Anal. Chem. 1993, 65,2686-2689
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CORRESPONDENCE
Molecular Basis of Peak Width in Capillary Gas Chromatography under High Column Pressure Drop Leonid M. Blumberg' and Terry A. Berger Hewlett-Packard Co., 2850 Centerville Road, Wilmington, Delaware 19808
Under normal conditions in GC (gas chromatography), the first peaks in the run are the narrowest and pose the highest demand for avoidanceof extra column peak broadening effects by detector volumes, time constants of detector electronics, etc. Furthermore, the widths of all the peaks in a chromatographic run are predetermined by the width of the unretained peak. Therefore, the width of the unretained peak can serve as an indicator of the speed of the entire analysis. In view of this, we made an attempt to derive a relation for the width of unretained peaks based on column dimensions and its operational conditions. We came to the unexpected and previously unknown conclusion that under high inlet-to-outlet pressure drop, the width of an unretained peak does not depend on the column diameter, but only on the column length and molecular properties of the carrier gas and the solute. A more accurate formulation of that result as well as its derivation and discussion is the subject of this report.
THEORY
The optimized columns, i.e. uout
- Vout,opt
(5)
are also considered. Variance of a Peak. In thin film capillary GC, under thecondition in eq 1, the variance of a peak can be expressed (see Appendix) as
G ( k )= 1
+ 6k f l l k 2
In eq 6 , while 11 is a property of only the mobile phase, Dout depends on the phase and on the solute. Substitution where Dm,outis coefficient of self-diffusionof the mobile phase at the column outlet conditions, allows the properties of the mobile phase, 7 and Dm,out,to be separated from the property of the mixture. Equation 6 becomes
Conditions for the Analysis. Typically, high-speed capillary GC is associated with small column diameters which frequency require high pressure drop, i.e. Pin >> Pout
(1)
That along with conditions M , >> M , YOUt
2 Yout,opt
(2)
(3)
constitutes the boundaries of analysis in this paper. In the latter inequalities, M , and M , are molecular weights of, respectively, the solute and the mobile phase while uOUtand uout,opt are, respectively, carrier gas outlet linear velocity and its V a n Deemter optimum.112 Also, in the analysis, a carrier gas is assumed to be ideal, and, when retained peaks are considered, the affect of the stationary phase film thickness on the plate height is ignored. The latter means that, in cases of a solute retention, thin film columns are assumed. Special attention in the analysis is given to overoptimized columns where Yout
>> Vout,opt
Reduced Diffusivity of a Solute. Ratio D,, in eq 7 , can be viewed as the reduced diffusivityof a solute in the mobile phase. It can be expressed via molecular properties of the mobile phase and the ~ o l u t e ~ , ~
Typically in GC, the solutes have much larger molecular weights compared to the carrier gas, eq 2. Equation 9 yields
Peak Width and Molecular Properties of Carrier Gases and Solutes. Assuming ideal gas conditions, the combination 7/(Dm,ou$out)in eq 8 can be further reduced. Due to the relations pD = 1.27 and M~ = 3RTiM = 3P/p known from the kinetic theory of ideal gases,5 one has
(4)
(1) Van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. S C ~1956, . 5, 271-289. (2) Golay, M. J. E. Gas Chromatography; Desty, D. H., Ed.; Butterworth: London, 1958; pp 36-55. 0003-2700/93/0365-2686$04.00/0
(3) Fuller, E. N.; Giddings, J. C. J.Gas Chromatogr. 1965,3,222-227. (4) Fuller, E. N.; Schettler, P. D.; Giddings, J. C. Ind. Eng. Chem. 1966, 58 (5), 19-27. (5) Moore, W. J. Physical Chemistry, Prentice-Hall: EnglewoodCliffs, N J , 1972. @ 1993 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
and eq 8 becomes
L2G(k)(1 + ( 1.2D,c(:
k ) 2 )
(11)
vout
Lower Bounds for Peak Widths. When the condition in eq 4 is met a column is overoptimized and eq 11becomes
Quantity T- represents the solute-specific lower bound for standard deviation of peaks. The solute dependence isrepresented by capacity ratio, k,for the solute and by ita reduced diffusivity D,, eq 7. Consider the quantity
It follows from eq 6 that G(k) 2 G(0) = 1. Also from eq 7 and refs 3 and 4 under the condition of eq 2, it follows that D, < 1. Therefore, shah < T- and quantity ~ - ~represents b the absolute lower bound for the standard deviation of unretained peaks. In a given carrier gas, no peak can have standard deviation below rminab. Comparison of eqs 12 and 13 indicates that
