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The column lengths necessary to achieve these efficiencies can be calculated employing equation (17). the results obtained shown as curves relating maximum column length to p[article diameter is shown in figure 3.
Graph of Column Length for Maximum Efficiency /Particle Diameter
Particle Diameter (micron)
It is seen from figure 3 that the column length ranges from just under a centimeter, which provides an efficiency of about 3500 theoretical plates to nearly 70 meters which provides an efficiency nearly 1.5 million
theoretical plates. Seventy meters is a long column, but not an impractical length. Th question remains is how long will the separation take?
The time taken to achieve these efficiencies when eluting the last peak at a k' value of 10 can be calculated employing equation (18). The results obtained are shown in figure 4 where the elution time obtained from columns of maximum efficiency are plotted against particle diameter
Graph of Log. Elution Time against Particle Diameter
Particle Diameter (micron)
On examination of the curve in figure 4 the problem associated with high efficiencies becomes apparent. The elution time for the short column having about 3500 theoretical plates is only just about one half minute. The elution time from the 1.5 million plate column, however, is about 54 days, a rather long time to wait for a chromatographic separation. It is also seen that the higher the efficiencies that are required (the more difficult the separation problem) the longer the separation time and this is inevitable as a result of practical limits to the column inlet pressure.
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2/E.Katz,K.L.0gan and R.P.W.Scott, o/r/r<m?to?T,270( 1983)51
3/J.H.Purnell and C.P.Quinn.in "Gas Chromatography I960" (ed. R.P.W.Scott), Butterworths London (1960) 184
4/ M. J. E. Golay," Gas Chromatography. 1958, (ed.D H.Desty) Butterworths, London, (1958)36
5/J.C.Giddings,"Dynamics of Chromatography",Marcel Dekker,New York, (1965)56
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Alternative Equations for Peak Dispersion
The Van Deemter equation remained the established equation for describing the peak dispersion that took piace in a packed column until about 1961. However, when experimental data that was measured at high linear mobile phase velocities was fitted to the Van Deemter equation it was found that there was often very poor agreement. In retrospect, this poor agreement between theory and experiment was probably due more to the presence of experimental artifacts, such as those caused by extra column dispersion, large detector sensor and detector electronic time constants etc. than the inadequacies of th Van Deemter equation Nevertheless, it was this poor agreement between theory and experiment, that provoked a number of workers in the field to develop alternative HETP equations in the hope that a more exact relationship between HETP and linear mobile phase velocity could be obtained that would be compatible with experimental data.
The Giddings Equation
In 1961, Giddings (1) developed an HETP equation of which the Van Deemter equation appeared to be a special case. Giddings was dissatisfied with the Van Deemter equation insomuch that it predicted a finite contribution to dispersion independent of the soiute diffusivity in the limit of zero mobile phase velocity. This concept, not surprisingly, appeared to him unreasonable and unacceptable. Giddings developed the following equation to avoid this irregularity.
H - —— + - + Cu .............................................. (1)
It is seen that when u >> E, equation (1) reduces to the Van Deemter equation,
H = A + - + Cu
It is also seen that at very low velocities, where è « E, the first term tends to zero, thus meeting the requirements that there is no multipath dispersion at zero mobile phase velocity Giddings also suggested that there was a coupling term that accounted for an increase in the 'effective diffusion' of the solute between the particles The increased 'diffusion' resulted from the tortuous path followed by the molecules as they twisted and turned through the interstices of the packing This process was considered to produced a form of microscopic turbulence that induced extremely rapid transfer of solute in the interparticulate spaces However, again at velocities where u » E , this mixing effect could be considered complete and the resistance to mass transfer in the mobile phase between the particles becomes very small and the equation again reduces to the Van Deemter equation. However, on consideration there is a difference: the Ñ term in the Van Deemter equation would now only describe the resistance to mass transfer in the mobile phase contained in the pores of the particles, and thus, would constitute an additional resistance to mass transfer in the stationary (staticmobile)phase. This concept has some indirect experimental support in the development of the form of f i(k') from experimental data given in the next chapter The form of f](k') is shown to be closer to the original form given by Van Deemter for f2(k') that is appropriate for the resistance to mass transfer in the stationary phase. It is not known for certain, but it is possible and likely, that this was the reason why Van Deemter et a! did not include a resistance to mass transfer term for the mobile phase in their original form of the equation.