this approach is the time saved in the analysis which results both from the ashing step and the final measurement. The use of the rapid ashing technique, in which benzenesulfonic acid is the key factor, reduces the elapsed ashing time from a minimum of 6 hours for the sulfuric acid-coking method to approximately 30 to 45 minutes. The transfer of the ash and the x-ray measurements require perhaps another 30 minutes, so that the entire determination, using duplicate samples, can be completed in well under 2 hours. This will be of great benefit in various refining operations, especially those where blending of feed stocks must be closely controlled t o prevent undue catalyst poisoning. The method is also well suited to multiple determinations; in this laboratory, one analyst can complete about 20 to 24 determinations in an 8-hour shift. Using the method as outlined, the limit of detection appears to be somewhat less than 0.1 p.p.m. when IO-gram
samples are taken. To gain an increase in sensitivity, samples of 100 grams can be employed; in this case experiments have shown that reproducible values in the region of 0.01 to 0.05 p.p.m. can thus be obtained, although the accuracy of such results cannot be verified by existing chemical methods. With the possible exception of neutron activation analysis, which is not practical for routine application in most laboratories, it seems reasonable to assume that the x-ray method provides the most accurate values obtainable for this concentration range. ACKNOWLEDGMENT
The authors express their appreciation to N. D. Coggeshall and D. T. Fanale for their encouragement and helpful suggestions, to M. S. Norris for reporting the ultraviolet absorption phenomenon described, and to R. L. Pivirotto for obtaining most of the x-ray data.
LITERATURE CITED
(1) Cavanagh, M. B., Roe, R. M.,
U.S.
Naval Research Laboratory Rept. No. 5158, July 25, 1958. (2) Davis, E. N., Hoeck, B. C., A N ~ L . CHEW27, 1880 (1955). (3) Dwiggins, C. W., Dunning, H. X., Ibid., 31, 1040 (1959). (4) Erdman, J. G., Mellon Institute of Industrial Research, Pittsburgh, Pa., private communication. (5) Hale, C. C., King, W. H., Jr., ANAL. CHEM.33, 74 (1961). (6) Horeczy, J. T., Hill, B. N., Walters, A. E. Schultze, H. G., Bonner, W. H., Ibid., 27, 1899 (1955). ( 7 ) Kang, C. -C. C., Keel, E. W., Solomon E., Zbid., 32, 221 (1960). ( 8 ) Nut. Bur. Standards (U.S.) Rept. N o . 7010, October 31, 1960. (9) Sandell, E. B., “Colorimetric Determination of Traces of Metals,” Third Edition, p. 668, Interscience New York, 1959. (10) Ibid., p. 928. RECEIVED for review November 1, 1960. Accepted December 15, 1960. Division of Petroleum Chemistry, 138th Meeting, ACS, New York, N. Y., September 1960.
Pred icti o n of Retention Te mperatu res in Programmed Temperature Gas Chromatography A Descriptive Equation and Computational Method ROBERT ROWAN, Jr. Esso Research and Engineering Co., linden, N. J. This paper covers the derivation and experimental study of an equation describing temperature programmed gas chromatography. It also describes a practical method of calculating predicted retention temperatures. According to the expression arrived at, the experimental variables are all related in a simple w a y except for one factor (function Phi) which contains exponential and exponential integral terms. The complexity of function Phi causes no difficulty in the use of the expression. With the aid of a single universally applicable chart, peak temperatures can b e predicted rapidly and with good accuracy. The method and examples of its use are presented, and data are given for construction of a chart.
