Zeroth-order assumption

It thus follows that it is permitted to express the concentration term in the rate equation as [C ] , i.e., as unity virtually, in the thermal explosion theory. This approach is refemed to as the zeroth-order assumption. 

This fact shows that the zeroth-order assumption holds for the rate of the oxidativcly-hcating reaction of the sawdust of every wood species at temperatures below about 180 C. At temperatures above about 180 t, however, the rate in general comes to accelerate gradually (Fig. 86). 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

The ambient temperature Tamb is ignored since T T amb. The assumption of a zeroth-order dissociation process implies that Thermal a Thus, A a A and E aE. 64 It should be noted that L thermal reaches a limiting value of A1 with increasing fluence. Fphoto increases with fluence, but as a very slow logarithmic function. 

Various refinements of the above model have been proposed for example, using alternative spherical potentials or allowing for nonspherical perturbations,and these can improve the agreement of the model with the abundance peaks observed in different experimental spectra. For small alkali metal clusters, the results are essentially equivalent to those obtained by TSH theory, for the simple reason that both approaches start from an assumption of zeroth-order spherical symmetry. This connection has been emphasized in two reviews,and also holds to some extent when considerations of symmetry breaking are applied. This aspect is discussed further below. The same shell structure is also observed in simple Hiickel calculations for alkali metals, again basically due to the symmetry of the systems considered. However, the developments of TSH theory, below, and the assumptions made in the jellium model itself, should make it clear that the latter approach is only likely to be successful for alkali and perhaps alkali earth metals. For example, recent results for aluminium clusters have led to the suggestion that symmetry-breaking effects are more important in these systems. ... 

The assumption that the standard free energy of adsorption is independent of coverage may be viewed as the "zeroth-order approximation. The first-order approximation will then be a linear dependence of AG° on 0 ... 

The first-order state /i) is therefore obtained simply by applying Fq to the zeroth-order state, with the caution of rejecting /o)(/ol Fq j/g), because the contribution coming from the state /q) is already includ in the basis set. We could go on that way, as this is just how Mori proceeds in his celebrated papers. However, to avoid the Hermitian assumption made by Mori, we build up a biorthogonal basis set. This means that the state /i) has to be associated with the corresponding left state (we assume (/q = (/o )... 

Let us return to the problem of solving the response of the quantum mechanical system to an external electric field. The zeroth-order wave function of the quantum mechanical system is obtained by use of any of the standard approximate methods in quantum chemistry and the coupling to the field is described by the electtic dipole operator. There exist a number of ways to determine the response functions, some of which differ in formulation only, whereas others will be inherently different. We will give a short review of the characteristics of tire most common formulations used for the calculation of molecular polarizabilities and hyperpolarizabilities. The survey begins with the assumption that the external perturbing fields arc non-oscillatory, in which case we may determine molecular properties at zero frequencies, and then continues with the general situation of time-dependent fields and dynamic properties. 

Notice from eqs. (A.23) and (A.24) that the slow variable assumption amounts to approximating the total thermodj namic force as a sum of (a) a zeroth-order term Feq[rp A] independent of A, which is the full force for an idealized infinitely slow (A = 0) process and (b) a first-order term -RA linear in A, which corrects Feq(Fp A) for the finite rate (A 0) of real processes. 

There are, however, three very important implicit assumptions in this model, apart from those of an ideal interface. Firstly, since desorption is only allowed to occur from a constant precursor state population (dn/dt = 0), it is effectively always a zeroth-order process. If a different order is observed, desorption is not the rate-limiting step. The second point is that this treatment is only appropriate for cases where the metal-metal bond energy (around the peripheries of the islands) is less than that for the metal- semiconductor, since for the opposite case the weaker adsorbate—surface bond will not prevent an atom desorbing once it has acquired sufficient energy to break the (stronger) metal—metal bond. Thirdly, no provision is made for possible diffusion of the adsorbate into the substrate during desorption. 

It is now generally recognized that the Nemst approximation for metal oxides is doomed to failure. The problem is in the assumption that the activity of the potential-determining ions on the surface is independent of the surface potential (zeroth order approximation). For example, as the bulk pH decreases... 

Both mechanisms assume the formation of a cage that prevents ROO from reacting with RH. This assumption is only a zeroth order approximation. Some radicals may escape the cage and start classical free-radical chemistry in solution. It is possible that the hydrocarbon loses a hydrogen atom, but is seems more likely that an electron and a proton are taken up by the manganese catalyst. 

The constant value of the chain contribution may be considered as a zeroth-order approximation, where it is assumed that the density of the chains is independent of the salt concentration. A motivation for this assumption is that the molecular area of the surfactant as a function of salt concentration in the systems used in Ref 46 was found [54] to be constant within the range of experimental error. 

The theory based on this assumption is expected to give zeroth order predictions of global polymer properties, and is usually called the blob theory. However, this nomenclature does not seem relevant to the author (see Section 1.4). Hence, in this book, we call it the Weill-des Cloizeaux theory. 

In a further step to generalize the Heisenberg Hamiltonians, one may decide that they will be spanned by all the possible OVB valence neutral determinants, without any assumption concerning their hybridization state. The carbon atom will be either s p, sp or p for instance. In the language of solid-state physics, one would say that the two bands s and p are both involved. The various zeroth-order energies of the determinants belonging to the model space are no longer d enerate, since in C2, for instance, the VB determinant Sa aJ a-Sb bJ b. 

Thus, the zeroth-order approximation (in k) of the Poisson equation is = 0. In other words, the LEN assumption is a consequence of the Poisson equation when 2. 1. Moreover, the first-order term of... 

A liquid state theory has been developed on the basis of an ideal liquid, which is a hard-sphere liquid. Usually, thus, a random disordered structure of liquid has been assumed. This is the basis for the description of liquid by the two-body density correlator, or the radial distribution function g r). Recent studies indicate this picture is not sufficient even for a hard-sphere liquid [46,47], The assumption of a disorder structure of a liquid is always correct as the zeroth order approximation. However, we believe that a physical description beyond this is prerequisite for understanding unsolved fundamental problems in a liquid state, which include thermodynamic and kinetic anomalies of water type liquids, liquid-liquid transition, liquid glass transition, and crystal nucleation. 

Waals bond vibrations in polyatom-atom species have usually been based on the assumption that the potential energy surface is adequately approximated as a superposition of atom-atom interactions. This assumption appears to give only a zeroth order approximation to the potential energy surface. 

The third and final item making a scaled experiment unacceptable is the assumption of pseudo-zeroth order conditions. If the depth of the 1 cm-diameter cell is reduced to 0.05 cm, with a concentration of 2.51 mg/ml, the total mass of monomer in solution is 1.25 10 g. A film grown to a thickness of 0.15 m, with the same diameter as the solution chamber and, having a density of 1.41 g/cm, consumes a mass of 2.M0 g of monomer, more than 10% of the original amount of monomer in solution. Clearly this would violate the assumption that the bulk and average concentrations do not change during the course of the experiment. 

In summary, the system under study is extremely complicated. When the experiment is scaled down to suppress convection it no longer conforms to the necessary assumptions used to describe and model the system in the laboratory, nor does it any longer operate under pseudo-zeroth order conditions. This makes it impossible to use the scaling approach to perform convection-free experiments in the laboratory rather than in space. It has been shown, however, that reductions of four or five orders of magnitude in the g-vector are sufficient to achieve a convection free environment in this system. These conditions are readily achievable on the space shuttle and on space station, making them ideal locations for this research. 

---