Plate, theory

When equilibrium is reached in the plate, the polymer concentration is Cs in the stationary phase and in the mobile phase. Their ratio is called the partition coefficient 

When the concentration is sufficiently low (cm c, overlap concentration), K does not depend on Cm but depends on the ratio of the chain dimension to the pore size. 

The partition ratio k is defined as the ratio in the number of molecules between the two phases and given as 

These different theories will be discussed briefly in the sections that follows  

Martin and Synge first proposed the plate theory in 1941, whereby they merely compared the GC separation to fractional distillation. Thus, the theoretical plate is the portion of the column wherein the solute is in complete equilibrium with the mobile and the stationary phase. 

of solute in Stationary Phase D Cone, of solute in Mobile Phase where, KD = Distribution coefficient. 

the distribution of a solute after V equilibrium (plates) may be defined by the expansion of the binomial in Eq. (a) below  

From the equalities shown in Equation 2.50 it follows that 

The concept of plate theory was originally proposed for the performance of distillation columns (12). However, Martin and Synge (13) first applied the plate theory to partition chromatography. The theory assumes that the column is divided into a number of zones called theoretical plates. One determines the zone thickness or height equivalent to a theoretical plate (HETP) by assuming that there is perfect equilibrium between the gas and liquid phases within each plate. The resulting behavior of the plate column is calculated on the assumption that the distribution coefficient remains unaffected by the presence of other 

Martin and Synge derived an expression for the total quantity, qn, of solute in plate n and the volume of mobile phase (carrier gas) that passes through the column. 

Equation 2.54 has the form of the normal error curve and from the geometrical properties of the curve we can show that 

As long as the sample occupies less than 0.5(n) theoretical plates, there will be no band broadening because of sample size As the total number of theoretical plates increases, within a column, the maximum space (in terms of theoretical plates) that should be occupied by the sample will also increase. However, the percentage of column length available for sample will decrease (because number of theoretical plates per column length increases), as shown in Table 2.3 

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We recall that in the Kirchhoff-Love plate theory the horizontal displacements depend linearly on the coordinate i.e. 

Fig. 13. Schematic of a laminate with the coordinates and ply notation used ia laminated plate theory.

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

Molecular Interactions, the Thermodynamics of Distribution, the Plate Theory and Extensions of the Plate... 

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... 

The concentration profiles of the solute in both the mobile and stationary phases are depicted as Gaussian in form. In due course, this assumption will be shown to be the ideal elution curve as predicted by the Plate Theory. Equilibrium occurs between the mobile phase and the stationary phase, when the probability of a solute molecule striking the boundary and entering the stationary phase is the same as the probability of a solute molecule randomly acquiring sufficient kinetic energy to leave the stationary phase and enter the mobile phase. The distribution system is continuously thermodynamically driven toward equilibrium. However, the moving phase will continuously displace the concentration profile of the solute in the mobile phase forward, relative to that in the stationary phase. This displacement, in a grossly... 

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... 

Figure 1. The Elution Curve of a Single Peak The Plate Theory...

Recalling the plate theory, it must be emphasized that (Vm) is not the same as (Vm)-(Vm) is the moving phase and a significant amount of (Vm) will be static (e.g., that contained in the pores). It should also be pointed out that the same applies to the volume of stationary phase, (Vs), which is not the same as (Vs), which may include material that is unavailable to the solute due to exclusion. 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

Equation (1) is the well-known Gaussian form of the elution curve equation and can be used as an alternative to the Poisson form in all applications of the Plate Theory. 

There is an interesting consequence to the above discussion on composite peak envelopes. If the actual retention times of a pair of solutes are accurately known, then the measured retention time of the composite peak will be related to the relative quantities of each solute present. Consequently, an assay of the two components could be obtained from accurate retention measurements only. This method of analysis was shown to be feasible and practical by Scott and Reese [1]. Consider two solutes that are eluted so close together that a single composite peak is produced. From the Plate Theory, using the Gaussian form of the elution curve, the concentration profile of such a peak can be described by the following equation ... 

Equation (10) also allows the peak width (2o) and the variance (o ) to be measured as a simple function of the retention volume of the solute but, unfortunately, does not help to identify those factors that cause the solute band to spread, nor how to control it. This problem has already been discussed and is the basic limitation of the plate theory. In fact, it was this limitation that originally invoked the development of the... 

Equation (18) displays the relationship between the column efficiency defined in theoretical plates and the column efficiency given in effective plates. It is clear that the number of effective plates in a column is not aii arbitrary measure of the column performance, but is directly related to the column efficiency as derived from the plate theory. Equation (18) clearly demonstrates that, as the capacity ratio (k ) becomes large, (n) and (Ne) will converge to the same value. 

Thus, from the plate theory, (Rr) the concept of resolution as introduced by Giddings, will be given by... 

In most mathematical processes, including the derivation of the plate theory, the assumption is made that the initial charge is placed on the first plate of the column. This is difficult to achieve in practice, as the charge must occupy a finite portion of... 

Vacancy chromatography has some quite unique properties and a number of potentially useful applications. Vacancy chromatography can be theoretically investigated using the equations derived from the plate theory for the elution of... 

Now, from the plate theory, this transient concentration change will be eluted through the column as a concentration difference and will be sensed as a negative or positive peak by the detector. The equation describing the resulting concentration profile of the eluted peak, from the plate theory, will be given by... 

Equation (25) can be extended to provide a general equation for a column equilibrated with (q) solutes at concentrations Xi, X2, X3,...Xq. For any particular solute (S), if its normal retention volume is Vr(S) on a column containing (n) plates, then from the plate theory, the plate volume of the column for the solute (S), i.e., (vs) is given by... 

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