I
for several years that programmed temperature gas chromatography (PTGC) is able to provide improved peak shape and important time savings when running wideboiling mixtures. Sometimes peaks appear which would otherwise be missed because of their broad low shape. Under proper conditions, the spacing T HAS BEEN KNOWN
510
ANALYTICAL CHEMISTRY
between members of homologous series is linear rather than logarithmic. These advantages have led to considerable use of the technique in spite of the fact that, until recently, calibration has had to be largely empirical. Various authors have reviewed the earlier work (3, 4). More recently, papers treating the theory of programmed temperature gas chromatography (PTGC) by Habgood and Harris ( 7 ) ,Dal Nogare and Langlois ( 5 ) ,and Giddings (6) have appeared. Dal Nogare and Langlois derived an equation which, upon numerical integration by means of a computer, yielded results in good agreement with experiment. Habgood and Harris derived a relationship between the definite integral of reciprocal retention volume and the ratio of heating rate to flow rate. They also proposed a method based on the equation by means of which, by graphical integration, emergence temperatures can be predicted. Most pertinent is the paper by Giddings (6), which appeared after the present work had been completed. In a purely theoretical study of the sub-
ject, Giddings derived an equation which describes PTGC in terms of fundamental and experimental (but not empirical) parameters. The equation developed in the present work was substantially the same as that of Giddings, although the approach was different in some particulars. This paper is presented as an alter. native treatment of PTGC. DISCUSSION
Derivation of Equation. The starting point is a relationship which can be easily derived from the partition coefficient (IO):
where
V = Apparent retention volume V , = Voidspace V , = Retention volume corrected for void space Absolute temperature 8 = Slope of straight line B’ = Constant which satisfies equation
T
=
If t, is the retention time (total time
minus dead time), uT the flow rate measured a t the column temperature, and u the flow rate measured a t flowmeter temperature &, then In In
+V
= In
( u T L / T )= In (ut,/&)
+ In ( B ' Q ) = 8 / T + In B In tr = 8 / T + In B - In u
(ut,) =
8/T
and tr
=
B / u exp. ( 8 / T )
(2)
We may write, by definition:
where t, = Time air peak emerges,* t, = Time sample peak emerges Substituting Equation 2,
DI
Note that u is treated as a constant; experimentally, it is held almost constant. Express T as a linear function of t: T=at+A
The factor in Equation 4 containing the exponential and exponential integral terms, here called function Phi (a), can be evaluated in a manner to be described. Expansion of carrier gas as the peak moves toward the column outlet has the effect of increasing the peak velocity. The results mill be independent of this effect, however, as long as the column length does not change. The assumption is made that the pressure drop and temperature effects on peak velocity do not interact, or that such interaction is negligible, if it occurs. Equation 4 is the basic relationship for finding Tv, the emergence temperature, since all quantities except @ are known. The problem becomes one of finding T , when @ is known. This is accomplished by means of a chart relating log @ and (+] - 42). Once we have the latter value, we can find 42 (+I is known), and T, = 8/42. A more detailed discussion of the method of temperature prediction will be found in a later section of this paper. Calculations. The most critical and difficult part of the computation is the evaluation of 9, starting with and c$~. All else is simple arithmetic, although sometimes laborious. The exponential integral term of Equation 4 can be expressed as the sum of two terms, as follows:
where A = Starting temperature (temperature a t time air peak emerges), and
a
=
Temperature rise rate.
Let O/T = 6,
d+ = [-&/(at
+ A)'] dt
and
where 41 = @/A, 42 = E)/@, cention temperature
T, = Re.
Fully integrated, Equation 3 yields an infinite power series which is hard to evaluate, since it a t first diverges, then converges. Equation 3 can, however, be integrated by parts:
The terms in function Phi can be evaluated by summing suitable data taken from tables or by using various approximations to calculate them. The 1V.P.A. tables (11) list values of ez and e-z a t 0.01-intervals of the argument up to X = 10. The same tables list, under the heading -Ei(-x), the function
Lrn7
x up to x
= 10. Above x = 10, however, the tables are not adequate, and x (or 4 ) is above 10 in a large fraction of the practical cases. When x is greater than 10, the following approximation is sufficiently good (11):
-Ei( -2)
* An
opposing viewpoint is that the void volume is being neglected and that to this extent the present treatment represents an approximation. The position taken here is that the derivation is rigorous. The fidelity of the final equation depends only upon how well Equation 1 holds, there being no approximations or assumptions in between. I n any event, the error would be negligible in almost all cases.
