Anal. Chem. 1998, 70, 3853-3856
General Equation for Peak Capacity in Column Chromatography Yufeng Shen and Milton L. Lee*
Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah 84602-5700
In this study, the peak capacities for packed capillaries used in various forms of chromatography were investigated. Experiments found that the peak width at halfheight (w1/2) linearly increased with increasing retention time when operating under isothermal, isobaric, and isocratic conditions in capillary gas chromatography, solvating gas chromatography, supercritical fluid chromatography, and liquid chromatography. From this relationship, a general peak capacity (n) expression was obtained. This expression covers conventional packed, packed capillary, and open tubular column chromatographies. With this general expression, it is possible to directly compare the separating capabilities of the various column types and chromatographic techniques. The peak capacity (n) has been used to describe the chromatographic separation capability.1,2 It is defined as the number of peaks that can be separated with a resolution of unity in a given time internal (t1-tn), and it is expressed as follows:
n)1+
∫ x4N dtt tn
t1
(1)
where N is the theoretical plate number. In column chromatography, the separation column can be either open tubular or packed, and the mobile phase can be compressible or noncompressible. Because of the small pressure drop in open tubular columns under typical operating conditions, uniformity of the mobile phase throughout the column can be assumed when such columns are used in gas chromatography (GC), supercritical fluid chromatography (SFC), and liquid chromatography (LC). Since the same kinetic processes exist in the various open tubular column chromatographies, the Golay equation3 can be used to describe peak broadening. In packed columns, the van Deemter equation4 can be used to describe the chromatographic process at any specific point in the column when a compressible mobile phase, such as a highpressure gas, supercritical fluid, or ultrahigh pressure liquid, is used. However, the pressure drop along the packed bed changes the mobile-phase physical properties, and therefore, apparent (1) Giddings, J. C. Anal. Chem. 1969, 39, 1027-1028. (2) Grushka, E. Anal. Chem. 1970, 42, 1142-1147. (3) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271-289. (4) Golay, M. J. E. In Gas Chromatography 1958; Desty, D. H., Ed.; Butterworths: London, 1956. S0003-2700(98)00242-X CCC: $15.00 Published on Web 08/13/1998
© 1998 American Chemical Society
parameters measured at the column outlet are typically used to characterize the overall chromatographic performance characteristics such as efficiency and retention. Even for the simple case of open tubular columns, the complex relationship between N and t in the Golay equation makes it impossible to directly integrate eq 1 to obtain an expression for n. Simplifications must be made to obtain n. On the basis of experimental data obtained for conventional packed column LC, Grushka treated N as a constant and obtained eq 2 to describe the effect of retention time and column efficiency on peak capacity.2
n)1+
xN tn ln 4 t1
(2)
However, the assumption that N is a constant for solutes having different retention gives incorrect values of n. In open tubular column GC, column efficiency greatly depends on the retention factor (k), and n has been expressed as follows:5
n)1+
k(N∞
∫
1 4
0
+ b/k + c/k2 + ...)1/2 dk 1+k
(3)
where N∞ is the number of theoretical plates when k f ∞. Equation 3 is suitable for open tubular column GC, SFC, and LC because all three involve the same peak-broadening mechanism (no A term and the same B and C term expressions). However, eq 3 has no physical significance, and it is difficult to use to obtain values for n. The packed capillary column is a specialized packed column that is finding increasing popularity for miniaturized separation systems. From studies involving packed capillary column SGC and LC, it was found that column efficiency also greatly depends on solute retention.6,7 The assumption that N is constant, therefore, is not always correct for these techniques. In this study, the relationship between n, N, and t was investigated. Open tubular column GC and packed capillary column SGC, SFC, and LC were used to study this relationship as examples of both open tubular and packed column chromatographies. Each involves a different degree of compressibility of (5) Krupcˇ´ık, J.; Garaj, J.; C ˇ ella´r, P.; Guiochon, G. J. Chromatogr. 1984, 312, 1-10. (6) Shen, Y.; Lee, M. L. Anal. Chem. 1997, 69, 2541-2549. (7) MacNair, J. E.; Lewis, K. C.; Jorgenson, J. W. Anal. Chem. 1997, 69, 983989.
