Programmed Temperature Gas Chromatography at Constant Pressure ROBERT ROWAN, Jr.' Analyfical Research Division, Esso Research and Engineering Co., linden, N . 1.
b A quantitative mathematical statement describing programmed temperature gas chromatography (PTGC) at constant pressure (as distinguished from constant flow) is developed and shown experimentally to be valid. The relation follows from the treatment of flow rate as a variable rather than a constant. The same treatment is shown to be applicable with advantage to ordinary PTGC where a flow controller is used. A method is described for the direct arithmetic calculation of predicted retention temperatures. This has not heretofore been possible. A comprehensive equation is presented which describes the location of a concentration maximum in the column at any time.
P
temperature gas chromatography has been treated from the standpoint of both theory and practice by a number of authors (1-6, 10,11). I n most such treatments, the gas flow rate has been assumed to be constant, although Giddings (4)made allowance for the variation of flow velocity with time. Constancy of mass flow is normally achieved in PTGC by means of a mechanical flow controller. These are quite effective, but even if they control flow, as measured a t the outlet a t constant temperature, with perfect constancy (seldom achieved), the effective flow in terms of retention volume will vary with column temperature through its effect on pressure drop. This is, of course, a consequence of the necessary application of the James and Martin correction factor (8) for gas expansion. The only way this can be avoided is to control both inlet and outlet pressures as functions of temperature, as discussed elsewhere (10). To make calculations involving a flow rate which is assumed constant, but actually' is not quite constant, it has been necessary to estimate a sort of weighted average flow rate. This introduces the possibility of errors and uncertainties. The theory and method of calculation set forth in this paper is based on the treatment of flow rate as a variable ROGRAMMED
1 Present address, New Mexico State University, University Park, N. M.
1042
ANALYTICAL CHEMISTRY
rather than a constant. This allows operation a t constant pressure, which also means constant pressure drop, and thus the James and Martin factor is always the same regardless of temperature. While this makes the calculation of predicted retention temperature more involved, it simplifies the derivation of a comprehensive equation which describes the position of a concentration band in the column at any time. This same treatment can also be applied to the case of ordinary PTGC, where a pressure controller is normally used. This is a point of some importance, especially from a practical standpoint. The paper also includes improvements in calculation techniques which make i t possible to compute predicted retention temperature directly, without the use of charts. EXPERIMENTAL
Two PTGC runs, A and B , were made on two different F & M Model 500 GC units. In one case, A , the usual flow controller was removed from the line so that the pressure remained constant at 22 p.s.i.g. during the run. Here the column was silicone SE-30 on firebrick. In run B, the flow controller was employed at 12.5 p.s.i.g. with a column of silicone nitrile on glass beads. In run B , column pressure drops were 6.5, 7.5, and 8.5 p.s.i. at 40, 80, and 120" C., respectively. In all cases, the carrier gas was helium, and the sample consisted of normal paraffins. Small (5-10 pl.) samples were used. In addition, certain data from a previous paper were recalculated, as will be explained. DISCUSSION A N D RESULTS
Derivation of Equation. It has been shown previously (10) that PTGC can be described by the following equation
= 9
where u
a
= carrier gas flow rate, measured
a t constant temperature (room) at the column outlet, ml. per minute = temperature rise rate, degrees per minute
B,e = constants in the equation describing V?,the void-corrected retention volume In V , =
T
=
A
=
T,
=
bl .-
=
qz 7
=
Te
+ In B
temperature of column at any time (" K.) starting temperature of program (" K.). retention temperature (" K.) @/A e/TI James and Martin correction for gas expansion
Throughout this paper, u shall mean the number of milliliters of carrier gas per minute isscing from the exit of the column a t room temperature and column exit pressure (1 atm.), uncorrected for pressure drop across the column. The term "flow rate" shall normally mean the same, except when more clearly specified. The term ''effective flow rate" or "corrected flow rate" shall refer to the flow rate corrected for pressure drop. The deviation of Equation 1 assumes that the effective flow rate is constant. As explained previously, this is not quite true, however, for as temperature rises, pressure drop must change, and this means that the correction factor for gas expansion also changes. The treatment is, therefore, inexact to the extent (several per cent) that effective flow rate varies during a run. We wish, therefore, to examine the case where inlet pressure and pressure drop remain constant during a run. With pressure drop constant, the flow through a porous medium is inversely proportional to the viscosity of the fluid, according to Darcy's law. This can also be derived from Poiseulle's formula for steady viscous flow through a capillary tube ('7). The fluid in this case is helium, and its viscosity is very close to being directly proportional to absolute temperature over the temyerature interval of most interest (0" to 300" C.) (6). This means that flow can be approximated by a relation such as the following u = -M +N T
where lli and N are constants.
