Anal. Chem. 2000, 72, 4667-4670
Constant Holdup Times in Gas Chromatography by Programming of Column Temperature and Inlet Pressure Tal M. Nahir* and Kathryn M. Morales
Department of Chemistry, California State University, Chico, California 95929-0210
The requirements for attaining constant holdup times in capillary gas chromatography (GC), while linearly programming both column temperature and inlet pressure, are illustrated. A scheme, which is based on a series of responses from isothermal and isobaric conditions, is proposed. The responses from two commonly used GC instruments, one with a mass-selective detector operating at near-vacuum pressure and another with a flameionization detector at ambient pressure, are analyzed. A significant deviation from Poiseuille flow is noted due to the use of helium as a carrier gas. Nonetheless, the experimental holdup times are approximately constant over a range of temperatures and pressures. Theoretical analysis reveals the spatial and temporal dependence of flow rates inside the column during the programmed runs. One of the essential parameters in chromatography is the time required for the mobile phase, or an unretained solute, to travel the length of a column.1 In gas chromatography (GC), this quantity, also known as the holdup time, is often determined by observing the retention time of a relatively inert and/or light compound such as N2, Ar, Ne, CH4, etc. Alternative approaches have employed calculations based on the retention of a series of n-alkanes.2 The holdup time is a fundamentally important quantity, which is also used for the calculation of capacity factors and retention indices of retained compounds.3 The magnitude of the holdup time can be adjusted by changing the temperature and pressure parameters during an analysis. Most modern gas chromatographs are capable of linear programming of both oven temperature and inlet pressure. When the temperature is increased, the viscosity of the gas inside the column increases, which leads to the slowing of the flow of the atoms and molecules toward the column outlet. In contrast, increasing the inlet pressure accelerates the elution of compounds because of a larger pressure gradient between the column inlet (injector) and the column outlet (detector). * Corresponding author. E-mail:
[email protected]. Fax: 530 898 5234. (1) (a) Gonzalez, F. R. J. Chromatogr., A 1999, 832, 165-172. (b) Didaoui, L.; Touabet, A.; Ahmed, A. Y. B. H.; Meklati, B. Y.; Engewald, W. J. High Resolut. Chromatogr. 1999, 22, 559-564. (2) (a) Quintanilla-Lo´pez, J. E.; Lebron-Aguilar, R.; Garcı´a-Domı´nguez, J. A. J. Chromatogr., A 1997, 767, 127-136. (b) Watanachaiyong, T.; Jeyashoke, N.; Krisnangkura, K. J. Chromatogr. Sci. 2000, 38, 67-71. (3) Gonzalez, F. R.; Nardillo, A. M. J. Chromatogr. A 1999, 842, 29-49. 10.1021/ac0004153 CCC: $19.00 Published on Web 08/30/2000
© 2000 American Chemical Society
The purpose of this work is to propose and experimentally confirm the possibility of simultaneously changing both column temperature and inlet pressure while maintaining a constant average rate of flow of unretained gas. This behavior is examined when the temperature and inlet pressure are programmed to increase in a monotonic fashion, when the column outlet is at either atmospheric pressure or near vacuum. EXPERIMENTAL SECTION The experiments were conducted on two similar HewlettPackard HP 6890 GC units; one chromatograph was connected to a HP 5973 mass-selective detector (MSD, near-vacuum outlet pressure), and another chromatograph was connected to a flame ionization detector (FID, ambient outlet pressure). The mobile phase in both cases was ultrahigh purity grade helium (Sierra Airgas, Sacramento, CA). For the measurements using the GC with a MSD (GC/MS), approximately 1-µL air samples were manually injected into a nominal 30 m (length) × 0.250 mm (inner diameter) × 0.25 µm (coating thickness) HP-5MS (cross-linked 5% phenylmethylsiloxane) column. Mass spectra for m/z ratios between 26.3 and 34.1 were recorded at 40 Hz. The identification of the major eluting compounds, i.e., molecular nitrogen and oxygen, was by their mass