The equation of the plate height is given as H = σ^2/L where σ is standard deviation in a normally distributed curve (representing width of analyte peak) and L is the length of the column. Why is the peak width represented as σ^2 and not 2σ?
It is an interesting question. First of all let us do a dimensional analysis, which is extremely important in mathematics to formulate any equation. Plate height should have units of length because theory is derived from distillation columns.
H = σ ^2 /L has dimensions of [length][length]/[length] = [length], which is dimensionally consistent.
If we use your suggestion H=2 σ /L = constant*[length]/[length], the plate height turns out to be a dimensionless parameter. So your suggestion to use 2σ /L turns out to be incorrect.
A deeper question is what is the difference between variance ( σ ^2) vs. standard deviation σ ? It turns out naturally, in diffusion of molecules, the parameter σ ^2 appears naturally in the equations. Liquid and gas chromatography is all about diffusion of the analyte in the stationary phase and the mobile phase. Whenever you see σ ^2, just realize that there is some diffusion process going on. You will see this in electrochemistry as well.
BTW, I am looking for the origin of modern efficiency equation N=(t/σ)^2 for some time. I cannot find who used it for the first time and why did he derive this? If any reader happens to know, that would be great. I have asked mathematicians but they could not interpret the meaning of the (mean squared/variance) as a measure of efficiency.
You can have a look at my article Figure 1, where I show how standard deviation is related to peak widths at various locations of the peak. This only applies to pure Gaussian shapes.
M. Farooq Wahab The explanation of the theory behind it made it easier to understand, thank you.
M. Farooq Wahab check this book:
Chromatography: Fundamentals and applications of chromatography and related differential migration methods - Part A: Fundamentals and techniques
E. Heftmann
eq.1.30 and 1.42.
regards,
GB
Grzegorz Boczkaj , I think the edition I have doesn't match with yours. I have chromatography, 5th edition: fundamentals and applications of chromatography and related differential migration methods, part A: fundamentals and techniques, I see that on page A10.
"At each stage during this process, a solute band has a bandwidth that can be characterized by the standard deviation of the band in length units, σ. It can be shown [2] that if L is the distance the band has migrated (i.e. the column length)
σ ^2 = H L (1.24)
where H is a proportionality constant defined as the column plateheight (also height equivalent to a theoretical plate or HETP). H depends on the solute, the operating conditions, and the column; from Eqn. 1.24, H = σ ^2 / L . A general goal in chromatography is to achieve narrow bands (small values of u), which means that we desire small values of H. The form of Eqn. 1.24 and the terminology now used ("plate height") derive from an early model of the chromatographic process [20], in which the column is mathematically divided into a number of equilibrium stages or "plates". The dimension of H is length, and its value for an efficient column is proportional to a characteristic distance within the column, i.e. the particle diameter for packed columns or the column diameter for open-tubular
(capillary) columns.
The plate height, H, is a measure of band broadening that is normalized for column length. To compare the performance of columns of different length, it is more useful to measure the column plate number, N. N can be defined for a band that is just about to leave the column
N = L2/σ2 (1.25),"
The question is why did early chromatographers define N this way? Nobody seems to know the reason. I have searched a lot and asked a lot.
Grzegorz Boczkaj , yes that was a different edition. The content is the same as I posted above. However the secondary question remains unanswered, as to why chromatographic efficiency was defined that way N=(tr/sigma)^2? Nobody seems to know. Every writer says it is defined this way.
M. Farooq Wahab ,
I think, that here you will find a direct explanation:
http://www.chromatographyonline.com/chromatography-fundamentals-part-v-theoretical-plates-significance-properties-and-uses?pageID=2
best regards,
GB
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