Mathematical models dilute solution theory

The complicated nature of the LLPTC reaction system is attributed to two mass transfer steps and two reaction steps in the organic and aqueous phases. The equilibrium partition of the catalysts between the two phases also affects the reaction rate. On the basis of the above factors and the steady-state two-film theory [60,63,64,68], a phase-plane model to describe the dynamics of a liquid-liquid PTC reaction has been derived. This model offers physically meaningful parameters that demonstrate the complicated reactive character of a liquid-liquid PT-catalyzed reaction. However, when the concentration of aqueous solution is dilute or the reactivity of aqueous reactant is weak, the onium cation has to exist in the aqueous phase. The mathematical model cannot describe this completely. When the onium cation exists in the aqueous phase, several important phenomena involved in the liquid-liquid reaction need to be analyzed and discussed. 

Thus, both transient and photostationary trapping experiments may be interpreted Quantitatively if an expression for the donor excitation function G (t) is available. In Section 2.2 we outline the development of an expression for G (t) for an aryl vinyl polymer in dilute solution. Here energy migration from a single monomer donor state to a single excimer trap state is analyzed by a one-dimensional model. We will present the analysis in more detail than the subsequent discussion of various many body theories because the treatment is straightforward and concise. This will allow the fundamental approach to be understood more clearly without the extensive mathematics required for the many-body treatment. 

What follows will concern electrolyte solutions as well as molten salts. In fact, as we will see later, within the framework of the McMillan-Mayer theory(l), there is no difference in the mathematical treatment of a dilute aqueous solution of a given electrolyte and the corresponding molten salt. Of course, the density, temperature and potential energy will be different, but in both cases, the model to be used will be the same. It should then not be surprising that the next section starts with a discussion of the McMillan-Mayer and Debye-Hiickel theories(2) for dilute systems of charged particles. The Debye-Hiickel theory (DH) has been the most successful theory of electrolyte solutions and some of the modern approximations are simple extensions of DH theory, which are statistically consistent. 

For some reactions use can also be made of statistical-mechanical equations either (rarely) alone or in combination with some of the quantities alluded to above. These are reactions taking place in systems for which we have a model which is at once realistic enough and mathematically tractable enough to be useful. An example is the calculation of the standard equilibrium constant of a gas reaction (and thence of the yield, but only if the gas mixture is nearly enough perfect) from spectroscopically determined molecular properties. Another example is the use of the Debye-Hiickel theory or its extensions to improve the calculation of the yield of a reaction in a dilute electrolyte solution from the standard equilibrium constant of the reaction when, as is usually so, it is not accurate enough to assume that the solution is ideal-dilute. 

Given this failure of the continuum model, it is evidently necessary to treat the solvent as an assembly of molecules. A hard-sphere model is the first approximation. Kinetic theory of diffusion in dilute gases, where the mean free path is much greater than the collision diameter, is well established it can be extended with some success to dense gases, where the two quantities are more nearly equal, and (more speculatively) to hard-sphere models of liquids, where they are comparable. For these highly mathematical theories the reader may consult more specialised works [14]. Analytical solutions are not always to be expected numerical solutions may be required. Computer-simulation calculations have had considerable success, and with the advent of fast computers have become a major source of understanding of real systems (cf., e.g.. Section 7.3.4.5). 

---