GASCHROMATOGRAPHY AT HIGHSOLUTE CONCENTRATION
rate of the rearrangement. Thus, according to Doering and Roth,* 2,6-octadiene was quite stable under conditions where other 1,8diolefins would rearrange readily. If their statement is interpreted to mean that less than 1% of 2,6-octadiene had reacted after 24 hr. at 230', the first-order rate constant would be less than lo-' set.-' a t 230'. Since there was no evidence of other reactions a t the given conditions, the rate constant for the dissociation at the central C-C bond of 2,g-octadiene would also be deemed less than lo-' sec.-'. This puts the lowest limit for the bond dissociation energy of the central C-C bond as approximately 48-50 kcal./
1283
mole, assuming 1014"15 set.-' for the A-factor of the unimolecular decomposition. The maximum stabilization energy assignable for the allyl radical is therefore 18 kcal./mole. Studies along this line are currently being undertaken in our laboratory.
Acknowledgment. The authors are grateful to Professor Hiroshi Tokuhisa for his interest and to Professor Susumu Kinumaki and his group for the spectral data. Thanks are also due Mr. Toshihiko Amatatsu, Mr. Tadashi Kunii, and Mr. Hisahiko Takahashi for their assistance in the experimental work.
Toward a Generalized Theory of Gas Chromatography at High Solute Concentrations
by D. L. Peterson and F. Helfferich Shell Development Company, Emeryuille, California (Receive1 October 22.
The working formula of gas chromatography is based on the assumption that the mobilephase velocity remains constant throughout the region of a solute pulse. Removal of this assumption leads to a more meaningful deduction of the usual result and eliminates contradictions otherwise encountered. In particular, the correct formula for the migration of solute concentration pulses in a column already equilibrated with a feed gas containing the solute at a high concentration results only when due allowance is made for the variation of mobile-phase velocity in the region of the pulse. The influence of this factor on the self-sharpening and nonsharpening character of concentration boundaries requires a new criterion, according to which it is possible for a nonlinear isotherm to produce a symmetrical chromatographic peak. Concentration-pulse migration-rate measurements at high solute concentrations make possible the measurement of nonlinear sorption isotherms. The method appears well suited to binary sorption equilibrium measurements, but less so for multicomponent equilibria.
Introduction Variations in solute concentration along the length of a chromatographic column are necessarily accompanied by changes in the velocity of the mobile phase. This occurs because sorbed solute advances only through the mobile phase, so that the total flux
of solute and carrier molecules must be greater where the " I n t sorbed is higher. While the working formulas of frontal analysis contain these different velocities preceding and following concentration fronts,l v 2 velocity differences in elution chromatography have ordinarily been presumed negligible and have Volume 69, Number 4
April 1966
1284
been omitted from derivations of pulse velocities. Indeed, only if the maximum concentrations attained during the entirety of passage of a pulse through a column are sufficiently small as to make this presumption plausible does the familiar relation3 between retention volume VR, stationary-phase volume vs, void volume V,, and concentrations of solute 1 in mobile and stationary phases, C1and €1, respectively, apply
D. L. PETERSON AND F. HELFFERICH
Influence of Equilibrium
We begin with the recognition that mobile-phase velocities will be different in zones of different solute concentration and consider first only the influence of equilibrium on migration rates of boundaries and pulses. We thus assume that concentrations and mobile-phase velocity change discontinuously a t a boundary between values which are everywhere constant within their respective zones (one-dimensional column and linear, ideal chromatography). I n all but one of the following cases this condition is equivalent to the two assumptions: (1) local equilibrium and absence of all diffusional effects (no boundary or band Otherwise, the higher velocity in the region of the pulse broadening due to kinetic factors); and (2) the ;soleads to a lower retention volume- than given by eq. 1. therm of the solute is linear in the range