Anal. Chem. 2001, 73, 684-685
Technical Notes
Correlation of Characteristic Thermal Constant and Elution Temperature in GC Leonid M. Blumberg*,† and Matthew S. Klee‡
Fast GC Consulting, P.O. Box 585, Hockessin, Delaware 19707, and Agilent Technologies, 2850 Centerville Road, Wilmington, Delaware 19808
In a temperature-programmed analysis, the solutes that elute at higher temperatures have generally larger characterisitic thermal constants, θchar. The change approximately matches the temperature-related increase in a carrier gas viscosity. Accounting for the effect allows reduction in the uncertainty of prediction of θchar by a factor or f ∼2 and, in a constant-pressure mode, description of a linear heating ramp by the same dimensionless rate for all solutes. In a recent publication1 we introduced characteristic thermal constant, θchar, and characteristic elution temperature, Tchar, of a solute migrating in a column. The former is the inverse of the rate of a relative reduction in retention factor, k, per unit of temperature at k ) 1 while the latter is elution temperature at which the solute elutes with k ) 1. Using θchar and void time, tM, as normalization factors, we defined dimensionless heating rate, r ) RTtM/θchar. Using r instead of actual heating rate (measured in units of temperature/time), RT, allows one to substantially simplify a search for the optimum heating rate, RT,opt. Indeed, an optimum dimensionless heating rate, ropt, can be the same for a broad class of mutually translatable2,3 methods. Once ropt is known, RT,opt for a particular method with its particular column dimensions, carrier gas type and flow rate, etc. can be found as
RT,opt ) roptθchar/tM
(1)
In practice, a given analyte mixture can consist of a large number of components, each having its own θchar. As a result, a single “universal” RT,opt does not exist. Nevertheless, it is useful to have a heating rate that can be considered optimal in the sense of providing a generally good compromise between RT,opt values for all components in a mixture. From that point of view, the smaller the spread of RT,opt values for the solutes in the mixture the better the overall heating rate optimizations can be achieved. Equation 1 shows that, for a fixed ropt and tM, the spread in RT,opt is
Figure 1. Characteristic thermal constants, θchar, of light hydrocarbons,5 n-alkanes,1 pesticides,1 and miscellaneous solutes6 versus their characteristic elution temperature, Tchar. A line represents the least-squares fit of the function eq 4.
proportional to that in θchar. However, tM may not remain fixed during a temperature-programmed analysis. In a constant-pressure mode4 of a temperature-programmed analysis, for examples, tM increases with temperature. This, depending on a temperature trend in θchar, might further increase or reduce the spread in RT,opt. In this report, we show that, in a constant-pressure mode, the combined effect of the temperature dependences in tM and θchar is a reduction in the spread of RT,opt. Particularly, for over 180 solutes (12 n-alkanes and 12 pesticides from our experiments and over 150 solutes from the literature5,6) examined a our previous study1, the correlation between tM and θchar reduces a relative standard deviation of distribution of RT,opt by a factor of ∼2. It is shown in Figure 1 that θchar in the above-mentioned 180 solutes are distributed between ∼23 and ∼46 °C. The distribution has relative standard deviation, RSTD ) 15.6%. One can observe, however, that θchar has a clear trend. It tends to increase with increase in Tchar. In search for a suitable approximation of θchar by a function monotonically increasing with temperature, T, we notice that tM is proportional3 to the carrier gas viscosity. This
†
