Fortune

Best ternary predictions are usually obtained for mixtures having a very broad two-phase region, i.e., where the two partially miscible liquids have only small mutual solubilities. Fortunately, this is the type of ternary that is most often used in commercial liquid-liquid extraction. 

Figure 16 shows observed and calculated VLE and LLE for the system benzene-water-ethanol. In this unusually fortunate case, predictions based on the binary data alone (dashed line) are in good agreement with the experimental ternary data. Several factors contribute to this good agreement VLE data for the mis-... 

In Equation (15), the third term is much more important than the second term. The third term gives the enthalpy of the ideal liquid mixture (corrected to zero pressure) relative to that of the ideal vapor at the same temperature and composition. The second term gives the excess enthalpy, i.e. the liquid-phase enthalpy of mixing often little basis exists for evaluation of this term, but fortunately its contribution to total liquid enthalpy is usually not large. 

The second term in Equation (15a) gives the enthalpy of mixing of the condensable components. It is difficult to estimate that enthalpy but fortunately it ma)ies only a small contri-... 

Fortunately, however, the technique used here does not depend on the magnitude of the variances, but only on their ratios. If estimates of the magnitudes of the variances are wrong but the ratios are correct, the residuals display the random behavior shown in Figure 3. However, the magnitudes of these deviations are then not consistent with the estimated variances. 

The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. 

Fortunately, the development of new exploration techniques has improved geologists understanding and increased the efficiency of exploration. So although targets are getting... 

Our experiences with the software developed within the ANDES project have shown that CBR is a helpful methodology for use in the interpretation of NDT data from field inspections. Because CBR systems can adapt to new situations they can cope with inspection of varying constructions in varying conditions. However, because CBR systems learn from classifications made by the operator this means that they will not be very useful for completely automatic interpretation. Fortunately, most of the NDT inspection requires the presence of an operator because of the required high reliability. 

As a general comment, it is fortunate for the study of monolayers that dissolving processes are generally slow enough to permit the relatively unperturbed study of equilib-... 

In principle, nucleation should occur for any supersaturation given enough time. The critical supersaturation ratio is often defined in terms of the condition needed to observe nucleation on a convenient time scale. As illustrated in Table IX-1, the nucleation rate changes so rapidly with degree of supersaturation that, fortunately, even a few powers of 10 error in the preexponential term make little difference. There has been some controversy surrounding the preexponential term and some detailed analyses are available [33-35]. 

It is perhaps fortunate that both versions lead to the same algebraic formulations, but we will imply a preference for the two-dimensional solution picture by expressing surface concentrations in terms of mole fractions. The adsorption process can be written as... 

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

Fortunately, the worst broadening interactions are also removed naturally in most liquids and solutions, or at least greatly reduced in their effect, by the tumbling motions of the molecules, for many of the broadening... 

In principle, every nucleus in a molecule, with spm quantum number /, splits every other resonance in the molecule into 2/ -t 1 equal peaks, i.e. one for each of its allowed values of m. This could make the NMR spectra of most molecules very complex indeed. Fortunately, many simplifications exist. 

Single-frequency decoupling is easy and rapidly carried out. However, it may be limited by the closeness of different multiplets. Also, it will not nonnally be possible to apply more than one frequency of decoupling irradiation at a time. Fortunately, these disadvantages do not apply to the equivalent multidimensional methods. 

Fortunately, for non-integer quadnipolar nuclei for the central transition = 0 and the dominant perturbation is second order only (equation Bl.12.8) which gives a characteristic lineshape (figure B1.12.1(cB for axial synnnetry) ... 

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

As written, Eq. (52) depends on all the (infinite number of) adiabatic electi onic states. Fortunately, the inverse dependence of the coupling strength on energy separation means that it is possible to separate the complete set of states into manifolds that effeetively do not interact with one another. In particular, Baer has recendy shown [54] that Eq. (57), and hence Eq. (58) also holds in the subset of mutually coupled states. This finding has important consequences for the use of diabatic states explored below. 

Hydrogen sulphide is a colourless gas, b.p. 213 K, with a most unpleasant odour the gas is very toxic, but the intense odour fortunately permits very minute concentrations of the gas to be detected. 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

There is still some debate regarding the form of a dynamical equation for the time evolution of the density distribution in the 9 / 1 regime. Fortunately, to evaluate the rate constant in the transition state theory approximation, we need only know the form of the equilibrium distribution. It is only when we wish to obtain a more accurate estimate of the rate constant, including an estimate of the transmission coefficient, that we need to define the system s dynamics. 

Hence, the main danger in the process of compiling test sets remains. Fortunately, there are some other approaches which, although they may look more sophisticated, do diminish the possibiHty of such incompatibilities. 

Fortunately, the theory of chemical reaction s n eed not always require such sophistication. In deed, sin cc n ticlel (atom s) arc m tich heavier than electrons vou can use classical mechanics lo tindcr-... 

In principle, Chen, given the flux relations there is no difficulty in constructing differencial equations to describe the behavior of a catalyst pellet in steady or unsteady states. In practice, however, this simple procedure is obstructed by the implicit nature of the flux relations, since an explicit solution of usefully compact form is obtainable only for binary mixtures- In steady states this impasse is avoided by using certain, relations between Che flux vectors which are associated with the stoichiometry of Che chemical reaction or reactions taking place in the pellet, and the major part of Chapter 11 is concerned with the derivation, application and limitations of these stoichiometric relations. Fortunately they permit practicable solution procedures to be constructed regardless of the number of substances in the reaction mixture, provided there are only one or two stoichiomeCrically independent chemical reactions. 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

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