Sources of error in measurement of retention times - ACS Publications

Aug 1, 1970 - Effect of signal noise on the determination of retention time of peak maximum. Computer simulation and experimental study. M. Goedert , ...
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Sources of Error in Measurement of Retention Times Michel Goedert and Georges Guiochon Dbpartement de Chimie, Ecole Polytechnique, 17 rue Descartes, Paris 5ime, France The influence of fluctuations of experimental parameters on the precision of retention time measurements was investi ated theoretically and experimentally. The coe ficient of error propagation of column outlet pressure, pressure drop through the column, and temperature were derived from existing theoretical relationships and found to be in agreement with experimental results. The problems associated with measurement were also investigated. Because the time when the signal is maximum must be determined, the noise of the signal may interfere. So the influence of signal to noise ratio and peak efficiency on the precision of the measurements may be important. Relationships are derived between these factors and the relative error. Experimental results are in agreement with theoretical predictions. With equipment specially designed and built in the laboratory, retention times were determined with a relative precision of 4 X at the 95% confidence level.

P

RETENTION TIMESin gas chromatography are measured for two main purposes : identification of unknown compounds or thermodynamics studies. In both cases, the accuracy of the measurement is of paramount importance. Whether the retention times are used directly or through use of relative retention data (retention indices), the more precise the results, the easier and faster the identification of the unknown, since the number of compounds which have the “same” retention data decreases with the range of error of measurement. Computers are more and more used in connection with gas chromatography. Aside from computations for quantitative analysis, they are more and more used for qualitative analysis. The results would be more efficient and less costly if the gas chromatographic equipment could give results with a higher precision. Gas chromatography is also much used to measure various parameters of interest in physical chemistry (solution thermodynamics, adsorption, kinetics, etc.). In most cases the primary results (retention times, volume flow rate) are now measured with a precision of a few per cent only. Many phenomena could be more thoroughly analyzed if more accurate data were available. We have made a theoretical and experimental investigation of the sources of error in the measurement of retention times. THEORETICAL

Usually the retention time is defined after the elution time of the peak maximum. Although this definition is used throughout this paper, it is valid only for symmetrical peaks, when some restrictive conditions are fullfilled ( I ) . For unsymmetrical peaks, in linear chromatography, the retention times should be obtained from the first moment of the zone (2, 3). The results discussed below can easily be extended to this case. (1) A. B. Littlewood, “Gas Chromatography,’’ Academic Press, London, 1962. (2) J. E. Oberholtzer and L. B. Rogers, ANAL.CHEM.,41, 1234 (1969). (3) J. Villermaux, Proceedings of 5th International Symposium on Separation Methods. Column Chromatography,Lausanne, 1969. 962

The random error of retention time measurements depends on two kinds of phenomena: fluctuations of the experimental parameters, and inaccuracy in the estimation of the initial and final events of this period, the injection of the sample and the elution of peak maximum. Intluence of Experimental Parameters on Retention Time. The retention time in linear chromatography is related to the experimental parameters by the equation (4) tR

4 L2

= ~

“

(

1

Pi3 - P03 + k’) (Pi2 -

Po2)2

Equation 1 shows that the retention time depends only on the column temperature and the inlet and outlet pressures and allows calculation of the effects of the fluctuations of these parameters on the error of measurement. Small variations, dx, d y , in the experimental parameters, x , y , result in small variations of tR, so that

Coefficients x/tR,btR/bX, and Y/tR, btR/by are called the error propagation coefficients. Noisy independent fluctuations of x , y , with relative standard deviations x , y , result in a random error in measurement of tR with a relative standard deviation given by

