Download (direct link):
Thus, °E =0 ,0c =0s+0v+0t2+aD+0R
Unfortunately, the magnitude of the variance contribution from each source will be different and the ultimate minimum size of each is often dictated by the limitations in the physical construction of of the different parts of the apparatus and consequently not controllable. It follows that equipartition of the permitted extra column dispersion is not possible. It will be seen later that the the maximum sample volume provides the maximum chromatographic mass and concentration sensitivity. Consequently, all other sources of dispersion must be kept to the absolute minimum to allow as large a sample volume as possible to be placed on the column without exceeding the permitted limit. At the same time it must be stressed, that all the permitted extra column dispersion can not be allotted solely to the sample volume.
The Alternative Axis of a Chromatogram
An elution curve of a chromatogram can be expressed using parameters other than the volume How of mobile phase as the independent variable. Instead of using milliliters of mobile phase, solute concentration in the mobile phase can be plotted against, time, or distance travelled by the solute band along the column and proportionally the same chromatogram will be obtained. This is illustrated in figure (1)
An Elution Curve Plotted on Different Axes
-- Time at a Fixed Distance Along
As the curves are describing the same chromatogram, by proportion, the ratio of the variance to the square of the retention, in the respective units in which the independent variables are defined , will all be equal.
2 2 2 0.
Consequently, —? = = -7
V2 l2 t2 r r
where ov, ox and ot are the standard deviations of the elution curves
when related to the volume flow of mobile phase,the distance travelled by the solute along the column and time, respectively, and Vp, 1 and tp refer to the retention volume,column length and retention time, respectively
Now, from the Plate Theory it has been shown that,
ov = Vn(vm+Kvs) and Vr =n(vm+Kvs)
n(vm+Kvs) _ °2
n2(vm+Kvs)2 |2 ,2
_ _x I2
The ratio, (-), the column length divided by the number of theoretical
plates in the column, has for obvious reasons become termed the Height Equivalent to the Theoretical Plate (HETP) and has been given the symbol
(H). However, it is seen that (H) is numerically equal to, which is, in
fact, the variance per unit length of the column. Thus, the function, is
the variance that the Rate Theory will provide an explicit equation to define and can be experimentally calculated for any column from its length and
column efficiency. It follows that the equations that give a value for, (H), the variance per unit length of the column, have been termed HETP equations.
To develop an HETP equation it is necessary to first identify the dispersion processes that occur in a column and then determine the variance that will result from each process per unit length of column. The sum of all these variances will be (H), the Height of the Theoretical Plate or the total variance per unit column length. There are a number of methods used to arrive at an expression for the variance resulting from each dispersion process and these can be obtained from the various references provided. However, as an example, the Random-Walk Model introduced by Giddings (5) will be employed here to illustrate the procedure.The theory of the Random-Walk processes itself can be found in any appropriate textbook on probability (6) and will not be given here but the consequential equation will be used
The Random Walk Model
The random-walk model consists of a series of steplike movements for each molecule which may be positive or negative the direction being completely random. After (p) steps, each step having a length (s) the average of the molecules will have moved some distance from the starting position and will form a Gaussian type distribution curve with a variance of <j2 .
Now according to the random-walk model,
î = s/p ............................................................... (1)
Equation (1) can be used in a general way to determine the variance resulting from the different dispersion processes that occur in an LC column. The application of equation (1) is simple, the problem that often arises is the identification of the average step and sometimes the total number of steps associated with the particular process being considered. As an illustration of its use, equation (1) will be first applied to the problem of radial dispersion that occurs when a sample is placed on an LC column in the manner of Horne eta! (7).
When a stream of mobile phase carrying a solute impinges against a particle, the stream divides and flows around the particle. Part of the divided stream then joins other split streams from neighboring particles, impinges on another and divides again. When a sample is placed on the column at the center of the packing, initially it is in a condition of non-