High Precision Measurements in Gas Chromatography A Study of Systematic Errors on the Determination of Retention Times Michel Goedert' and Georges Guiochon Laboratoire de Chimie Analytique Physique, €cole Polytechnique, 7 7, rue Descartes, Paris Veme, France
A comprehensive study of the sources of systematic errors in the measurements of retention times has been made. These sources include measurement errors, which are related to the problems of time measurements and are usually small and easy to correct and instrumental errors which are more difficult to correct and eventually limit the accuracy of the measurements. The main sources of systematic, instrumental errors are the retention of the "inert" peak (compounds which are easy to detect are not completely inert) and the effect of carrier gas volumes outside the column. These two factors are easy to correct in principle, but the random errors made in the determination of the parameters necessary to Calculate these corrections are the main limitations to the present accuracy of retention times measurements.
In a previous work, we have studied the sources of random errors in the measurement of retention times ( I ) . This theoretical and experimental study showed that the fluctuations of the inlet and outlet pressures and of the column temperature account for all the instrumental errors. A high precision gas chromatograph was built; the reproducibility of the measurements obtained with this equipment is in agreement with the predictions of the theoretical study ( I ) . Further studies have been made dealing with the problems of data handling in high precision work (2-41, the determination of the retention time from the signal using either peak maximum or first moment, and the effect of noise and density of measurement points on the precision of these determinations (5-7). As a result of this work, retention times can be measured with a relative precision of about 10 - 4 , provided suitable experimental conditions are realized. Measurements of retention times can serve two main purposes: qualitative analysis or thermodynamic studies. In both cases, the precision of the measurements is important but, of course, it is accurate measurements which are necessary. This requires precise measurements, the identification of all sources of systematic errors and either their suppression or the derivation of suitable corrections. The solution to this problem depends, in part, on the intended use of the measurements. In qualitative analysis, relative retentions are mostly used (retention indices are derived from relative retentions): lPresent address, c / o Professor R. S. Juvet, Department of Chemistry, University of Arizona, Tempe, Ariz. 85281. (1) M . Goedert and G. Guiochon, Anal. Chem., 42,962 (1970). (2) S. N. Chester and S. P. Cram, Anal. Chem., 43, 1922 (1971). (3) S. N. Chester and S. P. Cram, Anal. Chem.. 44, 2240 (1972). (4) M. Goedert and G. Guiochon, Chromatographia, in press. (5) M. Goedert and G. Guiochon J. Chromafogr. Sci.. in press. (6) M. Goedert and G. Guiochon Chromafographia, in press, 1973 (7) J. Wijvliet. Ph.D. Thesis, Eindhoven, The Netherlands, 1972.
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ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
where a1.2 is the relative retention of compounds 1 and 2, t H 1 and t ~ and 2 t , are the retention times of compounds 1, 2, and of a nonretained compound, respectively. Then only the factors which change the adjusted retention times, t R L - t , need to be accounted for; for example, carrier gas volumes outside the column such as connection tubes between column and sampling system or detector have no effect. In solution thermodynamics, the retention time is only an intermediate parameter in the determination of the specific retention volume, which is the only retention data of thermodynamic significance (8, 9). This requires not only the accurate determination of the absolute retention time, but also the measurement of the exact amount of stationary phase in the column and the determination of the carrier gas flow-rate, two other difficult problems. In our experiments in adsorption thermodynamics ( I O ) , we want to determine the variations of the adsorption energy with the temperature and to calculate the heat capacity of the sorbed molecule, which is a more precise test of the model of the sorbed molecule than the adsorption energy or entropy (11). We need only to study the variations with temperature of the column capacity ratio of the studied compounds:
which, in theory, makes necessary accurate knowledge of the absolute retention time of a nonretained compound and the adjusted retention times of the studied compounds. However, we have studied the systematic errors on the retention times of nonretained and retained compounds, and we have tried to calculate corrections for these errors. Opposite to what happened for random errors (121, it is not possible to make a general, theoretical study of this problem. Systematic errors depend very much on the equipment used even if the different possible sources are the same for all, and the derivation of a suitable correction will be mainly empirical as it involves complex functions of many parameters which are not amenable to exact calculations. Nevertheless, this study should allow one to understand how systematic errors originate and can be calculated and, partly, corrected. Finally, we are never sure that all the systematic errors have been identified A. B. Littlewood, "Gas Chromatography." Academic Press, New York, N.Y., 1962. S. Dal Nogare and R. S. Juvet, "Gas Liquid Chromatography," Interscience, New York. N.Y., 1962. C. Vidal-Madjar. M. F. Gonnord, M. Goedert, and G. Guiochon, to be published. C. Vidal-Madjar and G. Guiochon, Bull. SOC.Chim. F r . , 1971, 3110. G. Guiochon, M. Goedert, and L. Jacob, "Gas Chromatography 1970," R. Stock, Ed., The Institute of Petroleum, London, 1971, p 160.
and corrected, as there is a t present no independent check of the measurements which can be considered with confidence in its accuracy. J u s t as with random errors (12), we can distinguish systematic errors arising in the system of measurement or in the chromatograph itself. We shall discuss successively the various errors identified so far which belong t o these two types and derive a total correction. Finally, we shall discuss some preliminary results. In all cases, the error is given as the true value less the observed value. So, for correction of the observed value. the error calculated has to be added t o it.