dx, a t satisfactory intervals of
=
This formula (the first six terms) was used in this work until a better way was found. The equation should only be used when x is large, and too many terms should not be included, since it soon begins to diverge. The most satisfactory way found of evaluating @ was to calculate (by digital computer) the terms of the function by means of approximations for ez and
-E(-x) given by Hastings, Hayward, and Wong (8). These formulas are polynomial expressions containing a number of constants. A number of calculated values of e - $ and of - E i ( - x ) were checked against values taken from the W.P.A. tables, and there was no deviation before the seventh significant figure. While this degree of accuracy is not required, normal engineering accuracy is not sufficient because Phi is the difference between two numbers which are frequently close to being equal. Virtually all the calculations in this study were done on an 11311 Model 610 digital computer. EXPERIMENTAL TEST
OF THEORY
Experimental Procedure. Both isothermal and programmed temperature ( P T ) data lvere obtained with a n F & ILI, Nodel 202 gas chromatographic unit using a 10-foot Apiezon L (on firebyick) column. The carrier gas was helium. and sample size was about 5 pl. The flow control system of the F 8: M unit is arranged in the following manner: pressure regulating valve on cylinder + differential flow controller needle valve column. (Moore) A dial pressure gage was installed between needle valve and column so that pressure drop across the column could be read. It was necessary t o maintain pressure upstream of the flow controller above 45 p s i . Under these conditions flow, measured with a soap bubble flowmeter a t room temperature, did not vary over the temperature range by more than about 5%. Temperature rise rates were calibrated and found to be a few per cent low compared with values indicated on the unit. Isothermal data for CS to CIZ nparaffins are shown in Table I. AS is customary, these values of V , received the James and Martin correction for expansion of carrier gas (9). Tables of the correction factor, j , have been published (2). Retention volumes for programmed temperature runs were similarly corrected. Although flow rate was supposed to be constant, it did vary over a narrow range, as did pressure drop. The arithmetic average values of j and u within this range were estimated on the basis of the temperature range traversed for each compound, and the indicated corrections were applied. Validity and Utility. Results of the application of Equation 4 t o four runs on Cs t o C12 normal paraffins are shown in Table I1 and Figure 1. T h e method of prediction was briefly outlined in a previous section and rvill be described in more detail later on. Table I1 [results designated "Pre-
-
-
VOL. 33, NO. 4, APRIL 1961
511
Table I.
Temp., OK. 393 413 443 503
Isothermal Data and Calculated Constants for Normal Paraffins
Pressure Drop, P.S.I.G. 11.6 12.4 13.6 15.6
Flow, Ml./Min. 37.5 37.3 37.2 36.4
c 6
125 88.2 57.2 30.9 2699 0.0908
e*
B* Uncorrected data. Calculated on V , values corrected for pressure drop. Table II.
a 500
400
450
510
ACTUAL TEUP
Figure 1. Accuracy of temperature prediction
512
cs
ClO
Cll
c 1 2
2181 1222 550 184 4626 0.0118
2339 952 285 5055 0.0077
1745 438 5336 0,00679
1346' 670 5692 0.00514
Temperature, 473 K.
OK. Sample Air In. Peak 363 369
dicted (I)"] and Figure 1 show that at a rate of 3.2" per minute, the predicted temperature followed the actual temperature, but was in general 2' or 3" lower, although the bias was not the same throughout the range. At a rate of 8.3" per minute, the story was much the same except that the bias was about 10" to 12". The reason for the greater bias at the higher rate is not known, although it clearly has t o do with the flow rate or temperature rise rate, or both. As a speculation, however, it might be explained on the basis of a thermal gradient between axis and wall of the column, Le., the temperature a t the center lags behind. This thermal lag will, of course, be greater, the higher the rate.
400
CS 1072 646 317 118.4 4150 0.0194
529 335 180 76.1 3673 0.0322
Kind of Result" 1 Predicted (1) Predicted (2) Actual 2 3.2 363 365 Predicted (1) Predicted (2) Actual 3 8.3 413 419 Predicted (1) Predicted (2) Actual 4 3.2 413 414 Predicted (1) Predicted (2) Actual Predicted (1): Calculated in usual way-see text. Predicted ( 2 ) :
350
V,, M1.a
c7
A.ctual and Predicted Emergence Temperatures for Normal Paraffins in Programmed Temperature GC
Rate Deg./Min. 8.3
Run
a
C6 260 173.6 102.7 49.5 3164 0.0578
'K
5%
*Q
emergence
ANALYTICAL CHEMISTRY
Table 111.
Temperature of Emergence, O K . CS c, C8 CO ClO 401 422 447 473 497 520 409 432 457 .. 508 532 411 430 458 484 508 532 381 394 413 435 458 479 382 397 416 461 482 379 394 414 437 461 482 423 448 462 482 503 523 . . . 512 532 440 453 470 437 452 470 491 512 534 422 428 437 452 468 485 421 428 440 . . . 468 486 419 426 440 452 468 486 Calculated on basis of C8 as internal standard.
