Theoretical treatment Basic equations

Solute equilibrium between the mobile and stationary phases is never achieved in the chromatographic column except possibly (as Giddings points out) at the maximum of a peak (1). As stated before, to circumvent this non equilibrium condition and allow a simple mathematical treatment of the chromatographic process, Martin and Synge (2) borrowed the plate concept from distillation theory and considered the column consisted of a series of theoretical plates in which equilibrium could be assumed to occur. In fact each plate represented a dwell time for the solute to achieve equilibrium at that point in the column and the process of distribution could be considered as incremental. It has been shown that employing this concept an equation for the elution curve can be easily obtained and, from that basic equation, others can be developed that describe the various properties of a chromatogram. Such equations will permit the calculation of efficiency, the calculation of the number of theoretical plates required to achieve a specific separation and among many applications, elucidate the function of the heat of absorption detector. 

From this, the basic equation of the stationary-phase retention process can be derived, a number of assumptions and complex theoretical treatments being required. Taking as example the planetary centrifuge of Type J, the average cross-sectional area of a stationary-phase layer has been estimated for hydrophobic liquid systems, which are characterized by high values of interfacial tension y, low values of viscosity r], and low hydrodynamic equilibrium settling times ... 

In order to demonstrate the physical significance of asymjjtotic nonadiabatic transitions and especially the aiialj-tical theory developed an application is made to the resonant collisional excitation transfer between atoms. This presents a basic physical problem in the optical line broadening [25]. The theoretical considerations were mad( b( for< [25, 27, 28, 29, 25. 30] and their basic id( a has bec n verified experimentally [31]. These theoretical treatments assumed the impact parameter method and dealt with the time-dependent coupled differenticil equations imder the common nuclear trajectory approximation. At that time the authors could not find any analytical solutions and solved the coupled differential equations numerically. The results of calculations for the various cross sections agree well with each other and also with experiments, confirming the physical significance of the asymptotic type of transitions by the dipole-dipole interaction. 

Theoretical treatment of rodlike polymers is much easier than for flexible polymers since rodlike polymers can have only two kinds of motion, i.e., translation and rotation. Once the basic equation is set up, mathematical analysis is easy. However, important physics is included in an essential way in the problems of rodlike polymers. In particular, the importance of the orientational degrees of freedom and the peculiar nature of the topological constraints will be seen clearly in this syston. 

Despite the fact that thermodynamic E° values may be calculated under some conditions, an understanding of the basics of the technique of voltammetry actually requires a kinetic rather than thermodynamic theoretical treatment. In a voltammetric experiment, current flows in response to the reaction in Equation (8), being driven in either the forward or... 

Rectified diffusion was first recognized by Harvey et al The basic idea behind the theoretical treatment consists in coupling Tick s law of mass transfer and the equation of motion. After the preliminary attempt by Blake, who only considered the area effect, the shell effect was included, nd then different improvements were effected. Crum gave a comprehensive paper on the topic, and experimental studies were reported by Strasberg, and Gould. A summarized theoretical description was reported by Atchley and Crum. ... 

The stability of flow in open channels has been investigated theoretically from a more macroscopic or hydraulic point of view by several workers (Cl7, D9, DIO, Dll, 14, J4, K16, V2). Most of these stability criteria are expressed in the form of a numerical value for the critical Froude number. Unfortunately, most of these treatments refer to flow in channels of very small slope, and, under these circumstances, surface instability usually commences in the turbulent regime. Hence, the results, which are based mainly on the Ch<5zy or Manning coefficient for turbulent flow, are not directly applicable in the case of thin film flow on steep surfaces, where the instability of laminar flow is usually in question. The values of the critical Froude numbers vary from 0.58 to 2.2, depending on the resistance coefficient used. Dressier and Pohle (Dll) have used a general resistance coefficient, and Benjamin (B5) showed that the results of such analyses are not basically incompatible with those of the more exact investigations based on the differential rather than the integral ( hydraulic ) equations of motion. The hydraulic treatment of the stability of laminar flow by Ishihara et al. (12) has been mentioned already. 

The classical Marcus equation, derived from the basic model by the simplest treatment, in which all motions are treated by classical mechanics, is Equation (9.46), in which AG is the theoretical activation free energy and AG is the free energy of reaction ... 

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