Ca piIIa ry Programmed Tem peratu re Gas Chromatography Some Theoretical Aspects H. W. HABGOOD Research Council of Alberta, Edmonton, Alberta
W. E. HARRIS Deparfment of Chemistry, University of Alberfa, Edmonton, Alberfa
b Previously developed relationships for the retention characteristics and intrinsic resolution in programmed temperature gas chromatography are considered for the special case of capillary columns. Capillary columns have a free volume or dead space per gram of stationary phase about two orders of magnitude greater than that for packed columns. This restricts much of the effectiveness of temperature programming to very low values of the ratio, r/F, of heating rate to flow rate per gram of solvent. Even with low r/F the large dead space tends to b e detrimental, and with higher values it causes the intrinsic resolution to fall off rapidly. However, capillary columns permit very much larger flow rates per gram of solvent so that programs with lower r/F values than for packed columns are readily attainable. Two hypothetical compounds previously examined were chosen for purposes of illustration. It is shown how a significant decrease in analysis time without loss in resolution can b e expected with temperature programming of a capillary column.
To is the column temperature a t sample injection, and T E is the retention temperature-i.e., column temperature a t peak maximum. The specification of F and V Tin terms of 1gram of stationary phase, which was not included in the original presentation, adds a greater measure of generality without changing the relationship of Equation 1. The right side of this equation, for any given initial temperature is determined by the retention characteristics of the solute-solvent pair; the left side characterizes the particular program. This equation shows that the important quantity in PTGC is the ratio of heating rate to flow rate rather than either one alone. Resolution, defined (1) as the ratio of peak separation to average peak width, in PTGC may be expressed in the form
where n is the number of theoretical plates for each of the relevant chromatographic peaks and Ri is the intrinsic resolution. The intrinsic resolution is defined (4) as F
Ri
P
publications have discussed the general theory of programmed temperature gas chromatography (PTGC) in which the retention temperature of a solute is related to its isothermal retention volumes (7) and the factors contributing to resolution in PTGC (4). The fundamental relation between isothermal chromatography and PTGC may be given by
=
7
ATR o a"
(3)
REVIOUS
f
=
JT;R
dT VT
where r is the heating rate, F and VT are the carrier gas flow rate and isothermal retention volume, respectively, per gram of stationary phase corrected to mean column pressure and 0' C., 882
ANALYTICAL CHEMISTRY
where ATR is the difference in retention temperatures of the two solutes and V T Ris the isothermal retention volume a t the retention temperature. The intrinsic resolution is a quantity characteristic of the solute-solvent properties and of the program, but independent of the number of theoretical plates. The behavior of Ri as a function of r / F for a wide variety of solutesolvent interactions was summarized in Figures 2 and 4 and Table I1 of reference (4);it was found that R, frequently passed through a maximum with increasing r / F . This maximum tends to be more pronounced n-ith larger R, and to occur a t r / F values usually less than one. Comparison was made with isothermal intrinsic res-
olution, R ~ Tdefined , (4) as the ratio of the difference in retention volumes to the average retention volume, AVlV,,. Ri mas always greater than R ~ Ta t a temperature equal to the average retention temperature and as r / F approached zero the two became identical and equal to R $ T a t TO. Temperature programming may, of course, be applied to capillary columns (11) as well as to conventional packed columns. Two characteristics in particular, both derived from the relatively large gas to liquid ratio in capillary columns, make capillary PTGC significantly different from packed column PTGC. First, a capillary column is commonly operated under much higher flow rates per gram of solvent than are readily attainable with a packed column. For example, typical values might be in the ranges 50-5000 and 2-20 ml. per gram per minute, respectively. Thus, for heating rates which are convenient, it is possible to use r / F values lower by 10to 100-fold than with packed columns. Second, the larger dead space volume per gram of solvent, Vde, contributes proportionately more to the isothermal retention volumes, VT, of Equation 1. Values of V d s for capillary columns may range from 100-1500 (9) as compared with 3-10 ml. per gram for packed cohmns. A commonly used expression for VT is
where A and AH are constants. It was shown (4)that even for packed columns, omission of the dead space volume significantly affected the retention temperature and intrinsic resolution as calculated by Equations 1 and 3. Giddings (6, 6) has presented a number of useful approximations for cases where the dead space is a small part of the total retention volume. However, in capillary columns the dead space becomes a dominant term and as a rule cannot be neglected even in approximate calculations.