cs
c 1 1
(312
540 552 554 497 501 501 543 553 557 502 504 504
562 575 577 518 521 522 563 572 579 521 524 524
Calculated Ratios of Temperature Rise Rate to Flow Rate
Temp. Rise Rate/Flow Rate ( a / u ) , Deg./Ml. Runa 1 2 3 4 MeasMeasMeasMeasn-Paraffin uredb Ca1cd.c uredb Ca1cd.c ured* Ca1cd.c uredb Calcd." 0.363 0.126 0.084 0.123 0.109 0.331 cs 0.314 0.464 0.402 0,127 0,111 0,120 0.121 0.334 C.3 0.320 0,401 c, n- . _22.5 0.123 0.128 0.340 0.420 0.127 0.150 _ _ 0.428 .. _ ~ _ 0.430 0.128 0.138 0.326 0.433d 0.123 0.138 0.343 CS 0,126 0.138 0.350 0.433 0.131 0.135 0.338 0.430 CY 0.452 0.132 0.137 0.127 0,138 0.347 0.339 0.451 ClO 0.483 0.133 0.140 0.130 0,138 0,352 cn 0.342 0,464 0.512 0.135 0.148 0,131 0,145 0.348 0.352 0,454 Cl2 See Table I1 for conditions. * From measured values of a and u ( ressure corrected). c Effective value calculated from a k = ( 8 / B ) @assuming , 8 , T , and 8 to be correct. d Underlined values used in calculation of ''Predicted (2)" values in Table 11. ~
.-I
Q
It is possible to eliminate these biases by a calculation procedure which, in effect, uses one of the constituents of the sample (in this case C8 n-paraffin) as a n internal standard. The results thus obtained are designated Predieted (2) in Table I1 and plotted in Figure 2. This was done by computing for C8 from A , T,, e, and B only, using the expression in brackets in Equation 4. This value of 9 was then used to calculate a/u for C8, which value was assumed to hold for all the compounds in the sample. Ideally, of course, there is only one value of a/u, but in practice flow rate changes during a run and each constituent has its own effective a / u . Values of this ratio for other compounds, calculated in the
I
I
0 RATE 0 1 2 DEG;UIN 0 RATE
= B 3 DEGNIN
I 350
430
4%
I MO
550
ACTUAL TEMP, X.
Figure 2. Accuracy of emergence temperature prediction using octane as internal standard
same way as the Ce value, are shown in Table 111. For comparison, the measured values of a / u are also shown in Table 111. Referring to Figure 2, the small increase in deviation a t the extremes of the range is presumed to be due to changes in flow rate and pressure drop during the run. The selection of average values for these quantities overcompensates. on the one hand, and undercompensates on the other, since the effects are not linear. Apparently, these effects are not very serious. Pressure and Temperature Corrections. -1s was stated, the raw values of V , were corrected for pressure drop by, in practice, applying the correction to the flow rate. It is proper to use u, the flow rate measured a t a constant kmperature, for reasons shown in the derivation, since u is equivalent to uTIT (except for a constant factor). Another way to arrive a t retention volume measured a t column temperature ( =uTt,), which is what we must have ( I ) , is to correct the flow rate for temperature as well as pressure drop. I n this case, however, the flow rate is not normally constant and cannot properly be taken out from under the integral sign in the derivation. Nevertheless, it is interesting to observe the effects of the temperature and pressure drop corrections (abbreviated “P’and “T”), both together and separately, on predicted emergence temperatures and on the constants 8 and B. Certain isothermal V , data were treated in four ways: P corrected, T corrected. P T corrected, and uncorrected. Constants IT-ere computed for each treatment. I n calculations on programmed runs the flow rate value
+
Table IV.
