Rigor theory

One of the most commonly used constructs is a model. A model is a simple way of describing and predicting scientific results, which is known to be an incorrect or incomplete description. Models might be simple mathematical descriptions or completely nonmathematical. Models are very useful because they allow us to predict and understand phenomena without the work of performing the complex mathematical manipulations dictated by a rigorous theory. Experienced researchers continue to use models that were taught to them in high school and freshmen chemistry courses. However, they also realize that there will always be exceptions to the rules of these models. 

In general, the foUowing steps can occur in an overall Hquid—soHd extraction process solvent transfer from the bulk of the solution to the surface of the soHd penetration or diffusion of the solvent into the pores of the soHd dissolution of the solvent into the solute solute diffusion to the surface of the particle and solute transfer to the bulk of the solution. The various fundamental mechanisms and processes involved in these steps make it impracticable or impossible to describe leaching by any rigorous theory. 

Ion-selective electrodes (ISEs) with ionophore-based membranes allow for quantification of a large number of analytes in various matrixes. Tailoring of the composition of the membranes to comply with the analytical task, requires advanced theory of membrane response. Most of theoretical descriptions include nonrealistic extra-thermodynamic assumptions, in the first place it is assumed that some kind of species strongly predominate in membranes. Ideally, a rigorous theory of ISE response should be based on strict thermodynamics. However, real ISE membranes are too complex. Therefore, known attempts aimed at rigorous thermodynamic description of ISEs proved to be fraritless. 

It is one of the wonders of the history of physics that a rigorous theory of the behaviour of a chaotic assembly of molecules - a gas - preceded by several decades the experimental uncovering of the structure of regular, crystalline solids. Attempts to create a kinetic theory of gases go all the way back to the Swiss mathematician, Daniel Bernouilli, in 1738, followed by John Herapath in 1820 and John James Waterston in 1845. But it fell to the great James Clerk Maxwell in the 1860s to take... 

A more rigorous theory of kinetic isotope effects begins with the transition state equation k = (kTlh)K. Writing this for and ito leads to... 

Values of kH olki3. o tend to fall in the range 0.5 to 6. The direction of the effect, whether normal or inverse, can often be accounted for by combining a model of the transition state with vibrational frequencies, although quantitative calculation is not reliable. Because of the difficulty in applying rigorous theory to the solvent isotope effect, a phenomenological approach has been developed. We define <[), to be the ratio of D to H in site 1 of a reactant relative to the ratio of D to H in a solvent site. That is. 

A rigorous theory requires some smoothness conditions on the functions. 

Alternative methods, such as correcting the nonlinearity though the application of an appropriate physical theory as we described above, may do as well or even better than a Taylor series approximation, but a rigorous theory is not always available. Even in... 

It can also happen, and is perhaps more common, in the case of diffuse reflectance. In that measurement technique, absent a rigorous theory to describe this physical phenomenon, the concept of a variable pathlength is used as a first approximation to the nature of the change in the measurements. 

In order to estimate the region of this approximation applicability, it is necessary to examine macrokinetics of a polymeranalogous reaction with explicit allowance for the diffusion of a reagent Z into a globule. In this case, the profile of its constituent monomeric units will be fuzzy rather than stepwise (see Fig. 1). This brings up two questions. The first one is how this profile depends on kinetic and diffusion parameters of a reaction system. The second question is concerned with the effect of the profile shape on the statistical characteristics of the chemical structure of the products of a polymeranalogous reaction. A rigorous theory has been developed [22,23] which enables us to answer these questions. The main concepts of this theory are outlined in the subsequent Sections. 

The study of the scattering of light by colloidal systems has a long history. The Tyndall effect describes the scattering of light by suspended particles. In fact, the first rigorous theory was that of Rayleigh in 1871. 

Unfortunately, we are still so ignorant of the electrochemistry of the systems involved, that it is not possible to construct a detailed, rigorous theory to explain these interesting phenomena, which certainly merit further investigation. 

The solvophobic theory could be extended to the treatment of other special effects such as hydrogen bonding between eluite and species. present in the eluent. The predictive power of the theory may be improved by such extension. In addition a rigorous theory for treatment eluite interaction with surface silanols would be needed. 

The rigorous theory is well known [114], but approximations for cr and tr are used in the following discussion of results derived by means of the formalism given by Vauthier et al. [115]. 

Before being put into rigorous theory, chemical topology received much interest from many scientists and several contributions of utmost importance are worth mentioning. 

It is clear that the viscosity, thermal conductivity, and diffusion coefficients transport coefficients are defined in analogous ways. They relate the gradient in velocity, temperature, or concentration to the flux of momentum, energy, or mass, respectively. Section 12.3 will present a kinetic gas theory that allows an approximate calculation of each of these coefficients, and more rigorous theories are given later in this chapter. 

The previous section gave a simple treatment of transport properties based on the kinetic theory of gases. That approach has the advantages that it is very intuitive and mathematically tractable, and the final results are in approximate agreement with experiment and with more rigorous theory. To go beyond that treatment requires much more complex and specialized theory in statistical mechanics, molecular interactions and collisions [60,178, 269],... 

This section will give an introduction to the more rigorous theory of transport properties, but by necessity will not go through detailed derivation of all of the results. The goal here will be to sketch the physical and mathematical framework of the formulations. We then present results in enough detail to connect with the rigorous formulas used in computational evaluation of transport properties [211], which are summarized in the following section. 

The rigorous theory of species transport for mixtures containing more than two species is presented in Sections 12.7.2 and 12.7.3. A more approximate, but computationally less expensive, approach is discussed in Section 12.7.4. 

Expressions for the thickness of the diffusion layer, being approximations of rigorous theories... 

When a film is very thin, it may not be continuous, and conduction is subject to the percolation effect, whereby charge migrates by hopping or tunneling between island sites [50,51]. Such a process is activation controlled, and such thin films do not obey Ohm s law. The activation energy can be decreased by the presence of an applied electric field, making development of a rigorous theory difficult. The resistivity can be expressed by the relationship [5]... 

Thus the question of exclusion of the segment DFB in the rigorous theory of steady propagation takes on an outwardly completely different aspect. In fact, however, even in this more rigorous theory construction of a complete regime (with subsequent braking and cooling) in which relation (8) is... 

In the following section we shall begin to construct a rigorous theory of steady propagation of detonation, accounting for losses, which are necessary in all cases for determination of the boundary conditions. Before this we shall give an elementary proof of thq fact that the mechanism of ignition by a shock wave does indeed exclude tljie possibility of realization of the lower branch BGI. 

A more rigorous theory [40, 41] accounting for an internal field correction yields the following ratio of two field complex amplitudes ... 

The melting point of a compound is the temperature at which the solid and liquid phases are in equilibrium at one atmosphere pressure is specified because the melting process involves a change in volume and is therefore pressure dependent. Since the melting point can be determined easily experimentally, it is the most commonly reported physical property for organic compounds. However, in the absence of a rigorous theory of fusion, it is one of the most difficult to predict. 

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