THEORETICAL Errors of T i m e Measurements. There are various methods of obtaining retention time measurements, but for high precision determinations, only the first moment of the peak l.5) (retention time of the center of mass) or the time of the peak maximum (7) (from a parabolic least square fit of the data around the maximum of the signal) are used. Other methods have been studied elsewhere (1) Although convenient for many applications, they are more difficult to program or offer a high enough precision only for a much larger signal-to-noise ratio than the other two methods. The discussion of the systematic errors will be made in view of the application to one of these two methods of determination of retention times. The systematic errors arising in d a t a handling and associated with the density of points of measurements and the width of the integration limits have been previously studied I2-*5). It is easy to find conditions in which they are smaller than The systematic errors originating in time measurements are those associated with the clock, the definition of the starting time, the time control of the digital voltmeter, the effects of noise and of the time constant. All these errors are rather small and easy to correct. Measurement o f Time (€1). All systems of numerical measurements use a clock which periodically sends a pulse to the measuring equipment. This pulse initiates a process which finally results in the acquisition of the signal and the corresponding time. Most equipment uses the mains frequency as the fundamental frequency. This is correct for routine measurements as most electrical companies guarantee 86400 X 50 (or 60) periods a day and a standard deviation of the number of periods per second less than 0.4%. For very accurate measurements, it was preferable to use a quartz clock, which is much more stable for a short time, and not much more expensive. Because of design problems, however, it is not possible to adjust the fundamental frequency of the clock to better than 2-3 x of the desired value. Taking the rounded off value, as is usually done, leads to a systematic error, which is easy to calculate, as time or frequency can be measured very accurately. Thus the contribution is: (3) where uo and u are the required frequency and the actual frequency of the quartz used. Starting Time fez). It is important t h a t the time origin of any measurement be known accurately. It is also necessary t h a t this time origin be related to the injection of the sample into the column. This requires definition of the position of the starting time with respect t o the profile of the injection pulse a t the outlet of the sampling valve and the
residence time between the valve and the column inlet. This last correction will be discussed later, along with the instrumental errors. The definition of the starting time has been studied systematically by Cram (13). A position sensor on the sampling valve sends a pulse to reset the quartz clock during the movement of the valve. Usually the pulse is sent too early resulting in too long a retention time. This error, C Z , is determined from the variations of the retention time through a zero dead volume system, in stable experimental conditions, with the carrier gas flowrate. It is the limit a t large flow-rate of the first moment of the injected band. The capillary tube used for this determination has a low pressure drop. Control of the Digital Voltmeter l e a ) . This error results from the delay with which the order of measurement sent by the clock is performed by the volt meter. It depends on the design of the electronic systems used. Effect of the Signal Noise. If the noise is white, it has no systematic effect on the determination of the retention time by the methods used here f 5 ) , the opposite of what happens with other methods ( 1 ) . This has been checked by comparing the results obtained in computer simulation using random numbers and sequences of recorded baseline noise (5). Time Constant of the Electronic S.vstem ( ~ 4 1 .The flame ionization detector (FID) has a negligible time constant: as the total flow-rate through a typical FID is of several cm3/sec, the residence time in the flame is less than 1 msec and the time necessary to collect the ions is smaller ( 2 4 ) . The electronic system which is necessary to measure the detector current has a time constant which is usually much larger. This time constant modifies the peak shape which becomes unsymmetrical and increases the retention time 16, 1 5 261. It might seem a t first that this effect is adversary and should be reduced by using amplifiers with as small a time constant as possible. but the corresponding error can be accurately corrected, and the amplifier time constant has another effect which is quite beneficial: the reduction of noise by filtering. The first moment of the peak is increased by a n amount equal to the time constant, whether the peak is gaussian (17) or not (28). The increase in retention time of the peak maximum is equal to the time constant T , within less than if the ratio of T to the standard deviation of the peak is smaller than 0.25 (6, 15). As shown by a Fourier series analysis of a gaussian function (16), a time constant smaller than u/4 is necessary to limit the signal reduction below around peak maximum. It is, however, difficult to measure the time constant of a n amplifier with a precision better than 10% 161, because of the problem of the measurement itself, and because actual systems do not give a response following a first-order differential equation but rather a second-order one: strictly speaking there is no time constant. So the time constant has to be chosen small enough to reduce this contribution to the error (4-6). It has been shown also ( 5 ) that as long as conventional methods of handling chromatographic signals are used, the noise reduction which results from a larger time constant allows a better precision of the measurement of the
(13) T. H. Glenn and S. P. Cram, J. Chromatogr. Sci., 8, 46 (1970). J. E. Lovelock, Anal. Chem.. 33, 162 (1961). J. C. Sternberg, Advan. Chromatogr., 2, 255 (1966). G. McWilliamand H. C. Bolton,Ana/. Chem.. 41, 1755 (1969)
(14) (15) (16) (17)
E. Grushka.Ana/. Chem.. 44, 1733 (1972). (18) J. Villermaux, "Column chromatography," E. Kovats. Ed., Association of Swiss Chemists, AARAU (Switzerland), 1969, p 66.
ANALYTICAL CHEMISTRY, VOL. 45,
NO. 7 , JUNE 1973
1189
retention time. The optimum value of the time constant seems to be about of the peak standard deviation (6). Total Contribution of the Errors of Time Measurement. As these errors are systematic, their total contribution is simply their algebraic sum:
and the correction i s made by adding C M to the measurements. As results from the previous discussion, however, the correction is usually not known with great accuracy and the error on the correction may be larger than the correction itself, which may limit the accuracy of the measurements. The random errors ut on the various contributions to C M have been discussed above. They are independent so the error on the correction is: 0.v =
(Z0,2)”*
(19) P. Chovin, Informal Symposium on Gas Chromatography, Gas Chromatography Discussion Group: Manchester, April 1959. (20) R. Kieselbach, Anal. Chem., 35, 1342 (1963). (21) V . Maynard and E. Grushka. Anal. Chem., 44,1427 (1972).
