Detailed calculations

SO we may reasonably anticipate that we have just n potentially bonding orbitals derived from the n basis (these orbitals spanning subsets within the general P , D etc. functions available for the geometry of the particular cluster). Comparison with the transformations of the a (which give rise to 

Calculate G° for each species. The free energy of a substance, for which heat capacity data are available, can be calculated as a function 

For pure substances (i.e., solids, liquids, and gases) the heat capacity C is often expressed, as in Table 4.3, as a function of the absolute temperature  

For ionic substances, one has to use another method, such as proposed by Criss and Cobble in 1964 [1], to obtain the heat capacity, provided the temperature does not rise above 200°C. The expression of the ionic capacity in equation (Eq. 4.22) makes use of absolute entropy values and the parameters a and b contained in Table 4.3. 

Although these equations appear slightly overwhelming, they can be computed relatively simply with the use of a modern spreadsheet. 

For species O, the free energy of 1 mol can be obtained from G with Eq. (4.24)  

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Some detailed calculations have been made by Tully [209] on the trajectories for Rideal-type processes. Thus the collision of an oxygen atom with a carbon atom bound to Pt results in a CO that departs with essentially all of the reaction energy as vibrational energy (see Ref. 210 for a later discussion). 

Conceptually similar studies have since been carried out for the reaction of Ft atoms with Cl/Aii(l 11). More recently, quantum-state distributions have been obtamed for both the Ft + Cl/Aii(l 11)[, and M and Ft(D) + D (Ft)/Cii(l 11) systems. The results of these studies are in good qualitative agreement widi calculations. Even for the Ft(D) + D (Ft)/Cii(l 11) system [89], where we know that the incident atom caimot be significantly accommodated prior to reaction, reaction may not be direct. Detailed calculations yield much smaller cross sections for direct reaction than the overall experimental cross section, indicating that reaction may occur only after trapping of the incident atom [90]. 

One mol of each of the following is dissolved in water the final volume in each case is 1 0 L Identify the species that is present in greatest concentration in the respective solutions Detailed calculations are not necessary... 

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

Detailed Modeling Results. The results of a series of detailed calculations for an ideal isothermal plug-flow Langmuir system are summarized in Figure 15. The soHd lines show the form of the theoretical breakthrough curves for adsorption and desorption, calculated from the following set of model equations and expressed in terms of the dimensionless variables T, and P ... 

Many of these values are extrapolated see Ref. 5 for more details. Calculated value. 

The normal boiling point of /V-ethylaniline is 204°C. Therefore, steam distillation makes possible the distillation of /V-ethylaniline at atmospheric pressure at a temperature of 99.15°C instead of its normal boiling point of 204°C. Commercial appHcations of steam distillation include the fractionation of cmde tall oil (qv) (84), the distilling of turpentine (see Terpenoids), and certain essential oils (see Oils, essential). A detailed calculation of steam distiUation of turpentine has been reported (85). 

The resonance integral of the 7r-bond between the heteroatom and carbon is another possible parameter in the treatment of heteroatomic molecules. However, for nitrogen compounds more detailed calculations have suggested that this resonance integral is similar to that for a C—C bond and moreover the relative values of the reactivity Indices at different positions are not very sensitive to change in this parameter. 

Proper economic analysis will allow comparison of alternatives on a sound basis. Detailed calculations are beyond the scope of this section. The reader should review the material in the NACE Publication 3CI94, Item No. 24182, "Economics of Corrosion, Sept., 1994. 

The reader should refer to the chapter on pumps for more detailed calculation methods and for a description of common pumping equipment and pumping applications. The simple relationships provided here are useful in obtaining initial design specifications information in sizing a pump for an application. 

Equations for centerline velocity outside circular, square, rectangular and slot exhausts without and with flanges are presented in Table 10.3. These equations have been chosen to have as large an application as possible. For very detailed calculations it is recommended that the original research references be consulted. 09,40... 

Subsequently, in Chapter 4, we deal with cycles in which the turbines are cooled. The basic thermodynamics of turbine cooling, and its effect on plant efficiency, are considered. In Chapter 5, some detailed calculations of the performance of gas turbines with cooling are presented. 

