbital and spin momenta 1, and s,-, have the same energy, and the corresponding level is degenerated (4/ + 2) times. 

Common to these methods, is that only one point, that is, the maximum heat release rate of the thermogram, is used. Even if the method by itself is efficient in terms of simplicity, experimental work, and evaluation time, it represents a waste of information, in the sense that all the available information is not used in the procedure. Nevertheless, from the point of view of safety it is conservative, since it is based on the zero-order approximation. More complex methods are based on a kinetic analysis of the thermograms presented in Section 11.4.3 below. 

This procedure for estimating TD24 from the temperature at which the peak onset is detected in a dynamic experiment is justified, since the TM Rad is based on a zero-order approximation and at the beginning of the DSC peak, the conversion is close to zero. Thus, the heat release rate determined by the procedure is not affected by the rate equation, at least for non-autocatalytic reactions, and may be used for the estimation. 

Worked Example 12.1 TMRad from Zero-order Approximation... 

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