*
where D , and D o are the inlet and outlet volume flowrates, respectively, V, the volume of the tube, and j the compressibility factor (8, 9). Since the carrier gas mass flow-rate is constant in steady-state conditions:
(5)
The total relative error will usually decrease with increasing tr,, even if the time constant is kept proportional to t,,, because the contribution arising from sampling time is constant. Instrumental Errors. These errors are the most important, and the most difficult to correct. They depend on several parameters in the same time (local pressure, carrier gas flow-rate, temperature, . . .) and the theory of chromatography offers a limited help in their calculation, so that direct measurements give data on overall effects which are difficult to account for accurately. We are not even sure we have identified all the systematic errors, and there is no independent constant known with enough accuracy which could be measured by gas chromatography to check our results. We feel, however, that most possible sources of systematic errors have been studied. We have determined the contributions to the retention time of the transit time through carrier gas volumes outside the column, of the retention of the “inert” compound, of the column temperature fluctuations and of temperature gradients, and of spurious phenomena arising when the sampling valve is actuated. Transit through Carrier Gas Volumes outside the Column ( e l ’ ) . The retention time is the result of the sum of three contributions: the time spent by the molecules of the studied compound in the stationary phase, the time spent in the gas phase inside the column which is the same for all compounds, and the time spent in empty tubes or other volumes between the sampling valve and the column inlet, the column outlet, and the detector. The adjusted retention volume ( t R - t m ) is not affected by these volumes and the calculation of specific retention volumes does not require the experimental retention data to be corrected for this effect. However, the introduction of a noncorrected “inert” peak retention time in the calculation of some quantities of thermodynamic interest such as k ’ would result in an error. The existence of these volumes, the problems raised by their contribution and by their measurement have been discussed previously ( 2 5 , 19-21), but the need of an accurate correction makes a more detailed analysis necessary. Especially column upstream and downstream gas volumes should be determined separately, as the carrier gas velocity is different and changes differently with flow-rate. We shall
1190
assume that these gas volumes have a low pneumatic resistance compared to that of the column, so the pressure drop through them is negligible. The determination of the upstream ( V u ) and downstream ( v d ) carrier gas volume outside the column can be made from measurements of the retention time of an inert pulse, using a column of negligible inner volume but low permeability, such as a capillary tube carefully crushed with a screw between two metal plates. The retention time is then given by:
ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
J
P, and Po are the column inlet and outlet pressures, respectively. As v,, p,, Po can be measured with accuracy, vuand v d can be derived from a plot of Dot, us. Pi/Po. If
v, is negligible compared to vd, this plot is a straight h e of slope vu and ordinate v d . Alternatively, a plot of Dot, us. D o has a limit equal to V, vu vd when D o becomes very small. The correction is then given by:
+
+
Retention of the Inert Compound on the Stationary Phase (62’). t , in Equation 2 is the residence time of an inert compound which moves as the carrier gas but is not adsorbed or dissolved by the stationary phase, even if the carrier gas is. Unfortunately, no chemical compound is really inert, especially when a FID has to be used as detector. If t R , , is the actual retention time of the compound used to measure the retention time of the inert peak, the relative systematic error on t , is:
k,’ should be either smaller than the expected accuracy of t m or known with an absolute error smaller than that, which is difficult for accuracies better than and usually cannot be achieved by direct measurements. In gas liquid chromatography k,’ is related to the partition coefficient K , and the characteristics of the column by the equation:
Vl km’ = KmVm
(10)
where V , and V , are the volumes of the gas and liquid phase in the column. In gas-solid chromatography, we have the similar equation:
(11) where m a is the mass of adsorbent used to pack the column, A its specific surface area, and UA the adsorption coefficient. K , and Ua can be determined by independent experiments, using chromatographic or static methods, by extrapolation of the results obtained a t lower tempera-
tures, which is an inaccurate procedure, or preferably calculated from solution (22) or adsorption (23) thermodynamics. The correction to apply to the measurement t R , , is then: t2'
=
- k,'t,
=
- km' ~
1
+ k,'tR,m
(12)
It is usually quite significant as k,' is often around Temperature Gradient and Fluctuations (63'). The temperature to which the measurements are referred is the time average temperature, measured in the center of the coiled column. The value of the partition coefficient or k,' measured, however, is not the one corresponding to the average temperature, as the dependence between k' and T is not linear. A correction can be calculated knowing the enthalpy of dissolution or adsorption and the parameters of the column (24). If the equipment is well enough designed and controlled, the correction can be made negligible. E f f e c t o f the S a m p l i n g Value ( e l ' ) . During sampling, the carrier gas flow is interrupted when the valve is moved to injection position and again when it is moved back to sampling position. When the gas flow is temporarily stopped the flow-rate decreases and the pressure profile changes. An abrupt increase in the inlet pressure leads to a perturbation in the flow velocity which moves rapidly downstream and reduces the retention times markedly (25, 26). Fast commutation of the valve to the injection position and back to normal causes two perturbations which interfere and can cancel each other if they occur close together, or are simultaneous (limit for a zero injection time). To this transient effect is added the more simple effect resulting from the fact that, if under constant flow-rates u1 and U Z ,respectively (u1 > UZ),the retention times are tl and t~ ( t l < t z ) , when the flow-rate is abruptly changed after a time At < tl from u1 to U Z ,the retention time is given by (23):
(13) The two effects discussed should be added. The first one dominates a t very small and the second a t large values of At. The magnitude of these two effects depends on the pressure drops through the valve in the two positions. Ideally, they should be low and similar. The measurement correction can be made from systematic determination of the retention time of an inert compound when the time during which the sampling valve is kept in injection position is changed. Overall Instrumental Correction. The total systematic error originating from instrumental effects is given by the sum of the t i ' . For an "inert" peak:
For other compounds, the third term is zero. As these contributions are independent, the random error due to the lack of precision on the various correc(22) D. E. Martire, "Gas Chromatography 1966," A. 6. Littlewood, E d . , The Institute of Petroleum, London, 1967, p 21. (23) C. Vidal-Madjar, L. Jacob, and G. Guiochon, Bull. SOC.Chim. Fr., 1971, 3105. (24) M. Goedert and G. Guiochon, Anal. Chem., 45, in press. (25) L. Jacob and G. Guiochon, Nafure, 213,491 (1967). (26) L. Jacob and G. Guiochon, Bull. SOC. Chim. Fr., 1971, 4632.
