Short Communications 10081

Estimation of Extra-Column Contributions to Band Broadening in Capillary Columns

Olle Nilsson Department of Technical Analytical Chemistry, Chemical Center, Lund, Sweden

Key Words:

Chromatography, theory Extra-column Contribution Band broadening “abt” concept Taylor equation

In the preceeding number of this journal [l] it was pointed out that attempts to estimate extra-column contributions to band broadening, (b0,5)~,t, from a formula derived by Smuts eta/. [2] often yield imaginary values because of negative (b0.5)& inter- cepts. This paper describes an alternative method, which should be usable for capillary columns in the very common case where the “abt” concept is applicable [3], i.e. when the experimental peak width at half height, b0.5, is a reasonably linear function of the capacity ratio, k, according to

(1) bo.5 = bo + ak

Theory

In an investigation into the nature of b,, Kaiser and Rieder [4] found that a ten-fold change of injector volume did not affect the b,-value very much, and they concluded that “the reason for the large b,-value is not the capillary, nor the inlet system”. The simplest explanation of their results seems to be that for the non-retained samplecomponent (defined by its k in the chromato- gram) the diffusion-dependent column contribution to the band width, b,(k), is non-negligiblefor substances, whose k approaches zero in the chromatogram. With proper knowledge of the diffusion coefficient of the substance in the mobile phase, Dm, we are able to calculate this contribution by putting k = 0 in the Golay equation [5], which then reads

An equation of this type was first derived and tested by Taylor [S] and later extensively verified by Giddings and Seager [7]. It is now widely used in the experimental determination of diffusion coefficients with the aid of GC technique [8]. We choose to perform the calculations with D, values taken at the mean pressure of the mobile gas phase in the column, F, since

(3)

istheaverage linearvelocityofthemobile phase.The correction to D, is the same as for < i. e. simply

- L

tm u - -

-

(4)

where D, at normal pressure (P = 1 atm) can be estimated from literature data [9] through interpolation or T3” extrapolation. The resulting equation

(5) 8 In 2 . t’, 2 b, tm

b$(k) = L I - 24 D, t, L

implies that if the components of the sample were not retained, the column contribution to their peak broadening, b$(k), would vary only with their Dm values for a given gas velocity (or dead time, t,). The inner radius, r,, and the length, L, of the capillary are geometrical constants.

For a sample component at k in the chromatogram the total band contribution from the column is then obtained in accordance with the variance addition principle [lo] as

( b o d 2 = b$(k) + bz (6)

where b i is the contribution generated by the mass exchange reaction with the stationary phase [ll]. We can assume that the extra-column contribution at half height, hex, also adds quadratically [lo, 121, which givesfor the experimental band width

b8.5 = b f bi(k) + b: (7)

Combination with the experimentally found linear relation (1) for b0.5 yields

(8)

If the dependence of b$(k) on k is small we can, to afirst approxima- tion, write the component-dependence of b; as

(9)

b: + Pab,k + a2k2 = bzx + b$(k) + b i

b i = ask + a2k2

b$(k) = (b: - bSx) - (a, - 2ab,)k

where a, is close to 2ab,, which would then leave

(10)

for the substance-dependence of b$(k). Consequently, a plot of b$(k) calculated from eqn. (5) versus experimental k would give an intercept b:(k+,) which together with the experimental b,- value would permit the determination of bf from

(1 1) b& = b$ - b$(k+,)

Journal of High Resolution Chromatography &Chromatography Communications APRIL 1979 191

Short Communications 10081

Table 1

b:(k) from eqn. (5). b:(k-o) from eqn. (10) and b ix from eqn. (1 1). n-Alkane data of Kaiser and Rieder [4]. (L = 20 m, 2r, = 0.31 mm, temperature = 10loC, stationary phase = 0.23 p SE 52, carrier gas = N2).

