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

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The three contributions to dispersion are also shown as separate curves in figure 1. It is seen that the major contribution to dispersion at the optimum velocity, where the value of (H) is a minimum, is the multipath effect. Only at much lower velocities does the longitudinal diffusion effect become significant. Conversely, the mobile phase velocity must be increased to about 0.2 cm/sec before the resistance to mass transfer begins to become relatively significant compared to that of the multipath effect.

The values for the Van Deemter constants were found to be,

A=0.00117

6=0.0000175 cm2/sec and C=0.00250 sec.

It is seen that if the mean particle size was 0.00085 cm, thus as,

A=2*4

Then, 0.00117= 2X0.00085 or, A = 0 69

Giddings (5) determined theoretically that, for a weil packed column (A) should take a value of about 0.5 and thus the column used was reasonably well packed.

In a similar manner, Â = 2yDm Thus, 0.0000175 = 2yDm

Now Katz et 3/(2) determined the diffusivity of hexamethyl benzene in 4.83%w/v ethyl acetate in n decane and found it to be 1.17 10-5 cm2/sec,

Consequently, ó =0.747

11?

Giddings (5) also determined theoretically that for a well packed column (y) should be about 0.6 so again the longitudinal diffusion effect confirms that the column was reasonably well packed.

The HETP curve clearly shows, that for a packed column, the particle size has a profound effect on the minimum value of the HETP of a column and thus the maximum efficiency attainable. It would also appear that the highest efficiency column would be obtained from columns packed with the smallest particles. This will in due course be shown to be a fallacy, but what is true, is that the smaller the particle diameter the smaller will be the minimum HETP and thus, the larger the number of plates per unit length obtainable from the column. At this time It will suffice to point out that the total number of theoretical plates that can be obtained will depend on the length of the column which, in turn, must take into account the available inlet pressure.

The optimum mobile phase velocity can be obtained by differentiating equation 5 with respect to (u) and equating to zero,thus,

H = A + -+Cu

u

and

Equating to zero

— = 0 and consequently, --^ + C = 0 du , ,2

Thus,

(6)

Substituting for (B) and (C),

0.5

n2 (1+k )2 Ds

114

and letting Ds = ?Dm,

05

48óë2?(1+ê')2

(7)

It is seen from equation (7) that the optimum velocity is , directly proportional to the diffusivity of the solute in the mobile phase. To a lesser extent it also appears to be inversely dependant on the particle diameter of the packing (the particle size is an optional choice) and the film thickness of the stationary phase. The film thickness of the stationary phase is determined by the physical form of the packing, that is, in the case of silica gel, the nature of the surface and in the case of a reverse phase, on the bonding chemistry.

Now, if it is assumed that df « dp (which in practice will always be true)

Under these circumstances, it is seen that the optimum velocity is directly proportional to the solute diffusivity in the mobile phase and inversely proportional to the particle diameter of the packing. The quality of packing, obviously also plays a part, but to a very less significant extent.

Furthermore, when k'» 1, that is, for well retained peaks,

l62Dm

(8)

It is now possible to determine the factors that control the magnitude of Hmin. Substituting the function for Uopt from equation (6) in equation (5) an expression for Hmm is obtained.

or, Íòàï = À+2ò/ÂÑ .........

Substituting for A,Â, and C,

(9)

Hmin ~ 2 ß dp +2

2vDn

f 2 l + 6k'+l lk,2dP 8 +--

t 24(1+k')2 Dm *2(|+k')2DS

(10)

Noting that Ds= ?Dm, and simplifying,

’mm

= 2 A. dp+2

2y

(l + 6k' + I lk'2Jd

24(l+k')2

8 k’ n2 (1+k)2

(ii:

Again, assuming that, df « dp, and k'» 1, that is for well retained peaks,

Hfnin - 2Adp + 2

Bearing in mind the values of (A) and (y) for a well packed column are 0.5 and 0.6 respectively,

Hmin. = 2.48dp ...............................................................(12)

Thus, the approximate value of Hmin. for a well retained solute eluted from a well packed column and operated at the optimum linear mobile phase velocity, can be expected to be about 2.48dp. Furthermore, to the first approximation, this value will be independent of the nature of the solute, mobile phase or stationary phase. For the accurate design of the optimum columns for a particular separation however, this approximation can not be made, nevertheless, the value of 2.48 for Hmjn is a useful guide for assessing the quality of a column.

I 24

From d'Arcy's Law for fluid flow through a packed bed, at the optimum mobile phase velocity the length of the column is given by,

Substituting for uopt and Hmm from equations (8) and (12) respectively,

It is seen from equation (16) that, if an LC column is operated at its optimum linear velocity, the maximum efficiency obtainable for well retained peaks will be directly proportional to the inlet pressure available (P) and the square of the particle diameter of the packing. Thus, the larger the particle diameter, the greater efficiency attainable at a given pressure. This is because, as the particle diameter is increased the permeability is also increased allowing a longer column to be used. The permeability increases as the square of the particle diameter but as shown by equation (12) the variance per unit length only increases proportionally to the particle diameter. Thus, doubling the particle diameter will allow a column four times the length to be used but the number of plates per unit length will be halved. Consequently, the column efficiency will be increased by a

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