22L
2 0
2 2
2 4 1 / T I’KI
I
2 6 lo3
2 3
2 0
Figure 1. Variation of flow rate with reciprocal absolute temperature
The linear variation of viscosity with temperature does not normally hold aa well for other gases nor for helium over a wider temperature range. The dependence of gas viscosity on absolute temperature is best described by the Jeans relationship, according to which viscosity varies directly with Tn, where TZ is close to 0.65 for helium (9). I n addition to gas viscosity behavior, however, the accuracy of Equation 2 may well depend on other factors such as thermal expansion effects within the column. For this reason the validity of Equation 2 is best gaged on the basis of experiment. The linear dependence of u on 1/T is shown in Figure 1, where curve B pertains to He without a flow controller (Run A ) . One of the intermediate steps in the derivation of Equation 1is as follows
In previous discussions (IO), the factor j does not appear explicitly in the
Figure 2. VS.
Function
(41 - 421
R
3
.
0
0
1
,
,
, 41
equation because flow rate has been considered to have been corrected already for pressure drop. Here, however, since u is now a variable, and j is a constant, the two quantities must be separated. The constants B and 0 must be computed from V y values which have been corrected for messure dron. The role of j will be diskwed f u r t h k later in this paper. If we allow-u to vary with temperature, its value in terms of T (Equation 2) can be substituted into Equation 3
Table I.
-R
{
=
-
,
!
0.4 0.8 1.2 1 , 6 2.0 2.4
- 41
,
l
1
2.8 3 2
3.6 4 . 0
4 4
inside the integral sign. When the indicated integration is performed, the resultis
“-B=ive*-M[yd+ 3
(4)
..
Equation 4 can be rearranged
aB
iNe
=
Q
(1
-
g)
(5)
where R 0 =
A[-EFx’l }
4l
41-+a 0.1 0.2 0.3 0.4 0.6
0.6
0.8 1.0
1.2 1.4
1.6 I .8
2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8
-
6 5.949 5.896 5.840 5.782 5.722 5.660 5.529 5.390 5,242 5.087 4.924 4.755 4.580 4.214 3.829 3.430 3.018
7 6.949 6.896 6.840 6.783 6.723 6.662 6.532 6.394 6.248 6.095 5.936 5.770 5.598 5.239 4.862 4.472 4.071
8
7.949 7.896 7.841 7.783 7.724 7.663 7.534 7.397 7,253 7.102 6.944 6.780 6.610 6.256 5.885 5.501 5.107
9 8.949 8.896 8.841 8.784 8.725 8.663 8.535 8.400 8.256 8.106 7.950 7.788 7.620 7.269 6.902 6.523 6.133
10 9.949 9.896 9.841 9.784 9.725 9.664
8.279 7.916 7.539 7.153
11 12 10.949 11.949 10.896 11.896 10.841 11.841 10.784 11.785 10.725 11.726 10.665 11.665 ~. . . ~ 10.538 11.538 10.403 11.404 10,262 11.263 10.113 11.116 9.959 10.962 9.799 10.803 9.633 10.638 9.287 10.294 8.926 9.935 8.552 9.563 8.168 9.181 7.777 8.792
13 12.949 12.896 12.841 12.785 12.726 12.666 12.539 12.405 12.265 12.118 11.965 11.806 11.642 11.300 10.942 10.571 10.191
14 15 13.949 14.949 13.896 14.896 13.841 14.841 13.785 14.785 13.726 14.726 13.666 . _ .14.666 13.540 14.540 13.406 14.407 13 266 14.267 13.120 14.121 12.967 13.969 12.809 13.811 12.645 13.648 12.304 13.308 11.948 12.953 11.579 12.585 11.200 12.207 I
16 17 15.949 16.949 15.896 16.896 15.841 16.841 15.785 16.785 15.727 16.727 15.686 16.667 __ . 15.541 16.541 15.408 16.409 15.268 16.269 15.123 16,124 14.971 15.972 14.813 15.815 14.651 15.653 14.312 15.315 13.957 14.961 13.590 14.595 13.213 14.219
18 17.949 17.896 17.841 17.785 17.727 17.667 -. 17.542 17.409 17.270 17.125 16,974 16.817 16.655 16.318 15.964 15.599 15.224
19 18.949 18.896 18.841 18.785 18.727 18.667 18.542 18.410 18.271 18.126 17.975 17.818 17.667 17.320 16.968 16.603 16.228
-9:ioi io:iii ii-,52i i2:829 133835 14:sii i5:846
9.410 10.421 11.431 12.446 13.396 14.452 15.458 10.025 11.036 12.045 13.053 14.059 15.065
VOL 34, NO. 9, AUGUST 1962
1043
From a, (& - +J is found (IO). R is - &), using Figure then found from 2. A new value of @ is then calculated, and the same sequence is repeated. The third, or in many cases the second value of (+1 - +J yields an emergence temperature close enough for all practical purposes (a fraction of a degree) to that ultimately arrived at by unlimited cal- +J is culation. Finding T, from completely straightforward. We can - +2) and Then find +z,knowing
Table II. Predicted Retention Temperatures at Constant Pressure, Run A
n-Paraffin C, C, CI CS
Retention Temp., 'K. Calcd. Actual 366 383
366 383 402 429
...