We calculate the error propagation coefficients of column temperature, inlet and outlet pressures, and the over-all error in retention time measurements. The first two factors of Equation 1, column length L and permeability k , may be assumed to remain constant during an analysis. Both could increase with temperature but, because of more stringent requirements on thermal stability, this effect remains negligible. The carrier gas viscosity, 7, depends mainly on temperature; around atmospheric pressure the relative variation of the viscosity of the conventional carrier gases is about ‘ / l o 3 of that of the corresponding relative variation of pressure (5). The column capacity factor, k’, depends on both the column temperature and the average pressure. However, measurements by Desty (6) and others (7,8) show that the pressure dependence of k’ is very small: For a variation of l mb of the column average pressure, the relative variation of k’ would be of the order of 3 X 10-5 with carbon dioxide as carrier gas. With hydrogen or nitrogen, the effect is much smaller. Accordingly, it is neglected here. (4) G. Guiochon, “Chromatographic Reviews,” M. Lederer, ed., Vol. 8, p. 1, Elsevier, Amsterdam, 1967. ( 5 ) R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport

Phenomena,” Wiley, New York, 1962. (6) D. H. Desty, A. Goldup, G. R. Luckhurst, and W. T. Swanton, “Gas Chromatogravhv 1962,” M. Van Swaay, Ed., p 67, Butterworths, London; 1962. (7) D. H. Everett. Trans. Faraday SOC.,61,1637 (1965). (8j A. J. B. Cruickshank, B. W: Gainey, and C. L. Young, “Gas Chromatography 1968,” C.L.A. Harboun, Ed., p 81, Institute of Petroleum, London, 1968.

ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970

~~~

EFFECTOF PRESSURE VARIATIONS.Experimentally there are two methods of controlling the gas flow rate: to control separately either the inlet and outlet pressures, P i and Po, or the outlet pressure and the pressure drop (9). This last design is similar to the one used in conventional equipment, where the outlet pressure is atmospheric and the inlet pressure controller works by reference to atmospheric pressure. By differentiation of Equation 1, we obtain

~

Table I. Error Propagation Coefficients for Pressure on Air Retention Time. Pressure, bars Coefficient Inlet Outlet Pi 1.o 1.o 1.1 1.1 1.1 1.5 1.0

1 .o 1.5 1.5 2.0 4.0 4.0 m

a

- m

-2.95 -3.70 -2.14 -1.26 -1.50 -1

Po m

1.95 2.70 1.14 0.26 0.50 0

Inlet and outlet pressures controlled independently (Equation 4). ~~

This equation gives the error propagation coefficients of the inlet and outlet pressures when they are independently p in Equation 1 and difcontrolled. Letting Pi = Po ferentiating the new equation makes it possible to derive the error propagation coefficients of the outlet pressure and the pressure drop, in the second experimental case :

+

Table 11. Error Propagation Coefficients for Pressure on Air Retention Time. Pressure Outlet Coefficient drop, bars pressure, bars P Po 0 0.5 0.4 1.o 3.0 3.0

-1 -0.98 -0.99 -0.96 -0.91 -0.92

1.0 1.1 1.1 1.1 1.5

-1

m

0 -0.012 -0.013 -0.043 -0.085 -0.077 0

Outlet pressure and pressure drop controlled independently (Equation 5). Numerical results corresponding to various experimental conditions are given in Table I for the first method and in Table I1 for the second method. Comparison of these numerical data shows that the error propagation coefficients obtained are much smaller with the second method of pressure control, especially the coefficient for the outlet pressure: The effects of simultaneous, identical variations of the inlet and outlet pressures cancel each other almost completely. This result is different from the one obtained for the influence of inlet and outlet pressure fluctuations on the peak area (9), because the peak area depends on the outlet carrier gas flow rate, whereas the retention time depends only on the average flow rate, which is much less dependent on the outlet pressure (IO). According to Table 11, the column outlet pressure should be controlled within a range of variations larger by one order of magnitude than the fluctuations of the pressure drop. Although short-term fluctuations of atmospheric pressure are about 1 to 3 X (9), which would seem satisfactory, longterm drift may often exceed this value by a factor of 10. It thus seems better to control also the outlet pressure to achieve a day to day reproducibility of measurements, at least if a relative precision better than 10-3 is desired. EFFECTOF TEMPERATURE VARIATIONS.In Equation 1 only k ’ and q depend on the column temperature. The column capacity factor is given by

where AGO, ASo, and A H o are, respectively, the molar variations of free enthalpy, entropy, and enthalpy associated with the vaporization of the solute from the stationary phase. K is the partition coefficient, and V, and V Care the volumes of the mobile (gas) and liquid phases in the column. (9) M. Goedert and G. Guiochon, J . Chrornatogr. Sci., 7, 323 (1969). (10) I. Halasz, K. Hartmann, and E. Heine, “Gas Chromatography, 1964,” A. Goldup, Ed., p 38, Institute of Petroleum, London, 1964.

Differentiation of Equation 6 with respect to temperature gives

(7) In a not too broad range of temperature the viscosity is proportional to To.8,where T i s the absolute temperature (4). Accordingly,

Hence, the relative variation of retention time is given by tR

T

1

+ k’ RTZ

(9)

Numerical results given in Table I11 for different values of A H o show that the second term is much larger than the first and that they practically can never cancel. Usually, A H o increases with retention time and so does the effect of temperature fluctuations on the precision of retention time measurements. EFFECTOF OTHEREXPERIMENTAL FACTORS. No other experimental parameters appear in Equation 1. Admittedly, the retention time depends on the flow rate, but the flow rate depends on the inlet and outlet pressures and this effect has been taken into account in deriving Equation 1. In this derivation, two assumptions have been made. First, we are dealing with linear chromatography. Accordingly, the partition coefficient, and the retention time, are independent of the sample size. The second assumption is that the sample is injected as a small, narrow plug. The retention time may depend on the injection function if it is too broad (2); in the present lack of experimental data, the ratio of the width of the injection plug to the retention time should be kept smaller than the accepted error, or at least of the same order of magnitude. With an automatic system this effect could also introduce a systematic error (2).