of concentraSchay’ has called attention to this phenomenon, and tions contained in the boundary or band (no broadening Golay‘ has presented a criterion by which one may due to equilibrium factors). decide the maximum allowable sample size consistent Use will also be made of the following assumptions: with the precision with which VR is to be measured, (3) the range of solute concentrations covered in the such that the velocity effect may be neglected. band or boundary is negligible compared to the conIt is the purpose of this communication to show that centration of carrier; (4) the mobile phase is an ideal velocity variations, except in two very special cases, gas; ( 5 ) there is no influence of heat of sorption; and are never absent from chromatographic experiments, (6) decreases in the void volume due to stationaryand to discuss situations where they cannot be prephase volume increases are negligible (V, is constant). sumed so. Two cases are to be distinguished: (1) It will be possible to relax certain of these assumpthe use of large samples in a nonsorbed carrier gas, tions in some casesBas summarized in Table I. The and (2) the use of small concentration pulses or steps effects of removal of either assumption 1 or 2 form in a feed gas containing solute at high concentration. the subjects of later sections. We shall invariably The first case, of which use has been made in measureassume that: (7) the pressure drop is negligible; and ments of nonlinear isotherms from peak a ~ y m m e t r y , ~ (8) solute concentrations in a supported stationary is included in Golay’s* discussion and will only be inphase (g.1.c.) are not affected by the support. troduced here for purposes of comparison, Interest The derivations are concerned with gas-solid or in the second owes to the possibility of measuring sorpgas-liquid chromatography, in which the effects of tion equilibria at other than infinite dilution by simple mobile-phase velocity variations are far more signifiretention vo iume measurements, as opposed to deduccant than in liquid-phase Chromatography, for which tions from peak asymmetry. A description of the DeVault’s treatment3 will ordinarily be entirely ademore elegant tracer-pulse technique for determining quate. nonlinear isotherms directly from retention volumes, as well as a short critique of various other chroma(1) G. Schay, “Theoretische Grundlagen der Gaschromatographie,” tographic methods for such purposes, was presented Veb Deutsche Verlag der Wissenschaften, Berlin, 1961, Chapters 1 else~here.~.’ and 4. In the following, results anticipated from concentra(2) C. H.Bosanquet in “Gas Chromatography 1958,”D. H. Desty, Ed., Butterworth and Co., Ltd., London, 1958,p. 107. tion-pulse measurements are compared, within the (3) D. DeVault, J . Am. Chem. Soc., 65, 532 (1943). scope of linear, ideal chromatography, with those of (4) M.J. E. Golay, Nature, 202, 489 (1964). other chromatographic procedures for the simplest (5) See, for example, J. F. K. Huber and A. I. M ,Keulemans in example of only one sorbable solute in a nonsorbed “Gas Chromatography 1962,” hl. van Swaay, Ed., Butterworth, Inc., Washington, D. C.,1962,p. 26. carrier gas. Next, concentration-pulse migration in (6) F. Helfferich and D. L. Peterson, Science, 142, 661 (1963). the general binary and in multicomponent systems is (7) Tracer-pulse chromatography and possibly the influence of the considered. A study of the influence of mobilevelocity effect in chromatographic measurements at high solute conphase velocity on the criterion for self-sharpening bouncentrations appear to be contained in a treatment of F. I. Stalkup and H. A. Deans, A.1.Ch.E. J., 9, 107 (1963). daries follows. Finally, questions concerning the re(8) Assumption 4 can, of course, in any event be removed through the duction to practice of these idealized results are introduction of an equation of state for real gases, but it is retained taken up. here for simplicity and clarity. ~
The Journal of Phyaical Chemistry
GASCHROMATOGRAPHY AT HIGHSOLUTE CONCENTRATION
Table I :
1285
Underlying Assumptions of Derivations CaseFrontal analysie
Tracer pulse
Concentra-
X
X
X
Removable
Condition met Condition met Not required Condition met Condition met
X
X X
X X
Assumption
sample
Large aample
1. Local equilibrium and no diffusional spreading 2. Linear isotherm in range of boundary 3. Concentration change is small
X
X
X
X
X X X X X
X X
Small
X
4.
