# liquid chromatography column - Scott R.P.W.

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e*vvn

ÕÏ(|)=Õ0-|Ã

and for that portion of the sample on plate 2,

v _v e‘(v*,)(v+On

xn(2)~x0-^-

and similarly for that portion on plate 3

e-(v*2)(v+2)n

Xn(3)“X0~

n!

Consequently the composite elution curve for the sample on the first three plates will be given by,

Õï(1)+Õï(2)*Ë(3)-\>

e’V e’(v*,)(v+Dn e"(v*2)(v+2)n

n! n! n!

It follows that the elution curve for a sample initially occupying (r) plates will be,

Y Ð-Ã

ó =!$. ye-(v*p)fWirtn ¦¦n „.Za

Ï!ð-° ........................................ (1)

Equation (1) can be employed with the aid of a computer to calculate actual peak profiles should they be required It must be remembered, however, that

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(v) must be measured In plate volumes and volumes in milliliters must be converted to plate volumes in order to use equation (1) satisfactorily.

The curves resulting from a charge covering the first three consecutive plates of a column are shown in figure (2). The elution curve resulting from the contents of each induvidual plate are included together with the composite curve resulting from the total charge.

Figure 2 Elution Curves Resulting from the Injection of a Charge that Occupied the First Three Plates of a Column

Plate Volume (v)

To simplify the calculations the column was assumed to have only 100 theoretical plates.

Peak Asymmetry

The equation for the retention volume of a soiuie, that wab derived by differentiating the equation of the elution and given in chapter 2, can be used to obtain an equation for the retention time of a solute (tr) by dividing by the flow-rate (Q),(l).

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Thus,

Now the velocity of a band along the column (Z), is obtained by dividing the column length (1) by the retention time, (tr),

Consequently,

V 4n+ KVS

Thus the band velocity (Z) is Inversely proportional to (Vm + KVS) ,and for a significantly retained solute,

Vm<< KVS

Consequently, for any given column,

1 I '

It is seen from the above equation that the band velocity is inversely proportional to the distribution coefficient with respect to the mobile phase. It follows, that any changes In the distribution coefficient (K), will result directly in changes in in the band velocity (Z).

Consider the isotherms depicted in figure (3). Each curve represents a different isotherm relating the concentration of the solute in the stationary phase (Xs) to that of the mobile phase (Xm).

PqR represents a linear isotherm and, for this line, the distribution coefficient is given by,

Ü. = ia = ê,

Xm qn 1

Since PqQ is a linear isotherm, (Ki) is constant for all values of (Xs)then from equation (2) all concentrations in the band will travel at the same velocity and a symmetrical elution curve will be produced. This symmetrical curve is depicted as the normal peak (a) in figure (4). The symmetrical nature of the elution curve is to be expected from the plate theory.

Figure 3

Adsorption Isotherms

Now consider the situation where a large charge is placed on the column and the solute molecules, now in a relatively high concentration in the stationary phase, are no longer surrounded solely by solvent molecules but by solvent and solute molecules. Under these circumstances the interactive forces between the solute molecules and the molecules of the stationary phase are increased because the interactive forces between solute and solute are usually greater than the interactive forces between solute and solvent. Thus (K) will increase as (Xs) increases and the result will be an Isotherm of the form depicted by curve PrR. The distribution coefficient at the higher concentrations will now be,

Now as tr > tq and ro = qn, then,

K2 >Ki

Thus, from equation (2) it is seen that the higher concentrations of solute

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of solute. As a result, the peak will be distorted, and the peak maximum will be moved to a position of higher retention relative to the parts of the peak

where the solute is at a lower concentration. This results in a peak shaped of the form shown In figure (4b).

Figure 4

The Effect of Different Absorption Isotherms on Peak Shape

Conversely, in some liquid-solid systems, high concentrations of solute wlli cover the surface of the adsorbent with multilayers of solute molecules. As a result of the screening effect of the multilayers, the forces holding the outer layers of solute molecules on the surface will be significantly reduced relative to the forces holding the first or inner layers of solute molecules. This is because the outer layers will be either, experiencing only long range forces between them and the adsorbent surface or, be only interacting with the next layer of solute molecules. In any event, the effect will be the opposite to that of overload and the absorption isotherm will take the form of the curve PsS in figure (3b).

Thus, at high concentrations of solute,

Now since ts < tq and sm = qn,

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