( 4 ) Ibid., p. i93. (5) Brown, F., Hirst, E. L., Hough, L., Jones, J. K. N., Wadman. H., Nature 161, 720 (1948). (6) Demole, E., J . Chromatog. 1,24 (1958). ( 7 ) Demole, E., Chromatog. Revs. 1, 1 (1959). (8) Demole, E., J . Chromatog. 6, 2 (1961). (9) Gee, M., Ibid., 9, 278 (1962). (10) Gee, M.,Walker, H. G., Jr., ~ ~ N A L . CHEM.34, 650 (1962). ( I 1) Gee, M., Walker, H. G., Jr., Chem. & I n d . 829 (1961). (12) Hjrst, E. L., Hough, L., Jones, J. K. h ., J . Chem. SOC.928 (1949).
(13) Kircher, Henry W., ANAL. CHEM. 32, 1103 (1960). (14) Kuhn, R., Trischmann. H.. Low,, I.., Angew. .Chem. 67, 32 (1955). (15) Lemieux, R. U., Bishop, C. T., Pelletier, G. E., Can. J . Chem. 34, 1365 (1956). (16) Pastuska, G., 2. Anal. Chem. 179, 355 (1961). (17) Prey, V., Berbalk, H., Kausz, M., Mzkrochzm. Acta 968 (1961). (18) Ibid., 449 (1962). (19) Tate, M. E., Bishop, C. T., Can. J . Chem. 40, 1043 (1962). (20) Walker, H. G., Jr., Gee, M.,Mc-
Cready, R. M., J. Org. Chem. 27, 2100 (1962). (21) Wickbere. B.. “Methods in Carbohydrate Chumistry,” Whistler, R. L., Wolfrom, M. L., Eds., Vol. I, 31, Academic Press, 1962. RECEIVEDfor review October 1, 1962. Accepted January 16, 1963. Reference to a company or product name does not imply approval or recommendation of the product by the U. S. Department of Agriculture to the exclusion of others that may be suitable.
Plate Height of Nonuniform Chromatographic Columns Gas Compression Effects, Coupled Columns, and Analogous Systems J. CALVIN GlDDlNGS Deparfment o f Chemistry, University of Ufah, Salt lake Cify, Ufah
b A theoretical treatment is given for columns which have nonuniform properties along their lengths. The theory is applied to coupled columns and to gas compression effects in gas chromatography. It is shown by a simple two-segment model (in addition to a more detailed approach) that even with a lengthwise constant plate height, the observed plate height may appear as something entirely different from its constant local value. This refutes the argument that the velocity gradients accompanying gas compression are of no significance in determining plate height and resolution contributions of constant plate height terms.
C
are Often operated in such a way that the physical properties change from one end of the column to the other. I n gas chromatography, for instance, the finite pressure gradient required for gas flow causes a significant velocity gradient in the column. Different columns are occasionally coupled together to take advantage of several sorbents simultaneously or to seek advantages through a change in column diameter. Other nonuniformities may develop along a column’s length because of the failure to acquire a homogeneous packing. The object of this paper is to develop a simplified theory for those variations that occur strictly as a function of the distance along the column (this excludes such subjects as programmed temperature gas chromatography where the principal change occurs as a function of time), Of primary interest is the effect of gas compression in gas chromatography. HROMIATOGRAPHIC COLUMXS
The popular misconception that constant plate height terms (such as the gas phase terms, where changes in diffusivity and velocity exactly compensate) appear unchanged in a system with velocity gradients is refuted with the help of a simple two-segment model. G A S COMPRESSION
A number of authors have discussed the influence of gas compressibility on plate height and resolution in gas chromatography (GC) ( 1 , 2, 5-8). While the conclusions have not always been the same, widespread agreement exists on the point that plate height terms which are constant throughout the column ( A ,B/v, C,v) will contribute just this constant amount to the observed plate height whether a velocity gradient exists in the column or not. This view has been opposed by the author and his co-workers ( 6 , 8) who have derived the equation where
and
f,
=
9(P4 - l ) ( P- 1) 8 ( P 3 - 1)2
(3)
The term H , is the collective sum of all the previously mentioned plate height terms which remain constant in the column. This term is part of the local plate height-i.e., the plate height which exists locally in a_small section of the column. The term H , on the other hand, is the observed plate height, and
is defined in the usual way in terms of observable quantities, (4)
The terms f2 and fl, the latter of which is of main interest here, are the pressure correction terms, and are functions solely of the compression ratio, P = inlet pressurejoutlet pressure. If Equation 1 were written to agree with the majority of equations describing pressure and velocity gradient effects, it would be necessary to write f i = 1 in place of t h e f l value of Equation 3. Numerically, the two values differ a t most by 12.5%. However, this modest percentage is becoming more and more important as the agreement betrveen chromatographic theory and experiment becomes closer (4). In addition, because two conflicting conclusions have been presented, an incorrect set of principles must have been used to obtain the wrong conclusion, and such principles might lead to additional errors. Therefore, it is important to discuss further the differences that exist. Various arguments for fi = 1 are not difficult to find. I n a completely uniform column with no velocity gradients (such as the liquid systems for n-hich the plate height theory was initially intended), the observed plate height and the local plate heights are all identical. This concept is perhaps more meaningful if we say that such a column, if cut in two (while maintaining an equal flow velocity), would yield two observed plate height values (as defined in Equation 4) identical to each other and to the full-column value. VOL 35,
