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

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2 f2(k')d2 °RS ~ [i<T U .....................................................(9)

where (df) is the effective film thickness of the stationary phase.

By summing all the respective contributions from the different sources of dispersion the final Van Deemter equation is obtained,

2vDm fK^)d2 f2(k‘)d2

H = 2Xdp +~~0+fck') + - -u + -—-u............. (10)

U Dm Ds

Equation (10) is the basic form of the Van Deemter equation and will be expanded and discussed with other HETP equations in the next chapter.

References

(IXJ.C.Giddings," Dynamics of Chromatography Part Ã, Marcel Dekker Inc.New Vork,( 1965)22.

(2) AJ.P.Martin and R.LM.Synge, Biochem. 1941) 1358.

(3) J. J. Van Deemter,F. J. Zuiderweg and A. Klinkenberg, Chem. Eng. Sci. 5(1956)271

(4) A. Klinkenberg, "Gas Chromatography 1960 “,(Ed. R.P.W.Scott),

Butterworths, London/1960) 194.

(5) J.C.Giddings," Dynamics of Chromatography Part /",(1965)29

(6) W. Feller, “ Probability Theory and its Applications, Wiley,New York, (1950) Chapt. 14.

(7) D.S.Horne.J.H.Knox and E.McLaren, “ Separation Techniques in Chemistry and Biochemistry", (Ed.R.A.Keller), Marcel Dekker, New York,

(8) G. E. Uhlenbeck and L. S. Ornstein, Phys.Rev. 36(1930)823

Chapter 7

The Van Deemter Equation

The Van Deemter equation (1) was the first rate equation to be developed and this took place as long ago as 1956. However, it is only relatively recently that the equation has been validated by careful experimental measurement (2). As a result, the Van Deemter equation has been shown to be the most appropriate equation for the accurate prediction of dispersion in liquid chromatography columns. The Van Deemter equation is particularly pertinent at mobile phase velocities around the optimum velocity (a concept that will shortly be explained). Furthermore, as all LC columns should be operated at, or close to, the optimum velocity for maximum efficiency, the Van Deemter equation is particularly important in column design. Other rate equations that have been developed for liquid chromatography will be discussed in the next chapter and compared with the Van Deemter equation

Restating the Van Deemter equation,

?vn fi(k')dn f2(k’)d.2

H = 2Xdp+JL"1(4k,) + -ir-?-u + — Lu . (D

u um ug

in fact, in the original form, equation (1), was introduced by Van Deemter for packed 6C columns and consequently, the longitudinal diffusion term for

the liquid phase was not included and —^--(l+fo1), was replaced by,

Furthermore, as the equation was developed for GC, where the diffusivity of the solute in the gas was four to five orders of magnitude greater than in a liquid, Van Deemter considered the resistance to mass transfer in the

fi(k')d2

mobile phase to be negligible. As a result, the function,—-—-u, was also

Dm

not included.

110

The equation actually developed by Van Deemter took the form,

2óÎò f2(k')df

H = 2XdD + —^ + —-——u ........................... (2)

p u Ds

The form taken by f2(k') was considered by Van Deemter to be,

8 k' n2 (l+k')2

and thus, the complete HETP equation became,

H = 2ßdp + + Ë—k-y^-u ...................... (3)

P u ë (l+k') s

Equation (3), however, was developed for a gas chromatographic column and in the case of a liquid chromatographic column, the resistance to mass transfer in the mobile phase should be taken into account. Van Deemter et at did not derive an expression for f i(k ) for the mobile phase and it was left to Purnell (3) to suggest that the function of (k-), employed by Golay (4) for the resistance to mass transfer in the mobile phase in his rate equation for capillary columns, would also be appropriate for a packed column in LC. The form of f i(k') derived by Golay was as follows,

ïáê'-È Ik'2 24 (l+k')2

(It should be noted that the Golay equation for capillary columns will be discussed in detail in the next chapter.)

Thus, Van Deemter's equation for LC becomes,

2vDm l+fik'+llk'2 dn 5 [/ d?

H — 2Xdn + + o-^ + —7 9 n—^

u 24(l+k') Dm ë (l+k') DS

(4)

1

Equation (4 ) can put in a simplified form as follows,

H = A + -+CU

(5)

u

where,

A = 2Adp

Â - 2yDm

and

Equation (5) is a hyperbolic function which indicates that there will be a minimum value of (H) for a particular value of (u). That is a maximum efficiency will obtained at a particular linear mobile phase velocity. An example of an HETP curve obtained in practice is shown in figure I.

Figure 1

An Experimental HETP Curve for Hexamethylbenzene

0.003-1

-0.000

¦ H

¦ A

î B/u

x Ñè

• Wit)

-0.000 0.100 0.200 0.300 0.400 0.500 0.600

u (cm/sec)

112

The upper curve, which is the result of a curve fitting procedure to the points shown, Is the HETP curve. The column was 25 cm iong, 9 mm in diameter and packed with 8,5 micron (nominal 10 micron) Partisil silica gel The mobile phase was a solution of 4.8%w/v ethyl acetate in n-decane The minimum of the curve is clearly indicated and it is seen that the fit of the points to the curve is fairly good. As a result of the curve fitting procedure the values of the Van Deemter constants could be determined and the separate contributions to the curve from the multipath dispersion, longitudinal dispersion and the resistance to mass transfer calculated

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