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k’ – Retention or Capacity Factor

Similar to the α-value term in the fundamental resolution equation, the term with k' also approaches 1 at higher values of k', as seen by the data provided in Table 2. Therefore we can say that optimal k' values are also between 2 and 5.  Also, larger k' values will always lead to longer retention and analysis times.

Table 2.  Values of k2’ and k2'/(1 + k2').

k2' k2'/(1 + k2')
0 0
1 \(\dfrac{1}{2}\)
2 \(\dfrac{2}{3}\)
3 \(\dfrac{3}{4}\)
4 \(\dfrac{4}{5}\)
5 \(\dfrac{5}{6}\)

Remember that we defined k' earlier in the course as:

\[\mathrm{k_2' = K_2\left(\dfrac{V_S}{V_M} \right )}\]

So anything that alters any of the terms of this equation will lead to a change in k'.  As we already mentioned above, the most common way to change k' in liquid chromatography is to alter the constituents of the mobile phase.  This will change the distribution constant (K2), and if the correct change is made, make K2 larger and k' larger.  The analog to this in gas chromatography is to change the temperature.  Recall that:

\[\mathrm{K_2 = \dfrac{C_S}{C_M}}\]

We could consider what happens to this ratio as the temperature is changed.  The stationary phase is a liquid.  The mobile phase is a gas.  Hotter temperatures reduce the solubility of a gas in a liquid (warm soda will go flatter faster).  Thermal pollution refers to the reduction in the concentration of dissolved oxygen gas when warm water from a power plant or industrial cooling process is added to a river or lake.  So hotter temperatures will reduce CS, reduce K2 and reduce k'.  This would actually make the resolution worse.  So by cooling down a gas chromatographic column, we increase the retention factor and improve the resolution.  Of course, this also lengthens the analysis time, which may be undesirable.

If we examine the volume terms, we realize that it is difficult to change VM, the volume of the mobile phase. VM is the interstitial volume and is essentially fixed for a particular particle size. There is no way to crunch the particles closer together to reduce VM.  In gas chromatography, it is easy to alter VS, the volume of the stationary phase.  This would involve coating a thicker loading (e.g., instead of 3% liquid coating, raise it to 5%).  Changing VS in liquid chromatography really is not an option since we have bonded C18 groups and do not coat different thickness phases.  If we change the particle size from 5 µm to 3 µm, and keep everything else, we do create more capacity in the column (because the 3 µm particles have more surface area), but we create more plates as well.

Consider the chromatogram in Figure 45.  We observe two problems with this separation.  The first few compounds come out of the column very quickly and are not fully resolved.  The latter compounds are fully resolved but stay in the column too long and are very broad.

Figure 45. Chromatogram with non-optimal k’ values for the earlier and later eluting compounds.

What problem exists in this separation?  If we were to enlarge N to improve the resolution of the first few compounds, we would lengthen the retention time of the latter ones as well, which would be unacceptable.  If we changed α, it might resolve some of the overlapped constituents but might cause some that are already resolved to now overlap with each other.  We have a capacity problem (k') in this chromatogram.  The first few compounds do not have enough capacity and would benefit by spending more time in the stationary phase.  The latter few have too high a retention factor and are spending too much time in the stationary phase.  What we say is that a chromatogram has a limited peak capacity.  In other words, it is only possible to separate a certain amount of compounds within a fixed period of time in the chromatogram.  There is only some number of peaks that can be fit in side by side before they start to overlap with each other.

In gas chromatography, there is a way to address these problems during the chromatogram so that all of the constituents are chromatographed at a k' value that is more optimal.  If we lowered the temperature during the early portion of the chromatogram, we would raise the retention factor of these constituents of the sample and they would stay on the column longer.  If we raised the temperature during the latter portion of the chromatogram, we would lower the retention factor of these constituents and they would come off of the column faster.  This systematic raise in temperature during the chromatogram is known as a temperature program.  Most gas chromatograms are obtained using temperature programming rather than isothermal conditions.  The temperature program chromatogram for the same mixture is shown in Figure 46.

Figure 46. Chromatogram with more optimal k’ values.

There is an analogous procedure in liquid chromatography that is known as gradient elution.  In this technique, you start with a mobile phase that causes the constituents of the sample to have a high retention factor and then systematically vary the mobile phase during the run to lower the capacity of the constituents.  The exact nature of the changes that might be made to accomplish this will be discussed later in our unit.