Approximating Form

Other techniques such as X-ray diffusion or small angle neutron diffusion are also used in attempts to describe the size and form of asphaltenes in crude oil. It is generally believed that asphaltenes have the approximate form of very flat ellipsoids whose thicknesses are on the order of one nanometer and diameters of several dozen nanometers. 

We can now calculate the Donnan contribution to film pressure through the use of Eq. III-113 in the approximate form ... 

Fmri tier Oi hiltil theory is qualitative, so there is no need for great accuracy in th e calculation as Ion g as it produces the approximate form of tlie orbitals. 

Therefore the approximate form of the unknown function within this element is written as... 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

Extended Huckel provides the approximate shape and energy ordering of molecular orbitals. It also yields the approximate form of an electron density map. This is the only requirement for many qualitative applications of quantum mechanics calculations, such as Frontier Orbital estimates of chemical reactivity (see Frontier Molecular Orbitals on page 141). 

In this way every linear combination of two identical AOs gives two MOs, one higher and the other lower in energy than the AOs. Figure 7.13 illustrates the MOs from lx, 2s and 2p AOs showing the approximate forms of the MO wave functions. 

The quantitative relationship between the degree of adsorption at a solution iaterface (7), G—L or L—L, and the lowering of the free-surface energy can be deduced by usiag an approximate form of the Gibbs adsorption isotherm (eq. 9), which is appHcable to dilute biaary solutions where the activity coefficient is unity and the radius of curvature of the surface is not too great ... 

Equation 9 states that the surface excess of solute, F, is proportional to the concentration of solute, C, multipHed by the rate of change of surface tension, with respect to solute concentration, d /dC. The concentration of a surfactant ia a G—L iaterface can be calculated from the linear segment of a plot of surface tension versus concentration and similarly for the concentration ia an L—L iaterface from a plot of iaterfacial teasioa. la typical appHcatioas, the approximate form of the Gibbs equatioa was employed to calculate the area occupied by a series of sulfosucciaic ester molecules at the air—water iaterface (8) and the energies of adsorption at the air-water iaterface for a series of commercial aonionic surfactants (9). 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

When the stress that can be bom at the interface between two glassy polymers increases to the point that a craze can form then the toughness increases considerably as energy is now dissipated in forming and extending the craze structure. The most used model that describes the micro-mechanics of crazing failure was proposed by Brown [8] in a fairly simple and approximate form. This model has since been improved and extended by a number of authors. As the original form of the model is simple and physically intuitive it will be described first and then the improvements will be discussed. 

Many preliminary analyses of gas turbines are based on the assumption of a closed air standard cyclic plant, and for such analyses the use of tj as a thermal efficiency is entirely correct (as discussed in the early part of Chapter 3 of this book). But most practical gas turbines are of the open type and the rational efficiency should strictly be used, or at least its approximate form, the arbitrary overall efficiency tjq. We have followed this practice in the latter part of Chapter 3 and subsequent chapters even though some engineers consider this differentiation to be a somewhat pedantic point and many authors refer to tjo as a thermal efficiency (or sometimes the lower heating value thermal efficiency ). 

The temperature rise in the combustion chamber may then be determined from Eq. (3.33), in the approximate form (Tj T2) = (af + b). Strictly a and b are functions of the temperature of the reactants and the fuel-air ratio/, but fixed values are assumed to cover a reasonable range of conditions. Accordingly, the fuel-air ratio may be expressed as... 

Theessence of the procedure is that Halpin and Tsai [3-17] showed that Hermans solution [3-14] generalizing Hill s self-consistent model [3-13] can be reduced to the approximate form... 

Semi-empirical methods, such as AMI, MINDO/3 and PM3, implemented in programs like MOPAC, AMPAC, HyperChem, and Gaussian, use parameters derived from experimental data to simplify the computation. They solve an approximate form of the Schrodinger equation that depends on having appropriate parameters available for the type of chemical system under investigation. Different semi-emipirical methods are largely characterized by their differing parameter sets. 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

Fig. 6. Approximate form of vibration assigned to 1616 cm"1 Raman line. Also shown are the ring axes Ox Ox2Ox3. Reproduced from Polymer by permission of the publishers, Butterworth Co (Publishers) Ltd. (C)...