A useful rule of thumb is that the turbine work in a STIC plant is increased by a factor of about (1 + 25), since the specific heat of the steam is about double that of the specific heat of the dry gas. This is in agreement with the example given above and with the earlier detailed calculations by Fraize and Kinney [3]. (Their work was based on the assumption that the mixture of air and steam in the turbine behaved as a semi-perfect gas, with specific heats being determined simply by mass averaging of the values for the two components.)... 

These smaller effects are related to the smaller amount of water that can be injected for the intercooled cycle. Applying Eq. (6.19c) to the near optimum condition of Fig. 6.9 (tjdry = 0.5, with 5 = 0.1) yields (tjwet dry) 0.08. Applying the same equation to the near optimum condition of Fig. 6.10 (tjdry = 0.53, with S = 0.04) yields (tjwet Tory) 0.035. Both these approximate estimates are very close to the detailed calculations of the increases in thermal efficiency shown in the two figures. 

Corti and Manfrida [2] have also done detailed calculations of the performance of plant A2. They drew attention to the need to optimise the amines blend (including species such as di-ethanolamine and mono-ethanolamine) in the absorption process, if a removal efficiency of 80% is to be achieved and in order to reduce the heat required for regenerating the scrubbing solution. Their initial estimates of the penalty on efficiency are comparable to those of Chiesa and Consonni (about 6% compared with the basic CCGT plant) but they emphasise that recirculation of water from... 

Fig. 8.I I. Detailed calculation of a steam/TCR plant (after Lloyd 5 ). Princeton University Library.

Four parameters often used to determine a fireball s thermal-radiation hazard are the mass of fuel involved and the fireball s diameter, duration, and thermal-emissive power. Radiation hazards can then be calculated from empirical relations. For detailed calculations, additional information is required, including a knowledge of the change in the fireball s diameter with time, its vertical rise, and variations in the fireball s emissive power over its lifetime. Experiments have been performed, mostly on a small scale, to investigate these parameters. The relationships obtained for each of these parameters through experimental investigation are presented in later sections of this chapter. 

When a pressure vessel is not a sphere, or if the vessel does not fracture evenly, the resulting blast wave will be nonspherical. This, of course, is the case in almost every actual pressure vessel burst. Loss of symmetry means that detailed calculations... 

The overall frequency of the top event is calculated by combining together the constituent probabilities and frequencies of the various events in the fault tree using the appropriate logical relationships described by the AND and OR gates (the detailed calculation is given in CCPS, 1989b). 

Project management also influences the likelihood that staffing levels wiU be adequate for the tasks required. This latter factor, together with the extent to which appropriate jobs are assigned to individuals, and the complexity of the jobs, all influence the level of time pressure likely to be felt by the operator. The detailed calculations, which show how the probability of human error is influenced by changes in the sociotechnical factors in the situation, are given in Appendix 5A. 

FMO theory was developed at a time when detailed calculations of reaction paths were infeasible. As many sophisticated computational models, and methods for actually locating the TS, have become widespread, the use of FMO arguments for predicting reactivity has declined. The primary goal of computational chemistry, however, is not tc... 

It is essential to realistically establish the condensing conditions of the distillation overhead vapors, and any limitations on bottoms temperature at an estimated pressure drop through the system. Preliminary calculations for the number of trays or amoimt of packing must be performed to develop a fairly reasonable system pressure drop. With this accomplished, the top and bottom column conditions can be established, and more detailed calculations performed. For trays this can be 0.1 psi/actu-al tray to be installed [149] whether atmospheric or above, and use 0.05 psi/tray equivalent for low vacuum (not low absolute pressure). 

When the feed point is located by a match from tray-bytray calculation, the correct point can be established with greater confidence, but still alternate nozzles are suggested since even these detailed calculations can be off to a certain extent. 

The detailed calculations of Figure 8-54 present an example of the excellent performance analysis information that can be developed hy an orderly or systematic study of the variables in a multicomponent system. There are other variables to be studied as well. 

Detailed calculations regarding the design of the rotary screw compressor are beyond the scope of this handbook. Additional details can be found in other references [4,25,26,27]. 

A more detailed calculation requires performing the Gaussian integration over the disorder realizations close to the saddle-point configuration Eq. (3.25). One then finds the following expression for the average density of electron states per unit length,... 

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