Page 174 - Multidimensional Chromatography

166 Multidimensional Chromatography

this focusing phenomenon. However, the focusing effect of the spatial component of
the gradient tends to produce peaks that are narrower than they would otherwise be
in a pure temporal gradient. (If the direction of the spatial gradient is reversed from
what was described, then the peaks are defocused and broadened.)
Giddings wrote in 1963, ‘The plate height theory is particularly limited when
gradients exist in a column . . . These limitations do not, of course, make it neces-
sary to abandon the plate height theory as a measure of column efficiency, but they
do suggest that severe caution be taken in applying the plate height theory to new
theoretical areas’ (34). Giddings’ work at the time focused on GC. He explained the
effects of several non-uniformities in GC columns on apparent plate height. The non-
uniformities included pressure drop and the resulting velocity gradient on a GC col-
umn, inhomogeneous packings, and coupled columns. This work included the
development of the pressure correction factors for GC mentioned earlier, and an
explanation of the apparent plate height when dissimilar columns are coupled in
series. However, in all of this work, Giddings treated cases in which retention factors
were constant except when affected by the column.
There is a further complication in unified chromatography: even when conditions
are not programmed, the pressure drop on a column induces a spatial pressure gradi-
ent as in GC, but because the mobile phase is compressible and solvating, its strength
also changes spatially. This is a departure from the conditions that Giddings
described. Local retention factors are not constant in unified chromatography, in
general, except in the extremes (LC and GC). Local retention factors may vary in
unified chromatography even if the stationary phase is completely uniform, the
mobile-phase composition is constant, and no form of temporal programming is
applied.
Poe and Martire expanded Giddings’work and derived equations for the observed
plate height in general (35). Although these authors only claimed applicability to
GC, LC, and SFC, it appears that their equations are applicable to all of the forms of
unified chromatography mentioned earlier. Poe and Martire arrived at these equa-
tions by expressing Giddings’ equations in terms of density rather than pressure
(Boyle’s law is not followed by the non-ideal fluids often used as unified chromatog-
raphy mobile phases, although the density-volume product is always constant), and
by including the density influence on local retention factors and local plate height.
They reported the following:
ˆ 2 2 2
H 〈(H(1 k) ) z ((1 k) )〉 z (7.4)

where H is the local plate height on the column, k is the solute retention factor, is
the mobile-phase density, and the brackets, 〈〉 z, denote that the spatial average has
been taken along the length of the column. Note that k may change as a function of ,
H may change as a function of and of k, and will change with z on a column with
a significant pressure drop between the inlet and outlet if the fluid is compressible.
We should also note that equation (7.4) properly reduces to an agreement with
Giddings’ treatment in the GC limit.