NO. 3, MARCH 1963
353
This Ivould be true for any number of cuts in the column, and nith small enough segments n-e are entitled to call the resulting plate height the local plate height. Thus, the local plate height, u-hen constant in a uniform column, is equal to the observed plate height. The same conclusion may be said to apply to a constant local plate height in a nonuniform column such as exists in GC. The argument might follow the line that as long as the local plate height is constant there will be a fixed number, X, of plates in the column and thus the overall plate height, LIN, will remain constant. This, of course, is contrary to Equations 1, 2, and 3 because even if the local value, H,, were to remai: constant, the observed plate height, H , is predicted to change with the compression ratio P (it is a simple matter in practice t o vary P while maintaining H , constant). The m-eakness of the argument concerning the constant number of plates, or similar arguments based solely on the plate height concept, is that theoretical plates have no existence in reality, and one must carefully limit the range of phenomena t o which the concept is applied. The plate height theory is particularly limited when gradients exist in a column, as discussed elsen-here (3). These limitations do not, of course, make it necessary to abandon the plate height as a measure of column efficiency, but they do suggest that severe caution be taken in applying the plate height theory to new theoretical areas. This is why the author and his colleagues ( 5 ) investigated the basic peak spreading phenomenon itself in formulating Equations 1, 2, and 3. One of the primary goals of this paper is t o demonstrate with simplified concepts that the observed plate height is greater than the local plate height (assumed constant) when velocity gradients exist. 4 s mentioned previously, a set of terms represented collectively by H , remains constant in a gas chromatographic column although the flow velocity may be significantly greater at the outlet than at the inlet. Such a column may be arbitrarily divided into small segments for theoretical purposes. (These segments are in no may related t o individual theoretical plates.) The change in velocity can be assumed to occur discontinuously in passing from segment to segment. The effects we are seeking will then originate in the velocity change between segments. While the performance of the actual column will be described only as the segments become smaller, an approximate treatment, showing the direction and nature of the effect, can be made assuming the column to be divided into two equal segments. I n the first segment, length L/2, the gas flow will have a uniform velocity ul. This will 354
ANALYTICAL CHEMISTRY
constant, local value, but the magnitude change in the center, where the t\vo of the effect is exaggerated because segments join, to the outlet velocity only two segments are used. If three v2. The observed (or apparent) plate equal segments are assumed, with tl = height is given by Equation 4-i.e., = 2f3, similar reasoning shows fi = Lr2/t2. Theplate height in each fi = 1.074, a value somewhat closer segment will be H,. Thus if the segto the 1.03 given in Equation 3. As ments were disconnected, one would more segments are used, fl can be made find that this value could be observed to approach the latter limiting value for each segment directly in terms of the as closely as desired. segment lengths (L/2), the standard deviation in elapsed time (71 and T ~ ) , and the retention time ( t l and tp)-i.e., GENERAL THEORY H, = L rI2/2tl2and H, = L ~ ~ ~ / 2 t ~ ~ . However when the segments are joined, The above reasoning can be extended the cumulative effects of the two colto a very large number of segments umns must be considered. The length, to obtain the value applicable to real is, of course, equal to the sum of the columns. The method will be formuL/2 = L. The resegments, L/2 lated more generally by simply allom-ing tention or passage time is likewise for the effects of gas compression on the sum of the two segment values, flow velocity. Thus the treatment given t = tl t2. The peak quarter-width, below will apply to any column or colT , is not, however, the sum of 71 and ~ 2 , umn system where the local plate height but is obtained by the usual addition and zone velocity may vary arbitrarily rule for standard deviation, r2 = T > + T ~ ~ . from end to end. It mill be applicable [The preceding addition rule is valid t o columns in series, column diameter when the two sources are directly changes, inhomogeneous packing, temand independently responsible for peak perature gradients along the column, spreading. The velocity change beetc. tween segments does not invalidate this. Once again the column or column It is true that the velocity increase will system (as for columns in series) will widen