tions is given by the addition of the variances. For the inert peak:
For retained compounds, the second term ( u z ' ) is again zero but should be replaced by a temperature term (24). The corrections are usually important, especially for the inert peak and the error on these corrections'is the main contribution to the accuracy of the measurements of retention times. Total Error of Measurement. The total correction is obtained as the s u m of the corrections for the systematic errors of time measurements and of instrumental origin. They have to be calculated for each set of experimental conditions. The final accuracy of the measurements can be estimated from the sum of the variances of the different contributions. The repeatability of the measurements is not the determining factor in their accuracy which is, usually, limited by the inaccuracy of the corrections for the instrumental errors.
EXPERIMENTAL Chromatographic Equipment. The equipment used has been described in previous publications (1, 5). I t incorporates a n oil bath containing the column, the sampling system, and the base of the flame ionization detector. Temperature Control and Measurement. The temperature is controlled by a Melabs proportional controller; the temperature fluctuations are less than 0.01 "C in a n hour a t 80 "C and less than 0.02 " C in a n hour a t 180 "C. The temperature gradient is less than 0.05 "C. The temperature is measured using a platinum resistance thermometer with a high precision bridge (Melabs) and a quartz oscillator (Hewlett-Packard). Both have been calibrated by the Laboratoire National d'Essais, Paris. The second one is much more sensitive but, because of the quartz hysteresis, it is not more accurate than the first one. The absolute temperature is known with a probable error of about 0.05 "C. This is the drift of the, thermometer between two calibrations a t 18 months apart. Pressure Control. T h e outlet pressure is controlled by a Negretti and Zambra valve, which is temperature controlled ( * O . l "C) and works by reference to vacuum (0.01 Torr). The pressure fluctuations in the detector, where the absolute pressure is set a t around 1100 mbar, are around 0.3 mbar. The inlet pressure is controlled by a Texas Instruments pressure controller using a quartz Bourdon tube manometer. The controller works by reference to the detector pressure, thus keeping constant the pressure drop through the column (fluctuations -0.015 mbar in an hour). The carrier gas is helium and the flow-rate 0.146 cm3/sec. Column and Gas Circuit. The 2-m long, 2-mm i.d. stainless steel column is coiled (8 turns) and packed with 2.42 grams of graphitized carbon black, Sterling M T G (sample D i . specific surface area 8.7 m2/g). The gas circuit has been designed to limit the carrier gas volumes outside the column a s much a s possible. Sampling Ststem. The Carlo Erba rotating valve for liquid sampling (RSV 150) is actuated with compressed air. The sample volume (0.7 or 5 PI) has been measured by acid titration (271 and is known within about 1%. The sample is a mixture of helium, methane ( 1%) and benzene vapor. The sample pressure in the valve is about atmospheric. The dynamic method (28) is used to prepare a stream of helium and methane of adjustable concentration, which flows through a diffusion cell in which an adjustable amount of benzene vapor is added 129). The composition of the mixture is stable within 1%. The valve gives satisfactory results, but has to be serviced carefully, replaced after about 250 injections, dismantled, and
-
(27) C. L. Guillemin, J. Vermont, P. Juston. P. Ferrandine, A. A r t u r , and A. Peyron, "Advances in Chromatography 1970," A. Zlatkis. Ed., Houston, Texas, 1970, p 163. (28) L. Angely. E. Levart, G . Guiochon, and G. Peslerbe, Anal. Chem., 41, 1446 (1969). (29) J. M . McKelvev and H. E. Hoescher, Anal. Chem., 29, 123 (1957). ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
1191
Table I. Systematic Errors of Time Measurement Contributions of (sec) Ret. Samp. Time Starting time, period, sec sec const. Clocko time 1
1000
-1
+1.2
-0.030 250 0.2 -0.2 +0.25 -0.030 100 0.2 -0.2 f0.22 -0.030 100 0.2 -0.05 +0.22 -0.030 10 0.02 -0.005 +0.022 -0.030 a Actual quartz frequency 1.999600 f MHz instead of 2.0 MHz required, so the error is 2 X t ~ Because . of electronic desian the
f
7 r L
100
0 0
10
20
30
40
45
So
Figure 1. Top. Position sensor of the sampling valve and systematic error on the starting time. 1 , sampling position: 2, injection position, 6 is the adjustable angle between the light beam and the final position of the valve in the injection mode. The device shown on the valve can be rotated above the valve, around the valve shaft. Bottom. Variation of the systematic error with 6
Table I I . Approximate Values of the Adsorption Coefficient of Methane 1 O3 X U A , cm3/m2 T , "K
Measured
Calculated
256.5 260.5 265.9 272.7 276.6
9.83 8.84 7.71 6.53 5.96
10.1 9.5 8.7 7.9 7.5
Extrapolated
Calculated
373 393 41 3 433 453 473
1192
1.14 0.91 0.71 0.60 0.50 0.40
2.8 2.4 2.1 1.9 1.7 1.5
ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
Volmet. control
Total error, sec
+0.010
4-0.18
+0.010 +0.010
+0.03
+0.010 +0.010
0 f0.15 -0.003
U M ,sec
0.1 2.25 X 2.2 x 1 0 - 2 1.1 x 10-2 1 x 10-2
uM/ts
10-4 9 x 10-5 2.2 x 10-4 1.1 x 10-4 1 x 10-3
-is made one sampling period after the time window is open, which adds ts to the error. first measurement