428
The ratio R is a slowly varying function of 4 and ($J~- &) which approaches dl in the limit as (r#q - 42)approaches zero. Its value can easily be found by reference to a simple linear plot of R us. - 42),such as Figure 2, which is a family of gently curving lines, one for Data for constructing each value of such a plot are given in Table I. Equation 5 can be used to find T , in a practical case. All quantities are known except @ and R. These are both functions of and (+1 - +J, of which is known. To find +2, which leads to T, (TI= e/+*), it is merely necessary to find values of @ and R which satisfy Equation 5 and at the same time pertain to the same values of +1 and - +2). This is an iterative procedure in which Figure 2, to find R, and a plot of (6, - +2) us. @ (fO)to find - +2), may be employed. An alternative procedure is to use Equation 4 and to find the exponential integral term in a manner exactly analogous to that used for @ (IO). The calculation sequence is as follows: One first computes @ from Equation 5 on the basis of an assumed value of R. If it is not convenient to estimate R. it suffices to take of the absolute value of aB/Ne as a first approximation for @.
Table 111.
%-Paraffin
CS CS c 1 1 c 1 2 c 1 2
Table IV.
n-Paraffin C7 C.3 CO Cl" c 1 1 c 1 2
1044
T , = e/+*.
PTGC at Constant Pressure. Experimental confirmation of the theory just presented is provided in the results of calculations on Run A , where no flow controller was employed. The pressure drop in this case was constant, which, of course, means that j, the gas expansion factor, was also constant (at 0.544). The gas flow constants M and N were 53,170 and -47.9, respectively (see Figure 1). The results are shown in Table 11. These results were calculated on the basis that the temperature rise rate computed for heptane was correct, assuming no other error. In other words, n-heptane was used as internal standard. This is in accord with previous published results for ordinary PTGC (IO), where a similar calculation was also used. The true temperature rise rate was 4.3" C. per minute, and that necessary to bring the results into line was 5.6" C. per minute. The bias, if the former figure was used, was approximately -6" C. The factor j, in this calculation, is the same for all temperatures, and it could, if desired, serve as a constant multiplier for constants M and N in Equation 2 to make the equation describe corrected flow rate. If this were done, then
Comparison of Calculation Procedures for Predicted Retention Temperature Emergence Temp., O K.
Heat Rate, Deg./Min.
Actual
8.3 3.2 8.3 8.3 3.2
452 461 557 579 522
Calcd. Old Way With Without internal internal std. std. 453 461 553 572 521
Calcd. New Way With Without internal internal std. std.
448 458 543 563 518
453 461 555 575 522
448 459 544 571 5'70
Predicted Retention Temperatures at Constant Pressure, Run 6 Retention Temp., O K.
e 3319 4049 4609 5214 5022 5514
ANALYTICAL CHEMISTRY
B 0.00151 O.OOO361 0,000147 0.0000511 0.000147 0.0000706
Actual
Calcd.