the band as it passes between be divided into n equal segments. segments ( 6 ) ,but a given molecule a t the Extending the addition rules used prefront of the peak will still be ahead of viously, n-e have r 2 = Z r,2and t = Z = t , , one in the tail by the same time interval where the subscript i refers to the value after the velocity transition. The in segment i. The summation is over all constancy of the time interval during n segments. The apparent plate height, transition makes the above rule apEquation 3, is equal to plicable, but the lack of constancy in the actual peak dimensions invalidates the same rule applied to standard deviation in distance rather than time. with these substitutions. Xow suppose (See the appendix.) The previous that the local plate height, which in theoretical treatment (ti) was more this case may vary from point to point, complicated because units of distance is given in the usual way by Hi = were used in describing peak diL,rC2/tr2. I n addition, the segment mensions.] A combination of the forewhere retention time is t, = LE/Ravt, going addition rules gives R, is the ratio of the zone t o the gas velocity in segment i. Using rs2and t , B = U T 1 2 TZ*)/(h t2)2 (5) from these expressions, Equation 7 becomes Because H , = Lr12/2t12,r12 can be written as 2H,t12/L. The substitution I? = n ( 2 H , / R , 2 v , 2 ) / ( 2~ / R , U , ) (8) ~ of this and the analogous expression for This can be written in integral form r22into Equation 5 yields by noting that the limiting summation and integral are related by H = H , - 2't12 + lZ2) = H$, (6)
+
+
+
+
A
(tl
4-
t2)2
where .fl is the correction term appropriate to this case. Its value is unity only when the velocities, and thus the retention times, are equal in each segment. It is greater than unity otherwise, and thus l? > H,. As a specific example, suppose the inlet pressure of a column is one atmosphere above ambient, thus halving the velocity at the inlet compared to the outlet. Because retention time and velocity are inversely proportional, tl = 2t2. Substituting 2t2 into Equation 6 yields f l = 1°/sor ff = 1.111 H,. The apparent plate height clearly need not equal its
where z is the distance along the column axis. This type of conversion leads to
(10)
which is the desired expression. Xumerous special cases can be obtained from the last equation. One of the most interesting and relevant cases insofar as the present discussion is concerned is found when the local plate
height is consta.nt throughout the column system. The foregoing equation then becomes
(11)
If numerator and denominator are both divided by L2, they acquire the form of average values
fl = H(1/Rzvzj/(l/Rv)'
(12)
The nature of this equation is made more clear by the simplifying substitution, U = 1/Ru
B
=
H.p/.p
(13)
Figure 1, Simplified illustration of change in band length caused by a change in velocity of the carrier fluid
terpretation for fi being greater than unity is given in the last section. A treatment similar to the above can be applied to the plate height term for mass transfer in the liquid. The correction term, f2, is the same as that shown in Equation 2. COLUMNS IN SERIES
Thus I? is greater than H by the mean square of the variable quantity U divided by the square of its mean. This quantity is always equal to or greater than unity, and thus k 2 H. The equal sign holds only when U is constant so that under conditions of true variation in zone velocity, fi > H . The analysis of the preceding paragraph can be applied immediately to the problem of pressure induced velocity gradients using the set of constant plate height terms denoted by H,.Because R is constant, Equation 10 Wimplifies to
Another special case which may be of some interest is that in which two columns are joined in series. For simplicity we will assume that velocity gradients are negligible within each column, although a velocity change may occur between columns (due to a change in column diameter, packing density, etc.). The plate height will then be constant throughout each column, H l in the first and H Pin the second. Either the summation or integration formulas, Equations 8 and 10, may be used, although the latter is preferred. Each of the integrals from 0 to L is then obtained as the sum of two integrals, one over each column. This procedure gives
These integrals are easily evaluated by changing the variables t o absolute pressure, p . Since p v is a constant, the change from u to p is made in direct fashion. The change from z to p makes use of the pressure drop equation, ( 5 ) p 2 - p O 2= cz, ( z is here the distance from the column outlet, a fact which will not effect the previous equations for all of nhich z can be measured in either direction) and its differential, 2 p dp = cdz (c is a constant). The substitution of these expressions gives
I n a more special case, L1 may equal L2. I n that case the apparent plate height becomes
The evaluation of these simple integrals and the substitution of c = ( p z 2 po2)/Lyields
in exact agreement with the previous theory, Equation 3. -4s before, P = p , / p o . The value of fl is greater than unity whenever a pressure gradient exists in the column, pi > p,. The general proof that fl must be greater than unity was shown in connection with Equation 13. The physical in-
