Table I. Numerical Results (Last reported significant figure is approximate only)
DimensionlesE Time1.23
-3.0 -1.5 0
4.73
3.89 -2.12 -2.78
S1.5 +3.0 a
4
4 = 0.1
4 = 0.01 CR
2t/l/ur
K
a h 0
44.0 34.0 18.8 -40.8 47.0
0,0226
0.152 0,381
0,414 0,283
GR 0.313 1.11
0,570 -0.964 -0.766
a/qo
K
0.0887 0.377 0.846 0.717 '0.275
20.1
GR 0.0489 0.141
17.3 11.1
0 I0109
19.6 13.5
-0.142
-0.0670
1.0
=
4/Po
K
0 OM0 0,542
15.5 15.1
I
0,997 0,598 0.117
a
16.0
14.5
Insufficient significant figures t o calculate.
The present analysis does not include liquid phase diffusional resistance. If this lag is included, a somewhat higher O.ld,g value than -Dywould be required as the criterion. A rough correction could
The boundary conditions are: =
kt At
T
R; p
= go exp(-t2/2'")
= 0: 6q/6r = 0
(3)
The solution was accomplished with the Fourier transform. Upon inverting, the quantities of interest were obtained in integral form, and the integrations were performed numerically. The results were then "forced" into an equation of the form of Equation 1, ellowing K to be a function of time to observe how widely it varies. The results are shown in Table I. If one takes K to be a constant equal to 15, the linearized Equation 1 can be used to a fair degree of approximation, provided that C: is greater than about With rearrangement, this criterion is equivalent to
The dispersion (al)to be used in thk calculation should be one which typifies the pulse through most of the chromatographic column. Theory predicts that the dispersion will increase with the square root of length down the column, starting a t essentially zero at the top ( 3 ) . Thus a dispersion equal to the value a t the exit of the column will be conservative-Le., sharper than actual-for all but the upper '/g of the column. Using this, one obtains as the criterion
or in round numbers
This criterion will usually be satisfied in practice. For liquid chromat,ography, typical values are d, = 0.01 em., D,= lO-7sq. cm./sec. which means that the criterion will be satisfied if (rJexit is larger than 100 seconds. 0
ANALYTICAL CHEMISTRY
be made using a pseudo factor equivalent to the total resistance. However, if the criterion is satisfied with a severalfold margin, there will be no trouble in any event. CONCLUSION
parameter in the linearized Equation l u$ = spread parameter ("dispersion"), sec. (defined by Equation 3)
K
=
LITERATURE CITED
( 1 ) Glueckauf, E., Trans. Faraday SOC. 51, 1540 (1955). (2) Hamilton, P. B., Bogue, D. C., Anderson, R . A , , ANAL.CHEM.32, 1782
(1960).
RECEIVEDfor review July 11, 1960 Accepted September 14, 1960.
The linearized solid phase mass transfer equation
will ordinarily be satisfactory in elution chromatography. An approximate criterion for its use is that the pulse leaving the column satisfy the inequality
A severalfold margin should be allowed when liquid phase resistance is a significant part of the total resistance. NQMAENCLATURE
Roman Symbols d, = particle diameter, cm. D, = solid phase diffusion coefficient, cm./sec. B(~''o) ''= s(rlR)R' the gradient evaluated a t the surface N' = molar flux, moles/cc. solid sec. q = concentration in solid phase, moles/ cc. solid g = mean Concentration in sphere, moles/cc. solid q. = constant in Equation 3, moles/cc. solid qR = concentration in the solid phase a t the surface of the sphere, " . moles/ cc. solid R = particle radius, cm. r = radial position from center, cm. t = time, secs. Greek symbols
E
=
-
dimensionless pulse spread, dSDau8 2Ra
Corrections Surface-Active AI ky Oxide Condensation P ucts. Determinatio ethylene Glycols in Cationic Ethy I n this article by J. V. Killheffer, Jr., and Eric Jungerniann [ANAL. &EM. 32, 1178 (1960)l the first author's name should be J. V. Killheffer, Jr.
Infra red Qua w t it ative Analysis Data The symbol for micron, p, was incorrectly used in place of the spmbol for millimicron, mp, under the columns showing the analytical wave lengths and base line points and in the absorbance matrices of the following six infrared analytical methods: 68-126, 68-127, C8-128 [ANAL.CHCM.32, 730 (1960)l; CS-130 and C8-131 [Ibid., 32, 888 (l960)]; 68-135 [Ibid.,32,1212 (1960)l; and CS-144 [lbid., 32, 1723 (1980)l.