spectra, assuming that the signal is from a singly charged molecular ion M+. In the measurements on the instrument with a FID, commercial natural gas was bubbled into a small bottle containing approximately 10 mL of an n-alkane solvent, such as octane or dodecane, and approximately 0.1 µL of this solution was manually injected (usually with ∼1 µL of air) into the column. The elution time of the first peak, which was identified as methane in a separate measurement on the GC/MS, was recorded as the holdup time. The column used was a nominal 60 m × 0.250 mm × 0.25 µm HP-5 (cross-linked 5% phenylmethylsiloxane). The holdup times were determined by the built-in software for the apex of the CH4 (FID) and O2/N2 (MSD) peaks. Oven and column temperatures are reported according to the displayed values which were assigned to the GC through the software program. Split ratios were either 50:1 or 100:1, and no significant dependence of the measured holdup times on the split ratio was noticed. The calculation of theoretical holdup times with loss-modified Poiseuille flow followed the general outline presented by Cahill et al. (ref 14). A digital simulation was performed using 200 column Analytical Chemistry, Vol. 72, No. 19, October 1, 2000 4667
segments (each l/200 long, l ) 30 m), a time increment of 0.01 × (l/200)2, an outer to inner column diameter ratio of 1.25 (inner diameter of 250 µm), and maximum allowed error level of 0.001 Pa for the establishment of steady-state pressure profiles. On the basis of a least-squares linear regression analysis of the magnitude of the viscosity of helium between 50 and 270 °C (from Hawkes et al. (ref 9)), an exponential expression for the viscosity of helium: η(T) ) 0.371 × T0.698 µPa‚s was applied in the theoretical calculations. Pressures are reported vs ambient conditions, approximately 100 kPa throughout this investigation. RESULTS AND DISCUSSION Fundamental Relationships. The motion of inert gases through a capillary column is usually assumed to obey Poiseuille flow. Thus, the rate of transport of unretained gas is directly proportional to the pressure gradient inside the column and inversely proportional to the viscosity of the gas.4 Following wellestablished derivations of expressions for the flow rate and pressure profiles,5 the volumetric flow rate at any distance x from the column inlet is
f(x) ) πr2
πr4 dx ) dt 16lη(T)
pin(1 -
p2out/p2in)
x1 - x/l(1 - p2out/p2in)
(1)
where l is the column length, r is the column radius, η(T) is the temperature-dependent viscosity of the inert gas inside the column, and pin and pout are the pressures at the column inlet and the column outlet, respectively. The derivation of eq 1 assumes conservation of mass throughout the column and assumes that the flow rates and pressure profiles are established instantaneously upon any changes in T and pin.6 The general equation for the holdup time, tm, under isothermal and isobaric conditions is found by integrating eq 1 from initial to final elution conditions, i.e., from x ) 0 and t ) 0 to x ) l and t ) tm, respectively
tm )
p3in - p3out 32l2 η(T) 3r2 (p2in - p2out)2
(2)
The consequences of changing inlet pressures on holdup times and elution of retained compounds have also been discussed for isothermal programmed-pressure runs.7 It is evident from eq 2 that the holdup time depends on the viscosity of the gas of interest. The variation of carrier-gas viscosities with temperature has been reexamined recently (the change in η(T) with pressure is negligible under 5 atm).8 The magnitudes of the viscosities of the most frequently used carrier gases can be obtained using fairly complicated temperaturedependent expressions.9 The result for helium shows that its (4) Castellan, G. W. Physical Chemistry; Addison-Wesley: Reading, MA, 1964; pp 572-574. (5) Littlewood, A. B. Gas Chromatography; Academic Press: New York, 1970. (6) Leonard, C.; Sacks, R. Anal. Chem. 1999, 71, 5501-5507. (7) Vezzani, S.; Pierani, D.; Moretti, P.; Castello, G. J. Chromatogr., A 1999, 848, 229-238. (8) Hinshaw, J. V.; Ettre, L. S. J. High Resolut. Chromatogr. 1997, 20, 471481. (9) Hawkes, S. J. Chromatographia 1993, 37, 399-401.
