Initial Assumptions

The first step is to set out the assumptions concerning the defects that are likely to occur, using physical and chemical intuition about the system in mind. For illustrative purposes, take a nonstoichiometric phase of composition MX, nominally containing M2+ and X2- ions, with a stoichiometric composition, MXl 0. In this example the following is assumed  

The only ionic defects of importance are vacancies on metal sites, V, and 

MX can have an existence range that spans both sides of the stoichiometric composition, MXi.o, which is due to an imbalance in the population of vacancies. 

As well as vacancies, MX contains a varying population of electrons e, or holes, h, which can act to maintain electroneutrality when the solid becomes nonstoichiometric. At the stoichiometric composition of MXl 0, the number of electrons is equal to the number of holes, which are simply the normally present intrinsic electrons and holes. 

The most important gaseous component is X2, as is the case in most oxides, halides, and sulfides. The stoichiometric variation will be linked to the partial pressure of the surrounding nonmetal atmosphere. The nonmetal component will be gained at high pressures and lost at low pressures. These options correspond to oxidation and reduction. 

Since the first energy level calculations of the EM centres in silicon and germanium [28,34], many calculations have been undertaken to explain quantitatively the absorption and photoluminescence (PL) spectra associated with these centres in many semiconductors. The first part of this chapter is devoted to the presentation of the energy level calculations of EM donors and it is followed by the results of the calculations for EM acceptors. The modification to EMA, which is independent of the chemical nature of the centres, is also discussed. The chapter concludes with results of calculations of the oscillator strength (OS) for transitions between the ground states and the acceptor or donor states. 

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The reservoir model will usually be a computer based simulation model, such as the 3D model described in Section 8. As production continues, the monitoring programme generates a data base containing information on the performance of the field. The reservoir model is used to check whether the initial assumptions and description of the reservoir were correct. Where inconsistencies between the predicted and observed behaviour occur, the model is reviewed and adjusted until a new match (a so-called history match ) is achieved. The updated model is then used to predict future performance of the field, and as such is a very useful tool for generating production forecasts. In addition, the model is used to predict the outcome of alternative future development plans. The criterion used for selection is typically profitability (or any other stated objective of the operating company). 

An initial assumption is that the reaction takes place with maximum overlap between the HOMO on one molecule and the LUMO on the other. The HOMO ofphenylbutadiene has most density on the terminal carbon atom. 

The solution starts with an assumed arbitrary split of all the components to give estimates of top and bottom compositions that can be used to get initial end temperatures. The (Xj s evaluated at these temperatures are averaged with an assumed feed-stage temperature (assumed to be the bubble point of the feed) by using Eq. (13-36). The initial assumption for the split on i-Cs will be Dxp/Bxb = 3.15/16.85. As mentioned earlier, N usually ranges from 0.4N to 0.6N, and the initial value assumed here will be (0.6)(10) = 6.0. Equation (13-32) can be rewritten as... 

A computer solution was obtained as follows. The only initial assumptions are a condenser outlet temperature of 65 F and a bottoms-prodiict temperature of 165 F, The bubble-point temperature of the feed is computed as 123,5 F, In the initiahzation procedure, the constants A and B in (13-106) for inner-loop calcu-... 

The actual yield may be obtained from algebraic calculations or trial-and-error calculations when the heat effects in the process and any resultant evaporation are used to correc t the initial assumptions on calculated yield. When calculations are made by hand, it is generally preferable to use the trial-and-error system, since it permits easy adjustments for relatively small deviations found in practice, such as the addition of wash water, or instrument and purge water additions. The following calculations are typical of an evaporative ciy/staUizer precipitating a hydrated salt, if SI units are desired, kilograms = pounds X 0.454 K = (°F 459.7)/I.8. 

The temperature is now below the 265°F limit and consistent with the Section 1 temperature. At this point, the initial assumption for 2 sections can be considered a firm value. 

Step 1. Assume the composition of the liquid in the evaporator at equilibrium with its vapor to be 75 mol% propane and 25 mol% butane. This is the initial assumption. If it is correct, the composition of the initial charge can be checked. If it is not correct, the problem must be reworked with a new equilibrium assumption. The composition of the vapor in equilibrium with this liquid is determined from the following equation. 

In general, the kinetics of most allosteric modulators have been shown to be faster than the kinetics of binding of the tracer ligand. This is an initial assumption for this experimental approach. Under these circumstances, the rate of dissociation of the tracer ligand (pA t) n the presence of the allosteric ligand is given by [11, 12]... 

In order to estimate the value of the dissociation constant of the TaF72 complex according to the aforementioned scheme (61), TaF72 = F + TaF6, analysis of the available volume values was performed using a modified method developed by Frank and Foster [329]. This method is based on two main initial assumptions. The first is that the dissociation constant remains... 