of the solute a n is a negligible part of the total. For example, at 40' C. the dead space contributions are 4.4 and 4-40ml. for the packed and capillary
20
100
200 Tempe, a t ure,'C
350
4 GO
.-
Figure 1. Characte. urves for air and two hypothetical solute-solvent systems calcuiated for typical packed and copillory columns with an initial ternperature of 20" C. Ccnrpound I V r = 1 O-' exp (9000 RTJ Va. l273/n Compound Ft vr 10-8.' exp do,ooo/n~ vd# (273/n
-
+
Packed Column; vd. = 5 Capillary Cdumm V d l = 500 VT and V,il In ml. par gram of solvent
The purpose of this study is to xamine the implications of incresed olumn dead space on both retention haracteristics and resolution. PTCC CHARACTERISTIC CURVES
The relationship of Equation 1 may le expressed by characteristic curves in :hich r / P is plotted against retention emperature. -4. number of euch curves 3r typical packed columns have been hown previously ( 4 ) . For the hypohetical case of injtiaI temperature near bsoIute zero, these curves eneral similarity of sha / F is initially very c!ose to onsiderable range of temp hen increases at an accelerating rate. :he general region of this in ~ccurs at higher temperatures iigher the heat of sol oIute in the soIve tarting temperatures, the r/F origin is hifted to the starting temperature, and
+
stationary liquid and tne capuiary column 500 ml. per gram. The figure slso shows the air peak characteristic curvm for the two cases. The slope of the air peak curve which is also the limiting slope of a solute curve is much steeper, and the separation between the sir and solute curves is much greater for the packed column than for the capillary column. The theoretical treatment of Rowan (9) would s gest. that this separation of air an soIute curves wodd be independent of the dead space volume. This is clearly not true.
T
in relative contributions terms of Equation 4 to the tion volume. At low temperatures even the large dead space volume of a
columns, respectively, and both are small compared with the net retention volume for compound F of 4800 mI.; while at 200° C., the dead space volumes are 2.9 and 290 ml. and the net retention volume of 21 mI. is still large compared with the packed column dead space but small compared with the capillary EpaCe. Because of the above facta one would expect lower temperaturea to be generally preferable for capillary columns than for packed columns. Consequently for any given starting tcmperature lower r/F programs will be desirable for capillary PTGC. Furthermore, the normal operating limitations will generally require lower r/F programs for capillary columns. Thus, the value of r/F for a packed column may well be ns high as 10 with, for instance, a 3-meter column containing 7 grams of solvent heated a t 50' per minute with a fiow rate of 35 ml. per minute. While practical operating Iimib for capillary columns are not so well established in chromatographic literature, Desty and Goldup (2) have used column flow rates in the range M) to 5OOO ml. per gram pcr minute. Thus, a heating rate of 50' per minute would correspond to a maximum r/F of 1. RESOLUTION
As indicated in Equation 2, the resolution depends on the product of the intrinsic resolution and the square root of the number of theoretical plates. A major advantage of capillary columns over packed columns is that large values for n may be obtained rather easily; for example, 10,OOO pIates per milligram m compared with, say, 1W plates per gram i n a packed column. Early investigatore expected concomitant improved isothermal separations. However, as Purnell (8) has pointed out, the very large dead space which contributes to the large n value+ does not assist in the separat terms of the present treatm isothermal intrinsic resolution, AV/V.,, is emsller than for packed columns due to the'imcaeed contribution of dead space to V,,. The decreased intrinsic
gure rhaw the graphkal srotwsc for E and F,
InlMal temperakkr
vo
8, JUL
-
-r---
~
T
T
-
l 2 C
04
+
Capdlary
021
0
3 301
1
0 IO
0 9,
'IF, d e g Figure 3.