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Table 111. Error Propagation Coefficients. for Temperature on Retention Time k' T 'C AH, kcal/mole 0.8/P Coefficientc for k'o Over-all coefficient 4 0.23 x -0.54 x 10-2 -0.31 X 10-2 0.5 77 1.o 77 4 0.23 x -0.81 X lo-' -0.57 X lo-' 4 0.23 X 3.0 17 -1.22 x 10-2 -1.00 x 10-* 4 0.23 X -1.48 X 10-2 10.0 77 -1.25 X 10-e 10.0 127 4 0.2 x 10-2 - 1.14 X 10-2 -0.94 x 10-2 10.0 177 4 0.11 x 10-2 -0.89 X 10-2 -0.71 X lo-' 10.0 127 10 0.2 x 10-2 -2.88 X 10-2 -2.68 X lo-' 10.0 127 15 0.2 x lo-* -4.29 X lo-' -4.09 X 10-2 Error propagation coefficient for temperature related to absolute variation of temperature, other coefficientsrelated to relative fluctuations. b Equation 8. 0 Equations 1 and I.

0

The experimental parameters on which the response of the detector used depends seem to have no effect on the retention time. They should be controlled carefully, however, especially because of their possible influence on the detector noise. ERROROF MEASUREMENT. The retention time is given by a clock which is started at the time of the injection and stopped when the peak maximum is observed by the detector. The error of measurement can come from fluctuations of the clock, from delays between the physical events and the time when they are recorded, or from uncertainty as to the exact time when they take place. The first source of error is usually very small and may be neglected. The second is easy to determine from the characteristics of the chronometer. The last, which depends on the noise, is much more difficult to determine. TOTAL ERROR FROM FLUCTUATIONS OF EXPERIMENTAL PARAMETERS. The over-all error in the determination of the retention time can be calculated by combining the individual contributions, as shown by Equation 3 . If the calculation is carried out using the range of the fluctuations of the experimental parameters (respectively, p p , po, and p T for p , Pa, and T), we obtain what is practically the precision at the 95 confidence level (e):

The frequency of fluctuations of the various parameters should also be taken into account. Drift and long-term fluctuations result in random error. If the frequency of fluctuations is large compared to the inverse of the retention time, they do not affect the precision of the measurement but may result in systematic errors: Because Equation 1 is not linear in temperature or pressure, the experimental result does not correspond to the time-average value of the parameter. For example, when the column temperature fluctuates, the retention time is related to the time-average value of 9 (1 k'), which is not equal to the value of 9 (1 k") at the time-average temperature. For this reason the control of the experimental parameters should be the same for shortand long-term fluctuations. Mluence of Signal Noise on the Determination of Retention Time. Because of the noise of the signal at the peak maximum, there is a finite lapse of time during which it is not possible to determine whether or not the signal is constant,

+

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increases, or decreases. Only high frequency digital acquisition of the signal, followed by a least-squares fit of a theoretical curve on these data with the help of a computer or by the derivation of the time of elution of the peak center of gravity (first moment) (2), could reduce this uncertainty. It is, however, very useful to know, at least approximately, under what conditions this source of error is larger than those originating in the fluctuations of experimental parameters and thus when the use of a computer is worthwhile. The retention time may be given two definitions which are identical from the mathematical point of view, but different as far as the design of the equipment for measurements is concerned: The retention time is the lapse of time from the injection to the time when either the signal is maximum or the signal derivative becomes nil. We derive in both cases a relationship between the relative uncertainty on the retention time and the intensity of the noise. EFFECTOF NOISEON TIMEOF MAXIMUM PEAKHEIGHT, A rigorous calculation of the error made in determining the time of elution of peak maximum is very difficult and needs a detailed knowledge of the noise spectrum, which in turn depends much on the equipment used (type of detector, of amplifier, etc.). On the contrary, it is very easy to derive an approximate value, which is in excess of the actual one by a factor that depends only on the characteristics of the noise. The signal or peak height is given by the following equation ( I ) , which is valid as far as the peak is gaussian :

S is the response factor of the detector, C the concentration of the solute in the carrier gas at time t , rn the sample mass, and N the number of theoretical plates. We shall assume that the uncertainty on the retention time is the period of time during which the peak height is smaller than hM but larger than hM - i, where i is the noise (i is taken as equal to four times the standard deviation of the signal measurement). We also assume that this noise is the same on the base line and around the peak maximum. The absolute error, AtR, is then given by the equation

or practically, because AtR is very small compared to

tR,

Equation 13 is pessimistic, because we have neglected the fact that a small noise fluctuation arriving near peak maximum may result in a signal larger than the one obtained when a

ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970

