Simple Theories

For example, van den Tempel [35] reports the results shown in Fig. XIV-9 on the effect of electrolyte concentration on flocculation rates of an O/W emulsion. Note that d ln)ldt (equal to k in the simple theory) increases rapidly with ionic strength, presumably due to the decrease in double-layer half-thickness and perhaps also due to some Stem layer adsorption of positive ions. The preexponential factor in Eq. XIV-7, ko = (8kr/3 ), should have the value of about 10 " cm, but at low electrolyte concentration, the values in the figure are smaller by tenfold or a hundredfold. This reduction may be qualitatively ascribed to charged repulsion. 

Pack R T 1976 Simple theory of diffuse vibrational structure in continuous UV spectra of polyatomic molecules. I. Collinear photodissociation of symmetric triatomics J. Chem. Phys. 65 4765... 

Catalytic hydrogenation is mostly used to convert C—C triple bonds into C C double bonds and alkenes into alkanes or to replace allylic or benzylic hetero atoms by hydrogen (H. Kropf, 1980). Simple theory postulates cis- or syn-addition of hydrogen to the C—C triple or double bond with heterogeneous (R. L. Augustine, 1965, 1968, 1976 P. N. Rylander, 1979) and homogeneous (A. J. Birch, 1976) catalysts. Sulfur functions can be removed with reducing metals, e. g. with Raney nickel (G. R. Pettit, 1962 A). Heteroaromatic systems may be reduced with the aid of ruthenium on carbon. 

When the iateraction energy density is positive, equation 5 defines a critical temperature of the UCST type (Fig. la) that is a function of component molecular weights. The LCST-type phase diagram, quite common for polymer blends, is not predicted by this simple theory unless B is... 

Miscible Blends. Sometimes a miscible blend results when two polymers are combined. A miscible blend has only one amorphous phase because the polymers are soluble in each other. There may also be one or more crystal phases. Simple theory (26) has supported the empirical relation for the permeabihty of a miscible blend. Equation 18 expresses this relation where is the permeabihty of the miscible blend and ( ) and are the volume fractions of polymer 1 and 2. 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic dreoty of gases, and their interrelationship tlu ough k and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests tlrat this should be a dependence on 7 /, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to for the case of molecular inter-diffusion. The Anhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, tlren an activation enthalpy of a few kilojoules is observed. It will thus be found that when tire kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation entlralpy will be a few kilojoules only (less than 50 kJ). 

The chapter on equation-of-state properties provides the basic approaches used for describing the high-pressure shock-compression response of materials. These theories provide the basis for separating the elastic compression components from the thermal contributions in shock compression, which is necessary for comparing shock-compression results with those obtained from other techniques such as isothermal compression. A basic understanding of the simple theories of shock compression, such as the Mie-Gruneisen equation of state, are prerequisite to understanding more advanced theories that will be discussed in subsequent volumes. 

Most materials scientists at an early stage in their university courses learn some elementary aspects of what is still miscalled strength of materials . This field incorporates elementary treatments of problems such as the elastic response of beams to continuous or localised loading, the distribution of torque across a shaft under torsion, or the elastic stresses in the components of a simple girder. Materials come into it only insofar as the specific elastic properties of a particular metal or timber determine the numerical values for some of the symbols in the algebraic treatment. This kind of simple theory is an example of continuum mechanics, and its derivation does not require any knowledge of the crystal structure or crystal properties of simple materials or of the microstructure of more complex materials. The specific aim is to design simple structures that will not exceed their elastic limit under load. 

Draw the fire triangle and explain the simple theory of combustion. 

Fig, 5.15. A measured current-time pulse for shock-loaded Invar is shown. Time increases from left to right. The wave shape is closely predicted by the simple theory. Time from impact to peak current is about 1 fis. 

Smoluehowski also presented a simple theory of aggregation kineties assuming eollisions of perfeet eolleetion effieieney to prediet spherieal partiele size distributions in a uniform liquid shear field of eonstant veloeity gradient. The aggregation kernel is then expressed as... 

D. Henderson, M. Lozada-Cassou. A simple theory for the force between spheres immersed in a fluid. J Colloid Interface Sci 774 180-183, 1986. 

For any even vaguely realistic atomically constituted membrane it is unlikely that any theory will become available in the near future which will properly or reasonably describe the dynamic properties of the membrane, the fluids near it, and their passage, or selective passage, through it. Nevertheless, one should continue trying with simple models and simple theories [39-43], which show the way forward and can, as usual, be tested by the virtually exact results of molecular dynamics simulation. 