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Figure 1. A schematic description of the conditions during a linearprogrammed run. The initial temperature and pressure are T0 and pin,0, and the program parameters are g, °C/min, and gp, kPa/min. The isothermal and isobaric holdup times at T1, pin,1, and T2, pin,2 are identical. The measurements of holdup times during programmed runs (see Figures 3 and 4) were conducted at several Ti, pin,i combinations.
viscosity increases with temperature while exhibiting a slight negative curvature. The magnitude of the viscosity under normal GC analysis conditions was also given by simpler expressions involving exponential,10 quadratic,11 and linear12 temperature terms. Holdup Time at Near-Vacuum pout. The situation in which the pressure at the column outlet is near vacuum is explored first, since the equations here are somewhat simpler than when the outlet pressure is not set to zero. In particular, eq 2 simplifies considerably when pout ≈ 0, and the holdup time is linearly proportional to the inverse inlet pressure
tm )
32l2 η(T) 3r2 pin
(3)
Consider two isothermal runs at two different temperatures, T1 and T2, with corresponding constant inlet pressures, pin, 1 and pin, 2, as shown in the pressure-temperature plane in Figure 1. According to eq 3, an identical holdup time can be achieved in these two runs if the ratio between the viscosity and the inlet pressure in one run is the same as that ratio in the other run (for example, a higher inlet pressure requires a higher gas viscosity and, therefore, a higher temperature). Next, consider a run in which the column temperature and the inlet pressure are programmed to linearly change at respective rates of g °C/s and gp kPa/s. We assign the magnitudes of g, gp and the initial temperature and inlet pressure, T0 and pin, 0, respectively, such that at time t1 the program is at T1 and pin, 1 and at time t2 the program is at T2 and pin, 2. As shown in Figure 1, specifying the conditions for the two isothermal and isobaric runs determines the relationships among the program parameters through an equation of a straight line. Assuming that the viscosity can be adequately given as a linear function of temperature, η(T) ) aT + b, where a and b are constants, eq 3 requires that the relation (10) Jennings, W.; Mittlefehldt, E.; Stremple, P. Analytical Gas Chromatography; Academic Press: San Diego, CA, 1997; pp 116-120. (11) Vezzani, S.; Castello, G.; Pierani, D. J. Chromatogr., A 1998, 811, 85-96. (12) Ettre, L. S. Chromatographia 1984, 18, 243-248.
Figure 2. Holdup times for isothermal runs at five constant inlet pressures with pout ≈ 0; measured (circles) and calculated with (dashed line) and without (solid line) correction for leakage of carrier gas. Values are adjusted to show identical tm at 27 °C.
between the inlet pressure and the viscosity in isothermal and isobaric runs with identical tm be
pin,0 + gpti ) m[a(T0 + gti) + b]
(4)
where m is a constant which is equal to 32l2/(3r2tm) (changes in the physical dimensions of the column due to thermal expansion are negligible here13), and ti can be adjusted to yield the desired pin, i and Ti, including i ) 1 and 2 for the two constant-temperature and constant-pressure runs. Equation 4 implies that pin, 0 ) m(aT0 + b) and gp ) m(ag); dividing the last two expressions, to eliminate m, and rearranging yields
pin,0 )
gp b T + g 0 a
(
)
(5)
We now propose that if a set of g, gp, pin, 0, and T0 includes the conditions for any two isothermal and isobaric runs with identical tm, then we can obtain a constant holdup time throughout the programmed analysis (i.e., with both pin and T changing linearly with time). Thus, eq 5 specifies the desired relationship among the programmed-run parameters for achieving a constant holdup time throughout the experiment. Before proceeding to the analysis of actual measurements, an adjustment has to be made to the theory presented so far. Helium, the carrier gas used in this investigation, was shown to permeate through the fused-silica walls of capillary GC columns.14 Consequently, the assumption of conservation of mass throughout the column fails, and a loss-modified Poiseuille flow must be applied. Figure 2 shows the departure from the expected (nearly) linear relationship between isothermal holdup time and column temperature at constant inlet pressures. Experimental results from a series of relatively short programmed runs, which were obtained using the GC with a massselective detector at near-vacuum conditions, are shown in Figure 3. The purpose of these runs was to simulate the behavior of an unretained molecule in various stages during a single longer programmed run with the same g and gp. In each measurement (13) Tipler, P. A. Physics; Worth: New York, 1976; pp 427-429. (14) Cahill, J. E.; Tracy, D. H. J. High Resolut. Chromatogr. 1998, 21, 531-539.