C9 G o c11

c12

c13

Dm (crn2/s)(2)

0.087 0.080 0.073 0.070 0.067

%(k-0) (s2) b: (s2) b2x (s2)

87.4 88.2 106.8 146.0 154.4 276.0 224 1.20 1.20 1.16 1.11 1.10 1.05 1.07

b$(k) (S2)

0.201 0.196 0.192 0.191 0.190

0.198 1.28 1.08

0.205 0.200 0.196 0.195 0.194

0.202 1.12 0.92

0.332 0.31 9 0.307 0.302 0.298

0.324 2.19 1.87

0.678 0.641 0.605 0.591 0.577

0.91 5 0.860 0.806 0.785 0.762

5.01 4.64 4.27 4.1 2 3.96

2.69 2.50 2.31 2.23 2.1 5

0.656 1.28 0.62

0.883 2.31 1.43

4.80 5.86 1.06

2.57 2.96 0.38(3)

(1) Estimated through interpolation of reported excess pressure range 0.4 to 0.1 bar. (2) Estimated from ref. [9] through interpolation or T3h extrapolation (3) See text.

Results Application of eqn. (5) to the n-alkane data of Kaiserand Rieder [4] results in the bg(k) values of Table 1. Linear regressions according to (10) yield the b$(k-o) intercepts shown, with slopes (a, - 2ab0) varying from 0.01 to 0.1 while 2ab0 is of the order of 2 to 13. Thus, the approximation (9) seems to be justified in this case.

We see that the first six b ix values in Table 1 are of the same magnitude and show no significant dependence on flow rate. Such a behaviour is to be expected if b i x is determined mainly by the length of the syringe injection period [13]. The arithmetic mean is 1.1 6 s2 with a standard deviation of 0.43 s2, which means that the value b:x = 0.38 s2 obtained for the smaller injector volume (t, = 224 s in Table 1) is not significantly smaller than the other six b:,-values. The probability is greater than one in six for such a low value to occur. Thus Kaiser and Rieder seem to be correct in their first statement that b, is rather insensitive to this type of volume change. A comparison of b ix with bg(kd0) reveals that the second part of their statement, i. e. that b, is determined by external broadening, is correct only for high flow rates (low tm), when the external contribution b ix is at the most about 6 times the column contribution b;(kto).At low flow rates (high tm) the diffusion broadening in the column becomes important, leading in fact to an inverted ratio, the column contribution being about 6 times the external one.

These results are much more satisfying than those obtained earlier [l] according to Smuts eta/. [2]. Yet, the bgx-values of Table 1 have to be used with care, since there are still approxi- mations involved. The accuracy depends mainly on the applica- bility of the "abt" concept. For instance, it has been found [l 11 that bo(k) values from eqn. (5) are often a linear function of k/(l+k) rather than k. In the case of Kaiser and Rieder (Table 1) plotting b;(k) versus k/(l+k) would increase the intercepts 10 to 35 percent, but we do not see any reason for using those values in

combination with the present derivation. Upon exclusion of one or two of the latest peaks as in ref. [l], b$ is generally increasing more than b$(k-o), which leads to about 50 percent higher, and less constant, b:,-values. The uncertainty of the Dm-values is of minor importance.

References

[l] 0. Nilsson, J. HRC & CC 2 (1979) 147.

121 T. W. Smuts, T. S. Buys, K. de Clerk and T. G. Toit, J. HRC & CC 1 (1978) 41.

[3] R. E. Kaiser, Chromatographia 10 (1977) 323.

[41 R. E. Kaiser and R. Rieder, Chromatographia 10 (1977) 455.

[51 M. J. E. Golay, Gas Chromatography, Amsterdam 1958, D. H. Desty (Ed.), Butterworths, London, p. 36.

[61 G. Taylor, Proc. Roy. SOC. (London) A 219 (1953) 186.

171 J. C. Giddings and S. t. Seager, Ind. Eng. Chem. Fundarn. 1 (1962) 277.

[81 V. R. Maynard and E. Grushka, Advances in Chromatography 12 (1 975) 99.

[9] Landolt-Bornstein, "Zahlenwerte und Funktionen", 5. Teil, Bandteil a, Transportphanomene I, Springer-Verlag, Berlin 1969. pp. 550.

[I01 J. C. Sternberg, Advances in Chromatography 2 (1966) 205.

[I 11 0. Nilsson, Paper submitted to Chromatographia 7 March, 1978.

[I21 0. Nilsson, Chromatographia 10 (1977) 519

[13] G. Guiochon, Anal. Chem. 35 (1 963) 399.

MS received: 20 March 1979

192 APRIL 1979 Journal of High Resolution Chromatography & Chromatography Communications