The Rate Theory

There are two fundamental chromatography theories that deal with solute retention and solute dispersion and these are the Plate Theory and the Rate Theory, respectively. It is essential to be familiar with both these theories in order to understand the chromatographic process, the function of the column, and column design. The first effective theory to be developed was the plate theory, which revealed those factors that controlled chromatographic retention and allowed the... 

In a chromatographic separation, the individual components of a mixture are moved apart in the column due to their different affinities for the stationary phase and, as their dispersion is contained by appropriate system design, the individual solutes can be eluted discretely and resolution is achieved. Chromatography theory has been developed over the last half century, but the two critical theories, the Plate Theory and the Rate Theory, were both well established by 1960. There have been many contributors to chromatography theory over the intervening years but, with the... 

Dispersion in Columns and Mobile Phase Conduits, the Dynamics of Chromatography, the Rate Theory and Experimental Support of the... 

The theory that results from the investigation of the dynamics of solute distribution between the two phases of a chromatographic system and which allows the different dispersion processes to be qualitatively and quantitatively specified has been designated the Rate Theory. However, historically, the Rate Theory was never developed as such, but evolved over more than a decade from the work of a number of physical chemists and chemical engineers, such as those mentioned in chapter 1. 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

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... 

Analytical information taken from a chromatogram has almost exclusively involved either retention data (retention times, capacity factors, etc.) for peak identification or peak heights and peak areas for quantitative assessment. The width of the peak has been rarely used for analytical purposes, except occasionally to obtain approximate values for peak areas. Nevertheless, as seen from the Rate Theory, the peak width is inversely proportional to the solute diffusivity which, in turn, is a function of the solute molecular weight. It follows that for high molecular weight materials, particularly those that cannot be volatalized in the ionization source of a mass spectrometer, peak width measurement offers an approximate source of molecular weight data for very intractable solutes. 

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 computing ordinary short-term characteristics of plastics, the standard stress analysis formulas may be used. For predicting creep and stress-rupture behavior, the method will vary according to circumstances. In viscoelastic materials, relaxation data can be used in Eqs. 2-16 to 2-20 to predict creep deformations. In other cases the rate theory may be used. 

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]... 

At this point, it is important to stress the difference between separation and resolution. Although a pair of solutes may be separated they will only be resolved if the peaks are kept sufficiently narrow so that, having been moved apart (that is, separated), they are eluted discretely. Practically, this means that firstly there must be sufficient stationary phase in the column to move the peaks apart, and secondly, the column must be constructed so that the individual bands do not spread (disperse) to a greater extent than the phase system has separated them. It follows that the factors that determine peak dispersion must be identified and this requires an introduction to the Rate Theory. The Rate Theory will not be considered in detail as this subject has been treated extensively elsewhere (1), but the basic processes of band dispersion will be examined in order to understand... 

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. 

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). 

As the Plate Theory has two serious limitation, viz., first it does not speak of the separating power of a definite length of column, and second it does not suggest means of improving the performance of the column the Rate Theory has been introduced which endeavours to include the vital fact that- the mobile-phase flows continuously, besides the solute molecules are constantly being transported and partitioned in a gas chromatographic column . It is usually expressed by the following expression ... 

There have been relatively few applications of the rate theory to GPC, presumably because of the apparent complexity of this approach. One of the most widely quoted interpretations of the rate theory to GPC is that of Ouano and Baker (4). These authors have attempted to take advantage of the undoubted potential of the rate theory approach in constructing a model. They identified the key parameters in their model as the flow rate of the eluant, gel particle size, diffusion coefficient in the stationary and mobile phases and the partition coefficient for solute between phases. Although there is little doubt that the important parameters have been correctly identified, it is not immediately apparent how they are inter-related and hence how their coupled effect can be interpreted. A critical account of the various attempts which have been made to model the GPC process will be given elsewhere. 

The purpose of the Rate Theory is to aid in the understanding of the processes that cause dispersion in a chromatographic column and to identify those factors that control it, Such an understanding will allow the best column to be designed to effect a given separation in the most efficient way. However, before discussing the Rate Theory some basic concepts must be introduced and illustrated. 

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... 

The rate theory of Paton, as modified by Paton and Rang, rejects the assumption that the response is proportional to the number of occupied receptors, and instead proposes a relationship of response to the rate of drug-receptor complex formation. According... 

The fluctuation theory has received attention because it avoids some of the serious assumptions involved in the rate theory. The beginnings of fluctuation theory were presented by Einstein. Various workers since... 

Although HETP is a useful concept, it is an empirical factor. Since plate theory does not explain the mechanism that determines these factors, we must use a more sophisticated approach, the rate theory, to explain chromatographic behavior. Rate theory is based on such parameters as rate of mass transfer between stationary and mobile phases, diffusion rate of solute along the column, carrier gas flowrate, and the hydrodynamics of the mobile phase. 

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 ... 

The efficiency of a column is a number that describes peak broadening as a function of retention, and it is described in terms of the number of theoretical plates, N. Two major theories have been developed to describe column efficiency, both of which are used in modern chromatography. The plate theory, proposed by Martin and Synge,31 provides a simple and convenient way to measure column performance and efficiency, whereas the rate theory developed by van Deemter et al.32 provides a means to measure the contributions to band broadening and thereby optimize the efficiency. 

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