Two lower bounds, ~ - ~andbT-, eqs 13 and 14 (see also eq 16 for their relations), can be recognized. Equation 13 gives an absolute lower bound, 7hab,of the peak width. For a given column length and carrier gas, no peak in a GC run can be narrower than the absolute hwer bound, rminab,which does not depend on the column diameter, the nature of the solute in the peak, or its retention. The bound r h a bcan be expressed in terms of the molecular weight of the carrier gas (the fundamental property of the gas), its absolute temperature, and column length. Equation 13also suggests that the standard deviation of a peak can never fall under about 905% of the time defined as the ratio of the column length, L, to the mean square speed, ccm, of carrier gas molecules. If the components of a samplemixture and/or their capacity ratios (relative retentions) in a given column are known,then more accurate higher solute-specific bounds T-, eq 14, can be found for each component. For a given column length and carrier gas, a peak cannot be narrower than its solutespecific lower bound, ?-, which does not depend on the column diameter. Not only can the lower bounds for the peak widths be easily predicted, but their actual values as well. Specifically, if flow optimization, eq 5, for a given solute requires high column pressure drop, then the standard deviation, Topt, of the peak can be found from eq 15. This value, again, is independent of the column diameter. It is interesting to notice from eq 15 that for a given solute, the optimum peak width, Topt, exceeds its respective lower bound, T-, by about 41 5% That means that under high pressure drop conditions, there is no room for a significant peak width reduction beyond its Van Deemter optimum value. One can conclude that (seetheory and discussion below for more detailed conditions) The only column dimension which affects peak widths in high pressure drop capillary GC is column length. Change in the column diameter alone does not affect the peak widths. This conclusion can be illustrated by the following examples. To improve peak resolution achieved with a Van Deemter optimized thin f i i capillary column operatingunder high pressure drop, one can choose another capillary column with smaller diameter but with the same length, stationary phase, and phase ratio. If the new column is also optimized, then the solute will elute with the same peak width from both columns (although, in the new column, all peaks will be further apart from each other, and, therefore, better resolved). The same is true if both columns were overoptimized. To improve resolution, one can also increase column length while keeping its diameter, carrier gas mass flow, and other relevant conditions (see previous case) unchanged. In this case, peak widths will increase proportionally to the increase in column length (although, again, the time between the peaks will have even larger increase, so that the resoltuion will increase). Further discussion of the relations between column dimensions and resolution is outside of the scope of this report. Independence of the peak width from the column diameter in eqs 11and 13-15 can also be illustrated by the following analysis. For a given column length and solute capacity ratio, peak width is proportional to the quantity HIP. In this ratio, plate height, H,is proportional to column diameter, d,. Suppose that the column has optimum flow. Then Yopt2 ise inversely proportional to d , and in the relation rOpt2a HOpJ the column diameters in Hoptand ;opt2 cancel each other. Consider, e.g., reduction in column diameter. It tends to reduce statistical zone broadening in distance proportional to the reduction in plate height. However, increased column pressure causes reduction in the carrier gas average velocity
.
Also from eqs 5,11, and 12, one has
Due to the discussion following eq 13,the last three quantities relate as 7min9b
< ?mill < Topt
(16)
DISCUSSION Peak Widthand Column Diameter. The most important results of the theory are expressed in eqs 11 and 13-16. Equation 11 gives a general formula for the peak width (expressed via the peak’s standard deviation) under high pressure drop. It shows how the peak width relates to the molecular properties of the carrier gas and the solute at a given temperature. Equations 13-16 deal with special cases. The peak width also depends on column length but, interestingly (and unexpectedly), does not depend on column diameter. Below, we provide a more detailed discussion of this conclusion. A GC analysis depends on the combination of many conditions which, in this discussion, are assumed to remain constant (isothermal, isobaric, etc.) during a given run. One such condition is carrier gas flow. Increase in the flow, when all other conditions remain the same, can reduce peak widths and analysis time.6J However, there is a limit to the peak width reduction achievable via flow increase. In a capillary column, when the column flow gets larger and larger the outlet carrier gas velocity, vout, will, eventually, exceed its optimum vout,optand further reduction in the peak width will no longer be available, as eq 11, suggests. In other words, the lower bounds for the peak widths will be reached. (6)Schutjee, C. P.M.; Vermeer,E. A.; Rijks, J. A.; Cramere, C. A. J. Chromatop. 1982,253, 1-16. (7) Guiochon, G.; Cuillemin, C. L. Quantitative gas chromatography for laboratory analysis and on-line control;Elsevier: Amsterdam, 1988.