repolished. Otherwise, after several hundred injections, cracks and grooves begin to appear in these parts, resulting in a secondary injection when the valve is moved back to sampling position. Signal Acquisition a n d Handling. Detector. The flame ionization detector has a n isolated jet, surrounded by a coaxial collecting electrode a t 4 m m from the jet; the collecting voltage is 180 V (burner negative). The flow-rates of hydrogen and oxygen are respectively 1 7 and 400 cm3/min. Signa/ Measurement. The FID signal is measured using a Keithley amplifier and a Solartron digital voltmeter (LM 1480-3, Schlumberger) with a precision of f 1 digit. The measurement in BCD code is serialized and recorded on punched tape using a Facit fast puncher. The speed of the puncher limits the frequency of measurements to 5 Hz. A measurement is made every time the voltmeter receives a pulse sent by a quartz clock. To limit the length of punched tapes to handle and store, only parts of the chromatograms are recorded. during selected time windows. The clock (quartz frequency 1.999600 f Mhz. yearly drift less than 3 X Mhz in a two-year period) sends pulses a t a constant frequency, usually 5 Hz. to a logic system, which counts them, as they travel to a gate. At the beginning of the analysis, the gate is closed; after a given number of pulses have been received by the counter, the gate is open for another given number of pulses and then closed. This process can be repeated several times. It allows recording the signal only during the elution of peaks and knowing exactly a t which time every measurement has been made without a record of the time. The clock is automatically reset during the injection and the following operations are performed: sequential reading and printing of all the important parameters (temperatures, pressures, voltages, . . .) of the equipment; acquisition on punched tape of the CH4 signal during a time window centered on the expected retention time and 16 u wide ( a = standard deviation of the CHI peak): acquisition of the CsHc peak in similar conditions: and new injection and clock reset. The system to reset the clock uses a position sensor on the sampling valve as shown in Figure 1: a light pulse falls on the photocell before the valve reaches the injection position. Determination of t h e E r r o r s of Time Measurement. These errors have been determined as indicated above. Typical data obtained with the equipment described are given in Table I. Both t , and T are usually 0.20 sec. The calculation of the error in time measurement is explained in Table I. The standard deviation on this contribution is 10W5 x t,?,resulting from the fluctuations of the quartz frequency. As shown in Figure 1. the pulse is sent too early by the position sensor and the retention is too long by an amount depending on the angle 0, the pressure of the gas actuating the valve and the shear-stress in this valve. H has been set to about 8". resulting in a n error of -30 f 10 msec ( C f . Figure 1).The measurement of the signal is performed after a delay which results from the design of the voltmeter 14. .5). This apparatus is locked on the mains frequency to obtain better noise rejection. The measurement, which is an integration of the signal for 10 msec, begins when the mains voltage reaches a given preset value. If the order arrives later, the measurement is carried out during the following period. So, if the main frequency drifts or if the frequency of the pulses sent by the quartz clock is not a n exact multiple of the mains frequency. series of measurements are made with constant sampling period. t,,, and these series are separated by intervals t.< =t 20 msec. I t has been shown i.5) that the resulting error is equal to the sum of a systematic error equal to +10 f 0.1 msec and a random error with a maximum standard deviation of 5 msec.
tmDo OJ5
fm3)
a
0;lO 0
02
0.4
0.6
q8 Do (cm3s-’)
Figure 2. Determination of t h e outer column gas volumes. a. Plot of t, X Do VS. Pi/P,. b. Plot of t~ X D o V S . D o
-
The total correction is -0.004 sec for methane ( t H 80 sec) and between +0.036 and 0.012 sec for benzene ( t H between 280 and 160 sec); these corrections are thus practically negligible in both cases. Table I also gives the random error made on the determination of the correction t A . This error is of the same order of magnitude as the repeatability of the measurements. D e t e r m i n a t i o n of the I n s t r u m e n t a l Errors. These errors have been derived in the specific case of the determination of the retention time of methane (“inert” peak) and benzene on graphitized carbon black. Transit through Carrier Gas Volumes outside the Coiumn. Figure 2a shows the plot of D o t , us. P,IP, obtained when using a 20-cm long, 0.25-111111 i.d. capillary tube in place of the column ( V , 10 11). The dispersion of the d a t a results from the very short retention times (less t h a n 1 sec), t h e strongly skewed, tailing peaks, and the noisy base line due to the use of a small time constant. The slope of the straight line on Figure 2a is very small, hence Vu. The downstream dead volume is estimated to about 110 pl from the ordinate of the straight line. These results are in agreement with those derived from Figure 26 ( V , + V , = 120 plj. Thus Vu is less than 10 p1 and its contribution to the correction (Equation 8) negligible while V, is 110 & 20 pl. T h e random error on t h e correction t l ’ (0.02/0,) is relatively important as the carrier gas flow-rate is about 0.15 cm3/sec. Retention of the “Inert” Compound, Helium is used as carrier gas because the effect of non-ideal behavior of the gas phase is then minimal. Furthermore helium is practically not adsorbed on graphitized carbon black above ambient temperature. The least adsorbed gas should he used. hut neon and argon cannot he detected with a FID. For hydrogen, a signal is obtained only with very large samples and with a n Hz flow-rate to the flame very different from the optimal one, so it is questionable whether the retention times of hydrogen peaks could he considered accurately as equal to t,. Methane gives a large signal. The 5-pl sample of helium with 1% methane (-20 ng) gives a peak height about 3 x lo-” A (peak width -18 sec), i . e . . a signal-to-noise ratio larger than 100, which is satisfactory for the signal acquisition system
-
r,j).