large fluctuation occurs at time tR - AtR. The actual value could be calculated by combining spectral analysis of the noise and the peak profile. Nevertheless, Equation 13 leads to some interesting conclusions. We see first that the precision could be increased only by increasing either the plate height or the signal to noise ratio; second, that because of the square root in the RHS of Equation 13 a very important increase in these factors will result in a modest improvement of the precision. As shown in Table IV, it is difficult under very good conditions to reach a precision better than 10-4. Conversely, the retention time of very small peaks may be measured with a relatively good precision. It is not possible to increase the maximum peak height much, because the sample size is limited, especially in thermodynamic studies such as the determination of the activity coefficient at infinite dilution. Obviously, the experiments should be carried out at the optimum flow rate of the Van Deemter equation, since h.w as well as N is then maximum ( I ) . Care should be taken to work with the best columns available. Because of its opposite effect on the plate number and the maximum peak height, an increase in the column length, L, will not change the precision appreciably: At and constant sample size, h.w is proportional to both l / V R ; hence, it is approximately proportional to l/&. The relative error on the retention time will thus be proportional to L-”4: There is not much to gain in precision by increasing the column length. The only practical method of improving the precision on retention time is to reduce the signal noise and use a very sensitive detector. Equation 13 is valid for a mass-flow detector as the flame ionization detector as well as for a concentration detector as the catharometer, whereas Equation 11 is valid only for concentration detectors (11). Thus the use of a flame ionization detector, or eventually an electroncapture detector, instead of a catharometer, may allow either more precise measurements of retention time (larger signal to noise ratio for a given sample size) or more meaningful measurements (smaller sample size for a given signal to noise ratio). Care should be taken, of course, when using a mass-flow detector that the peak height increases with increasing flow rate (11) and accordingly that the product N x h.+, is optimized. EFFECTOF NOISEON TIME OF ZERO SIGNAL DERIVATIVE. When using a digital integrator, such as the Infotronics CRS’s, to determine the retention time, the logical process is different: The signal variation is differentiated and the derivative is compared to a value set by the operator. Ideally, this value should be zero, but this is impossible because of the base line noise which would otherwise give erratic measurements, printing as many peaks as there are noise spikes. Accordingly, the retention time given by these instruments is the time when the signal derivative reaches a negative value which can be set depending on the experimental conditions. The time measured is larger than the true retention time. A correction should and can be calculated to the measurement; it is not negligible: Using a low frequency emf generator with a large signal to noise ratio, we have determined that this bias is at least 2z of the peak width, when the integrator is well tuned, a critical procedure, and may be much larger. In addition to this systematic error, a random error arises also from the signal noise which results in a noise of the signal

dz

(11) I. Halasz, ANAL.CHEM., 36, 1428 (1964).

Table IV. Effect of Noise On Time of Maximum Peak Height Signal to noise ratio, hM/i 2.5 102 103 108

Plate number, N 2 x 103 2 x 103 2 x 103 2

x

AtRltR,

x

2

0.3 0.1 0.01

106

On Time of Zero Signal Derivative Retention Plate time, hM/i’,asec number, N tR, sec AtRftR, 10 2 x 103 600 3 102 2 x 103 600 0.3 103 2 x 103 600 0.03 103 2 x 103 60 0.003 103 2 x 106 3 x 103 0.0015 Using recording system (9) with 0.3-sec response time (attenuation 3 dB at 3 hz), Gow-Mac 9285 catharometer, and HewlettPackard 6102 A power supply, signal noise is 10 to 20 WVand noise of signal derivative 1 to 2 pV/sec.

derivative. The time when the derivative reaches the level of detection may thus be observed either too early or too late. This effect is easy to calculate when the noise of the signal derivative, i’, is known. There is a relationship between signal noise i and i’, but this again depends on the noise spectrum and varies from one equipment to the other. Determination of Systematic Error. The derivative of the signal is

if hl’ (hl’< 0)is the set value of the signal derivative which triggers the measurement of the retention time, the correction to apply to the measurement is given by

Equation 15 has to be solved numerically for atc, but usually 6tc is small enough to allow neglecting the exponential factor, taken as equal to unity. Determination of Random Error. When the signal derivative is sufficiently near the trigger value, hl’,so that the difference, h’ - hl’,is smaller than i’, the range value of the noise derivative, the logic circuit may detect “peak maximum” at any time, depending on the random appearance of the noise. The error on the retention time is thus given by the equation

Ih’ - hl’ I = i’ (16) When h’ is near hl‘, we may expand it in powers of At (At is small compared to the correction 6tc,since hl’has to be much larger than i ’ ) : h’ =