Analytical Chemistry, Vol. 70, No. 18, September 15, 1998 3853
the mobile phase. A general expression for n was deduced and used to study the separation capabilities of the various forms of column chromatography. EXPERIMENTAL SECTION Materials. Spherical ODS particles having 10-µm diameter and 80-Å pores were purchased from Alltech (Deerfield, IL). Fused-silica capillary tubing was purchased from Polymicro Technologies (Phoenix, AZ). Column connections were made using PEEK tubing and zero dead volume unions (Valco Instruments, Houston, TX). SFC grade CO2 was purchased from Scott Specialty Gases (Plumsteadville, PA). Chemicals used were purchased from Sigma (St. Louis, MO) and Aldrich (Milwaukee, WI). Preparation of Packed Capillary Columns. A CO2 slurry packing method was used to prepare long packed capillary columns containing 10-µm particles.6,8 A steel screen (2-µm pores, Valco) was used to support the packed particles. Using CO2 at room temperature and 40 atm, the particles were slowly packed into the column. At ∼200 atm, the column was fully packed, and it was conditioned at room temperature and 250 atm in an ultrasonic bath for 30 min. Packed Capillary Column Chromatography Experiments. SGC experiments were carried out using the instrumentation previously described in ref 6. A Lee Scientific model 600 SFC/ GC instrument (Dionex, Salt Lake Division, Salt Lake City, UT) was used. SFC grade CO2 was used as the mobile phase. A manual liquid injector (Valco instruments) with a rotor volume of 0.2 µL was used for introduction of samples. A 10 cm × 15 µm i.d. fused-silica capillary was used as an injector split line. A 10 cm × 50 µm i.d. fused-silica capillary was used to connect the separation column to the flame ionization detector (FID). SFC experiments were carried out using the same instrumentation as for SGC. SFC grade CO2 was used as the mobile phase. The packed capillary column was cut to 1.5 m in length. A restrictor (pinhole) was made at the end of the 50-µm-i.d. connecting tubing. An FID was used for detection. Open Tubular Column Experiments. An HP-5890A series gas chromatograph (Hewlett-Packard, Avondale, PA) was used. Helium (Scott Specialty Gases) was used as the carrier gas. The experiments were performed using 250-µm-i.d. (20- and 30-m lengths) fused-silica capillaries coated with 0.25-µm SE-54 stationary phase (0.25-µm df, Supelco, Bellefonte, PA). Samples were introduced into the column by split injection with a split ratio of 100:1. Data Acquisition. A microcomputer equipped with EXCEL (Microsoft, Redmond, CA) was used for data acquisition and presentation. RESULTS AND DISCUSSION Relationship between Column Efficiency and Solute Retention in Packed Capillary Column SGC, SFC, and LC. In our previous studies of column efficiency in packed capillary column SGC, it was found that the column efficiency was strongly dependent on the solute retention factor (k).6,9 Figure 1 shows (8) Malik, A.; Li, W.; Lee, M. L. J. Microcolumn Sep. 1995, 5, 361-367. (9) Shen, Y.; Lee, M. L. Chromatographia, submitted. (10) Ettre, L. S. Pure Appl. Chem. 1993, 65, 819-872. (11) Shen, Y.; Malik, A.; Li, W.; Lee, M. L. J. Chromatogr. 1995, 707, 303-310.
3854 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998
Figure 1. Experimental relationship between column efficiency (N), effective efficiency (Neff), and retention factor (k) for packed capillaries in SGC. Conditions: 4.0 m × 250 µm i.d. fused-silica capillary containing 10-µm ODS particles (80-Å pores), CO2, 250 atm column inlet pressure, FID. Test solutes: n-hydrocarbons from n-octane to n-hexadecane (to was determined from extrapolation of retention factor versus carbon number of homologous n-hydrocarbons). (b, O) 190 °C; (9, 0) 160 °C. Table 1. Linear Relationships between w1/2 and t for Column Chromatography
SGCa SFCb LCc OTGCd
experiment 1
experiment 2
w1/2 ) 0.0082t - 0.0442 R2 ) 1.000 w1/2 ) 0.0244t - 0.0288 R2 ) 0.992 w1/2 ) 0.00571t - 0.0199 R2 ) 0.999 w1/2 ) 0.0092t - 0.0026 R2 ) 1.000
w1/2 ) 0.0076t - 0.0364 R2 ) 0.997 w1/2 ) 0.0241t - 0.0272 R2 ) 0.998 w1/2 ) 0.00629t - 0.0019 R2 ) 0.998 w1/2 ) 0.0087t - 0.0021 R2 ) 1.000
a Conditions: 4.0 m × 250 µm i.d. fused-silica capillary containing 10-µm ODS particles (80-Å pores), CO2, 250 atm column inlet pressure, FID. Test solutes: n-hydrocarbons from n-octane to n-hexadecane (to was determined from extrapolation of retention factor versus carbon number of homologous n-hydrocarbons). Experiment 1, 160 °C; experiment 2, 190 °C. b Conditions: 1.5 m × 250 µm i.d. fused-silica capillary containing 10-µm ODS particles (80-Å pores), CO2, 400 atm column inlet pressure. Experiment 1, 80 °C; experiment 2, 100 °C. Other conditions are the same as in SGC. c Conditions: see ref 7. The experimental data for hydroquinone, catechol, and 4-ethylresorcinol were used for data analysis. Experiment 1, 1300 bar; experiment 2, 4100 bar (1bar ) 0.987 atm). d Conditions: 20 m × 250 µm i.d. fusedsilica capillary column coated with 0.25-µm SE-54 stationary phase, He carrier gas, FID. Test solutes: n-hydrocarbons from n-octane to n-hexadecane (to was calculated from extrapolation of retention factor versus carbon number of homologous n-hydrocarbons). Experiment 1, 130 °C; experiment 2, 160 °C.