319 329 345 360 375 388
321 .O 331.6 345.7 360.4 371.6 386.9
j would not appear in Equation 5. In all cases, of course, pressure-drop-corrected retention volumes must be used to compute 8 and B. At constant pressure, this correction does not affect the value of e, but does affect B. Application of New Equation to Ordinary PTGC. As already stated, there is a n inevitable variation in flow rate in so-called "constant flow" PTGC. It happens, however, t h a t even in this case flow rate is a linear function of reciprocal absolute temperature, provided one first corrects flow for gas expansion. Such a relationship is shown in Figure 1, curve A . I n this case, constants M ( = 6110), and N ( = 10.7) already include the factor j, so t h a t i t must not be included in further calculations which make use of these constants. Curve A , Figure 1 is based on data taken from a previous paper (IO), where calculations of predicted retention temperature were made by Equation 1 on PTGC runs using a IO-foot Apiezon-L column with helium as carrier. The linear dependence of corrected flow on 1/T makes it possible to recalculate these results using Equation 5. Representative values are shown in Table 111. The principal point to observe here is that the new calculation method gives agreement with experiment generally as good or better than the old, which verifies the soundness of the theory. The C,? n-paraffin case was the worst of all in point of agreement, and some improvement is to be noted. Another application of Equation 5 to ordinary PTGC involved Run B. I n this case, M = 2645 and N = 8.81. The results are shown in Table IV. It is interesting to note here that the measured temperature rise rate (3.45' per minute) was employed, in contrast to other cases where an internal standard had to be used. In this case, the constants e and B for CI1 and C12paraffins were determined over a different temperature range than the other compounds, and they are thought t o be less accurate. In fact, the most difficult part of this work is to obtain good values of these constants. Although the variation of retention volume with reciprocal absolute temperature is represented as linear, there is in fact a slight curvature, more or less, depending upon the solute-solvent system. In other words, AH is not completely independent of temperature. Direct Arithmetic Calculation of Emergence Temperature. Previous calculations of emergence temperature involved the use of a chart and were in general not convenient. It was not possible t o use computer methods because no direct, rigorous, analytical solution exists for T,, the emergence temperature. However, a method has now been developed for
solving this problem very satisfactorily in a devious way. The problem is to find 92 in a = -e-4s - - + e-41 62
represented aa follows: Note that this sequence d ~ e r slightly s from that d e scribed previously, where only charts where used.
(6)
d1
where +z = B/T,(B = constant; 9 and +1 are known). It was found that, in the general case, if
then lnf(z)
=
Ctz
+ Cl(z)l’z + Co
is an excellent approximation, where C2, C1, and Co are constants. These constants can be found by solving several simultaneous equations containing selected, known values of z and In f(z), treating C2, C1, and COas unknowns. Thus: @ = e-*!(&) - e-+lf(&) = e - eln I (e) - e - +I eln f(+d = exp KCz - l)& CI(&Yz COI- Y where Y is the +1 term, which can be computed, since +1 is known. Accordingly :
*
In (a
+ Y)=
+
(C2
-
+
1) &
+
CI
(#22)‘/*
+ co
Solving the quadratic : (&)”Z
=
The third value of is quite close to the ultimate value, even when no flow controller is used-i.e., constant pressure operation. If a controller is used, then the value of R has much less effect on the final result and the second value of 92 is adequate. The direct calculation of from 9 and bll described in a previous section, fits in well with machine computation. Thus all operations in the above sequence except finding R can be done well by machine. R can, of course, be determined from a plot such as Figure 2, and this yields satisfactory accuracy. However, the use of a graph does not fit in with machine computation very well. To solve this problem, an equation has been evolved for the numerical calculation of R: R
and
e
T, = -
42
Results of application of this calculation procedure in some illustrative cmes are shown in Table V. The above calculations were made with an IBM Model 610 computer, using a single set of constants (Cz= 0.15777, Cl = - 2.18336, Co = 0.55290). The computer is convenient, but not wsential; the same calculations can be done with a log-log slide rule with an accuracy of (usually) 1 in the second decimal place-i.e., one can distinguish between 9.01 and 9.02. The accuracy of the computation can be improved by selecting constants to cover only a limited range of 9 values. While the accuracy shown is quite sufficient for the present purposes, this point is being emphasized because the function in question is fairly common, and the method of finding one limit of the definite integral may be useful elsewhere. So far as the author knows, the method, including the approximation, has not been described elsewhere. Machine Computation of Emergence Temperatures. The use of the method just described along with Equation 5 for determination of d2 and - &), and thence T,, can be
=
+I
- (0.5712 - 0.00116 $1)
A+
-
where A+ = By this empirical relation, R can be computed with an error of no more than 0.02 in the range = 6 to 20 and A@ = 0 to 3.2. The IBM 610 computer was programmed for calculation of #z,making use of the equations and techniques discussed. The calculation is automatically repeated as many times (as many iterations) as desired. A suitable criterion of completion is constancy of (+1 - 9Jl although as stated before, this is hardly needed. The calculation is self-correcting-Le., i t will approach the true value of #2 from either side.
