The Rate Theory Equations

Before progressing to the Rate Theory Equation, an interesting and practical example of the use of the summation of variances is the determination of the maximum sample volume that can be placed on a column. This is important because excessive sample volume broadens the peak and reduces the resolution. It is therefore important to be able to choose a sample volume that is as large as possible to provide maximum sensitivity but, at the same time insufficient, to affect the overall resolution. 

An examination of the rate theory equations for film and particle diffusion kinetics reveals an overall dependence of the rate of reaction on 0 and respectively for gel ( homogeneous ) exchangers. For a macroporous exchanger the rate of exchange, under particle diffusion control, may be either independent of particle size or vary as depending upon whether diffusion within the gel microsphere structure or macropores respectively is rate controlling. 

In summary, the rate theory provides the following equations for the variance per unit length (H) for four different columns. 

Equations that quantitatively describe peak dispersion are derived from the rate theory. The equations relate the variance per unit length of the solute concentration... 

Rate theory An alternate method available involves the manipulation of the rate theory based on the Arrhenius equation. This procedure requires considerable test data but the indications are that considerably more latitude is obtained and more materials obey the rate theory. The method can also be used to predict stress-rupture of plastics as well as the creep characteristics of a material, which is a strong plus for the method. 

In flow-induced degradation, K is strongly dependent on the chain length and on the fluid strain-rate (e). According to the rate theory of molecular fracture (Eqs. 70 and 73), the scission rate constant K can be described by the following equation [155]... 

Chapter 4, from the rate theory that the variance per unit length of a column, or the HETP, was described by the following equation (1). 

In this equation it is the reaction rate constant, k, which is independent of concentration, that is affected by the temperature the concentration-dependent terms, J[c), usually remain unchanged at different temperatures. The relationship between the rate constant of a reaction and the absolute temperature can be described essentially by three equations. These are the Arrhenius equation, the collision theory equation, and the absolute reaction rate theory equation. This presentation will concern itself only with the first. 

In this section, you learned how to relate the rate of a chemical reaction to the concentrations of the reactants using the rate law. You classified reactions based on their reaction order. You determined the rate law equation from empirical data. Then you learned about the half-life of a first-order reaction. As you worked through sections 6.1 and 6.2, you may have wondered why factors such as concentration and temperature affect the rates of chemical reactions. In the following section, you will learn about some theories that have been developed to explain the effects of these factors. 

Individual variances. This is how the Rate Theory provides an equation for the final variance of the peak leaving the column. As an. example the principle of the summation of variances will be applied to extra column dispersion... 

In the rate theory of gas-solid chromatography, the equation for h has essentially the same terms except that Cj, replaces C . Ck is a term characteristic of adsorption kinetics. Equation... 

The rate theory of Grote and Hynes [149] included the non-Markovian (memory) effects by considering the following generalized Langevin equation (GLE) for the dynamics along the reaction coordinate ... 

We begin with the transition state theory result for the rate constant, Equation A 1.40. We shall need to consider bimolecular processes, A + B—+ C, for which the proper adaptation of Equation A1.40 is Equation A2.1. The quantity A El is... 

C. Resistance to Mass Transfer In the plate theory, it was assumed that the transfer of solute molecules between the mobile phase and the stationary phase was instantaneous. In the rate theory, it is accepted that there is a finite rate of mass transfer. In addition, molecules of the same species may spend different lengths of time in the stationary and mobile phases (Fig. 1.15). Resistance to mass transfer is represented by the C term of the van Deemter equation. 

Many modifications and refinements were proposed to improve the utility of this equation for different situations. Golay3 showed that the A term was not needed for open tubular columns in GC, and Giddings4 proposed a complicated coupled term in his extensive treatment of the theory. Huber5 and others noted that a fourth term was necessary for LC to account for mass transfer in the liquid mobile phase. The exact terms in these variations of the rate theory are given in Table 1 mainly for their historical value. 

Comparing these statistical rate theory equations, with Eqs. (6.283) and (6.284), we obtain the following phenomenological coefficients... 

Golayf ° and Giddings, respectively, described a modification of the rate theory for capillary columns (hollow tube with inner wall coated with liquid phase) and the random walk, non-equilibrium theory. The former derived an equation to describe the efficiency of an open tubular column, while the random walk theory describes chromatographic separation in terms of statistical moments. The non-equilibrium theory involves a rigorous mathematical treatment to account for incomplete equilibrium between the two phases. ... 

As evident from the preceding considerations the fit of experimental rate data to a particular rate theory equation does not necessarily prove the rate controlling mechanism. In the absence of previous knowledge of a system it is desirable to seek further mechanism qualifying data. 

The open-tubular column or capillary column is the one most commonly used in gas chromatography (GC) today. The equation that describes dispersion in open tubes was developed by Golay [1], who employed a modified form of the rate theory, and is similar in form to that for packed columns. However, as there is no packing, there can be no multipath term and, thus, the equation only describes two types of dispersion. One function describes the longitudinal diffusion effect and two others describe the combined resistance to mass-transfer terms for the mobile and stationary phases. 

The rate theory examines the kinetics of exchange that takes place in a chromatographic system and identifies the factors that control band dispersion. The first explicit height equivalent to a theoretical plate (HETP) equation was developed by Van Deemter et al. in 1956 [1] for a packed gas chromatography (GC) column. Van Deemter et al. considered that four spreading processes were responsible for peak dispersion, namely multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase, and resistance to mass transfer in the stationary phase. 

The rate theory takes into account the finite rate at which the solute can equilibrate between the mobile and stationary phases. The shapes of the bands that are predicted by the rate theory depend on the rate of elution, the diffusion of the solute along the length of the column, and the availability of different paths for the solute molecules to follow. The value of the HETP now depends on v, the volume flow rate of the mobile phase, according to the van Deemter equation, Eq. 14.7 ... 

A mathematical model is set up using the rate theory approach and mass balance equation for a solute over a differential volume in each of the two phases. The differential equations forming the model are to be solved using the Laplace transformation [53, 54]. 

To apply this equation to different situations and not exclusively to gas chromatography, it needs to be modified. The rate theory was developed and showed that term A was negligible and that term C corresponds to the sum of mass transfer in the stationary phase and in the mobile phase. 

In the early 1990s, in their studies of proton transfer in solution using Marcus rate theory Equation (5), Hynes and coworkers16 17 noticed the following limitation. If Q is the tunneling distance, it can be shown that the tunneling matrix element that appears in Equation (5) has the form A e- e. For typical electron transfer reactions... 

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