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

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the column efficiency. These assumptions are exactly analogous to those employed In the development of the theory of distillation

Said (2) developed the equation of the elution curve in the following way. Consider the equilibrium existing in each plate, then :-

Xs=KXm ..........................................(1)

Where, (Xm) and (Xs) are the concentrations of the solute in the mobile and stationary phases respectively and (K) is the distribution coefficient of the solute between the two phases.

(It should be noted that (K) is defined with reference to the stationary phase, i.e. Ê = Xs/Xm .thus the larger the distribution coefficient, the more solute is distributed in the stationary phase)

(K) is a dimensionless constant and thus in liquid/liquid systems, (Xs) and (Xm) are conveniently measured as mass of solute per unit volume of phase. In liquid /solid chromatography an alternative method of measurement could be, mass of solute per unit mass of phase. However, in LC the difference between a liquid/liquid system and a liquid/solid system is moot. In fact, in practice, the reverse phase system in LC is more often considered liquid/liquid than a liquid/solid system.

Equation (1) merely states that the general distribution law applies to the system and that the adsorption isotherm is linear. At the concentrations normally employed in liquid chromatographic separations this will be true. It will be shown later that the adsorption isotherms must be very close to linear if the system is to have practical use, since nonlinear isotherms produce asymmetrical peaks.

Differentiating equation (1),

dXs = KdXm .......................................... (2)

Consider three consecutive plates in a column, the (p-1), the (p) and the (p+1) plates and let there be a total of (n) plates in the column. The three plate are depicted In figure (1) Let the volumes of mobile phase and stationary phase in each plate be (vm) and (vs) respectively and the concentrations of solute in the mobile and stationary phase in each piate be Xm(p-l), Xs(p-i), Xm(p), X$(p), Xm(p+1), and Xs(p+i) respectively. Let a volume of mobile phase, dV, pass from plate (p-1) into plate (p) at the same time displacing the same volume of mobile phase from plate (p) to plate (p+1). In doing so, there will be a change of mass of solute in plate (p) that

will be equal to the difference in the mass entering plate (p) from plate (p-1) and the mass of solute leaving plate (p) and entering plate (p+1). Thus bearing in mind that mass is the product of concentration and volume, the change of mass of solute in plate (p) 1s:-

dm = (Xm(p-l)-Xfn(p))dV .....................................(3)

Figure 1

Three Consecutive Theoretical Plates In an LC Column.

Plate (p-1) Plate (P) Plate (p+1)

Ym Vm Vm

^m(p-1) ^m(p) Y m(p*D

Vs Vs vs

Y ë s(p-1) ^s(p) X s(p+l)

Now if equilibrium is to be maintained in the plate (p), the mass (dm) will distribute itself between the two phases, which will result in a change of solute concentration in the the mobile phase of dXm(p) and in the stationary phase of dXs(p).

Thus, dm = vsdXs(p) ¦ vmdXm(p) ....................................(4)

Substituting for dXs(p) from equation (2),.

dm ~ (vm + Kvs)dXm(p)

(5)

!6

Equating equations (3) and (5) and rearranging,

dHn(p)_Xm(p-l)~Xlp)

dV " vm+Kv,

Now, to aid in algebraic manipulation it is necessary to effect a change of variable. The volume flow of mobile phase will now be measured in units of, (vm + Kvs), instead of milliliters. Thus the new variable (v) can be defined where,

v= --V—ã........................................... (7)

Kn+Kv9)

The function (vm + Kvs), Is given the name 'plate volume' and thus, for the present, the flow of mobile phase through the column will be measured in 'plate volumes' instead of milliliters. The 'plate volume' is, and can be defined as, that volume of mobile phase that can contain all the solute that is in the plate at the equilibrium concentration of the solute in the mobile phase.

Differentiating equation (7),

rk/=_

(vm+Kvs)

dV ,0, dv=7-ã .......................................... (8)

Substituting for dV from (8) in (6)

dV

=’4 r\) ......................................W)

Equation (9) is the basic differential equation that describes the rate of change of concentration of solute in the mobile phase in plate (p) with the volume flow of mobile phase through it. Thus, the integration of equation (9) will provide the equation for the elution curve of a solute from any plate in the column. A simple algebraic solution to equation (9) is given below and the resulting equation for the elution curve from plate (p) is as follows:-

19

_X0e~vvp

^m(pl ^.i

V'-

Where Xm(p) is the concentration of the solute in the mobile phase leaving the (p)th plate and X0 is the initial concentration of solute placed on the 1st plate of the column

Thus, the equation for the elution curve from the (n)th plate which is the last plate in the column (that is, the equation relating the concentration of solute in the mobile phase entering the detector with volume of mobile phase passed through the column), is given by:-

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