ChromFAQ:TheoreticalPlates

From ChromFAQ

What are Theoretical Plates? . . . and why are they only "Theoretical"?

An early conceptual model of the chromatographic process used the same mathematical formalism used for distillation calculations. A plate represents a stage of discrete equilibration between two phases. In the case of distillation, plates are often physical as shown here (http://journeytoforever.org/biofuel_library/ethanol_motherearth/meCh7.html#7-3). In the case of chromatography, there are no physical plates, but we can consider the number of plates which would have been required to produce a given peak as a characteristic of that peak. Hence the term "theoretical plates". The mental process goes something like this:

Imagine a series of beakers, each of which contains equal volumes of two immiscible solvents (water and methylene chloride, for example). Suppose that you had 10 of these beakers lined up in a fume hood. Start by putting a sample of some compound in beaker #1; it will (presumably) partition between the two phases, with the exact distribution determined by the partition coefficient. For convenience, we can assume a partition coefficient of 1 (in other words, equal amounts of sample in both phases).

Now, take the aqueous (upper) phase out of the last beaker and discard it. Transfer the aqueous phase from the next-to-last beaker to the last beaker, transfer the aqueous phase from the second-to-last beaker to the next-to-last beaker, and so on . . . , finish by adding fresh aqueous phase to the first beaker. Your sample is now contained in the first two beakers.

Repeat. Repeat. Repeat . . .

At the end of 5 transfers, the sample is spread among the first five beakers, with the highest concentration in the third beaker.

If you keep repeating, the sample will slowly move down the chain of beakers. If you were to plot the concentration of sample in the last beaker as a function of the number of repeats, as shown on the left, and you had a large number of beakers, you would get an approximately Gaussian distribution ("normal distribution"). The number of repeats required to get the high-point of the peak to the last beaker depends on the partition coefficient of the analyte (in the figures shown here, the partition coefficient is assumed to be 1; that is, the analyte has equal solubility in both phases). For any given partition coefficient, the width of the distribution depends on the number of beakers (which represent discrete equilibration stages) in the series. For any peak, we can calculate how many beakers would be required to generate the observed peak profile. That number is referred to as the "plate number" or "theoretical plate number".

With the exception of CounterCurrent Chromatography used primarily for preparative work, the "plate model" of chromatography has long been superseded by more complex models as a tool for quantitative predictions, but the plate number has continued as a way of characterizing chromatographic peaks because it is a direct function of the column length: doubling the length of the column doubles the plate number. The plate number depends in a more complex fashion on stationary phase particle size (generally, smaller particles => higher plate numbers), flow rate, temperature, and analyte molecular weight.

So how do I measure the plate number?

First of all, the standard calculations only work for isocratic chromatograms. If you use the same computation on a gradient separation, the resulting number is completely meaningless. That said, the concept of plate number (the number of equilibrations between the two phases) is useful in gradients. See the gradient section for more details.

Alright, on to the computation. First of all, we are assuming that the peak shape is reasonably well described by a Gaussian distribution ("normal distribution"). The Gaussian distribution is characterized by two values: the mean and the standard deviation σ (sigma); for a chromatographic peak, the mean is the retention time of the peak (t_R). The plate number, N is given by:

N = (t_R/σ)²

Retention time is easily measured. Prior to the development of computerized data systems, measuring σ was impractical. As an alternative, the baseline width of the peak (defined as the distance between the intersections of the tangents to the peak inflection points with the basline) was assumed to be 4σ (which is exactly what it would be for an ideal Gaussian peak). This gives the familiar formula:

N = 16(t_R/w_b)²

used in chromatography textbooks. Drawing tangents to the leading and trailing edges of a peak is relatively easy using chart paper and a ruler. Estimating baseline width is much more difficult for a computerized data system. It is much easier to calculate the distance from the leading edge to the trailing edge at some particular fraction of the peak maximum height (the width at half-height is the most commonly used of these measurements). For a Gaussian peak, w_0.5 is equal to 2.3 times σ. Plugging this into our original definition gives:

N = 5.54(t_R/w_0.5)²

In the ideal case (a perfectly Gaussian peak), all three forumulas will give the same value for N. In the real world, peak shape is seldom ideal, and plate numbers based on different formulas can vary tremendously. The greater the deviation from ideality, the greater the variation in plate number. In general, measurements made closer to the top of the peak will generate larger plate numbers than will measurements made near the bottom. When describing column performance, it is important to specify not only the plate number but also the formula used.

What about plate numbers in gradient chromatograms?

The concept of theoretical plates -- the average number of equilibrations undergone by sample molecules as they go through the chromatographic system -- is equally valid in isocratic and gradient HPLC. The computation, however is quite different. In fact, the plate number cannot be calculated from a single gradient chromatography run. Results from at least two runs (for linear gradients) with different gradient steepness are required, and even then the computation is far from simple. In most cases, it is safe to assume that the isocratic plate number for a given column can be applied to a gradient separation.

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