Methane. however, is adsorbed by graphitized carbon black ( 1 1 , 30. 3 1 1 and even in the temperature range 100-200 “ C the
correction is not negligible. Table I1 gives the values of the adsorption coefficient of methane obtained by two different methods. First they have been calculated by statistical thermodynamics i l l . 2.9. 30). The adsorption coefficients obtained from these (30) A. V . Kiselev and Ya A . Yashin, ”La Chromatographie gaz-solide.” Masson, Paris, 1969. (31) R . S. Hansen and J. A . M u r p h y , J . Chem. Phys., 39,1642 (1963).
Table 111. Capacity Ratio of Methane on Graphitized Carbon Black 10‘ X k‘ T, “C
100 120
140
Extrapolated 0.34 0.27 0.21 0.18 0.15 0.12
160 180 200 am^ = 2.42 grams. A = 8.7 mz/gram.
Calculateda 0.84 0.72 0.63 0.57 0.51 0.45
calculations are usually in excellent agreement with the experimental results (within 10% or so). The other series of values has been obtained by extrapolation of measurements made at lower temperature (256-276 “K) where hydrogen and methane can be resolved. Although the use of other equipment (with a thermal conductivity detector) and of a different column introduces some errors, the main source of trouble is obviously the extrapolation of d a t a in a large temperature range. when there are some reasons to think that a t the very low values observed here, the logarithm of the adsorption coefficient is no longer proportional to ’/T (31). It is therefore not surprising that the theoretical values are larger than those extrapolated from the measurements. Table I11 gives the values of k’(CH4) derived from the adsorption coefficients in Table 11. In view of the theoretical reasons discussed above (31) and the general agreement between the experimental results and the calculations of statistical thermodynamics i l l , 30, 32). we consider that the calculated d a t a are a better approximation of k’ and t h a t they are accurate within 25%. Unfortunately this is a large correction and the error on the estimation of k’(CH4) is the main source of residual error in our experiments. Temperature Gradient and Fluctuations. The temperature gradient in the whole oil bath is 0.05 “C, h u t near the center of the bath where the column is placed, the gradient is smaller, so that the maximum temperature difference between the column center where the temperature sensor is located and the various points in the column is f 0 . 0 1 “C. From d a t a given elsewhere (241, if a 1-m long column is coiled 4 turns, the variation in k ’ resulting from such a gradient is -0.2 X for k,‘ = 10, which is negligible here. The effect on t , is smaller. The effect is still smaller for a n 8-turn coiled column, 2-m long, as the one used here. (32) E, V. Kalashnikova, A. V . Kiselev. R. S. Petrova, K . D. Scher-
bakova, Chrornatographia, 4, 495 (1971). ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
11931
Table IV. Accuracy of Retention Times Measurements Methane Random errors A. Fluctuations of instrumental parameters Pressure drop Outlet pressure Temperature 13. Errors of measurements Noise Density of points of measurements Total contribution of random errors Repeatability of experimental results Errors on the corrections Time measurements Outer column gas volumes Retention of methane Flow perturbations Total
Benzene
2 x 10-5 2.7 x 10-5 1 x 10-5
2 x 10-5 2.7 x 10-5 1.1 x 10-4
1.5 x 10-5
1.0 x 10-5
0.5 x 10-5
0.5 x 10-5 0.4 x 10-4
1.15
(0.7 x 10-4)
(1
i x 10-4 1.6 x 10-3 2.5 x 10-3 6 X
x 10-4
x 10-4)
9 x 10-5 5.5 x 10-4 0 2.2 x 10-4 3 x 10-3
6 X
x 10-3
6 X
3
Total error
Table V. Experimental Conditions of the Measurements and Results Methane peaka Series
Pressure drop, mbar
No.
T , "C
1 2 3 4 5 6 7 8 9
125.012 129.954 135.020 140.028 145.050 149.981 154.966 159.972 125.001
a
Outlet pressure. mbar
Flow-rate, cm3/sec
Signal to noise ratio
t ~sec ,
1144.70 1135.50 1138.40 1137.20 1137.20 1137.40 1137.50 1137.70 1137.30
0.1481 0.1481 0.1474 0.1470 0.1464 0.1459 0.1454 0.1449 0.1482
250 260 260 270 270 260 260 260 260
82.107 82.408 82.686 82.949 83.263 83.563 83.844 84.145 82.276
1447.40
... ... ... ... ...
... ... ...
Benzene peak*
(104 tu) t
M
d
0.8 1.o 0.6 0.6 0.9 0.9 0.8 0.5 0.4
Signal to noise ratio
125 140 150 160 170 180 180 200 130
(104t~)/ tR, sec
tR
283.87 258.36 236.34 21 7.70 202.07 188.78 177.25 167.30 284.1 5
f l o
0.8 1.1 0.8 1.2 1.4 0.8 1.3 0.5 1 .o
Sample size, 77 ng ( - 16%), Sample size 80 ng ( - 3 % ) .
Table V I . Influence of Sample Sizea on Retention Times Methane Series No.
Sample size, ng
Benzene Sample size, ng
to)/ d 1 0 fR
( i o 4 3
h/ib
tR,
sec
hlib
tR,
sec
250 500 222.06 250 84.948 1.5 75 95 260 222.08 0.6 80 180 84.968 222.03 15 25 50 3.7 13 84.934 asample size 0.7 pl. 1% in the sample gas corresponds to 5 ng C H 4 or times the standard deviation of the base-line signal). 10 11 12
(1043tu)/ d 1 0 tR 0.7 1.3 3
Maximum peak height over base-line noise ( 4
24 ng CsH6, respectively.
Table V I I . Correctiona of the Measurements of Methane Retention Times NO.
Measured retention time, sec
1 2 3 4 5 6 7 8 9
82.107 82.408 82.686 82.949 83.263 83.563 83.844 84.145 82.276
Series
Errors of measurements, msec elb
4-216 .