the experimental relationship between N and k for packed capillary column SGC. The relationship between effective column efficiency (Neff) (see ref 10) and k was added to the figure. Even when k > 5, N significantly decreases with increasing k. The relationship between N, Neff, and k in Figure 1 is similar to observations in open tubular column GC.5 We directly investigated the peak broadening in SGC as measured at the column outlet. It was found that a linear relationship exists between the peak width at half-height (w1/2) and the elution time (t). Table 1 lists the experimental results under SGC conditions. The R2 values were all greater than 0.99. Increasing the temperature from 160 to 190 °C slightly changed the slope of the line. Particles with various pore sizes and chemically modified surfaces9 were also tested, and a relationship
between w1/2 and t similar to that in Table 1 was always observed. This relationship can be described as
w1/2 ) at + b
(4)
where a and b are constants. It is noteworthy that different homologous compound series have specific values of a and b, and the difference in these values depends on the compound properties. The relationship between w1/2 and t in SFC was investigated using packed capillaries, and the results are also listed in Table 1. A linear relationship was again observed. For LC, experimental data reported by MacNair et al.7 were used. It was found that, for different solute types, some deviation from linearity existed (R2 300 peaks with a resolution of unity. The operating temperature is an important factor affecting the peak capacity in SGC. Using eq 2 to calculate the peak capacity, >100% error was obtained. Figure 3 shows the calculated peak capacities of packed capillaries for specific time intervals according to eq 6 using experimental data for a and b under SFC conditions. Because of Analytical Chemistry, Vol. 70, No. 18, September 15, 1998
3855
Figure 4. Calculated peak capacities (n) for packed capillary column LC according to eq 6. Experimental conditions to obtain a and b constants are the same as in Table 1. (b, O) n calculated for 4100 and 1300 bar, respectively.
excessively long analysis times experienced when longer columns were used in SFC, a 1.5-m capillary column containing 10-µm particles was used. Using 400 atm column inlet pressure, >130 peaks can be separated in 60 min with a resolution of unity. An increase in peak capacity for SFC depends on the maximum pressure that can be imposed at the column inlet. In this study, the maximum pressure was limited to 400 atm. Figure 4 shows the calculated peak capacities of packed capillaries for specific time intervals according to eq 6 using experimental data of MacNair et al. for ultrahigh-pressure LC.7 The peak capacity in LC greatly depends on the column inlet pressure. Using a 0.66-m capillary column containing 1.5-µm particles with 1300 bar (1 bar ) 0.987 atm) column inlet pressure, ∼200 peaks can be resolved in 60 min. However, when the pressure is increased to 4100 bar, ∼300 peaks can be separated with a 0.52-m capillary column containing 1.5-µm particles.
3856 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998
Figure 5. Peak capacities (n) for open tubular column GC and packed capillary column SGC. (b) 250 cm × 250 µm i.d. fused silica capillary containing 10-µm ODS particles (80-Å pores), 160 °C, other SGC conditions are the same as in Table 1; (0, ]) n for 20- and 30-m open tubular column lengths at 160 °C, respectively; other open tubular column GC conditions are the same as in Table 1.
Increasing the pressure decreases the dead time and improves the peak capacity, although a loss in column efficiency occurs. Figure 5 shows the calculated peak capacities of open tubular column GC for specific time intervals according to eq 6 using experimental a and b values. Using a 2.5-m capillary column containing 10-µm particles, SGC provides a separating capability similar to a 20-30 m × 250 µm i.d. open tubular column in typical GC.
Received for review March 4, 1998. Accepted June 18, 1998. AC9802426