Fast GC Consulting. Agilent Technologies. (1) Blumberg, L. M.; Klee, M. S. Anal. Chem. 2000, 72, 4080-4089. (2) Snyder, W. D.; Blumberg, L. M. Proceedings of 14th International. Symposium on Capillary Chromatography; ISCC92: Baltimore, 1992; pp 28-38. (3) Blumberg, L. M.; Klee, M. S. Anal. Chem. 1998, 70, 3828-3839. ‡
684 Analytical Chemistry, Vol. 73, No. 3, February 1, 2001
(4) Blumberg, L. M.; Wilson, W. H.; Klee, M. S. J. Chromatogr., A 1999, 842, 15-28. (5) Laub, R. J.; Purnell, J. H. J. High Resolut. Chromatogr. 1988, 11, 649-660. (6) Snijders, H.; Janssen, H.-G.; Cramers, C. A. J. Chromatogr. 1995, 718, 339355. 10.1021/ac001035k CCC: $20.00
© 2001 American Chemical Society Published on Web 12/30/2000
allows one to approximate tM in a constant-pressure analysis with a typical carrier gases (helium, hydrogen, nitrogen, argon) as7,8
tM ) (T/Tref) 0.7tM,ref
(2)
where tM,ref is a value of tM at some a priori selected reference temperature, Tref (Tref can be initial temperature in a temperature program analysis, a standard temperature of 273.15 K, etc.). Combining eq 1 with eq 2, one has
RT,opt ) (T/Tref)- 0.7roptθchar/tM,ref
(3)
In view of this relation, it is convenient to approximate the temperature trend in θchar by a least-squares fitting of the function
θchar,fit ) (Tchar/Tref)0.7θchar,ref
(4)
where θchar,ref is a value of θchar,fit at Tref. In eq 4, both Tchar and Tref are expressed in degree kelvin while θchar,ref and, hence, θchar,fit are in degree celsius. For the solutes from our previous study,1 the standard deviation of relative departures, (θchar - θchar,fit)/ θchar,fit, of θchar from θchar,fit (both for the same Tchar) is less than 7.2%, which is more than a factor of 2 lower compared to the previously mentioned RSTD in relation to the average θchar. Equation 1 can be modified to reduce the uncertainty in prediction of RT,opt by eliminating the correlated dependencies in tM and θchar. Replacing θchar in eq 3 by θchar,fit from eq 4, one has (7) Touloukian, Y. S.; Saxena, S. C.; Hestermans, P. Viscosity; IFI/Plenum: New York, 1975. (8) Hinshaw, J. V.; Ettre, L. S. J. High Resolut. Chromatogr. 1997, 20, 471481.
RT,opt ) roptθchar,ref/tM,ref
(5)
One can conclude that accounting for the trend in θchar with the temperature increase in a temperature-programmed analysis has at least two benefits. First, it reduces the spread of RT,opt values for the solute in a mixture. For example, RSTD of the distribution of RT,opt values for all solutes from our previous study1 relative to RT,opt in eq 5 is ∼7.2% compared to ∼15.6% for the distribution of θchar for the same solutes. Second, in a constant-pressure analysis, because the trend of the changes in θchar compensates the temperature dependence in tM, all parameters in eq 5 are constants. That means that a single-ramp temperature program in a constant-pressure mode is a suitable implementation of an optimum temperature program with a fixed optimum dimensionless heating rate, ropt. Equation 5 also suggests a simple way of finding of RT,opt for a single-ramp temperature program with a given ropt. Suppose that initial temperature, Tinit, of a single-ramp temperature program is chosen as a reference temperature, Tref, in eq 5. It is likely that a statistical average, θchar,init, of θchar values for the solutes having characteristic elution temperatures close to Tinit would be close to the θchar,init that should replace θchar,ref in eq 5. The latter becomes
RT,opt ) roptθchar,init/tM,init
(6)
suggesting that an optimum dimensionless heating rate, ropt, in a constant-pressure analysis can be implemented as a single-ramp temperature program with the heating rate calculated from eq 5 where tM,init is void time at initial temperature, Tinit, and θchar,init is statistical average of characteristic thermal constants for the solutes having characteristic elution temperatures, Tchar, that are close to Tinit. Received for review August 30, 2000. Accepted November 17, 2000. AC001035K
Analytical Chemistry, Vol. 73, No. 3, February 1, 2001
685
