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mass transfer in the stationary phase for these two particular solvent/stationary phase/solute systems. Overall, however, all the results in figures (5),(6) and (7) support the Van Deemter equation extremely well.
Katz et al (1) also examined the effect of particle diameter on the value of the overall resistance to mass transfer constant (C) They employed columns packed with 3.2ö, 4.4ö., 7 8ö, and 17.5ö, and obtained HETP curves for the solute benzyl acetate in 4.3%w/w of ethyl acetate in n-heptane oh each column. The data was curve fitted to the Van Deemter equation and the values for the A, Â and Ñ terms for all four columns calculated. According to the Van Deemter equation the (C) term should be linearly related to the square of the particle diameter.
A graph relating the value of the (C) term with the square of the particle diameter is shown in figure (8).
Figure 8 Graph of (C) Term against Particle Diameter 5quared
Particle Diameter Squared (X O.OOOOOOI)
Figure 8 shows the predicted linear relationship between the resistance to mass transfer term and the square of the particle diameter. The linear correlation is extremely good and it is seen that there is, indeed an intercept on the (C) term axis, at zero particle diameter, which confirms the existence of a small, but significant, contribution from the resistance to mass transfer in the stationary phase.
The Effect of the Function of (k) on Peak Dispersion
Reiterating the Van Deemter equation,
H = 2X.dp +
u Dm D$
where the symbols used have the meaning previously ascribed to them
If it is assumed that the column is operated at relatively high velocities, such that the contribution from longitudinal diffusion is no longer significant, then
H - 2Xd,: f - r-—p è + —LU ...................... (7)
P Dm DS
Assuming that the diffusivity of the solute in the stationary phase (Ds) is simply related to the diffusivity in the mobile phase (Dm), i e. Ds =
Substituting for (Ds) in equation (7)
(fi(k')d2 + f2(k')d2/(;
Inserting the expression for fi(k') recommenaed by Purnell (9) and the expression for f 1 Ck‘) as derived by Van Deemter (2) and rearranging,
H-2X.dp = — um
1+6k' + l Ik'2 ? 8 k' d„ +
24(1 + ê)2 P ë2(| + ê'2) ^
Dividing throughout by
i + 6k' +11k
24(1 + ê')'
oa fi iiA
- • V "I
+ 6k' + l ik'
, ana rearranging,
n2 (l + áê' + I lk,2j
It follows that, if suitable experimental data was available, a graph
24(1 + k')
l + 6k' + i Ik'
(l + 6k'-
, should provide
a straight line. Katz and Scott (10), in their work involving the development of a method for the measurement of solute molecular weight from chromatographic data, generated sufficient data to test the relationship given in equation (10). Furthermore,the equation could be tested against the two alternative values for the capacity factor (k ) calculated by employing the fully permeating dead volume, or (k'e) derived by employing the excluded
,Dm 24(1 + k) dead volume. The graph relating, (H-2Xdp) 111
should provide a straight line.
(l + 6k' + i ik'2)
Figure 9 Graph of (H-2Xdp)—
+ 6k' + l Ik'2
(l + 6k' + l lk'2)
(k‘ values calculated from the retention time of the fully permeating unretained solute)
It is seen that a linear curve is not obtained and the use of (Þ values derived from the fully permeating dead volume can not be used in the kinetic studies of LC columns. In contrast, the linear curve shown in figure (10), is the same basic graph but, in this case, the ordinate values were calculated using (k'e) values based on the excluded dead volume.
Wk;)" 1 + 6 ¦ 11 ê;»
The straight line confirms that the excluded dead volume must not only be used for measuring mobile phase velocities but in kinetic studies of LC columns and LC column design it must also be employed for the measurement of capacity factors.
In summary, it can be said that all the dispersion equations give a good fit to experimental data but only the Van Deemter equation, the ãï^ñÈïë÷ equation and the Knox equation give positive and real values for the constants in the respective equations. The basically correct equation appears to be that of Giddings but, over the range of mobile phase velocities normally employed in LC, the Van Deemter equation is the simplest and most
appropriate to use. The Van Deemter equation appears to be a special case of the Giddings equation, which simplifies to the van Deemter equation when the mobile phase velocity is close to, or around, the optimum mobile phase velocity. The form of the Van Deemter equation and, in particular, the individual functions contained in it, are well substantiated by experiment. The Knox equation is obtained from an empirical fit to experimental data and the individual functions contained in the equation are not all substantiated by experiment.