Suppose novi that the zone velocity, Ru, is JI times as great in segment 1 as segment 2 . This could arise through a difference in the nature or the amount of stationary phase, in which case R1 = MRs, or it could arise through a difference in the cross sectional areas of the columns, in which case u1 = M u 2 . I n either case the last equation reduces to
This equation shows that the plate height of the second column, in which the slow zone velocity is found, is weighted by a factor M 2 times larger than the plate height of the first column, Thus the slow-velocity segment or segments are more important in establishing the over-all plate height. It is completely irrelevant whether the slow column is first or last since the sequence can be changed without affecting the chromatographic performance. Although H P is weighted by M2, i? is not the lyeighted average of the two
segments or columns. I n agreement with the preceding paragraphs, ft is greater than the weighted average except where M = 1. This is particularly noticeable when 1cf >> 1 and B approaches twice the weighted average, or 2H2,in value. PHYSICAL INTERPRETATION
The explanation behind the mathematical treatment showing the apparent plate height to be greater than its local value is not difficult to find. The explanation is of particular interest where pressure gradients are involved in gas chromatography, for it is here that conflicting views have been presented. Part of this explanation is negative-Le., the belief that apparent and local plate heights should be equal is based on the plate height model and is questionable to begin with. On the positive side, however, a direct explanation can be given for these effects. To magnify the gas compression effects so that they are most noticeable, let us assume that the column outlet is under vacuum and consequently the velocity a t the outlet is nearly infinitely greater than a t the inlet. The column can then be divided roughly into two regions (not necessarily equal in length), the first of which involves normal f l o ~rates and the second of which involves a very fast flow. The local plate height, H,, is constant throughout. Yearly all of the residence time is spent in the first region where flow is not abnormally rapid. Because H, = L,T,2/t,2 is constant and consequently the ratio T J ~ , is constant for a given segment length, T mill be negligible wherever t is, namely in the final, high-flow region of the column. The following conclusion can be drawn regarding the terms of the apparent plate height, k = L ~ ~ j t The ' . T~ and t2terms will be contributed entirely by the first region of the column, The length however nil1 be contributed by both regions. Therefore ft will approximately equal (L1 L2) T I 2 / t l 2 = L1TI2/tl2 L2T12/t12. The first term of the last expression, L 1 ~ 1 2 j t 1is2 , by itself the local plate height, H, (assuming near uniformity in each region), and thus I? = H , L$T12/t12. Thus k > H,. This inequality arises, therefore, because the rapid-flow regions of the column are contributing to the length, giving a proportional increase in k,but they do not contribute to the terms in r 2 / t 2 , which normally would diminish a t a rate sufficient to compensate for the increase of L. ( 9 / t 2 is of such a character that additional contributions actually diminish its value since t increases more rapidly than T.) The above reasoning applies also when the outlet
+
+
+
VOL 35, NO. 3, MARCH 1963
0
355
js not under vacuum, but the arguments are not so clear cut nor the effects so great in magnitude. APPENDIX
Figure 1 can be used t o demonstrate the principle that while a band may broaden when it encounters a velocity increase, the time interval between front and rear remains essentially constant except for the slight amount of increase due t o normal band-spreading processes. The figure shows a column containing an incompressible fluid which is speeded up when forced into a column of smaller cross-sectional area, A?. The final velocity of the band is larger than the original velocity in the ratio v2/v1 = A1/A2. The volume of the band is constant because the fluid is incompressible, and the final band length divided by its initial value is &/l1 = Al/ilz. From the above expression we have lz/ll = v2/vl, showing that the
band has broadened considerably. The passage time of the band before the transition is Atl = ll/Rvl and after the transition it is &/Rv2. The expression just preceding this shows that ll/vl = 12/v2or Atl = At?, and consequently that the zone spreading in time units is constant. This argument, applicable to columns linked in series, may be extended to include gas compression effects in gas chromatography. If a transition occurs from a region where the pressure is p l to a region of pressure p?, and p1 > p?, the entire mass of gas, along with the contained band, expands in volume by the ratio p1/p2. The velocity also increases by this ratio. Because the cross-sectional area is constant, this means that the band increases in length, 1, such that 12/11 = p l / p z = vz/vl. The passage time of the band is first At, = ll/Rvl and after the transistion is Atz = 12/Rv2. Because 12/11 =
vz/vl, these passage times are once again equal to one another. LITERATURE CITED
(1) Bohemen, J., Purnell, J. H., J . Chem. SOC.1961,360.