Figure 3. Holdup time as a function of temperature at the time of injection for a series of programmed runs with near-vacuum outlet pressure. The series with tm near 0.75 min was obtained when gp/g ) 0.50; one set with gp ) 2.0 kPa/min and g ) 4.0 °C/min (circles), and one set with gp ) 4.0 kPa/min and g ) 8.0 °C/min (triangles). The program for tm near 1 min used gp ) 1.5 kPa/min and g ) 4.0 °C/min. The longest runs were with gp ) 1.0 kPa/min and g ) 3.4 °C/min. The temperature at the time of injection and indicated b/a values, corresponding to the parameters in the linearized viscosity expression, were used to calculate the inlet pressure at the time of injection for each run according to eq 5.
of holdup time, the injection was at a temperature Ti ) T0 + gti and an inlet pressure pin, i ) pin, 0 + gpti, which correspond to conditions existing at some time ti during the long programmed run (Figure 1). For instance, the series of holdup times near 1.0 min was obtained at g ) 4.0 °C/min and gp ) 1.50 kPa/min, with initial conditions in the long programmed run of T0 ) 50 °C and pin, 0 ) 100 kPa (vs ambient). The first measurement was taken when ti ) 0, i.e., at 50 °C and 100 kPa. The second holdup time was measured with initial conditions of 100 °C and 118.8 kPa, which correspond to the situation 12.5 min into a long programmed run beginning with 50 °C and 100 kPa. The rest of the data at higher temperatures were obtained in a similar fashion. Our results show that assigning a b/a ratio of 210 yields the most consistent results in the range between 50 and 150 °C. At higher temperatures, the holdup time is somewhat shorter than predicted using Poiseuille flow. This observation can be directly related to the permeation of helium out of the column and the noticeable curvature in Figure 2 at comparable temperatures. To further substantiate the validity of the proposed approach, consider the nature of the flow rates inside the column during the programmed analysis. From eq 1, the volumetric flow rate inside a capillary column at any distance x from the column inlet reduces to
f(x) )
pin πr4 η(T) 16lx1 - x/l
(6)
in the case of near-vacuum outlet pressure. Clearly, as long as the ratio of inlet pressure to viscosity is experimentally arranged to be constant, the flow rate at x is fixed, which immediately implies that the holdup time is unchanged throughout such a programmed run. Note that this is exactly the requirement stipulated in eq 4, which resulted from proposing that the programmed-run conditions were an extension of the conditions needed for two isothermal and isobaric runs with identical holdup times. Analytical Chemistry, Vol. 72, No. 19, October 1, 2000
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Holdup Time at Nonzero pout. In many instances, including when common detectors such as flame ionization (FID), thermal conductivity (TCD), and electron capture (ECD) are used, the pressure at the end of the column is not near vacuum. To achieve constant holdup times during a programmed run, we next apply an overall approach similar to that shown above when pout ≈ 0. From eq 2, the proposed condition for a linearly programmed run, which includes a set of isothermal and isobaric conditions with identical holdup times, is
((pin,0 + gpti)2 - p2out)2 (pin,0 + gpti)3 - p3out
) mη(T0 + gti)
(7)
where, again, m ) 32l2/(3r2tm). This relationship is similar to eq 4, except that the term containing the pressure parameters (on the left side) is more complicated. The magnitudes of g, gp, pin, 0, and T0 can be determined from eq 7 by least-squares curve fitting.15 The goal in this procedure is to find the best set of programmed-run parameters, entered on the left side of the equation, which would match the viscosity on the right side of the equation. We first chose an initial column temperature, T0, and a desired temperature gradient, g. In addition, pout is given from measuring the ambient pressure, and r and l are from the manufacturer’s specifications (actual dimensions are probably different, but this discrepancy only causes a consistent shift in the magnitude of tm). To