This identifies the time between jumps t and the time of a jump t and breaks the initial assumption of the model, which considers jumps as instantaneous x [Pg.219]

The restriction t < follows from the initial assumption concerning the nonnegativity of the coefficients B P, Q). It is clear that D[P) = 0. 

In general, a preparation of mixed monolayer can be realized by either a kinetic control or a thermodynamic control (Figure 1, left). Kinetic control is based on a suggestion that for an initial deposition step the desorption rate is ignorable in comparison with the adsorption rate. In this case, the concentration ratio of the adsorbed species A and B on the surface corresponds to the ratio of products of their adsorption rate constant ( a or b) and concentration (Ca or Cb) A aCa/A bC b. The validity of the initial assumption on low desorption rate means that the total surface coverage obtained under kinetic control is essentially lower than 100%. This non-complete coverage does not disturb most of optical applications of the... 

This is little different from calculated value and shows that initial assumption that no ethylene was condensed or dissolved was reasonable so report ethylene in liquid as trace . 

Reactor conversion. Earlier in Chapters 5 to 7, the initial choice of reactor type, operating conditions and conversion was discussed. The initial assumption for the conversion varies according to whether there are single reactions or multiple reactions producing byproducts and whether reactions are reversible. Consider the simple reaction ... 

Determine the temperature of the fluid leaving the volume element using the overall energy balance (13.2.65) and compare this result with the initial assumption. 

The high temperatures and pressures created during transient cavitation are difficult both to calculate and to determine experimentally. The simplest models of collapse, which neglect heat transport and the effects of condensable vapor, predict maximum temperatures and pressures as high as 10,000 K and 10,000 atmospheres. More realistic estimates from increasingly sophisticated hydrodynamic models yield estimates of 5000 K and 1000 atmospheres with effective residence times of <100 nseconds, but the models are very sensitive to initial assumptions of the boundary conditions (30-32). 

But the DC input current (0.67A) cannot flow through the switch during the OFF time. Therefore it must be getting diverted as shown, charging the input capacitors in the process. Note that if this diverted current were not 0.67A, it would amount to introducing an AC component in the supposedly DC input current (which in turn would imply imperfect decoupling, which is contradictory to our initial assumption). 

Of the parameters which occur in Eqns. 11.24 and 11.25, M and M are in the particular position that they are of virtually atomic character. Furthermore, the initial assumptions concerning M and M have indi-... 

Depending on the relative amount of reagents, three types of products may be obtained in the reaction of hydroxymethylphosphines with amines acyclic aminomethylphosphines, 1,3,5-diazaphosphorinanes, and 1,5,3,7-diazadiphosphacyclooctanes. Indeed, the interaction of phosphorus-containing diols with 1 mol of amine gives eight-membered heterocycles. The initial assumption about the formation of 1,3-azaphosphetidines... 

In the present model, a maximum of two localized states arises, which depends on the initial assumptions that only one adatom orbital interacts... 

This relationship is drawn in Figure 7.6 as a function of the 87Sr/86Sr ratio in the input magma for the dates t= 1760, 1820 and 1880. Provided the initial assumptions on the magmatic regime are valid, the magma chamber is smaller than 1 km3, o... 

Here it has been shown that the conclusion about is related to the mathematical approximation used in interpreting the data by way of Method I it is easy to show that the second and third conclusions are also dependent on the initial assumptions in Method I. Rearrangement of Equation 28 and substitution of the constant capacitance constraint yields a relationship between i/>q and In a + ... 

For ethanol, the same procedure yields ER(EtO ) = -9.2 kJ mol-1 and s(EtO-H) = 449 kJ mol-1. This value is still some 48 kJ mol-1 higher than the value obtained for phenol, questioning our initial assumption that the bond strengths should be similar in both compounds. Yet we have forgotten two import issues the resonance stabilization of the phenoxyl radical and the hyperconjugation of the ethoxyl radical. 

The classical method of solution is effective and the iterates are quite insensitive to the initial assumptions, if properly chosen as described in books on unit operations. No claim is made that the precedence scheme of Fig. 21 is either better or quicker than the classical method. However, the procedure... 

For large spills which undergo an RPT, one must reconsider the initial assumption that only with LNG of an appropriate composition would one achieve an RPT. Thus, the concept of a large-scale, coherent RPT was developed. In these cases, the superheat concept may still play an important role, i.e., to trigger the RPT. Other triggering sources are, however, also possible. Current concepts of large-scale RPTs require the following set of circumstances ... 

If the conclusion seems to be in error, and the chain of reasoning is valid, we have then demonstrated that one (or more) of our initial assumptions must be erroneous. For example, if one of the premises was the implication If a ketone is present then the compound is an alkane , then the conclusion that would follow from the observation a ketone is present would be obviously false. Since the PC is logically correct (in fact defines logically correct ) any erroneous conclusions must arise from an erroneous axiom, and cannot be the fault of the calculating procedure used. This allows us to focus our attention on the assumptions and frees us from having to worry about the procedure. 

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