(2) Desty, D. H., Goldup, *4.,Swanton, W. T., 1961 International Gas Chromatography Symposium, East Lansing, Mich., June 1961, p. 83 of preprints. 1 3 ) Giddings. J. C.. AKAL.CHEW.34. 722 (1962). (4) Giddings, J. C., hlallik, K. L., Eikelburger, M., Ibid., 1026. ( 5 ) Giddings, J. C., Seager, S. L., Stucki, L. R., Stewart, G. H., I b i d , 32, 867 (1960). (6) Kkselbach, R.,Ibid., 3 3 , 2 3 (1961). i'ii, Littlewood. A. B.. "Gas Chromatography, 19p8," D. H. Desty, ed., p. 35, Academic Press, K e x York, 1958. (8) Sterart, G. H., Seager, S. L., Giddings, J. C., h A L . CHERT. 31,1738 (1959). RECEIVEDfor review October 8, 1962. Accepted January 14, 1963. Kork supported by a Research Grant S o . GM 10851-06 from the National Institutes of Health, Public Health Service. Y
l
~
\
Rapid Gas Chromatographic Method for Screening of Toxicological Extracts for Alkaloids, Barbiturates, Sympathomimetic Amines, and Tranquilizers KENNETH D. PARKER, CHARLES R. FONTAN, and PAUL L. KIRK School of Criminology, University of California, Berkeley, Calif.
b A gas chromatographic method is described for the screening of toxicological extraction residues for alkaloids, barbiturates, sympathomimetic amines, and tranquilizers. Retention data are given for 41 alkaloids chromatographed on an SE-30 column at five temperatures. Representative members of other groups of compounds chromatographed on the same column are listed. With a sufficient elevation of temperature these groups emerge under the same conditions, enabling the toxicologist to examine a tissue extract rapidly for the presence of microgram quantities of a large number of acidic, basic, and neutral organic compounds. Two Carbowax 20M (alkaline and nonalkaline) columns are compared to determine their usefulness in the screening of toxicological extracts. A solid injector which allows concentration of the extractives and elimination of the solvent is described.
T
toxicological extracts have been screened for nonvolatile organic compounds by the tedious process of systematic color and crystal RADITIONALLY
356
ANALYTICAL CHEMISTRY
tests (17). Ultraviolet absorption and its change with pH have been used extensively as a systematic approach to the presumptive identification of compounds to which they are applicable (6, 16). Serious limitations are imposed, however, on the use of ultraviolet absorption for the examination of mixtures, because a composite spectrum of all absorbing compounds is obtained. Paper chromatography has been used most successfully to screen tissue extracts (17). The resulting fractions, located and characterized by an array of chromogenic reagents, may, depending upon previous treatment, be eluted for further examination. Many different chromatographic solvent systems are required, however, for adequate separation within the acidic, basic, and neutral fractions that result from toxicological extractions. Brackett (4) used a single generally applicable solvent system and attained unique characterization of many compounds by chromatographing simultaneously the parent compound and various derivatives formed from it on the chromatogram at the origin by the addition of oxidizing, reducing, and other reagents. Anders and hhnner-
ing (1) illustrated a similar approach applicable to gas chromatography. Derivatives were formed on the column from a number of alkaloids and steroids. As with paper chromatography, gas chromatography of alkaloids (9), barbiturates (6, 7 , 13, 1.4)) sympathomimetic amines ( l a ) ,and tranquilizers (11) requires very different operating parameters for adequate separations within the different groups. This paper presents a single gas chromatographic system of general applicability to the screening of toxicological extraction residues for the groups of compounds specified above. An injector designed for the introduction of dry samples is also presented. EXPERIMENTAL
Apparatus. Figure 1 shows the construction of the solid injector. T h e components were chosen t o make the injector gas-tight when assembled. The sample carrier was adapted from a cleaning wire supplied with the B-D Yale hypodermic needle, 22 G, 3 inches. The plunger handle was formed by inserting one end of the sampling wire into a B-D Yale 24 G, I-inch, hypodermic needle and crimping
