of the carrier gas. The solubility of the molecules is again assumed to be 50 %. The carrier movement represents the distance moved by carrier gas molecules that were present at the instant of sample injection.
Each colored trace shows the distribution of molecules and the corresponding peak shape obtained for a different number of equilibria (N). In practice, the narrower peaks would be much higher: the scale of the vertical axis is not the same for each trace. When a column operates under optimum conditions, a peak experiences more equilibria as it passes through the same length of column, resulting in narrower peaks that are easier to separate.
Figure 2.9 Effect of Having More Equilibria.
The gold curve in Figure 2.9 is a smoothed version of the 1:4:6:4:1 distribution we obtained with five equilibria. This embryonic peak would appear in the first two millimeters of a regular packed column and would take less than one fourth of a second to form.
Continuing the jerky mechanism for more equilibria would be tedious, but luckily it can be done mathematically. Figure 2.9 also displays the curves for 50, 500, and 5,000 equilibria, plotted as if they occurred along the same length of column. They look just like chromatogram peaks! And that's exactly what they are. You have just witnessed how the standard peak shape forms.
In chromatography theory, the plate number (N) is the number of equilibria achieved by a column. Of course, the notion of separate equilibria occurring in the column is just a theoretical concept. One might argue that the gas and liquid never quite achieve equilibrium because the gas is constantly moving. True; and that leads to another theory of chromatography, which assumes equilibrium never happens. Nevertheless, even that theory ends up with a plate number to express the efficiency of a column, its ability to produce narrow peaks.
The plate number is a useful parameter for evaluating column performance. As is evident from Figure 2.9, anything that increases the plate number must be a good thing, because it reduces peak width and narrower peaks provide better separations.
Some conclusions
Figure 2.9 contains a lot of information. Let's pause for a moment and see what it can tell us.
Identical molecules – different behavior
Notice that the propane molecules do not all spend the same time in the column, even though they are identical to each other. They enter the detector at different times.
The variable arrival time of identical molecules at the detector is the root cause of chromatogram peak shape.
Clearly, some of the propane molecules are delayed longer in the liquid phase than others are. It's a crap shoot in there, a totally random process where some get left behind and others don't. As a result, the peak shape follows the standard bell‐shaped curve that is found in many random processes. You saw this happen in Figures 2.5–2.8.
All peaks are symmetrical
The statistical bell‐shaped curve is named for Carl Friedrich Gauss, the acclaimed German mathematician who derived its mathematical equation. The two major theories of chromatography both deduce that all the peaks in the column should be Gaussian in shape.
Thus, our ideal peaks in Figure 2.9 are perfectly symmetrical. Unfortunately, real peaks are not perfect, and they may be slightly or grossly asymmetric due to other mechanisms occurring in the columns or in the instrument itself.
More equilibria – narrower peaks
On a given column, the peaks always get narrower as the number of equilibria increases. This is a very important observation because narrow peaks are highly desirable. It's so much easier to separate narrow peaks from each other than to separate wide peaks from each other. Therefore, to ensure that the peak widths are as narrow as possible, we adjust the column operating conditions − particularly the carrier gas flow rate − to generate the maximum number of equilibria.
In practice, each peak exhibits a different plate number. We can calculate the plate number of a selected peak from chromatogram measurements of its width and retention time. This calculation is used to evaluate column performance and to optimize column operating conditions. We'll look at it later.
It follows that a certain length of column must be necessary to generate each equilibrium. You can calculate it if you wish: just divide the column length by the plate number. In most columns the answer will be about 0.5 mm. We call this the plate height (H). It's a very important parameter for optimizing the performance of a column.
More equilibria – taller peaks
The apex of the peak gets higher when the plate number increases. It actually gets very much higher because the peak area remains constant (the vertical axis in Figure 2.9 is not drawn to scale). A higher peak in the graph means there are more molecules in the detector and a larger detector output signal, making the peak easier to measure. The chromatogram baseline always suffers from a little detector signal noise, so a higher peak has a better signal‐to‐noise ratio.
Retention at the apex
The apex of the peak is where the most propane molecules are. It's the highest concentration of propane molecules in the carrier gas. With symmetrical peaks like these, it's also the position of the average molecule. Since the recorded peak retention time should represent an average molecule, the elapsed time at peak apex is our best estimate of its retention time.
More equilibria – same retention time
In Figure 2.9, the retention time of the propane peak is always the same, regardless of the plate number. The plate number of a column affects only the peak width; it has no effect on the peak retention time.
In Figure 2.9, the peak apex has traveled along the column exactly half the distance that the carrier gas has traveled. This remains true during all of its journey through the column. When the carrier gas reaches the end of the column, as depicted, the propane peak has reached the mid‐point. At that time, it's only halfway through the column.