gm.ml
1.0
0
'
Intrinsic resolution for compounds
E and F
of Figure 1 with packed and capillary columns The behavior of R, in PTGC for capillary and packed columns is consistent with these observations. Intrinsic resolution as a function of r / F for the pair EF a t a starting temperature of 20" C. is shown in Figure 3. The curve for the packed column is the same as shown previously (4) except that i t has been ehtended one decade lower in r / F . The intrinsic resolution at the starting temperature is slightly lower for the capillary column, reflecting the contribution of the dead space to the retention volume. The divergence from the packed column curve only becomes pronounced for r / F values corresponding to retention temperatures where the dead space is a major part of the total retention volume. I n this region the capillary R, rapidly becomes very small. It is of interest with this particular pair to note that extending the calculations for the packed column one decade lower in r ' F from the previously published curve, revealed a maximum in R, which !vas not apparent before. I n terms ot present practice with packed columns, this very low value of r / F would be difficult to attain. Comparison of Figure 2 with the more complete set of characteristic curves published previously ( 4 ) suggests that one can expect improvement in intrinsic resolution for capillary PTGC analysis of these two compounds by use of initial temperatures somewhat lower than 20' C. Restriction of the program to low r / F will still be necessary as implied by Figure 1. Giddings has suggested (6) as an alternative criterion for good resolution that the column "dead temperature'' should be kept low. The dead temperature, which is the temperature rise to the air peak, tends for any given column to be roughly proportional to r / F and, as already noted, R, has its maximum value a t a low, although not necessarily zero, value of r/F. On the basis of this dead temperature criterion Giddings suggested that a particularly low r / F value would be desirable for capillary PTGC operation. Such, of course, is the conclusion of the present study and the example chosen shows 884
ANALYTICAL CHEMISTRY
clearly that R, decreases rapidly as the dead temperature increases with increasing r / F . Howeyer, the dead temperature has less quantitative usefulness and the limiting value of 2Cb.30" C. suggested by Giddings seems to have little significance for either type of column alone or in a comparison of the two types. For example, in the present case the respective values of Ri for capillary and packed columns are 0.89 and 0.55 for a dead temperature of 3" C.; 0.69 and 0.41 for 10" C.; and 0.39 and 0.30 for 30" C. A N APPLICATION
Desty, Goldup, and Swanton (3) have recently reported a remarkable analysis of the gasoline fraction of Ponca City crude oil into 122 components up to n-nonane. The analysis was carried out a t 25" C. with a 270meter glass capillary column, 0.015 cm. i.d. coated with squalane. AIthough not specifically stated by the authors, we have estimated the total weight of liquid in the column as 11 mg. and the average flow rate as 0.12 ml. per minute (0.87 ml. per minute at the outlet with an inlet pressure of 150 p.s.i.g. The authors reported the flow rate as 0.5 ml. per minute but it ?vas not clear which conditions this referred to; the value stated above appears more consistent with the rest of their paper). These values correspond to a dead space of 450 ml. per gram and a flow rate of approsimately 10 ml. per gram per minute. From the data given in the paper the retention times for octane and nonane may be estimated as 410 minutes and 1240 minutes, respectively. From previous measurements by the same authors (2) of the specific retention volume of heptane on squalane a t 25" C. the isothermal retention volumes of octane and nonane at 25' C. may be estimated as 4450 and 12,900 ml. per gram, respectively. These are close to the values for the hypothetical compounds E and F , 5650 and 14,500. Thus, the calculations of this paper may be used to estimate the probable effect of applying temperature pro-