hi‘

+ h” (6tc) At

(1 7)

or by differentiation of Equation 14: (18)

Usually hl’ is small, so that h may be taken as equal to hM, and 6t, is small compared to the peak standard deviation, ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970

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Table V. Range of Fluctuation of Experimental Parameters Fluctuations Av value 1 hour 10 hours 1.1 2.5 x 10-4 2.5 x 10-4 0.4 4 x 10-4 4 x 10-4 52 0.005 "C 0.01 "C 190 2 x 10-4 2 x 10-4

Parameters Outlet pressure, atm Pressure drop, atm Temperature, "C Bridge current, mA

Table VI. Comparison between Experimental and Theoretical Error Propagation Coefficients. Error propagation coefficient Parameter Calcd Measd Temperature" -0.014 -0.015

100 hours 3.5 x 10-4 7 x 10-4 0.03 "C 5 x 10-4

this size, the values obtained are, respectively, 2.8 X 10-3 and 5 x 10-3. Consequently, the method of measurement has to be chosen depending on experimental conditions.

I

Outlet pressure Inlet pressure Outlet flow rate

2.7 -3.7

...

2.4 -3.6 -0.92

EXPERIMENTAL

We have used a chromatograph built in our laboratory, which has been described in detail (9).

I1

Outlet pressure -0.01 -0.02 Pressure drop -0.99 -0.90 Sample size ... 0.02 Bridge current 0 0.001 I. Inlet pressure and outlet pressure controlled (Table I). 11. Outlet pressure and pressure drop controlled (cf. Table 11) a Temperature coefficient related to absolute variation of temperature. Experimental Conditions. Column 2 meters long, 6 rnm4.d. packed with Molecular Sieve 5 A 200 to 250 microns. Carrier gas flow rate 60 cm3/min. Inlet pressure 1500 mb. Outlet pressure 1100 mb. Temperature 325 "K. Sample nitrogen k' = 6.2. Retention time t R = 330 sec. N = 700 theoretical plates. Noise i = 10 p V , i = 2 pV/sec.

u. Since N i s equal to t R 2 / U 2 , the first term in h" is negligible and Equation 16 becomes

When the signal to noise ratio becomes small, the approximative Equation 19 is no longer valid and the relative error is obtained by combining Equations 16 and 18. In Equation 19 or its analogs, it/hMis related to the signal to noise ratio as discussed above, since for a given equipment i t is proportional to the noise. COMPARISON OF Two METHODS OF MEASUREMENTS. If we compare Equations 13 and 19 or the numerical data given in Table IV, we see that for a given signal to noise ratio the second method of retention time measurement may be more precise than the first, when the retention time is small and the signal to noise ratio is large. It will become less precise for compounds with large retention times. This comes from the fact that for a given plate number and peak height, peaks with large retention time are much broader and the variation of the signal derivative near peak maximum is much slower than for peaks with small retention times. Also, the precision in retention time measurements from signal derivative is more sensitive to variations of the signal to noise ratio. For example, with a flame ionization detector in good experimental conditions i = 10-13 A and i t % 5 X 10-13 A per second. If the retention time is 250 seconds and the column efficiency 2500 plates, the peak height for a 1-pg hydrocarbon sample is about A. The first method of retention time measurements allows a precision of 2.8 X and the second one of about 5 X With a sample 1 / 1 ~ ~ 966

The chromatograph is placed in a liquid bath, the temperature of which is controlled within 0.01 "C between 40" and 130 'C, using a Melabs CTC-1A proportional regulator, and is measured with an accuracy of 0.1 "C, using a Melabs CTB-1 bridge and a calibrated platinum resistance. The sampling valve (Microtek 713107-G-1 six-port valve), the column (2 meters long, 0.6 cm in i.d., Molecular Sieve 5 A), and the detector (catharometer Gow-Mac 9285) are in direct contact with the liquid, except that the detector is wrapped in an asbestos cloth to minimize the effect of high frequency temperature noise from the heater. The carrier gas is hydrogen. The absolute outlet pressure and the pressure drop through the column are controlled using two Negretti and Zambra Type R 182 pressure controllers. These controllers work usually with reference to the atmosphere. Because the normal short-term fluctuations and of atmospheric pressure are of the order of 2 X may exceed 5 X this would limit the precision attainable. It is possible to control the outlet pressure at a value of about 1.100 mb with fluctuations of about 0.2 mb only by using vacuum (2 X 10W mb from a mechanical pump) as reference pressure, The stable outlet pressure thus obtained is also used as reference pressure for a second pressure controller which controls the inlet pressure. All pneumatic connectors, except those between the sampling valve, the column, and the detector, are made with large tubes to prevent time delay in transmission of pressure fluctuations, and flow-rate variations due to temperature changes via their effect on gas viscosity. Table V shows the range of fluctuation of the main experimental parameters. Most of the series of experiments need less than 10 hours. The sample is either pure nitrogen or a mixture of argon, nitrogen, carbon monoxide, and methane. Argon was prefered to oxygen, which has almost the same retention time on Molecular Sieve 5 A because of its inertness toward CO and CHI. The adsorption enthalpy of nitrogen on Molecular Sieve 5 A at ambient temperature is about 4 kcal per mole. The signal noise is 10 to 20 pV and the noise of the signal derivative is 1 to 2 pV per second. EXPERIMENTAL RESULTS

