values (Table I) are exceedingly large, ranging from 100 to 500. I n columns of excessive diameter (sometimes referred to as preparative columns) some unusual phenomena occur which require unusual modification to the usual plate height expression of Equation 2 (7). For one thing plate height acquires a dependence on column length. I n order to test whether the present results fall in this category, we have calculated the “transition length,” Ld, for several columns, where (4)
and d, is the column diameter (the criterion presented by Ld is adequate despite the high v values). With one exception these lengths are the order of lo4 cm., much greater than the column length. Thus the columns do indeed fall into the special category, characterized by a lack of equilibrium across the lateral dimensions of the column and by unusual plate height effects. I n order to make some comparisons with the column efficiency which should be obtainable in a small bore tube, we have calculated theoretical plate height values assuming m = 20 (a column of roughly 1 mm. diameter). Equation 11 has been used, neglecting the first two terms on the right because of their small contributions. Estimates of the X, and wI parameters are those obtained from random walk theory ( 2 ) . The results of this calculation are shown by the solid cuve in Figure 1. This curve in no sense represents a conceptual minimum; it should roughly reflect typical experimental values. The experimental work reported here usually, but not always, exhibits a higher plate height than the theoretical curve. Evidence from Gel Permeation Columns. T h e previously cited work of Smith and Kollmansberger (ZO), while not directly comparable to the above data fcr the reasons stated, is nevertheless indicative on several points. First of all it offers evidence in support of the plate height equations used here; specifically, a plot of plate height us. velocity is concave down as predicted by the coupling theory of Equation 2 (the authors also note a correspondence with film diffusion theory). Second, rough values of reduced plate height can be calculated by estimating particle
1000
ANALYTICAL CHEMISTRY
size. Assuming d, = 60p (a value, often used in gel permeation work), and noting that their plate height is typically about 500p,it is concluded that the reduced plate height is of the order of 10. This is roughly consistent with several, total plate numbers mentioned by Moore and Hendrickson (13). These values are comparable with those obtained from good-but not the very best-gel filtration columns. Conclusions. There is no inherent reason why a majority of gel filtration columns cannot be improved in efficiency if enough is understood about the fundamental mechanisms in operation. One element of improvement may involve a reduction in column diameter. This requires a compromise with sample capacity that must ultimately be judged by the operator. Packed columns down to a fraction of a millimeter have been used successfully in gas chromatography (14). With proper detection methods a like diameter could perhaps be applied to gel filtration. Figure 1 shows that, in contrast to most forms of gas chromatography, plate height is not greatly increased at high relative velocities-Le., velocities greatly exceeding that corresponding to minimum plate height a t about v = 1. The coupling phenomenon ( 2 ) , described in the last term of Equation 2, is partly responsible for a slow increase in plate height with velocity in both gas and liquid forms. I n contrast to gas chromatography, gel permeation (perhaps in common with most liquid chromatography) has a relatively small stationary phase term which makes no significant contribution until Y >> 100. Thus high relative velocities can be, and have commonly been used leading to an increased speed separation. The absolute flow velocity is, of course, less than that in gas chromatography, but not as much smaller as indicated by the ratio of those velocities which yield a minimum in the plate height curve. LITERATURE CITED
( 1 ) Boyack, J. R., Giddings, J. C., ANAL.CHEM.36, 1229 (1964). ( 2 ) Giddings, ,J; C., “Dynamics of Chro-
matography,
Marcel Dekker, Inc.,
New York, 1965. (3) Giddings, J. C., J. Chromatog. 13,
301 (1964). (4) Giddings, J. C., J. Gas Chromatog. 1. 12 (1963). (5) ’Giddings,’J. C., J . Phys. Chem. 68, 184 (1964). (6) Gosting, L. T., “Advances in Protein
Chemistry,” M. L. Anson, K. Bailey, J. T. Edsall, eds., Vol. 11, p. 429-554, Academic Press, New York, 1956. (7) Hawkes, S. J., Giddings, J. C., ANAL. CHEM.36, 2229 (1964). ( 8 ) Horowitz, S. B., Fenichel, I. R., J . Phys. Chem. 68,3378 (1964). (9) Knox, J. H., ANAL.CHEM.38, 253 (1966). (10) Knox, J. H., McLaren, L., Ibid., 36, 1477 (1964). (11) Laurent, T. C., Killar‘der, J., J. Chromatog. 14, 317 (1964). (12) Moore, J. C., J . Polymer Sci. A2, 835 (1964). (13) Moore, J. C., Hendrickson, J. G., J . Polymer Sci. C3, 223 (1965). (14) Myers, M. N., Giddings, J. C., ANAL.CHEM.38, 294 (1966). (15) Ogston, A. G., Trans. Faraday SOC. 54, 1754 (1958). (16) Porath, J., “Advances in Protein
Chemistry,” C. B. Anfinsen, M. L. Anson, K. Bailey, and J. T. Edsall, eds., Vol. 17, p. 209, Academic Press, New York, 1962. (17) Porath, J., Pure A p p l . Chem. 6, 233
f\____,. 1 Qfi.?) (18) Porath, J., Flodin, P., Nature 183, 1657 (1969). (19) Porath, ‘ J., Flodin, P., “Protides of the Biological Fluids,” H. Peeters, ed., 10th ed., p. 290, American Elsevier, New York, 1963. (20) Smith, W. B., Kollmansberger, A., J . Phys. Chem. 69, 4157 (1965). (21) Sternberg, J. C., Poulson, R. E., ANAL.CHEY.36, 1492 (1964).
RECEIVEDfor review January 31, 1966. Accepted May 2, 1966. This investigation was supported by Public Health Service Research Grant GM 10851-09 from the National Institutes of Health.
Correction Cont in uo us Ohmic Pola r iza tion Compensator for a Vo It a mmetric A p p a rat us Utilizing Operational Amplifiers I n this article by Dirk Pouli, James
R. Huff, and James C. Pearson [ANAL. CHEW 38, 383 (1966)] on page 383 a n error appears in the legend for Figure 2 . “RL 100 kiloohms” should read “RL Load resistor.”
