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An Introduction to Chromatographic Separations

Lecture 34

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Substitution of 1,2 and 5 in equation 4 we get:

L/tR = L/tM * {1/(1 + k’)}

 

Rearrangement gives:

tR = tM (1+k’) (6)

This equation can also be written as:

k’ = (tR – tM)/tM

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The Selectivity Factor

For two solutes to be separated, they should have different migration rates. This is referred to as having different selectivity factors with regard to a specific solute. The selectivity factor, , can be defined as:

 

= kB’/kA’ (7)Therefore, a can be defined also as:

= (tR,B – tM)/ (tR,A – tM) (8) For the separation of A and B from their mixture, the

selectivity factor must be more than unity.

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The Shapes of Chromatographic Peaks

 

Chromatographic peaks will be considered as symmetrical normal error peaks (Gaussian peaks). This assumption is necessary in order to continue developing equations governing chromatographic performance. However, in many cases tailing or fronting peaks are observed.

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Gaussian peaks (normal error curves) are easier to deal with since statistical equations for such curves are well established and will be used for derivation of some basic chromatographic relations. It should also be indicated that as solutes move inside a column, their concentration zones are spread more and more where the zone breadth is related to the residence time of a solute in a chromatographic column.

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Plate Theory

Solutes in a chromatographic separation are partitioned between the stationary and mobile phases. Multiple partitions take place while a solute is moving towards the end of the column. The number of partitions a solute experiences inside a column very much resembles performing multiple extractions. It may be possible to denote each partitioning step as an individual extraction and the column can thus be regarded as a system having a number of segments or plates, where each plate represents a single extraction or partition process.

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Therefore, a chromatographic column can be divided to a number of theoretical plates where eventually the efficiency of a separation increases as the number of theoretical plates (N) increases. In other words, efficiency of a chromatographic separation will be increased as the height of the theoretical plate (H) is decreased.

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Column Efficiency and the Plate Theory

 

If the column length is referred to as L, the efficiency of that column can be defined as the number of theoretical plates that can fit in that column length. This can be described by the relation:

 

N = L/H

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The plate theory successfully accounts for the Gaussian shape of chromatographic peaks but unfortunately fails to account for zone broadening. In addition, the idea that a column is composed of plates is unrealistic as this implies full equilibrium in each plate which is never true. The equilibrium in chromatographic separations is just a dynamic equilibrium as the mobile phase is continuously moving.

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Definition of Plate Height

From statistics, the breadth of a Gaussian curve is related to the variance 2. Therefore, the plate height can be defined as the variance per unit length of the column:

H = 2/L (9)In other words, the plate height can be defined

as column length in cm which contains 34% of the solute at the end of the column (as the solute elutes). This can be graphically shown as:

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The peak width can also be represented in terms of time, t, where:

v = L/tR

When L approximate , tR approximates t where t is the time segment that corresponds to a distance .

v = L/tR = / t (10)

t = /(L/tR) (11)The width of the peak at the baseline, W, is related to t

by the relation:W = 4t where 96% of the solute is contained under the

peak.

= L t/tR

= LW/4tR (12)

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2 = L2W2/16tR2

2 = HL

H = LW2/16tR2

 

N = 16(tR/W)2 (13) Also, from statistics we have:

W1/2 = 2.354 t (14)2 = LH

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= L t/tR

= L (W1/2/2.354)/tR

2 = L2 (W1/2/2.354)2 /tR2 (15)

Substitution in equation 9 gives:

LH = L2 (W1/2/2.354)2 /tR2

H = L (W1/2/2.354 tR)2

N = L/H

N = 5.54 (tR/W1/2)2 (16)

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Asymmetric PeaksThe efficiency, N, can be estimated for an asymmetric chromatographic peak using

the relation:

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N = 41.7 (tR/W0.1)2 / (A/B + 1.25) (17)

 

Draw a horizontal line across the peak at a height equal to 1/10 of the maximum height.

Where W0.1 = peak width at 1/10 height = A + B

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Rate Theory

Band Broadening

 

Apart from specific characteristics of solutes that cause differential migration, average migration rates for molecules of the same solute are not identical. Three main factors contribute to this behavior:

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Longitudinal Diffusion

Molecules tend to diffuse in all directions because these are always present in a concentration zone as compared to the other parts of the column.

  This contributes to H as follows: 

HL = K1DM/V 

Where, DM is the diffusion of solute in the mobile phase. This factor is not very important in liquid chromatography except at low flow rates.

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This contributes to H as follows:

 

HL = K1DM/V

 

Where, DM is the diffusion of solute in the mobile phase. This factor is not very important in liquid chromatography except at low flow rates.

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Resistance to Mass Transfer

Mass transfer through mobile and stationary phases contributes to this type of band broadening.

 1. Stationary Phase Mass Transfer This contribution can be simply attributed to

the fact that not all molecules penetrate to the same extent into the stationary phase. Therefore, some molecules of the same solute tend to stay longer in the stationary phase than other molecules

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Quantitatively, this behavior can be represented by the equation:

 Hs = K2 ds

2 V/Ds

Where ds is the

thickness of stationary phase and Ds is the diffusion coefficient of solute in the stationary phase.