For every series of analyses the average value of the measurements (lR)and their standard deviation (u) are calculated. The relative dispersion of the results is given as a t / t R , where t is the value of the Student factor corresponding to the number, n, of analyses made and to a confidence level of 9 5 z . Because the number of analyses always exceeds 40, t is practically equal to 2.0. The precision on the average value, iR,is equal to a t / i R

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Table VII. Precision in Determination of Retention Time Fluctuations Influence on x 10-4 reproducibility X 10-4 Parameter (in 2 hours) Measd coeff Calcd coeff Outlet pressure 2.5 0.03 0.05 Pressure drop 4.0 4.0 3.6 Temperature 0.3 1.4 1.5 Measurement 0 . 1 sec 2.0 2.0 Total error e = dF2 4.7 4.5 Experimental reproducibility 5 Same experimental conditions as for Table VI. 100

Determination of Error Propagation Coefficients. These coefficients were determined from the variations in the retention time of nitrogen for small changes in the experimental parameters around the values selected for optimal performance of the equipment (9). The results obtained are given in Table VI with the values derived in this particular case from the theoretical calculations discussed earlier, using Equations 5 and 9. The agreement between the two series of results is excellent. The coefficient for sample size is not nil, which shows that the sample size used in these experiments is too large. This is in agreement with a previous result (9),that the peak height is not proportional to the sample size in the size range used here, although peak area was proportional to sample size: The column is overloaded with a l-cm3 nitrogen sample (pressure 1500 mb). The volume of the sample loop used with the Microtek sampling valve cannot be reduced much below about 1 cm3: The length of this loop is at least 5 cm; the pressure drop through it in the sample gas flow line or the carrier gas flow line should be kept small enough to prevent perturbation of the carrier gas flow rate when the sample is injected and to allow proper working of the sample pressure controller (9). The pressure in the sample loop has to be near the column inlet pressure to prevent leaks between the two lines and to reduce fluctuations of pressure and flow rate during the injection. The origin of the effect of variations of the bridge current on the retention time is not known. At any rate, the influence of variations of the bridge current on the precision of the retention time measurement may be kept small enough to be negligible. Precision in Determination of Retention Time. From the error propagation coefficients and the range of fluctuations of the experimental parameters, Equation 10 makes it possible to calculate the precision. The results are given in Table VII. There is excellent agreement between the precision calculated from either the measured or the calculated error

Table VIII.

300

500 t R

(JeC)mo

900

1100

Figure 1. Variations of dispersion of retention times measurements with retention time t = Student factor Experimental results and theoretical curye Experimental conditions. Cf. Table VI propagation coefficients (4.5 and 4.7 X lod4, respectively) and the reproducibility of the experimental measurements (5 x As may be seen in Table VII, the major sources of error are the fluctuations of the pressure drop and the too large definition of the clock used in this work (0.1 second). The errors due to the noise which may be calculated from Equations 13 and 19 are, respectively, 18 and 1 X lo-'. Since the measurements have been made using the Infotronics, the contribution of the noise to the error is small and the results obtained are also in good agreement with the theoretical prediction regarding the effect of the noise. Influence of Retention Time on Precision. We have measured the retention times of four gases, Ar, Nz, CO, and CH4, under the same experimental conditions, from analysis made on a gas mixture (Table VIII). The reproducibility improves with increasing retention time, except for carbon monoxide which has a much smaller peak height than the first three gases. The two methods of flow control give similar results in this case because, in the experimental conditions of this experiment, the reproducibility is not controlled primarily by the fluctuations of the experimental parameters but by either the time definition of the clock or the noise effect on the logic circuit which determines the retention time. This is demonstrated in Figure 1, which shows the variations of the dispersion (a X t / t R ) with the retention time for the various compounds of the gas mixture analyzed. The theoretical curve is obtained by combination of the various sources of error following the law of variances :

0 is given by Equation 10. The contribution of the error of

Influence of Retention Time on Reproducibility Reproducibility Inlet press.Pressure dropcontrolled (10-4) controlled (10-4)