used in Equation 4 had the same (corresponding) corrections as were used in obtaining 0 and B in each case. Clearly, this has to be the case, since anything else would amount to changing units of measurement in the middle of a calculation without adding a conversion factor. I n the case of programmed runs the pressure drop correction was that corresponding to the arithmetic mean of the temperature limits, Le., ( A T,)/2. The base flow rate within the narrow range of flow variation was similary chosen, and the temperature correction, when applied, was ( A T,)/2&. The result for n-octane are shown helorn :
+
+
Rate, Deg./Min. 8.3
Predicted Actual Predicted -4ctual
3.2
0 1 0 15 0 2 0 3
6.0 . (5)7366 .14)1144 . (-2ji579 ,
. (4)3645
0 4 0 5
,
0 6
0 8
(4)4908
. ( 3j1402 (3)2085
1 0 1 2
1 1 1 2 2 2
(4)25$2
3959 0.0452
The results show clearly that it does not make much difference in the accuracy of prediction which correction is used, or whether we use any a t all, provided only that the same procedure is used in both programmed and isothermal runs. Applicability to Different Flow Rates. To test the effect of variation in flow, the gas rate n-as cut approximately in half (from about 37 t o about 19 ml. per minute). Predicted and actual emergence temperatures were as follows for three n-paraffins:
4 0 8 0 4 8
(3)2911
. (3)4013 . (3)548G
. (337405 , (2)1376 . (23‘2552
3 2 3 6 4 0 4 4
e--di
9.0
10.0
. (5)1986 . (5)3079
.(6)5584 , (6)8649 .(5)1191 . (5)1909 . (5)2723 .(5)3646 . (5)4695 . (5)723i ,1411052 , (4jis76
.(6)1621 . (6)2509 .(6)3452 . (6)5524 . (6)7866 . (5)1051 . (5)1351 . (5)2075 . (5)3002 . (5 j 4 m . (515725 .(5)7697 . (4)1024 .(4)1352 . (4E327 . (4)3969 .(4)6i57 .(3)1153
. (7)4824
. (5)9745
(4)1308 . (4)1688 .14)2616 14)3826
,
(4)5404
(4)7471 .(3)1018 (3)1375 (3)1845 (3)3294 (3)5808
,
(412027
.(4)2740 . (4)3668 .(4148i6 1438523 . (3ji480 , (3)25i3 .(3)4500
cs 508
510
ClO 557
562
P+T 475
435 3796
0 0661
Csto Ciz normal paraffins. The change in 0 is practically linear up to G o , after which the curve falls off. It is probable that the linear relationship should continue beyond Cl0, however. The constants for CIO, CII, and C I ~ were calculated from a different portion of the In V? us. 1 / T plot than the lower members of the series. This suggests that, in general, e values for higher, or missing, members of a series could be obtained by extrapolation. The constant, B, may be obtained in a similar manner, although the curve is not
Values of Function Phi (@) for Reference Charta
8.0
.i5j68ii
435
e
7.0 . i5 14246
Correction P T Peak Temp., OK. 474 473 484 435 435 437 3517 4150 0.0194 0.1838
473
B
450 457
I n this case no corrections to flow rate were used. The heating rate was 8.3’ per minute. These results show that the method takes proper account of flow rate. Further evidence is found in the fact that the method of Habgood and Harris, m-hich is similar in principle, was tested over a much wider range of flow and found to be valid ( 7 ) . Variations of e and B with Carbon Number for Normal Paraffins. Figure 3 shows the variation of the constants e and B with carbon number for
None
dl 41 - 4%
Ce Predicted, OK. Actual, OK.
. (6)2332 .(6)3113 . (6)3995 .(6)6114 ,1638818 . (ij1668 .(5)2233 .(5)2956 .(5)3884
.(5)6607 .(4jiii2 .(4)1865 .(4)3129 . (4)5266
.o 12.0
11
. (7)1468 .(7)7062 .(7)9417 . (6)lZOi .(6)1843 ,16’r2650
. (Sj4~582
. (6)6645 . (6)8’764
.(5)1117 .(531!333 . (5j3221
. (5)5340
.(5)8844 . (4)146i
13.0
14.0
. (833009
. (9)9534
____
15.0
. (8)4526 . (839612 . (7ji533 . (7)2176 . (7)2899 . (7)3711 . (7)5654 .(7)8113 . (6)1123 . (6)1517 . (6)2018 . (6)2654 . (6)3461 . (6)5793 .(6j9576 . (5)1573 . (5)2579 . (5)4230
. (9 j3052 . (9)6882 .(8)1169 .(8)1773 , (8)2531 . (8j3485 . (8)4683 . (8)6191
. ($)7709 . (0)1396
(6)2350
. (6)3393
. (7)1048 . (7)1728 . (7W309 . (i)4531 .( i ) i 2 8 2 . (6)1168 . (6)1874
B-ml
a + = = - - -
42 01 Sumbers in parentheses indicate number of zeros before first significant figure.
VOL. 33, NO. 4, APRIL 1961
513
,
1
5
Figure 3.
6
)
7
,
/
8
I
10 11 CARBONNUMBER
9
I?
