Variance per unit length of column

With effects causing asymmetrical spreading assumed negligible, the plate height h (or HETP, height equivalent to a theoretical plate) for a gaussian distribution may be defined as variance per unit length of column ... 

In Section 6B-2, we pointed out that the breadth of a Gaussian curve is described by the standard deviation a and the variance chromatographic bands are usually Gaussian and because the efficiency of a column is reflected in the breadth of chromatographic peaks, the variance per unit length of column is used by chromatographers as a measure of column efficiency. That is, the column efficiency H is defined as... 

Assuming a Gaussian profile, the extent of band broadening is measured by the variance or standard deviation of a chromatographic peak. The height of a theoretical plate is defined as the variance per unit length of the column... 

The different dispersion processes (1, 2, 3,...) that occur in a column will now be considered theoretically, their individual contributions to the variance per unit length of the column (Hi, H2, H3...) evaluated and then summed to provide an expression for the total variance per unit length of the column (H), i.e.,... 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

HD(m))5 the variance per unit length of the column contributed by longitudinal... 

Equation (15) gives the variance per unit length of a GC column in terms of the outlet pressure (atmospheric) the outlet velocity and physical and physicochemical properties of the column, packing, and phases and is independent of the inlet pressure. However, equation (13) is the recommended form for HETP measurements as the inlet pressure of a column is usually known, and is the less complex form of the HETP expression. 

Assuming the minimum variance per unit length of a packed column is (2dp), then... 

Figure 5, Graph of the Variance per Unit Length of the Column (H). against the Reciprocal of the Diffusivity...

Now, it is of interest to determine if either the resistance to mass transfer term for the mobile phase or, the resistance to mass transfer term in the stationary phase dominate in the equation for the variance per unit length of a GC packed column. Consequently, taking the ratio of the two resistance to mass transfer terms (G)... 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

Equation (9) is the basic form of the Van Deemter equation that describes the variance per unit length of a column in terms of the physical properties of the column contents and the distribution system. 

Consequently, (ax) the variance per unit length of a column is numerically equal to the height equivalent to a theoretical plate (H) and thus, equation (9) becomes... 

Hence the term "HETP equation" for the equation for the variance per unit length of a column. 

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. 

Van Deemter also introduced a constant (y)into the Longitudinal Dispersion contribution to variance to account for some packing Inhomogeneity and so the expression for the Diffusion contribution to the variance per unit length of the column became,... 

Where (H) is the variance per unit length of the column for the given solute, (k) is the capacity factor of the eluted solute,... 

The equations discussed in the previous chapter, that described the variance per unit length of a solute band after passing through an LC column, were all significantly different. It is, therefore, necessary to identify the specific equation that most accurately describes the dispersion that takes place, so that it can be employed with confidence in the design of optimized columns. The different equations were tested against an extensive set of accurately measured experimental data by Katz et at (1) and, in order to identify the most pertinent equation, their data and some of their conclusions will be considered in this chapter. 

In a packed column, the individual solute molecules will describe a tortuous path through the interstices between the particles, and some will randomly travel shorter routes than the average and some will travel longer routes. Consequently, those molecules taking the shorter paths will move ahead of the mean and those that take the longer paths lag behind the mean which will result in band dispersion. Van Deemter et al. derived the following function for the multipath variance contribution (oif) to the overall variance per unit length of the column (o ) ... 

Van Deemter derived the following expression for the variance contribution by the resistance to mass transfer in the mobile phase (cr ) to the overall variance per unit length of the column aff. 

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