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
0.8
1
f 0.6 % 0.4
02
0
Co(unn damstet, rrm
Molecdar wwg lM
Figure 1. Reduced diffusivity, Dr,eq 10, of hydrocarbons in He (lower line), H2(mkMleline) and In N2(upper line). All data for collision diameters in eq 10 are taken from refs 3 and 4. dm: 2.88’13 = 1.42, 7.07’/3 = 1.92, 17.Q113= 2.62 for He, H2 and N2, respectively, and ds = (3.96 20.46n)’13 where n Is hydrocarbon number.
+
0 0
0
0
0
/
0
0.5,
0
c
4
0
W 0
.c
0
c
/
4
-
-
c
ot. 0
¶
1
2
3
4
5
Capacity ratb
Figure 2. Effective peak width (arealheight) of Mecane vs its capacity ratio at 100 OC for different carrier gases: middle line, He; lower line, H2; upper line, N2. Effective peak width was calculated as (2?r)’/2~wt = 2.5rwt, where rwt was calculated from eqs 15 and 13. Data for reduced dlffusivity Dr,eq 10, were the same as in Figure 1. L = 10 m.
(in spite of the mass flow increase). Reduced average velocity of the carrier gas proportionally reduces average velocity of migration of the solute. Reduced solute average velocity compensates for the reduction in the statistical zone broadening in distance resulting in unchanged peak width in time. Graphical illustration of this discussion can be found in Figures 1-3. (Many commercially available integrators report peak width in units of w,ff = areaiheight, or effective peak width.8 For Gaussian peaks, w,ff = ( ~ T ) ~ / To ~ T facilitate . the comparison of the peak width values in Figures 2 and 3 with the experimentally available data, effective peak width is used in Figures 2 and 3.) Notice that the simplicity of eqs 13and 14can be beneficial beyond the immediate needs of GC. For example, at k = 0, eq 14 becomes rmin = ~ ~ i , , ~ b $ D ~ l /Together z. with eq 10, it can be used as the basis for the alternative techniquelo of measurement of diffusion coefficients and collision diameters of gases. Boundaries for the Application of the Theoretical Results. This theory requires high pressure drop, eq 1,and, therefore, strictly speaking, is valid only asymptotically when the ratio Pi,IPOutapproaches infinity (assuming, of course, (8) Blumberg, L. M. Anal. Chem. 1984,56 (9), 1726-1729.
(9) Giddings, J. C.; Seager, S. L.; Stucki, L. R.; Stewart, G. H. Anal. Chem. 1960,32, 867-870. (10) Maynard, V.R.;Grushka, E. Adu. Chromatogr. 1975,12,99-140. (11)Touloukian, Y. S.;Saxena, S. C.; Hestermans, P. Viscosity: IFIiPlenum: New York-Washington, 1975.
Figure3. Effective peak widths, wan(rislng curves), and inlet pressures, PI, (fallingcurves), vs column diameter, dc,for Mecane at 100 OC with optimum flow, eq 5,of He (solid lines), H2(long dashes), and N2(short dashes). we,, was calculated as wan = ( 2 ~ ) ~ / ~ 7where ,,, rapt was calculated from nonsimplified eqs 17 and 18 with DM calculated from known relations‘ and t) taken from experimental data.ll L = 10 m. The horizontal line marks the level where Ph = 2PM = 2 atm. The vertical line at dc = 0.21 mm marks the intersection of curve (wenfor He) with the horizontal line. At this point, = 0.31 s. To the left of this line (at dc I 0.2 mm) w.w,He only sllghtly depends on dc. Moving further to the left, Ph rapidly increases and the curve becomes practically flat (independent of dc)at = 0.24 s. k = 1.