13
14
15
IC
Variation of 0 and B with carbon number Normal paraffins [+I
linear. Alternatively, B may be calculated from a single l’, value a t constant temperature, if 8 is known. The constant 0, as mas pointed out by Porter, Deal, and Stross (IO) and others, is a function of the latent heat of vaporization and the excess partial molal free energy of solution of component in substrate. The former quantity ( A H v / R ) predominates in controlling the value of 8, a t least in the case of nonpolar solutes and solvents. Constant B is also related to fundamental physical quantities. I n certain favorable cases these constants might be derived from published information. Probably they had best be determined esperimentally, however, as was done here, in view of the uncertainties and possible cumulative errors involved in assembling terms piecemeal. The constant, 8,is independent of the circumstances of measurement, being a function only of the nature of solute and stationary liquid. The constant, B, will similarly be independent of esperimental conditions provided the retention volume per gram of substrate is the quantity considered. In this case the equation Till be: rf,
=
aB,G U 8
\There B , = Constant when retention volume per gram of stationary phase is used. G = Total weight of stationary phase, grams. Another Relationship Describing PTGC. It was shown previously t h a t t h e two equations: 514
ANALYTICAL CHEMISTRY
- 62,
Figure 4. Chart of log In
0
(ut,) = -
T
+
111 H
iETq
+ In B’ T are essentially equivalent e x e p t for the constants B and B’ ( B = B’Q). Because of the additional factor T, in the second equation, however, it yields a different result when used as the starting point for a derivation such as that leading to Equation 4. Assuming that U T is constant, we arrive a t : In
Calculations according to Equation 5 Lvere made on some of the same data already treated. The d u e s of ?V shown below, when compared with those obtained by Equation 4,leave no doubt that Equation 6 is valid. In each instance the numbers in parentheses refer to the constant-u case, n-hereas those without parentheses refer to the constant-ur case.
$ vs.
- qz)
The constant-u case here was calculated on data having no corrections a t all. This is the situation nhich comes closest to a constant t~~~since the pressure drop and temperature corrections tend to cancel each other. That the cancellation in this instance is not complete is slionn by the fact that 8 values 2’” for the “no correction” a n d “P cases are not equal, ab they nould be othermise. The good correspondence of W values for the two cases does not depend upon constancy of UT. This i. shown by the equally good agreement between other, similar pairs of values calculated from P-corrected data (e.g., 1.334 us. 1.327) and (P ?“‘-corrected data (e.g., 1.186 vs. 1.181). T-corrected data ncre not treated. For the calculation by Equation 5 to be fundanmitally sound, hon ever, it is necessary that be constant. The only way this can be true is for the pressure drop to be coiitrolled nith time so
+
+
IT.’
Rate
Start, OK.
369
Deg./Min. 8.3
365
3.2 0
B‘
n-Paraffins
CS
CS
ClO
1.246 (1.244) 1.006 (1,011) 3425 (2981)
1.295 (1.287) 1.065 (1,062) 4406 (3959) O.OOOO369 (0.0452)
1.273 (1.272) 1.059 (1,056) 5312 (4859) 0.0000147 (0.0182)
0.000109
(0.1321)
that pressure change and temperature effects exactly cancel each other: j X T/Q = 1
where j is the James and Martin correction factor. If pi and PO are inlet and outlet pressures, respectively, p i / p o = b, and =
2= dT
3 .__ (b2 - 1) 2 (b3 - 1) 3 ib
++12)1 2
2Q b(b
I n other words, the variable, b, must be controlled along a line whose slope is described by the above equation. This line is not straight, but is almost so. KOattempts have been made to operate in this manner, although the control problem does not appear to be especially difficult. K h e n programmed runs can be made in this way, it will be possible to achieve the simplification suggested by Equation 5 . It may he that the use of Equation 5 will provide a method of predicting emergence temperatures which is superior to that described here. Further work will ckcide this point. Method of Prediction of Emergence Temperature. A brief outline of t h e method was given previously. It will be described in more detail in t h e following paragraphs. The quantity a for a given peak can be found by Equation 4,(rearranged) : a = -aB 8 ZL
Here, everything is known except a. The problem is to find T,. the peak temperature, from a. This is done by means of a chart
of log us. ( $ I - & ) , a n example of which is shown in Figure 4. Values of a t suitable intervals of and 62)are shown in Table IV. These numbers were computed as previously explained. They have been rounded from values accurate in the sixth significant figure, so that the maximum error in the numbers given is one part in 2000. This is considerably greater accuracy than can be made use of here. It will be observed (Figure 4) that the linear distance between two given lines of constant $1 is the same for any value of ($1- $2). Also, the distance between lines of constant $1 changes very slightly as changes. This means that interpolation between these lines is practically linear. For example, if linear interpolation is used for a point halfway between two lines, the error is about one part in 200. -4step-ise procedure for this calculation is given below. It is assumed that 0 and B have already been determined from two or more isothermal runs. 1. Calculate 0
(+
3
= -
2. Find$ll+l = @ / A ) 3. Refer to the chart and find a. Move along horizontally to a point representing $1, bet-xeen two chart lines of constant (integral) $1. .4t this point, which satisfies both + and o1, read off ($1
- 93).
