1686
Vol. 61
NOTES
smaller cross-section than the molecule of nitrogen, can penetrate into the interstitial regions of our graphitized carbon while nitrogen cannot. We have no very convincing explanation to offer as to why the nitrogen should cause swelling in the graphite studied by McDermot, et al., although it does not do so in our sample. It may be noted that they obtained a wide hysteresis loop in the region of high relative pressures while we did not. This suggests that their graphite sample was porous, while ours was non-porous. Thus the edges of the graphitic layers between which a gas molecule might penetrate would possibly be more accessible than in our non-porous material. I n order to account for adsorption isotherms of the type shown in Fig. 2, i t was suggested in an earlier publication* that two dimensional patches of adsorbed ammonia may be initiated at oxygen com lexes on the carbon surface. Since all oxygen has presumabyy been removed from the particular sample under study here without substantially altering the form of the ammonia isotherm, it seems more probable that the patch initiating sites are topographical irregularities in the nearly homogeneous carbon surface.
EQUILIBRIUM OPERATION O F CHROMATOGRAPHIC COLUMNS WITH LONGITUDINAL DIFFUSION: FINAL FORM FRONTS
Consider steady feed of a solution of constant composition to a long packed column in which no gross radial concentration gradients exist. Let C = concn. of solute in the fluid stream, mass per unit
vol. of soln. n = amount of solute sorbed in packing, mass per unit vol. of packed bed v = velocity of fluid through interstices of bed 2 =: distance along bed, measured from upstream end
D = effective difusivity in axial direction CO = concn. of solute in feed 01
t
= fractional void volume in bed = time
n(c) represents the equilibrium relation between n
and c. It can be shown by a material balance over a section of the column and assuming local equilibrium that Introducing the variables
BY E. N. LIGHTFOOT Department of Chemical Engineering, The University of Wisconsin, Madison, Wisconsin Received September 9 , 1967
Axial mixing has long been recognized as of importance in chromatographic columns, but has not often been considered in theoretical developments. Lapidus and Amundson' and Bastian and LapidusZ have considered longitudinal diffusion for the case of linear isotherms. A solution is given below for non-linear isotherms and equilibrium operation in those cases where a final form front is approached. Such fronts may be expected under conditions of constant feed in the sorption of solutes exhibiting convex isotherms (e.g., Langmuir) or elution of those exhibiting concave isotherms (e.g., in the Donnan exclusion of an electrolyte from a n ion exchange resin) if the column is reasonably long. Work of Lapidus3 suggests that the asymptotic form is approached quite rapidly in practice. Axial mixing in a packed bed is due to a number of mixing processes, but can be described by a law similar to Fick's law for molecular diffusion in a long packed bed of sufficient tube-to-particle diameter ratio. This is the approach used below. The solution offered has been found useful, e.g., in the elution of glycerol from Dowex 50 ionexchange resin. (1) L. Lapidus and N. R. Amundson, T H IJOURNAL, ~ 66, 984 (1952). (2) W. C. Bastian and L. Lapidus, ibid., 60, 816 (1956). (3) L. Lapidus, "Ion Exchange," No. 14, Cberu. Eng. Prog. Symp.
Belies.
6 = tvZ/D
equation 1 becomes
For large values of X under the above-mentioned conditions the concentration profile will asymptotically approach one with the following characteristics (1) C = C ( X ) ; (2) there is no net flow of solute relative to a cross-section of the column a t constant X . By use of condition 1 equation 2 may be reduced to a simple ordinary differential equation. By use of condition 2 this may be integrated to give
where subscripts 1 and 2 indicate any given points in the column. Position relative to the upstream end a t any time may be simply determined by an over-all material balance. This solution appears to be useful for correlation of data in a number of important cases, as will be indicated in a subsequent paper. It is also useful for determining equilibrium adsorption isotherms. This is especially important since the work of McHenry and Wilhelm4 and others on axial mixing throws considerable doubt on the previous column methods for determining equilibrium isotherms. (4) K. W. McHenry and R. H. Wilhelm, A . I . Ch. E . J . , 8 , 83 March (1957).
1687
Dec., 1957 ADDITIONS AND CORRECTIONS
+
VOL. 59, 1955 Clarence Postmus and Edward L. King. The Rate Law for the Forward and Reverse of the Reaction Cr(OHz)e+++ SCN- = Cr(OH& NCS + + HzO.
readhji. Ineq. (3.2.15), x s h o u l d r e a d
Page 1221. In footnote (17), line 2, for “-13 and -14” read “ -9.,, and EDWARD EDWARD L. KING
qp = d(a,)d = (
+
+
the right-hand side of eq. (2.2.19), wi w i should read wj wi. Insert a brace, 1, after 2 in the middle part of eq. (3.2.1). On the left-hand side of eq. (3.2.2), hij should
+
VOL. 61, 1957 Wien Effect in
2 a2sin hela.
In eq. (7.15),
2
should read
n=l
E.I n the first n=l
(5.2.6),
I n eq. (4.3.3),
qlp
= d(ap) =
should read
E.I n eq. i=l
=1
8
8
should read Zj
. . . should read 8
8
Pages 198 ff. The following corrections should be made. I n the second equation of eqs. (3.2), e - W d should read eiWo). I n the second equation of section 4, 9-j should read q - j . I n eq. (6.8) the term 2 a3 sin h-la should read 1
...
ap) =
E. Ineq.(3.2.21), i=l
g=l
d( -aD)=
Lars Onsager and Shoon Kyung Kim. Simple Strong Electrolytes.
8
8
E.
At the end of the twelfth line in
i#j
the first colum of page (227), xipshould read X P j e In the denominator of eq. (7.2.13), 460 should read 48o.-LARS ONSAQER.
0.D. Bonner and Linda Lou Smith. A Selectivity Scale for Some Divalent Cations on Dowex 50. Page 326. I n col. 1, line 13 from the end, the trade name PAN represents the compound ( 1-( 2-pyridyl-azo)-2-naphtho1.-0. D. BONNER.
term on the right hand side of eq. (9.12), $g(su) should read P g (8”). I n the fifth line on the right-hand column of page (213), w218 should read ~ ~ 2 3 On . the right-hand column of in the fifth line from the bottom should page (214), P ~ z * read PL3*and wz13 in the secnd line from the bottom should ONSAGER. read w~~~.-LARS
E. G. McRae. Theory of Solvent Effects on Molecular Electronic Spectra. Frequency Shifts.
Lars Onsager and Shoon Kyung Kim. Effects in Mixed Strong Electrolytes.
Page 566. I n Eqs. 14 and 14’ for the numerical factor, G. MCRAE. “2.13 X 10-m” read “2.13 X lO-8O/hc.-E.
The Relaxation
Pages 215 ff. The following corrections should be made. I n the first line below eq. (2.2.1),fji(v) should readPji(r). On the right-hand side of eq. (2.2.17), kil should read ki. I n eq. (2.2.19) k i j should read hji. I n the denominator on
Norman E. Weston and David P. Shoemaker. Comment on NasPbal.
A Further
Page 1448. In the title line the formula should read “Na31Pb~.JJ-DAvID P. SHOEMAKER.