A Comprehensive Equation. The incorporation of flow rate in Equation 5 as a variable quantity and the discovery that the equation holds for ordinary P T G C as well as constant pressure PTGC, make i t possible to write down a comprehensive relation which describes the position of a peak in the column a t any time. The constant 0 is independent of the amount of solute in the column, while B is directly proportional to this quantity. We shall define B, as the value of B per gram of solute in the column of length L containing G grams of solute. Let SL be the distance from column inlet to the location of a certain peak a t temperature T’ in the linear program. Then, assuming uniform packing, sB,G is the value of B pertaining to the initial distance SL for this substance. Likewise, (1 - $)BUGis the value of B pertaining to the rest of the column. We may now write
using Equation 4 as a model, where = e/T‘, and a1 pertains to the limits 9’and +l. To find the position of a peak a t any selected temperature TI, where A < T’ < T,, it is necessary to evaluate the right-hand side of Equation 10 and solve for s. It follows also that we can determine T‘ for a given s. This is, of course, the problem which is the subject of this paper. The equation describing the peak behavior in the outlet portion of the column is
9’
a(l
- s) BUG =
%Ne
dz
e-+
- M L , 7 d+ (11)
where GZ pertains to the limits
and
9’. If we add Equations 10 and 11, after condensing terms, we have Equation 4, which pertains to the whole column. This is equivalent to joining two segments of column end to end.
Table V.
Test of Direct Calculation Method for Predicted Retention Temperature Temperature #2 True Calcd. True” Calcd. dl Arbitrary values 7.000 6.000 5.996 1o.Ooo 9.000 9.002 10.000 8.000 8.003 15.000 14.000 13.998 15.000 11.800 11.798 17.000 15.000 14.998 17.000 12.200 12.198 Values baaed on actual data 8.668 8.030 8.032 394 393.9 11.370 9.497 9.499 437 436.9 13.749 10.863 10.862 524 524.0 15.594 10.904 10,904 522 522.0 “True” so far 88 this calculation ia concerned-i.e., the theoretically perfect answer.
VOL 34, NO. 9, AUGUST 1962
1045
If the form of Equation 5 is desired (and this ie really easier to work with), the procedure is straightforward. Two values af R are involved: R1 (pertaining to +1 - +’) and.&*(pertaining to +’-+J. R*
= &@I
+ Rr@a
The factor j does not appear, for it is incorporated in the constants M and N in the manner described previously in this paper. The constants e and B,, as always, must be calculated from gasexpansion-corrected retention volumes. This may be done conveniently by obtaining the necessary effective flow ratea
from Equation 2 using the j-corrected M and N valuea. LITERATURE CITED
(1) Dal Nogare, S., Langloie, W. E., ANAL.CHFiM. 32, 767 (1960). (2) Fryer, J. F., Habgood, H. W., Harris, W.E.,Ibid., 33, 1515 (1961). (3) Giddings, J.. C., J . Chromatog. 4, 11 (1960). (4) Giddings, J. C., Proceedings of GC Symposium, p. 41, Michigan State University, East Lansing, June 1961. (5) Habgood, H. W., H a m , W. E., ANAL. CHEM.32, 450 (1960). (6) Handbook of Chemistry and Phpica,
39th ed., p. 2045, Chemical Rubber Publishin Co., Cleveland, 1955. J.. Rev. 81%.Imts. 32. 1 17) Hoae. . (1960.’ (8) JameB, A. T., Martin, A. J. P., Biochem. J . 50, 679 (1952). (9) Partington, J. R., “An Advanced Treatise on Physical Chemistry,” Vol. I, p. 847, Longmans, Green & Co., London, 1949. (10) Rowan. R.. ANAL. CHEM.33. 510 ‘ (igsi). (11) Sad, A. S., Proceedings of GC S pogium, p. 65, Michigan State niversity, East Lansing, June 1961.
8.