Gas-phase ideality 5. No heat of sorption 6. V , is constant 7. No pressure drop 8. Inert support in g.1.c. 9. Constant mobile-phase velocity 10. Equal equilibrium isotopic distribution
X X X X
X X
tion pulae
X
X X X
Superceded X
Binary Mobile Phase with One Sorbable Component. In this section we consider a binary feed gas consisting of one sorbable c'omponent (the solute) and one nonsorbable component (the carrier). A . Small-Sample Pulse in a Pure Carrier. It will be instructive to consider first an ordinary chromatographic measurement at vanishing solute concentration, which technique is distinguished below by the term "small-sample case." It is pictured in Figure la, in which the shaded region shows the mobile- and stationary-phase concentration profiles C1 and c1 of an idealized pulse of sorbable solute in moles per unit volume of the respective phase.9 The broken outline shows the pulse after it has advanced a distance dz along the column. In this one instance, the usual additional assumption is tentatively made that: (9) the mobile-phase velocity v, within the boundaries of the pulse is equal to the velocity z10 ahead of and behind the pulse. The rate of migration of the pulse, or what is equivalent in the present approximation, of the leading boundary, is obtained by writing the mat4erial balance around the element of column volume whose length is dz. If the length and void volume of the column are, respectively, L and V,, then the amount of solute entering this element in the time dt during which the boundary traverses dz is C1V,vodt/L, and the increase within it is (CIV, C,Va)dz/L. Since none leaves, these two quantities must be equal and we have for the velocity of the boundary
+
dr
a ) SMALL SAMPLE
A
d) TRACER PULSE
I
---
e) C O N C E N T R A T I O N PULSE I
-
////ryo
/ / / /
/
-
vi / / / / /
~~
-
///I/ "0 / /
//
/
Illustration of solute concentration profiles in the mobile (upper half) and stationary (lower half) phases and of their progress through a n element of column length dz.
The retention volume is VR = v g h / v b
(3)
Together, these equations lead to eq. 1 or in terms of the adjustJedretention volume VR' VR - V,, to VR'
=
ClVs/c,
(4)
B . Large-Sample Pulse in a Pure Carrier. This case, which differs from the small-sample case only through its divestiture of the assumption of constant velocity, (9) The use of these concentration units is arbitrary, and any other choice may be made so long as the units are consistent.
Volume 69, Number 4
April 1965
D. L. PETERSON AND F. HELFFERICH
1286
is illustrated by Figure lb. of concern are
The amounts of solute
entering : C1V,v id t/ L leaving: none ClV,)dz/L increasing: (CIV,
+
The corresponding amounts of carrier are entering: (C - C1)V,vidt/L leaving : C‘V,vodt/L increasing : - CIV,dz/L where C is the total molar concentration in the mobile phase. Equating the net flux to the increase, we obtain for the solute (5) and for the carrier
vb
=
210
Combined with eq. 2 this leads to the requirement c1 = 0, whereas the solute is supposed to be sorbable. On the other hand, there arises no contradiction a t any point in the derivation of eq. 8 if C1 is assumed to be small. These considerations verify the necessity that v, > vo no matter how small the size of a sorbable solute sample. That is, the sorbed molecules provide a continual source and sink within the band; coherence of the band exists, and advance of sorbed molecules occurs only i f the mobile-phase velocity i s higher within the band than outside it. C. Frontal Analysis in a n Initially Solute-Free Column. In frontal analysis, the different velocities on each side of the boundary persist a t the respective ends of the column (Figure IC). I n terms of the effluent volume VR~’, which moves through the column with velocity vo, its value from eq. 9 is VRo’ = ccclvs/Clc
Elimination of v i leads to
2’b
=
ClV,
+
ClVgVO Cl(C - Cl)V,/C
(7)
clV,/cl
(11)
CcvR,’/C
(12)
It may be noted that vRo’ =
!??.!?(1 Cl
-
$)
(8)
