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radial equilibrium, but as a result of this process the sample spreads across the column during passage through the column and eventually achieves radial equilibrium. In the very early work in liquid chromatography, relatively low inlet pressures were employed and thus, samples could be injected on the column by turning off the pump and injecting the sample with a syringe through an appropriate septum device This method of injection often resulted in radial equilibrium never being achieved by the solutes before they were eluted. The introduction of the sample valve, however, aids in establishing radial equilibrium early in the separation but unless some special spreading device is employed at the front of the column, it will not necessarily occur at the point of injection. The stream splitting process is depicted in Figure (2A).
The Mechanism of Radial Dispersion
À/ The Stream Splitting Process Â/Illustration of the Typical Radial
If a particular molecule is considered passing round a particle, it will suffer a lateral movement that, from figure (2B), can be seen to be given by,
Lateral Movement/Particle = -^-cose
dn f rnsft dn
It follows that the average lateral step will be, ~ I-de = —
l * ë ë
Employing the random walk function, the radial variance is given by:-
o2= (Number of Steps)X(Step Length)2 ........................ (2)
Now assuming one lateral step is taken by a molecule for every distance (jdp) that it moves axially, then, (n) the number of steps is given by:-
_ 1 j^p
where (1) is the distance travelled axially by the solute band.
Thus, substituting for (n) in equation (2),
d \2 aP
_ 2?p_ j*2
In practice the value of (j) will lie between 0.5 and 1.0, but, for simplicity the value of (j) will be taken as unity.This implies that one lateral step will be taken by a given molecule for every step travelled axially equivalent to one particle diameter.
Thus, 0 = ^r
or, î n = — — ............................................ (3)
Consider a sample injected precisely at the center of a 4 mm diameter LC column. Equation (3) allows the calculation of the distance traveled axially by the solute band before the radial standard deviation of the sample of solute is numerically equal to the column radius. That is, the band has now spread evenly across the column and the solute is in radial equilibrium.
For the conditions given above or = 0.2.
Thus, substituting this value in equation (3),
, .0.5 ldp
Using equation (4) the distance that a solute band must pass along a column before a sample, injected at the center of the packing, is evenly spread across its diameter, was calculated for columns packed with different sized particles.
The results are shown as a graph relating length against particle diameter in figure (3).
Graph of Column Length Traveled by Solute against Particle Diameter
Particle Diameter (micron)
It is seen that with the particle diameter range normally employed in liquid chromatography (i.e. 2-25 micron) it is likely that radial equilibrium would never be achieved for those column lengths commonly in use in LC today, if the sample was placed symmetrically at the center of the packing. However, if the column packing is completely homogeneous throughout the column length, then the column efficiency should not be impaired. Unfortunately, ideal packing conditions are not always achieved and channeling often occurs, under which circumstances lack of radial equilibrium could result in the column efficiency being reduced with consequent loss in resolution
Thus, to ensure radial equilibrium, either It must be achieved on injection by the use of a suitable sample distribution device, or by employing narrow bore columns where radial equilibrium is more quickly achieved. The latter alternative, however, will depend on the resolution required and will be discussed later under LC column design.
Dispersion Processes that take Place in an LC Column
There are four basic dispersion processes that can occur in a packed column that will account for the final band variance. They are namely, The Multipath Effect, Longitudinal Diffusion, the Resistance to Mass Transfer in the Mobile Phase and the Resistance to Mass Transfer in the Stationary Phase. All these processes are random and essentially noninteracting and, therefore, provide individual contributions of variance that can be summed to produce the final band variance. Each process will now be discussed individually.
The Multipath Process
In a packed column the solute molecules will describe a tortuous path through the interstices between the particles and obviously some will travel shorter paths than the average and some longer paths. Consequently, some molecules will move ahead of the average and some will lag behind thus causing band dispersion. The multipath effect is illustrated in figure (4)
Figure 4 Dispersion by the Multipath Effect
The Multipath effect can also be used to demonstrate the use of the Random Walk Model.