gramming to the octane and nonane peaks of their analysis. Figure 3 indicates that in order to maintain the intrinsic resolution a t a value as high as the isothermal value at 20" C. or 25' C., the program should have an r / F of 0.01 or smaller. Because of the narrow diameter and great length of the column the isothermal F value which was used, 10 ml. per gram per minute, could not be significantly increased LTithin the limitations of the apparatus. This is a relatively Iow value for a capillary column and thus for r / F to be below 0.01, r should be less than 0.1" per minute. This rate while low, is readily attainable experimentally. The portion of Figure 1 corresponding to the very low r / F region is shown in Figure 2 . For a n r / F of 0.01 and a n initial temperature of 25" C., the retention temperatures for E and F are found by graphical estimation ( 7 ) to be 50' C. and 64' C., respectively. These correspond to retention times of 250 minutes and 390 minutes as compared with the isothermal times 410 and 1240 minutes. As noted above, the intrinsic resolution is as good for this program as for isothermal analysis a t the starting temperature. Over-all resolution depends also upon the ?quare root of the number of plates. The total number of plates in the capillary column would be expected from the work of Scott (IO) to be somewhat increased with an increase in temperature. Consequently, the over-all resolution of these trro peaks, and probably therefore of the intermediate peaks also is likely to be, if anything, slightly better than that obtained isothermally. Thus, the application of a heating rate of 0.1' per minute without any other change in the operating conditions used by Desty Goldup, and Sn-anton would reduce the analysis time from over 20 hours to 6l/2 hours without loss of resolution. If such an analysis were to be carried out on a routine basis, temperature programming a t a low rate beconies very attractive. The example just considered is a particularly favorable case for capillary PTGC. Similar favorable behavior, such as found in PTGC for packed columns, would be expected in other cases where the much larger dead space yolume of the capillary column is still small compared to the isothermal retention volumes. Retention temperatures will normally be appreciably higher than they would otherwise have been without the large dead space volume. This tends to restrict capillary PTGC to relatively low initial temperatures and relatively lorn r / F programs. On the other hand, programs with much 1oLver r / F values are possible with capillary columns than can be obtained easily with packed columns. It would seem, therefore, that there is a definite
pIace for temperature programming in capillary column gas chromatography involving analysis of mixtures having a wide boiling range of constituents. LITERATURE CITED
(1) Ambrose, D., et al., “G?: Chromatog-
raphy-Edinburgh 1960, R. P. IT. Scott, ed., p. 429, Butterworths, London, 1960. (2) Destv, D. H., Goldup, A., “Gas Chrom”atography-Edinburgh 1960,” R. P. IT.Scott, ed., p. 162, Butterworths, London, 1960.
(3) Desty, D. H., Goldup, A., Swanton, ~ ~ r ~ ~ t ~ ~ ~ ~ 83, 1961,
(4) Fryer, J. F., Habgood, H. W.,Harris, W. E., ANAL.CHEW33, 1515 (1961). (5) Giddings, J. C., J . Chromatoq. 4, 11 (1960j.
(6) Giddings, J. C., ISA Proc.: 1961 Int. Gas Chromatog. Symposium, Preprints, p. 41, 1961. ( 7 ) Habgood, H. IT.,Harris, IT;. E., ASAL. CHEM.32. 450 11960). (8) Purnell,’ J. H., k a t u r e 184, 2009 (1969).
Failure of the Eddy Diffusion Concept Gas Chromatography
(9) Rowan, R., ANAL. CHEX 33, 510 y (1961). ~ ~ ~ ~ ~ u n ~ g (10) Scott, R. P. W., J . Inst. Petrol. 47, 284 (1961). (11) Teranishi, R., Nimmo, C. C., Corse, J., ANAL.CHEM.32, 1384 (1960). RECEIVEDfor review February 26, 1962. Accepted ,May 4, 1962. Joint contribution from the Department of Chemistry, University of Alberta, and No. 172 from the Research Council of Alberta. Presented to the 45th Annual conference of the Chemical Institute of Canada, Edmonton, May 1962.