Approximate form of velocity profile In turbulent region... 

A simple approximate form of the relation between u+ and y+ for the turbulent flow of a fluid in a pipe of circular cross-section may be obtained using the Prandtl one-seventh power law and the Blasius equation. These two equations have been shown (Section 11.4) to be mutually consistent. 

As well as being attracted to the nucleus, each electron in a many-electron atom is repelled by the other electrons present. As a result, it is less tightly bound to the nucleus than it would be if those other electrons were absent. We say that each electron is shielded from the full attraction of the nucleus by the other electrons in the atom. The shielding effectively reduces the pull of the nucleus on an electron. The effective nuclear charge, Z lle, experienced by the electron is always less than the actual nuclear charge, Ze, because the electron-electron repulsions work against the pull of the nucleus. A very approximate form of the energy of an electron in a many-electron atom is a version of Eq. 14b in which the true atomic number is replaced by the effective atomic number ... 

Tables IV and V contain appropriate balance equations for nonisothermal free-radical polymerizations and copolymerizations, which are seen to conform to equation 2k. Following the procedure outlined above, we obtain the CT s for homopolymerizations listed in Table VI. Corresponding CT s for copolymerizations can be. obtained in a similar way, and indeed the first and fourth listed in Table VII were. The remaining ones, however, were derived via an alternate route based upon the definitions in Table VI labeled "equivalent" together with approximate forms for pj, which were necessitated by application of the Semenov-type runaway analysis to copolymerizations, and which will subsequently be described. Some useful dimensionless parameters defined in terms of these CT s appear in Tables VIII, IX and X.

Keenan has made an investigation of the exchange reaction between Pu(IV) and Pu(in) in perchlorate media. The isotopic method was used with an a energy analyser to separate the tracer activity ( Pu) from that normally present from the major constituent ( Pu). Tributylphosphate extraction of the Pu(IV) formed the basis of the separation method. It was shown that the rate law has the approximate form... 

Tsoucaris, decided to treat by Fourier transformation, not the Schrodinger equation itself, but one of its most popular approximate forms for electron systems, namely the Hartree-Fock equations. The form of these equations was known before, in connection with electron-scattering problems [13], but their advantage for Quantum Chemistry calculations was not yet recognized. 

Thanks to eq. (2.7), the electronic density expression given by eq. (2.5) can be cast into the (obviously approximate) form... 

The current increases first exponentially and then levels off. The same dependence follows from Eq. (34.27). At not large deviations of the electrode potential from the equilibrium potential (i.e., at not large overpotentials, r = Eq - E), the approximate form of Eq. (34.27) is as follows ... 

The potential of a chromatographic system to provide a certain separation can be estimated from its separation number, S SH, also referred to as the spot capacity in TLC. The separation j nund>er in TLC is defined as the number of spots that can be. completely separated (R, > 1) between R 0 and R, ° 1 (6). It is calculated in an approximate form by equation (7.16) and morej exactly by equation (7.17) with b, and b, as defined in Figure 7.4. 

Consider next the case of binary rupture when A = 0, s = 1, and br(xly) = 2. Assuming that fragments once eroded do not fragment further, an approximate form of the evolution equation for clusters larger than size e is... 

As mentioned in Section 2, the CPs of solids have to be calculated on the quasi-particle scheme. In order to calculate the quasi-particle states, non-local and energy-dependent self-energy in Equation (13) must be evaluated in a real system. In practice, the exact self-energy for real systems are impossible to compute, and we always resort to approximate forms. A more realistic but relatively simple approximation to the selfenergy is the GWA proposed by Hedin [7]. In the GW A, the self-energy operator in Equation (12) is... 

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