,
.
... 4-217 .
..
... ... ... 4-216
€4
=
T
-20
1194
-200
... ... ...
... ... ... ... ... ... ... ...
... ... ...
... ..,
corrections in msec. 2 X the error on k ' ( C H 4 ) . UT = ( U M ~-t U a All
+ (3
fz
... -3 .. .. .. ..
-4
UM
el'
f4'
22 ...
-743
-168
...
..,
-746 -748 -751 -754 -757 -759 -742
... .. . , . . ... ... ... . .. ...
.
. .
.
.
, ,
. .. .., ... . .. ...
+ t s (Cf. Table i ) .
Including ~4- u ~ ) ' /u~=; standard devia-
tR A
Zci -4 ...
Instrumental errors, msec
ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
61'
+ f4'
-911 -911 -914 -916 -919 -922 -925 -927 -910
UAc
202 200 198 196 194 192 190 189 203
Corrected t R sec
81.196 81.489 81.771 82.032 82.337 82.642 82.918 83.216 81.358
kCH4'
tm,
rnsec
(62')
- 564 - 549 - 533 -518 - 504 -491 - 479 - 468 - 566
Total d error
t m , sec
80.63 80.94 81.24 81.51 81.83 82.15 82.44 82.75 80.79
103 X U T / t m 2.5
... ... 2.4
... 2.3
... ... 2.5
tion of the measurements in each series.
-
Table VI1 I. Correctiona of the Measurements of Benzene Retention Times Instrumental errors (msec) Error of measurements (msec) Series Measured €1' €4' 2€,' No. t R . sec €3'' €4 = 7 €2 + € 3 Zri UM 1
283.87 +557 -20 -200 $337 22 2 258.36 552 ... ... 332 3 236.34 547 ... ... 327 ... ... 324 4 21 7.70 544 5 202.07 540 ... ... 320 6 188.78 538 ... ... 31 8 7 177.25 535 ... ... 315 ... ... 31 3 8 167.30 533 ... ... 337 9 284.15 557 a A l l corrections in msec. Retention times in sec. b 2 X tR ts (Cf. Table I ) . t s is different for CH4 (0.2 sec) and CsH6 (0.5 sec). UT =
+
No.
r, " c
ku'
1 2 3 4
125.012 129.954 135.020 140.028 145.050 149.981 154.966 159.972 125.001
2.457 2.135 1.858 1.624 1.427 1.259 1.114 0.988 2.454
5 6 7 8 9
-=[(%) dk' + dt,* + dt k'
(tR
-
kc
"1
2.513 2.185 1.902 1.663 1.462 1.291 1.143 1.014 2.510
-740
-170
-910
...
...
...
... ...
...
-920
...
... ... ...
...
... ... ...
... ...
-930
149
... ...
... ...
(UM*
+
... ...
u12
each series.
Table IX. Corrected and Uncorrected Values of k' for Benzene Series Uncorrected Correyted (kc'
UA
+
-
ku')/ kc', %
...
Total errorC
283.30 257.78 235.75 217.10 201.47 188.18 176.63 166.68 283.58
0.53 0.60 0.64 0.71 0.77 0.80 0.88 0.91 0.54
145
..
,
147 , , ,
_.
,
k'calcdb
2.8 2.9 3.0 3.1 3.2 3.4 3.6 3.8 2.8
2.508 2.185 1.904 1.667 1.463 1.292 1.142 1.012 2.508
'I2
dt,,
dtR
total random errors
The range of the fluctuations of temperature is f l x "C at 80 "C, f l X "C a t 140 "C and f 2 X "C at 180 "C, in half a day. This includes the drift as well as the noise, b u t the analysis of temperature records shows t h a t most of the fluctuations have a period between 100 and 200 sec, depending on the temperature. As the programmer actuates the sampling valve independently of the temperature controller, and these fluctuations have a period smaller than the retention time of benzene, the effect on the average of 10 independent measurements is smaller than b u t this introduces a random error of 1.1x 1241. This is somewhat less t h a n the maximum effect t h a t a fluctuation of 2 X "C might have: a variation of 2.6 x l o m 4for a compound whose adsorption enthalpy is approximately 9 kcal/ mole 11). The residual drift (about 5 x "C in a day) can cause a maximum relative variation of the retention time of 1.4 x 10-4. These corrections have been neglected as they are much smaller than the random error. Effect of the Sampling Value. Fast commutation of the sampling valve to the injection position and back to sampling position was necessary to ensure a narrow injection plug. Otherwise desorption of benzene from some part of the valve makes a tailing injection. Furthermore. this fast commutation reduces the effect of the perturbation of the carrier gas velocity profile due to the change in pneumatic resistance of the valve. Figure 3 shows the variation of the retention time of methane with the time during which the valve remains in the injection position. Curve 1 'was obtained with a sampling volume of 0.7 pl, corresponding to a very narrow bore. The carrier gas flow-rate changes by 24% in steady state conditions. The dotted line corresponds to t h e effects of the perturbations of the velocity profile during the movements of the valve (261, while the straight line corresponds to the effect described by Equation 13. Curve 2 was obtained with a sampling volume of 5 pl. The flow-rate in steady state conditions changes by about 270 when the valve is moved to the injection position. This illustrates the necessity to compromise between a very small sample size, which is necessary to make meaningful thermodynamic measurements and a relatively large sample volume. Accordingly the sample is a mixture of helium, methane (1701, and benzene vapor (0.2 to 107~).