obtain a good fit, the optimized variables in eq 7 were the initial inlet pressure, pin, 0, and the pressure gradient, gp. A crucial issue for the success of the fitting procedure was to obtain an accurate relationship between the viscosity, calculated from experimental holdup time measurements, and the displayed column temperature. In an attempt to compensate for the leakage of helium through the column walls, “effective” values of η(T) were calculated from eq 2 using actual measurements of holdup times during a series of isothermal and isobaric runs. For example, using a series of isothermal measurements of holdup time at a constant inlet pressure of 300 kPa, the values of η(T) were calculated. Even if the actual response of the chromatograph differs significantly from theoretical predictions (we noticed an enhanced negative curvature in the dependence of the viscosity on temperature above ∼175 °C), this operating curve could be used for obtaining the desired fit such that the holdup time was maintained approximately constant throughout the programmed runs. To simulate the elution during a long programmed run, samples were injected at progressively higher temperatures and pressures according to the run parameters, as explained above for the near-vacuum case (for Figure 3). The results for several short programmed runs are shown in Figure 4. The analysis of flow rates is more complicated in this case when pout is not zero. To illustrate the nature of the change in flow rates during a temperature-and-pressure-programmed analysis, consider the conditions in two locations at different distances from the inlet, x1 and x2, with the latter further from the inlet. (15) Harris, D. C. J. Chem. Educ. 1998, 75, 119-120. (16) Gonzalez, F. R.; Nardillo, A. M. J. Chromatogr., A 1997, 779, 263-274. (17) Strobel, H. A.; Heineman, W. R. Chemical Instrumentation: A Systematic Approach; Wiley: New York, 1988; p 898.
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Figure 4. Holdup time as a function of temperature at the time of injection for a series of programmed runs with ambient outlet pressure. All results were obtained with T0 ) 50 °C and g ) 4.0 °C/min. Using the least-squares curve-fitting procedure, the series with tm near 2.1 min was obtained when pin, 0 ) 326.4 kPa and gp ) 2.18 kPa/min; the program for tm near 2.9 min used pin, 0 ) 229.0 kPa and gp ) 1.52 kPa/min; and the runs with tm near 3.7 min were with pin, 0 ) 175.7 kPa and gp ) 1.16 kPa/min.
The ratio of the respective flow rates at any instant during the run is given using eq 1
f(x2) f(x1)
)
x1 - x1/l(1 - p2out/p2in) x1 - x2/l(1 - p2out/p2in)
(8)
If the programmed inlet pressure increases with time, then this ratio increases with time. Under these circumstances, the only way to maintain a constant average flow rate through the column (and therefore a constant holdup time) is for the flow at the inlet to be gradually decreasing with time. This is in contrast with the unchanging flow-rate profiles in the near-vacuum outlet pressure case. CONCLUSIONS The results presented here show that the magnitude of the holdup time can be maintained constant throughout a programmed temperature analysis by compensating for viscosity effects through adjustments in the column inlet pressure. In the case of near-vacuum outlet pressure, the discrepancy between actual and predicted holdup times could be related to a nonlinear dependence of the viscosity on the temperature and the leakage of carrier gas through the column walls. The use of nitrogen as a carrier gas is expected to substantially reduce the nonlinearity of the results.14,16 However, N2 is not the preferred choice for most analyses because of the significant decrease in separation efficiency (i.e., large HETP) at high flow rates.17 In addition, one current limitation to achieving constant holdup times is the instrumental design which normally allows for only linear gradients. More flexibility in pressure programming should produce a better match to the curvature in the holdup time vs temperature plot. Received for review April 10, 2000. Accepted July 13, 2000. AC0004153
