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

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The Horvath and Lin Equation

In (1976) Horvath and Lin (8) and (9) introduced yet another equation to describe the value of (H) as a function of the linear mobile phase velocity (u). Their equation is given as follows.

The equation of Horvath and Lin was very similar to that of Huber and Hulsman and, in fact, only differed in the magnitude of the power function of (u) in their (A) and (D) terms. These workers were also trying to address the problem of a zero (A) term at zero velocity and the fact that some form of "turbulence' between particles aided in the solute transfer across the voids between the particles. All the above equations will be tested against experimental data in the next chapter.

The Golay Equation

The basic equation describing the dispersion that takes place in an open tubular column was developed by Golay (8) for gas chromatography but is equally, and directly, applicable to liquid chromatography. The Golay equation differs in one important aspect from the equations for packed columns in that, as there is no packing, there can be no multipath term or coupling factor and thus, contains only two functions. One function describes the longitudinal diffusion effect and the other the combined resistance to mass transfer terms for the mobile and stationary phases.The Golay equation takes the following form:-

Where (H) is the variance per unit length of the column for the given solute, (k)’ is the capacity factor of the eluted solute,

(K)the distribution coefficient of the solute between the two phases. (Dm) is the diffusivity of the solute in the mobile phase,

(D$) is the diffusivity of the solute In the stationary phase,

(r) is the radius of the column, and (u) is the linear velocity of the mobile phase.

H = - Ac - + - + Ñè + Du2/3 E è

(6)

u

(7)

129

If the solute is unretained (i.e. ê-0 ) then the the Golay equation reduces to,

,2

í = +

U 24 Dm

(8)

Taking a value of 2.5 XIO-5 for Dm (the diffusivity of benzyl acetate in n-heptane) equation (8) can be employed to calculate the curve relating (H) and

(u) for an uncoated capillary tube. The results are shown in figure (3).

Figure 3

Graph of H against U for a Capillary Column ( Unretained Peak)

Linear Velocity (cm/Sec)

It is seen that the Golay equation produces a curve identical to the Van Deemter equation but with no contribution from a multipath term, it is also seen that the value of (H) is solely dependent on the diffusivity of the solute in the mobile phase and the linear mobile phase velocity. It is clear that the capillary column can, therefore, provide a simple means of determining the diffusivity of a solute in any given liquid.

The Golay equation (equation 7) can be DUt in a simplified form in a similar manner to the equations for packed columnsr

B r H = - + Cu

u

(9)

!30

Where,

and

Ñ =

(l + 6k‘ + I lk'2)r2

~ ~-

24(l + k') Dm 6(1 + k') K2D

k'3r2

The form of the HETP curve for a capillary column is the same as that for a packed column and exhibits a minimum value for (H) at an optimum velocity.

Differentiating equation (2) with respect to (u)

— = + C du h2 +

Thus, when

H - Hpnin, then, —ó + Ñ — Î u

and

(10)

Substituting for (B) and (C) in equation (10)

uopt

2Dn

k‘3r2

-,05

(l + 6k' +1 lk'2]r2

_ + ^— 24(l + k') Dm 6(1+ ê) Ê Ds

or,

uopt.

_

K2^(l + 6k' + 11k'2 j + 4k';

0.5

(11)

13

It is seen that, in a similar manner to the packed column, the optimum mobile phase velocity is directly proportional to the diffusivity of the solute in the mobile phase. However, in the capillary column the radius (r) replaces the particle diameter (dp) of the packed column and consequently, (uopt) is inversely proportional to the column radius.

Now,

nmin.

Â

uopt

CUopt

B_

Ãâ

.c?

or,

Hmm — 2 i/ÂÑ

(12)

Again substituting for (B) and (C),

'mm

= 2

(2Dm)

(l + 6k' + 1 Ik'2 jr2

-?- + ?

24(l + k') Dm 6(l + k') K2DS

k'V

^min. — 2r

Ã77 (l + 6k'

+11k'^

,,3

12(1 +k')2 3(1+ k')2K2^

(13)

Equation (13) shows that the minimum value of (H) is solely dependant on the column radius (r) and the thermodynamic properties of the solute/phase system. As opposed to the optimum velocity, the minimum value of (H) is not dependent on the solute diffusivity.

In addition it Is seen, from equation (12), that the expression for uopt is very similar to that for a packed column but the expression for Hmjn. is much simpler as it is devoid of the (A) term from the multipath effect.

Due to the relative simple, and precisely defined, geometry of the capillary column it contains no arbitrary constants involved with the packing quality of the column such as (y) and (Ë). As a consequence, the properties of the

Ó p.

capillary column can be explored further in a relatively straightforward manner.

From Poiseuille's Equation, describing the flow of liquid through an open tube,

It is interesting to note from equation (14) that when a column is run at its optimum velocity, the maximum efficiency attainable from a capillary column is directly proportional to the inlet pressure and the square of the radius and inversely proportional to the solvent viscosity and the diffusivity of the solute in the mobile phase. This means that the maximum efficiency attainable from a capillary column increases with the column radius. Consequently, very high efficiencies will be obtained from relatively large diameter columns.

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