1 o3 U T l t R
-910 145 u ~ ) ' ' u~ ,= standard deviation of the measurements in
l o 3 X dk'lk'''
2.3 2.3 2.4 2.4 2.5 2.5 2.6 2.6 2.3
Corrected t R , SeC
lo3 X A k ' l k ' '
-2.0 0 +2.0 +2.4 +1.0 +0.8 -0.9
-2.0 -0.8
bt'calculatyd
on tm and t~ respectively, I e , UT in Tables VI1 and V l l l from Equation 19 Log, k' = -10 322 + 4475 2 / T C A k ' = k
trn
calc,j
- k,
:sec.j
f
89
88
87
66
Figure 3 Variation of the retention time of methane with the time spent by the sampling valve in the injection position
The m e a s u r e m e n t s are n o w m a d e u s i n g t h e 5-pl s a m p l i n g v o l u m e , but t h o s e reported h e r e were o b t a i n e d w i t h the 0.7-p1 v o l u m e . The c o r r e c t i o n is d e r i v e d f r o m t h e s l o p e of t h e s t r a i g h t line o b t a i n e d o n F i g u r e 3. W h e n t h e three points (numbers 3 t o 5 ) c o r r e s p o n d i n g to t h e t r a n s i e n t a d j u s t m e n t of the f l o w - r a t e are r e j e c t e d , a l e a s t square fit o n a l i n e a r e q u a t i o n gives:
t,
=
+ 0.184 At
86.16
ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
(16) 1195
Table X. Long Term Variations of k’ and AH
AH>
Analysis group
T
1
124.999
25 a
125.001
Kcal/rnole
k’
Initial value Final value Initial value Final value
2.525
2.522
-8.84
2.514 2.510
-8.89
One month later.
The regression coefficient is 0.997 and the standard deviation 0.073. For curve 2, the result is:
t , = 86.16
+ 0.008 At
(16a)
As the injection time is always 1 sec in all the experimental work, the error is derived from the equation: =
-0.18 sec
(17)
The error made on the correction can be considered equal to the difference between the limit value (86.16 sec) and the value measured for A t = 0.3 sec, L e . : 04’= 0.055 sec (18) Ucerall Znstrumental Correction. The total correction is usually around -1.5 sec for the methane (t.q * 80 sec) and -0.5 sec for benzene. The absolute standard deviation on this correction is around 0.2 sec (-0.25%) for the methane and 0.15 sec (-0.07%) for benzene. The final accuracy of the measurement results can be estimated from the different contributions shown in Table IV. At present the determining factor is not the repeatability of the measurements but the inaccuracy of the corrections due to the retention of methane and to the carrier gas volumes outside the column. The last one appears to be the most difficult to reduce in practice.
RESULTS AND DISCUSSION As an example of the results obtained and of the problems of high precision measurements in gas chromatography, we shall report and discuss the results of a series of measurements of retention times of methane and benzene on graphitized carbon black between 125 and 150 “C. In all the following, the retention times are the peaks first moments (18) calculated as explained above. The (5). error introduced by the software is smaller than The experimental conditions and the results are given in Table V. In each case, 10 measurements have been made. Only the average value and the dispersion are given. The repeatability of these measurements, which is the ultimate check of the stability of the equipment, is excellent and could hardly be improved. Effect of Sample Size. In Table VI the retention times of methane and benzene are given for different sample amounts of these two compounds. To prevent benzene condensation in the sample line, no measurement has been made using partial pressures larger than 80 mbar. The retention time and benzene adsorption coefficient do not change when the sample size is increased tenfold. A sample size of 25 ng corresponds to a surface coverage ratio of
1196
ANALYTICAL CHEMISTRY, VOL. 45,
NO. 7,
JUNE 1973
about a t column outlet, which is quite satisfactory for measurements of adsorption coefficients at zero coverage. Correction of the Measurements. The various corrections together with the random errors are given for the retention times of methane and benzene in Tables VI1 and VIII, respectively, using the equations derived in the previous sections of this work. The corrections can be made on the average value of each series of measurements, because all the experimental parameters are stable during any series. The values of k’ derived from the corrected retention times are given in Table IX together with the “uncorrected” k’, derived from experimental retention times and the error on the corrected values. The correction is almost constant. The measurements have been made in the order given in the Tables. The last one is made a t the same temperature as the first one. If a correction is made for the slight temperature difference, the two values of k’ differ by 4 x 10-3 or 0.16% which is less than the error, so there is no appreciable change in the column during a series of analyses. There is a slight trend, however, to an increase in the adsorption enthalpy and a decrease in k’ on the long term as shown by the data in Table X. This effect however is small, if significant. It is currently under investigation and the results will be reported later. Discussion of Results. A least-square fit of the values of kc’ given in Table IX has been made to determine the best,coefficients of the linear plot:
Log k’ = a
+ b/T
(19)
The value of AH derived from b (AH = -Rb) is 8.89 kcal/ mole. The coefficient of regression is 0.999 and the standard deviation 1.77 x The adsorption enthalpy of benzene on Sterling MTG is thus 8.89 f 0.03 kcal/mole a t the 95% confidence level. The same value of the adsorption enthalpy is obtained, whether the result of series g is used or not, although this series repeats the initial temperature (125 “C). The corrections themselves have only a slight effect as the adsorption enthalpy is decreased by 0.04 kcal/mole, i . e . , 0.45%. This result although small is significant. The fact that the corrections don’t change much the value of the adsorption enthalpy results from the relative constancy of the correction on k’ (Cf.Table IX): this corresponds to a mere shift of the plot parallel to the Log k’ axis, with little change in AH and a small increase in the adsorption entropy. Although this might seem to correspond to an excellent linear fit of the data, it is interesting to observe that there is definitely a slight curvature of the plot, as shown in Table IX, where the difference between the actual data and those calculated from the best coefficients of the linear function are given. This shows that the precision of the results is sufficient to give at least an order of magnitude of the heat capacity of the sorbed molecule. More systematic work is in progress in this field (10). Received for review October 31, 1972. Accepted January 26, 1973.
