(1.15) The time tder can be calculated from the values following from the time distribution of the concentration of the eluted substance; the substance concentration distribution is characterized by values t , and A t (see Fig. 1.1). If the limits, + 4 x A t , are applied to function Y, the following relationship can be written for tder:
v,,, = u . tdet = ii . TI . r2 . tder
( t L - t l ) = +4dr = z . tdet (1.16)
td,,/dt = 8/z (1.17)
Equation (1.17) expresses the relationship between the separation conditions and the effective volume of the detector; it is evident that the fraction equals zero for an ideal detector. If t,,,/At 4 0, then the concentration distribution in the detector changes in comparison with the conditions at the column outlet, described by equation (1.5).
The newly established distribution of function St (see Fig. 1.5) is given by the fol- lowing equation [62]:
When expressions for Pax are compared, it is observed that the maximum on the concentration distribution shifts by fdec/2 on passage of the gas through the detector.
I 1
'det 1
FIG. 1.5.
Conversion of function cs into funkction S t .
L!?
t I
A
- v[m[/min]
FIG. 1.6. The shift in the elution time as a function of the effective volume of the detector and the carrier gas flow-rate.
TABLE 1.2
THE TIME, tR[sec], OF PASSAGE OF GAS THROUGH THE EFFECTIVE VOLUME O F THE DETECTOR AT VARIOUS FLOW-RATES
0.5 1.2 3 6 12 30 60 120 240 600
1 .o 0.6 1.5 3 6 15 30 60 120 300
2.5 0.24 0.6 1.3 2.4 6 12 24 48 120
5 0.12 0.3 0 . 6 1.2 3 6 12 24 60
10 0.06 0.15 0 . 3 0.6 1.5 3 6 12 30
25 0.06 0.12 0.24 0.6 1.2 2.4 4.8 12
50 0.06 0.12 0.3 0.6 1.2 2.4 6
I00 0.06 0.15 0.3 0.6 1.2 3
I50 0.10 0.2 0.4 0 . 8 2
The shift in the elution time, A t , , is inversely proportional to the gas flow-rate under constant experimental conditions, i. e., at a constant effective volume of the detector:
t R = - + A 'dec 21
26
where A is an experimental constant (see Fig. 1.6). The shift in the elution time in- creases with increasing detector volume; some values are given in Table 1.2.
Because of the finite values of the effective volume of the detector and the gas flow-rate and because of the necessity of measuring the concentration using a sensor that yields an electric signal, the measured signal is distorted. This distortion is called
L a 5 4
O - 5 - 4 - 3 - 2 - 7 0 1 2 3 4 5 Af
FIG. 1.7. Time constant of the device.
FIG. 1.8. Distortion of the response of a detec- tor with laminar flow as a function of changes in the tder/dt ratio [19].
the time constant of the device, T. When it is assumed that each signal value corre- sponding t o instantaneous concentration, cs, is reached by a step, then the given signal is, in practice, reached only after a certain time (Fig. 1.7) which is approximately
TABLE 1.3
DISTORTION OF THE DETECTOR SIGNAL AS A FUNCTION OF VARYING tder/dt RATIOS FOR LAMINAR AND TURBULENT FLOW CONDITIONS
Laminar Turbulent Maximum
shift Y
7; %
-~ _ _ _ _ . ~ _ _ _ -
tder = z , df -
area width height area width height
8 62 270 20 100 - 24 1.66
4 74 150 50 100 - 40 I .35
2 89 106 84 100 154 60 1 .o
1 97 101 96 100 124 78 0.5
0.5 99.2 100.4 99 100 110 91 0.25
0.25 99.8 100.2 99.7 100 107 97.4
0 100 100 100 100 100 100
__ ~~
The shift of the maximum on the elution curve, d t ~ = y . d t
equal to three times the time constant. Owing to this distortion. the maximum of the output function, S""", is shifted with respect to the maximum of the input function, c;lnx. When the time constant increases, not only is the output function maximum shifted to longer times along the descending branch of the ideal Gaussian curve, but it is generally distorted.
The time integral of the output function, S", within limits ( t , . t 2 ) has been defined as the response. Therefore, distortion of the signal causes distortion of the detector response, which is thus not identical with the concentration distribution at the column outlet. Figs. 1.8 and 1.9 show changes in the detector response caused by changes in the t,,,/dt ratio. Mass transport also plays a role during the passage of the eluted substance through the detector. If there is a concentration gradient of the eluted substance in the direction of the flow, i.e. if the mixture is transported by laminar flow alone and is not stirred, then the probability that all the eluted substances reach the sensor decreases with increasing detector volume and hence the response decreases (Fig. 1.8 and Table 1.3). If there is no concentration gradient in the detector along the direction of flow, i.e., the mixture is stirred 'as a result of turbulent flow,
0.8
0.51 0.4 0.3 t
0.21 0.1 c
(4 L 0 L 8 12 16
A t
20
- t
-t FIG. 1.9. Distortion of the response of
a detector with turbulent flow as a func- tion of changes in the f&t/dt ratio [19].
FIG. 1.10. Changes in the response of a detector with a large effective volume as a function of the carrier gas flow-rate.
then there is always a certain average concentration of the eluted substance in the effective volume of the detector; when the effective volume of the detector is increased or the flow-rate decreased, the response is distorted, as is shown in Fig. 1.9.
Much attention has been paid to the dependence of signal distortion on the flow-rate of the mobile phase. When the carrier gas flow-rate increases, tdet decreases
28
more rapidly than the elution peak width, A t , and therefore the distortion decreases.
An increase in the flow-rate leads to a decrease in the influence of the effective volume of the detector and the time constant of the device is determined by the time constants of the other components of the apparatus, such as the recorder. Response distortion for these cases is shown in Fig. 1.9.
The largest shift in elution times is exhibited by devices that employ diffusion processes and those with large effective volumes of the detector, operating at low gas flow-rates. Large cells were employed in older types of thermal conductivity detectors; the average concentration is then measured. An increase in the flow-rate results in an initial increase in the peak height, corresponding to the maximum number of moles, N ; , present in the detector. On increasing the flow-rate, tdet is increased and hence the elution curve area is decreased (Fig. 1.10). It is thus desirable that the measuring device should have as low a time constant as possible and thus also the smallest possible effective volume of the detector.
In the measuring device the signal S" is converted into an electrical impulse. This impulse is usually a voltage formed on the amplifier resistor, R, through which the current corresponding t o changes in the effective volume of the detector passes. The current is sometimes measured directly using a current follower. Therefore, the output signal can always be converted into a current and has the dimension of [A]. As the response is defined as the time integral of the signal, it has the dimension of electric charge, [ C ] .
The separation of the eluted substance at the column output is described by its concentration distribution. Therefore, the gas chromatographic experiment requires evaluation of the relationship between the measured response (signal) and the con- centration of the eluted substance.