To
go
further:
According to the plate theory, the
chromatogram allows us to calculate some interesting properties about the column and the
compounds.
Click to see how to calculate the number of plates of
the
column, or the
partition coefficient
of the compounds.
How to calculate the number of plates of a column?
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Using a mathematical developpement, based on a step-by-step equilibration identical to what is
shown
in the table above, it can be demonstrated* that the chromatographic peak of a given
compound follows this equation:
A chromatographic peak derived from the plate theory is not exactly a gaussian peak. But when
the
number of column plates is big enough (i.e. n>100), the chromatographic peak becomes very close
to a
normal gaussian peak, as shown in the graphics below:
As we know from statistics courses, a gaussian peak is characterized by its mean μ and its
standard
deviation σ.
In the case of our chromatographic peak, it can be shown** that the gaussian
approximation of the chromatographic peak has these parameters:
Where P, F and n
have the same meanings as above.
By dividing the 1st equation by the
2nd one:
In summary, we need to know the mean
μ
and the standard deviation σ of one of the chromatogram peaks to calculate the number of column
plates n.
The mean of a gaussian peak is simply the distance between the peak summit and the origin of the
X
axis. However, the standard deviation can not be measured directly but can be approximated by
measuring the peak Full Width at Half Maximum (FWHM), which is equal to 2.355 times the standard
deviation***.
Therefore:
This equation allows you to estimate
the
number of plates of a column thanks to a chromatogram.
You can use a ruler to measure the
equation parameters as shown in the example below:
You can now go back to the
chromatogram
you obtained in the last part of this application and measure the number of plates. Then, you
can
compare this value with the number of plates you have initially entered at the top of the page.
References:
* Said
AS.
Theoretical-plate concept in chromatography. AIChE J. 1956 ; 2(4) : 477-81
** Scott
RPW.
The Plate Theory of Chromatography. 1st ed. Integritext UK; 2014
***
en.wikipedia.org/wiki/Full_width_at_half_maximum
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The plate theory and the partition coefficients:
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We will start by defining the Capacity Factor (or the Retention Factor), generally noted as k':
As we have seen in the last section about the number of plates, the retention time of a compound
is
given by the equation of the mean of the gaussian peak:
And for an unretained compound (with a partition coefficien K=0) the equation becomes:
By substituting tR and t0 in the equation of k' and doing some very simple
maths:
This equation tells us that, according to the plate theory, we need to know vm and
vs (which are the volumes of mobile phase and stationary phase for a plate) in order
to
calculate the partition coefficient K of our compound from the
chromatogram.
vm
and vs are not always precisely known, but what is interesting is to compare two
retention factors k' of two compounds, as:
Therefore, by measuring the dead time t0 and the retention times tR of two
compounds, we can calculate the ratio of their partition coefficients:
You can now go back to the
application
and measure the ratio of the partition coefficients of two compounds from your
chromatogram.
Don't forget to inject an unretained compound in order to measure
t0!
Then, you can compare this value with the partition coefficients you have
initially entered at the top of the page.
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