that the flow remains laminar). In practical calculations, though, one should take into account that most of the components in all equations have errors. Thus, the empirical equations for diffusivity alone can produce errors4of 110%. The errors in the analytical results from the theory fall within that range when Pi, is 2 to 3 times higher than Pout, Figure 3. Also included in the derivation of the general eq 11 were the assumptions that the carrier gas was ideal and thin stationary phase films were used. The former may exclude application of the theory to techniques, such as SFC (supercritical fluid chromatography), where behavior of the compressible mobile phase may significantly deviate from that of an ideal gas. The low film thickness can be essential for the validity of eq 11 for peaks with medium retention. However, it has no affect on the width of unretained or highly retained peaks.2 Specifically, the film thickness does not affect the value of the absolute lower bound, ~ ~ i n , ~ eq b a 13, , for the peak width as that bound can be reached only by unretained solutes. Also, the film thickness does not affect the value, T~~,(D,,O) = Tmin,ah/(Dr)1/2, eq 14, of the more accurate, solute-specific lower bound. Other conditions in the theory, eqs 2 and 4, do not affect its general result, eq 11. Equation 4 was used in the derivation of the lower bounds, eqs 13 and 14, for the peak width. It can be viewed as the column overoptimization condition when these lower bounds can be achieved. Again, from the practical point of view, T in eq 11 gives a good approximation of the lower bounds in eqs 13 and 14 as long as column flow exceeds its Van Deemter optimum value by a factor of 2 or more. Unlike all other conditions, the requirement in eq 2 that the molecular weights of a solute be larger that the molecular weight of the carrier gas may not be under the control of the operator and can be violated in practice (very light molecules migrate in, say, nitrogen). In the realistic GC, though, it almost never happens, especially with hydrogen or helium as a carrier gas. It must also be noticed that even the violation of this condition does not affect the expressions of the main results, eqs 11 and 13-16. Nor does it affect the conclusion of independence of peak widths and their lower bounds on the column diameter. It can only affect the accuracy of calculations if the simple formula, eq 10, was used instead of eq 9 for the calculation of the reduced diffusivity of the extremely light solutes.
ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
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CONCLUSION The key conclusion in this report is: When optimization of a column flow in capillary GC requires high pressure drop, widtha of all peaks depend on column length but do not depend on column diameter, as long as the column flow remains equal to or greater than ita optimum value. Expressions relating the peak widths and their lower bounds to the molecular properties of carrier gas and the solutes to be separated are derived and their implication is discussed.
SYMBOLS subscript indicating mobile phase subscript indicating minimum subscript indicating optimum subscript indicating column outlet subscript indicating solute diffusivity of a solute in a mobile phase, length2/time self-diffusivity of a mobile phase, length2/time reduced diffusivity of a solute, 1 column internal diameter, length collision diameter of the mobile phase molecules, length collision diameter of the solute molecules, length plate height, length capacity ratio, 1 column length, length molecular weight, mass/mole column efficiency, 1 pressure, (mass/length)/time2 column inlet pressure, (mass/length)/time2 gas constant (8.31431 (J/K)/mol) temperature, temperature hold-up time, time carrier gas velocity, length/time average carrier gas velocity, lengthhime effective peak width, time mean square speed of carrier gas molecules, length/ time carrier gas viscosity, (mass/length)/time density, mass/length3 variance of a peak, time2
Equation 18for H is valid for thin film columns only. Under condition eq 1, eqs 17 become
Combining the last two of these relations, one has
(20) At the optimum carrier gas velocity, both additive terms on the right-hand side of the relation for H, eq 18, are equal to each other, i.e.
-mout - G(k)vout,optd,2 Vout,opt
96(1 + k)2Dout
which allows for the following transformations of H in eq 18
H=-- 2 0 0 , Vout,opt
2
Vout,opt
Yout
+
G(k)voutd,2 - G(k)vou,optd, ~ou,opt 96(1 + k)2Dout 96(1 + k)2Dout Vout
+
Substitution of eqs 20 and 21 in the expression for 72, eq 19, yields
G(k) 96(1 + k)2Dout( l +
vout,opt
2569L = 9vouQoutd,2
APPENDIX The following relations can be found in refs 7 and 9 and other text books on capillary GC
RECEIVED for review January 25, 1993. Accepted June 23, 1993.