Example: $1 = 9.31 + = 1.67 X 10-6 Distance between ($1 = 9.0) and ($1 = 10.0) is67.5mm. 0.31 X 67.5 = 21.0mm. At a point 21 mm. down from $1 = 9.0, find a. This defines one and only one point on - $2) = the chart. This point lies on 0.88. 4.
From
($1
- +2), find $2.
$2, find T,(T, = &/e). 6. From T,, find t,, the time
5. From
[t, =
( T , - A ) / a ] ,if t, is desired rather than T,.
If it is desired t o use the internal standard technique, a value of which is computed from e, B, A , and T , only can be obtained by reversing the calculation procedure [Le., start with ($1-$2)]. This value of can then be used to compute a n effective a/u. Thus it is clear that the method requires only the most elementary arithmetic operations, in connection with the chart. LITERATURE CITED
(1) Ambrose, D., Keulemans, A. I. SI., Purnell, J. R., , ~ N A L . CHEM.30, 1582 ( I 958). (2) “Chromatographic Data,” J . Chromatog., 2 , D33-L)45 (1959). (3) Dal Nogare, S., A s . 4 ~ . CHEW 32,
1F)R (1960). (4) Dal‘Nogare, S., Harden, J. D., l b i d . , 31,1829 (1959). ( 5 ) Dal Xogare, S., Langlois, W. E., Ibid., 32,767 (1960). (6) Giddings, J C., J . Chromatog. 4, 11 (1960). ( 7 ) Habgood, H. LY., Harris, W. E., A r a ~ CHEM. . 32, 450 (1960). (8) Hastings, C., Hayward, J. T., Wong, J. P., “Amroximations for Digital Comouters.”‘ Princeton Cniv. press.. 1955: (9) James. James, A. T.. T., Martin. Martin, -4.J. P.. P., Biochem. J :. 50, 679 (1982j. (1982). (10) Porter, P. E., Deal, C. H., Stross, F. H., J . Am. Chem. SOC.78, 2999 (1956). (11) “Sfpe, Cosine, and Exponential Functions, Federal Works Agency, W.P.A. Sponsor: Satl. Bur. Standards, recent edition, Government Printing Office, Washington, D. C. \
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RECEIVEDfor review hugust 30, 1960. Accepted January 30, 1961. Division of Analytical Chemistry, 138th Meeting, ACS, Sew York, X. Y., September 1960.
A Vapor Detector Based on Changes in Dielectric Constant J. D. WINEFORDNER, D. STEINBRECHER, and W. E. LEAR University of Florida, Gainesville, Fla.
b A sensitive stable vapor detector based on response to changes in dielectric constant i s described. This detector consists of a variable capacitor mounted in a special cell which allows gas to flow between its plates. The capacitor i s part of the resonant circuit of a Clapp oscillator, the output of which i s beat against that of a reference oscillator. It i s shown theoretically and verified experimentally that if the difference frequency between oscillators i s adjusted to zero with a pure carrier
gas flowing through the cell, the presence of an impurity vapor in the cell produces a difference frequency which i s a linear function of the amount of an organic liquid introduced into the cell via the carrier gas. Experimental results show that the response of the detector i s extremely rapid and i s nearly insensitive to high carrier gas flow rates and variations in flow rate. It should be possible to use this detector for monitoring flow streams, particularly as applied to gas chromatog ra phy.
T
describes a vapor detector which is sensitive to changes in dielectric constant. The detector consists of a special cell constructed to allow gas flow betn-een the plates of a capacitor m-hich is part of the tank circuit of a 70 mc. per second Clapp oscillator. The output of this oscillator is beat against that of a reference oscillator and the difference frequency is a direct-reading determined by frequency meter. When the difference frequency is adjusted t o zero with a pure carrier gas flowing through the HIS PAPER
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