Clearly, proximity and orientation play a role in enzyme catalysis, but there is a problem with each of the above comparisons. In both cases, it is impossible to separate true proximity and orientation effects from the effects of entropy loss when molecules are brought together (described the Section 16.4). The actual rate accelerations afforded by proximity and orientation effects in Figures 16.14 and 16.15, respectively, are much smaller than the values given in these figures. Simple theories based on probability and nearest-neighbor models, for example, predict that proximity effects may actually provide rate increases of only 5- to 10-fold. For any real case of enzymatic catalysis, it is nonetheless important to remember that proximity and orientation effects are significant. 

The shapes of the monomeric molecules of the Group 2 halides (gas phase or matrix isolation) pose some interesting problems for those who are content with simple theories of bonding and molecular geometry. Thus, as expected on the basis of either sp hybridization or the... 

Resonance theory tells us that molecules which cannot be adequately represented in terms of a single Lewis structure are likely to be unusually stable. What the simple theory does not tell us is the magnitude of the effect, the so-called resonance energy. This can be assessed via molecular modeling. 

It is interesting to note that all the simple theories (such as Hiickel jr-electron theory) have now reappeared as options in these very same packages Thus, very many scientists now routinely use computational quantum chemistry as a futuristic tool for modelling the properties of pharmaceutical molecules, dyestuffs and hiopolymers. I wrote the original Computational Quantum Chemistry text as... 

The simple theory of fractional precipitation has been given in Section 2.8. It... 

Typical SAH are weakly charged flexible networks. A simple theory was proposed to describe these networks, in which the ionic swelling pressure includes only an ideal (Donnan) contribution [4,101], In this approximation, the swelling equilibrium is described by a system of equations including Eq. (3.1) and the... 

The original Reynolds analogy involves a number of simplifying assumptions which are justifiable only in a limited range of conditions. Thus it was assumed that fluid was transferred from outside the boundary layer to the surface without mixing with the intervening fluid, that it was brought to rest at the surface, and that thermal equilibrium was established. Various modifications have been made to this simple theory to take account of the existence of the laminar sub-layer and the buffer layer close to the surface. 

In the Taylor-Prandtl modification of the theory of heat transfer to a turbulent fluid, it was assumed that the heat passed directly from the turbulent fluid to the laminar sublayer and the existence of the buffer layer was neglected. It was therefore possible to apply the simple theory for the boundary layer in order to calculate the heat transfer. In most cases, the results so obtained are sufficiently accurate, but errors become significant when the relations are used to calculate heat transfer to liquids of high viscosities. A more accurate expression can be obtained if the temperature difference across the buffer layer is taken into account. The exact conditions in the buffer layer are difficult to define and any mathematical treatment of the problem involves a number of assumptions. However, the conditions close to the surface over which fluid is flowing can be calculated approximately using the universal velocity profile,(10)... 

A number of new results have been obtained in extending and refining London s simple theory, taking into consideration quantitative spectral and thermochemical data. Some of these results are described in the following paragraphs. 

The comparison of these results with the simple theory of conjugated systems [Pauling and Sherman, J. Chem. Phys., 1, 679 (1933)] not straightforward because of non-orthogonality of the canonical structures. If we assume that the double bond character... 

The form of the functions may be closely similar to that of the molecular orbitals used in the simple theory of metals. If there are M interatomic positions in the crystal which might be occupied by any one of the N electron-pair bonds, then the M functions linear aggregates that approximate the solutions of the wave equation with inclusion of the interaction terms representing resonance. This combination can be effected with use of Bloch factors ... 

A Simple Theory of Resonating Covalent Bonds in Metals... 

This simple theory is unsatisfactory, in that the rate of change of the difference in free energy of liquid and crystalline lead predicted by the Clausius-Clapeyron equation leads to a temperature scale for Fig. 8 four... 

An obvious refinement of the simple theory for cobalt and nickel and their alloys can be made which leads to a significant increase in the calculated value of the Curie temperature. The foregoing calculation for nickel, for example, is based upon the assumption that the uncoupled valence electrons spend equal amounts of time on the nickel atoms with / = 1 and the nickel atoms with J = 0. However, the stabilizing interaction of the spins of the valence electrons and the parallel atomic moments would cause an increase in the wave function for the valence electrons in the neighborhood of the atoms with / = 1 and the parallel orientation. This effect also produces a change in the shape of the curve of saturation magnetization as a function of temperature. The details of this refined theory will be published later. 

---