How the plate height is assumed to be the standard deviation?

Question

Fundamentals of Analytical Chemistry book states that the plate height can be thought of as the length of column that contains a fraction of the analyte that lies between $L$ and $L - \sigma$ [1, p. 870].

But what I know is that $H = \sigma^2 / L$ (At the retention time), so, how can plate height be equal to $L - (L - \sigma)$?

$H$ is the plate height. $L$ is the column length. $\sigma$ is the standard deviation.

Reference

1. Skoog, D. A.; West, D. M.; Holler, F. J.; Crouch, S. R. Fundamentals of Analytical Chemistry, 9th ed.; Brooks Cole: Belmont, CA, 2013. ISBN 978-0-495-55828-6.

Answer 1

I feel that you misunderstood what the authors were trying to say in that textbook. Forget about everything for the time being. Imagine if you injected a single analyte band into an HPLC column of length L at a constant flow rate. You waited for some time, and now the band is exiting the column. For simplicity, assume that the band is rectangular. Ask yourself, how much average distance did the analyte band travel? Indeed, it must have traveled a distance L if it has to exit the column. Most modern HPLC columns are 10 cm long; assume analyte traveled an average of L=10 cm.

Now a band cannot be infinitely thin inside a chromatographic column. The analyte spreads in space, i.e., its width increases as it passes through the column due to diffusion and other processes.

So let us say the band is now a 0.05 cm wide rectangle when it reached the end of the column. Half of it is inside the column, and half of it has exited the column. What width of the band is inside the column, and what width is outside the column? You would say 0.05/2 cm of the band is inside the column, and 0.05/2 cm of the band is outside the column.

1. Total distance traveled by the portion of the band which has exited = 10 + 0.05/2 cm
2. Total distance traveled by the portion of the band which is inside = 10 - 0.05/2 cm

Now in chromatography, as you have seen, peaks are not rectangles, but they are Gaussians. Furthermore, Gaussians are characterized by their "standard deviation," which is a measure of their spread. If you apply the sample logic

0. The mean (=center of the Gaussian) has traveled a distance of 10 cm or L
1. The portion of the Gaussian which is still inside traveled a distance L-1*standard deviation
2. The portion of the Gaussian which exited the column traveled a distance L+1*standard deviation.

Note that conventionally a full Gaussian band at the base is considered to have a width of 4xstd. dev.

Answer 2

You are right that the description in SWH&C is a bit nebulous and potentially misleading. What the book attempts to do is to explain the meaning of $\sigma$ as it relates to the peak shape and peak area. The catch is that you can't report a single value of $\sigma$ as representative of the efficiency of a column, since it would not be constant under all circumstances. But you can report one value of H (for a given flow rate and solute/mobile phase) and this will be representative. Then if H and L are known the standard deviation of the Gaussian peak shape in distance units can be computed as $\sigma=\sqrt{LH}$.

One might wonder why $\sigma$ is not constant, or why plate height is not proportional to $\sigma/L$, as the textbook might appear to suggest, so I will explain why this is.

The reason is that one would like to have a property that depends on the composition of the column but not on its dimensions (or the elution time). If you double the length of a column, then a solute will take twice as long to elute (if the mobile phase flows at the same rate), but the width of the peak will increase $\propto\sigma \propto t^{1/2}\propto L^{1/2}$, meaning $\sigma/L\propto L^{-1/2}$, so $\sigma/L$ is not a property that is independent of the length of the column. On the other hand, since $\sigma^2 \propto t \propto L$, $\sigma^2/L$ is independent of L and of the elution time (provided the flow rate is constant).

Additional relations that allow you to draw these conclusions are $L\propto\mu t$ and $\sigma=\sqrt{2Dt}$, where $\mu$ is the mobility of the solute and D its diffusion coefficient. D is a constant property for a given solute and column phases, and $\mu$ is proportional to the flow rate and phases, but not dependent on column dimensions. Therefore $D/\mu$ is independent of column dimensions or of the elution time, and by extension so is $\sigma^2/L$.

Note this only covers the effect of diffusion of the solute in the mobile phase (the efficiency is also a function of the flow rate for other reasons).

For more information on how H is determined in practice you might want to look at a GE Healthcare application note (Ref 1) or the reference therein (Ref 2).

References

1. GE Healthcare Application note 28-9372-07 AA

2. Hagel, L. et al. Handbook of process chromatography 2nd ed., John Wiley and Sons, Inc., New York (2008).