I
.
r
RECEIVEDfor review March 8, 1962. Accepted June 1, 1962.
Effect of Substituents on Relative Retention Times in Gas Chromatography of Steroids B. A. KNIGHTS and G. H. THOMAS Department of Anatomy, Medical School, University of Birmingham, Birmingham 7 5, England
b A number of steroids have been chromatographed on columns coated with QF-1-0065 (a fluorinated silicone). The results have been expressed as the logarithm of the retention times relative to cholestane (log r). Evidence is presented to show that the log r values can be estimated from the additive contributions of the individwl substituents together with that of the steroid nucleus to which they are attached. The log r contribution for a substituent depends not only upon its chemical nature, but also on its position in the molecule and its stereochemical configuration.
D
the past two years the separation of steroids by gas chromatography has been accomplished successfully by a number of workers (9, 6-11, 13-28). The substances analyzed include in addition to derivatives of androstane and pregnane, vitamins Dz and DB @3), corticosteroids (17), Cn to C2g sterols @) and their methyl ethers (6), sapogenins (18), bile acid esters as their trifiuoroacetates (go), and estrogens as their acetates (92). Further, gas chromatography has been used to determine the concentration of cholesterol and squalene in blood (14) and the 17-ketosteroids in urine (9). Lipsky and Lrtndowne (13) studied the effect of both polar and nonpolar stationary phases on the relative retention times of androstane and pregnane derivatives. Horning, Vanden Heuvel, and Haahti have also used a number of phases of differing polarity. These range from the nonpolar methylsilicone gum SE30,through a variety of URINO
1046
ANALYTICAL CHEMISTRY
polyesters to the fluorinated silicone QJ?-1-0065 (16). The latter had a remarkable &ty for ketones compared with alcohols. Thus, 5a-pregnane-3,20dione had relative retention time, 5.93, (cholestane = LO), whereas 3&20@dihydroxy-5a-pregnane had relative retention time, 1.94, and the corresponding 3-hydroxy-%ketone, 2.98. Using SE-30,the relative times were 0.72, 0.67, and 0.67, respectively, and for neopentylglycol succinate 7.20, 6.47, and 6.52. The most polar stationary phase for diols and diones so far used, appears to be ethyleneglycol isophthalate (10, 11). Mixtures of this ester in varying proportions with SE-30 have been used to obtain intermediate retention times without reducing the relative separation between steroids (10). In view of this, QF-1-0065 appeared to be a suitable stationary phase for studying the effect of substituents on the relative retention times of steroids. The results of this investigation are presented. EXPERIMENTAL
A Pye argon gas chromatograph with strontium-90 ionization detector was used. Chromatography was effected on a 4-foot column, 6/sAnch i.d., packed with acid washed Celite 545 (85-100 mesh) coated with 6% QF-l0065. The temperature of the column was 250’ C. The gas flow rate was 55 ml. per minute (inlet pressure 16 p.s.i. argon), Samples were introduced as solids from the end of a glass rod and were not preheated or flash-heated. The time of emergence was measured from the negative air peak, and the relative retention times (r) were calculated using cholestane as standard. The efficiency of the column was about
1600 theoretical plates (for androst-4ene3,17dione). Where the same compounds have been investigated, the results agree well with the values recorded by VandenHeuvel, Haahti, and Horning (16) using 1% QF-1-0065 as the stationary phase at 195-202’ C. RESULTS AND DISCUSSION
Relationship between Structure and Relative Retention Time of Steroids. The concept developed by Bate-Smith and Westall (1) that RM values for the components of a molecule contribute additively to its paper chromatographic mobility, has been applied only recently to steroid analysis (3-6,19). Clayton (7) has demonstrated that the retention time r of a polysubstituted steroid in which intramolecular group interactions are negligible can be expressed as r =rnXk,Xkr,Xk
where r, is the retention time of the unsubstituted nucleus and l ~ . . s , .~ are group retention factors for a series of noninteracting groups a t positions, a,b,c . . of the nucleus. We wish to show that the results obtained for gas chromatography of steroids are also amenable to this type of quantitative treatment, and that the logarithm of the retention time of a steroid is made up of the additive contributions of the substituents together with that of the steroid nucleus to which they are attached. To demonstrate the accuracy with which such values can be determined, the log r contributions for the methyl group at C-10 were calculated by subtracting from the log r values of the compounds listed in Table I, the log r values of the corresponding 19-nor-
..
.