Compound Peak height, mV Retention time, sec 19 Argon 137 11 15 284 Nitrogen 8 11 Methane 484 4 1064 3.5 Carbon monoxide 9 Experimental Conditions. Mixture of argon (14.9Z), nitrogen (25.2Z),carbon monoxide (23.3 Z), methane (36.6%). as for Table VI.

15

7 5 5

Other conditions

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Table M. Specifications for Precise Measurements of Retention Time Precision achieved 10-2 10-4 Stability 2 . 5 X 10-1 2 . 5 x 10-3 Outlet pressure Pressure drop 5 . 5 x 10-3 5 . 5 x 10-6 Temperature 1.0 x 10-3 1.0 x 10-6 Error of measurement 5 . 0 x 10-3 5.0 x

Figure 2. Variations of dispersion of retention times measurements with signal to noise ratio Experimental conditions. Cf. Table VI

measurement (0.1 second) is 0.02 sec2 because the error is made twice, when starting and stopping the chronometer, the two errors being independent. This curve fits the experimental points well, the discrepancy coming largely from the approximations made in deriving Equation 19, and in determining the noise of the signal derivative. The minimum of the curve results from the opposite variation of the two main contributions to the error: At short retention times the effect of the time definition of the clock (0.1 second) becomes important. At large retention times, the noise effect is predominant. Influence of Signal to Noise Ratio. Figure 2 shows the variation of the dispersion of the measurements with the signal to noise ratio. The noise is between 10 and 20 pV. Samples of hydrogen of various concentrations in methane were injected, resulting in signals between 150 and 20 mV. The dispersion decreases by a factor of only 6 for a hundredfold increase in the signal. This is a somewhat smaller effect than expected, because the noise effect is only one of the sources of error, while the total effect is given in Figure 2. As shown in Tables IV and VII, the fluctuations of the experimental parameters become the major source of error at large values of the signal to noise ratio. Also, it is possible that the signal noise is larger near peak maximum than on the base line. CONCLUSIONS

The measurement of retention times with a precision of a few parts per 10,000 is possible at least in favorable cases, and the requirements for control of experimental parameters to reach any desired level of reproducibility can be determined. Table IX gives the specifications of equipment that would give measurements with dispersions of lo+ and respectively, on the assumption that the noise contribution is negligible. To obtain results precise within 1 %, it is not necessary to control the outlet pressure. The inlet pressure should be

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controlled within 0.5 %, which needs a good quality pressure controller. The column temperature should be constant within about 0.4 "C, which is difficult to achieve with an air bath. The time measurement itself should be precise within 0.5%. It is doubtful from our experience that such an accuracy could be reached when a ruler is used to measure lengths on chromatograms : Paper elasticity and hygroscopicity, fluctuations of the paper rate, etc., may alter the relationship between time and length. Consequently, it is not as simple as would appear at first glance to measure retention times within 1%. Some care and equipment more advanced than most available commercial equipment are necessary. A precision of 10-4 seems extremely difficult to achieve even with the most sophisticated equipment available. The outlet pressure control is no problem. Admittedly, the temperature control (0.0035 "C) is difficult to reach, needs complex equipment, and restricts the temperature range in which it is possible to work, but the pressure drop control is the most demanding. It needs not only the best pressure controller available but a careful design of all the equipment and temperature control of all the pneumatic circuit. Last, the noise should be kept small enough to allow a precision of 10-4; Equation 19 shows that this is not so easy. It is thus probable that measurements with a precision of or better could be carried out only by using a computer to calculate the retention time from the whole peak shape. This, however, will not solve the problems of controlling the temperature and pressures. Retention time measurements are often not of direct interest. The evaluation of many thermodynamics data (partition coefficients, activity coefficients, adsorption coefficients) needs also the measurement of other data such as temperature, volume flow rate, mass of stationary phase, etc., which should be made with a comparable precision. Furthermore, in most cases, precision in this field is nothing: Accuracy is necessary. Accordingly, the results described here are only one step in the long, difficult road toward accurate measurements of thermodynamics data. ACKNOWLEDGMENT

We thank Laurent Jacob, Ecole Polytechnique, Paris, for fruitful discussions and Guy Preau for technical assistance. RECEIVED for review November 14, 1969. Accepted April 20, 1970. Work carried out with the financial help of the Com1'Energie i Atomique, Paris, France, under research missariat ? contract 10.830-II/B 6.

ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970