An expression equivalent to eq. 7 was derived by G01ay.~ Equation 8 reduces immediately to the smallsample result when C1 is negligible compared with C. If this is not true, the actual retention volume is lowered from that obtained with a small sample by the niobilephase mole fraction of solute C1/C. Because of the variation of C1 across a real band and with the length of column traversed, a criterion of negligible influence of v , > vo on a retention volume measurement must involve the initial, maximum value of C1 at the column entrance. As a derivation of eq. 1 for a small sample, that represented by eq. 5 through 8 with C1 0, it is the front boundary which becomes diffuse and the rear boundary which is selfsharpening. The criterion for a self-sharpening rear boundary, if v, = uo, is (d2Cl/dC12) > 0. From eq. 22 it is seen that, with isotherms of such curvature, the occurrence of higher velocities at higher concentrations counteracts the effect of equilibrium on the boundary shapes. In the special example of a linear isotherm, with (d2Cl/dCI2)= 0, all regions of a band would migrate with equal rate if the mobile-phase velocity were constant. Spreading and sharpening through equilibrium effects would then be absent; the front as well as the rear boundary would retain their original shapes. It is apparent, then, that allowance for the velocity variation in the mobile phase will alter the self-sharpening criterion, and that an absence of sharpening and spreading through equilibrium effects will be found with a particular nonlinear isotherm shape; i.e., it i s possible for a nonlint,ar isotherm to produce symmetrical peaks. The criterion for self-sharpening boundaries is presented below for the simplest case of a solute pulse in a nonsorbed carrier flowing through an initially solutefree column, and its significance considered ; more general cases are thereafter briefly outlined. The criterion for self-sharpening front or rear boundaries for a single solute pulse in a nonsorbed carrier follows immediately from the formula for the retention volume derived without the limitation that u, = uo. I t is required, then, that eq. 8 for a large sample applies with the same value of VR’ to all concentrations if the boundary is neither to sharpen nor to spread.I5 This means simply that it is in ternis of cl(C - C1)/CIC, and not of that the criterion is to be drawn. The same is true in the frontal analysis case (in an
cl/C‘,,
The Journal of Physical Chemistrv
(23)
Then front boundaries are
and rear boundaries, if
where €1 is the isotherm under test.I6 The difference between the exact criteria 24 and 25 and those derived with v, = vo is greatest when the distribution coefficient C1/Cl is high and the total mobile-phase concentration C is low, and when the mole fraction of solute in the mobile phase Cl/C is large. The former conditions prevail particularly at lower column pressures, and the latter one in frontal analysis. It seems probable that under the conditions of gas-phase elution chromatography, the term on the right-hand sides of relations 24 and 2.5 could attain high values. The practical meaning of this circumstance is that the sort of band asymmetry characterized by a steeper leading than trailing boundary, and generally associated with isotherms having negative curvature, can also be produced by isotherms having appreciable positive curvature. In either event, reductions in C1 will not completely eliminate the asymmetry even if such a reduction is carried to the point where the isotherm is practically linear. If an experimental lower limit on C1exists in the range of a nonlinear isotherm, then relations 24 and 25 imply that the resulting asymmetry in the chromatogram may be reduced by increasing the column pressure.” The self-sharpening criterion for concentration steps and pulses of a single solute in a nonsorbed carrier (in partially presaturated columns) follows from eq. 14, which implies that the balance point occurs when (15) Strictly, this assertion requires a demonstration that the migration velocity Bb(c1) of’each portion of the boundary, e.g., that with extreme concentrations CI, C1, and CI XI, SI?,, is solely ‘a function of CI. This is ensured by eq. 17. (16) It is. of course, possible that conditions 24 or 25 apply only over a part of the range of concentrations occurring within the respective boundary. The rather complicated sharpening behavior of boundaries in such cases has been discussed by Tudge (A. P. Tudge, Can. J . Phys., 39, 1600, 1611 (1961)) with the assumption that L’, = ao. One can readily extend Tudge’s results to the case where n , # BO with the aid of conditions 24 and 25.
+
+
GASCHROMATOGRAPHY AT HIGHSOLUTE CONCENTRATION
where SCl = C1 - C? and SCl = Cl - Cl0, Cl0 and €10 denoting here the concentrations ahead of the step. The criterion for a self-sharpening leading boundary is then d2C1
-dCi2