Of
J. CALVIN GlDDlNGS and RICHARD A. ROBISON Department of Chemistry, University of Utah, Salt Lake City, Utah F A critical examination of the status of e d d y diffusion in gas chromatography i s made in terms of past experimental results and some new experimental work. Previous experimental anomalies, such as negative eddy diffusivities, are examined in the light of both old and new theoretical concepts, and possible explanations are given for the departure from the expected pattern. A summary of this work strongly indicates the failing of the classical eddy diffusion concept. This i s confirmed b y further experimental results using an inert glass b e a d column in which the plate height i s sometimes less than the particle diameter. An alternative to the classical concept, the coupling theory of eddy diffusion, i s in satisfactory agreement with past and present results.
T
HE widely accepted role of eddy diffusion as one of the major sources of peak spreading in chromatography is subject to grave doubt. This vievi has, of course, been expressed before, especially in the light of numerous experimental results which have not agreed with the eddy diffusion concept. Whatever doubt has arisen, hoaever, has apparently failed t o stimulate a n active concern with alternatives to the eddy diffusion concept and explanations for the anomalous and sometimes contradictory experimental results. I n only a few cases have workers attempted t o explain the reasons for the rather severe departure of their results from those predicted by eddy diffusion theory. The object of this paper is t o summarize and put in perspective the explanations n hich have been made, and, with some new suggestions, t o explore the relation-
ship between previous experimental anomalies and the theory of eddy diffusion. I n addition, experimental n-ork which bypasses some of the previous experimental difficulties is presented. Eddy diffusion is a name given to the spreading of chromatographic peaks which is a direct and sole result of the interaction of mobile fluid with the solid support. The phenomenon depends entirely upon the geometrical arrangement of the support particles and is thus nonchemical in nature. It arises from the nonequivalence in velocity of various flow paths which a fluid and its contained solute follow in migrating through the porous support. The less tortuous, higher velocity paths lead some solute to a position in advance of the bulk of solute and vice versa. Because a random distribution of flow paths exists, the resulting band spreading is random, and the subsequent concentration profile (providing one starts with a narrow pulse) is gaussian. There is no exact theory of eddy diffusion for the simple reason that the enormous geometrical complications of porous materials defy mathematical tractability. This is not t o say that exact relationships have failed t o appear (an example, using similitude principles, relates eddy diffusion t o particle diameter without assuming simple geometries), but simply that a purely theoretical calculation of the value of eddy diffusivity is now out of the question. The fact that eddy diffusion is, under some circumstances, a real phenomenon is demonstrated b y the existence of a t least three approximate derivations which all agree as to the order of magnitude of the effect. These derivations employ a similitude principle (23), a mixing stage model ( 5 , 25) (very much
like a plate height model), and a random walk model (9). They all predict a plate height contribution, A, which is proportional to the particle size (for geometrically similar packing) and independent of all nongeometrical parameters such as flow velocity and temperature. I n addition, i t is generally assumed that A is equal or larger than a single particle diameter (5, 15). These two sentences summarize the results of what it e will call the classical eddy diffusion concept of chromatography. In mathematical form A A
= 2Xdp
(1)
> d,
(2)
n here d, is the mean particle diameter and h is a geometrical constant of order unity. Only one theoretical treatment has been given with results in wide variance with Equations 1 and 2. This theory, which predicts A t o be a function of flow velocity and t o be capable of approaching zero a t low velocities, is of interest in view of the failure of experimental results to conform to Equations 1 and 2 . For lack of a better name we will call this the coupling theory of eddy diffusion (see later). I n approximate mathematical terms
A=
1 1
(3)
A =0
A = 2Xdp
(4)
v-+O
v + m
1
2Xd, + c,u
where v is the mean gas velocity and C,v is the nonequilibrium or mass transVOL 34, NO. 8, JULY 1962
* 885
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r
e
