Anal. Chem. 2000, 72, 4080-4089
Characteristic Thermal Constant and Dimensionless Heating Rate. The Links to Optimum Heating Rate in GC Leonid M. Blumberg*,† and Matthew S. Klee‡
Fast GC Consulting, P.O. Box 585, Hockessin, Delaware 19707, and Agilent Technologies, 2850 Centerville Rd., Wilmington, Delaware 19808
An initial step in the quest of deriving a generalized approach to optimization of a temperature program in gas chromatography is presented. Central to this is the introduction of a dimensionless heating rate, r. As a first step to defining r, a characteristic thermal constant, θchar, defined as θchar ) -dT/dk at k ) 1, where T and k are, respectively, column temperature and solute retention factor, is introduced and evaluated for our own experimental data and for thermodynamic data from the literature. It was determined that, for silicone stationary phases with a phase ratio of 250, θchar ranged from about 23 °C for low molecular weight hydrocarbons such as dimethylpropane to about 45 °C for high molecular weight pesticides such as mirex. It was also found that, for a particular solute and a stationary phase type, a 2 orders of magnitude increase in the film thickness caused only about a 2-fold increase in the characteristic thermal constant. Using θchar as a fundamental temperature unit in GC and void time as a fundamental time unit, a dimensionless heating rate is introduced and its potential utility for the evaluation of the separation-speed tradeoffs in a temperature-programmed GC is demonstrated. This study is driven by the need to find an optimum heating rate in a temperature-programmed GC (gas chromatography) that represents the best tradeoff between quality of separation and speed of analysis. Some aspects of this issue were previously investigated by several workers1-5 and are addressed in some commercial software packages.6,7 However, no specific recommendations for the optimum tradeoff were developed. Here, revisiting the problem, we do not yet present the complete †
Fast GC Consulting. Agilent Technologies. * Corresponding author. (1) Giddings, J. C. In Gas Chromatography, June 13-16, 1961; Brenner, N., Callen, J. E., Weiss, M. D. Eds.; Academic Press: New York, 1962; p 57. (2) Giddings, J. C. J. Chem. Educ. 1962, 39, 569. (3) Harris, W. E.; Habgood, H. W. Programmed Temperature Gas Chromatography; John Wiley & Sons: New York, 1966. (4) Littlewood, A. B. Gas Chromatography. Principles, Techniques, and Applications; Academic Press: New York, 1970. (5) Goedert, M.; Guiochon, G. Anal. Chem. 1970, 42, 962. (6) Resteck Corp. Pro EzGC; Resteck Corp.: Bellfonte, PA, 1995. (7) LC Resources. DryLab 2000; LC Resources: Walnut Creak, CA, 2000. ‡
4080 Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
universal process of finding the optimum heating rate, but rather, we make a step in that direction by removing one of the major roadblockssthe fact that a single optimization criterion can lead to an intractable range (from fractions to thousands of degrees Celsius per minute) of optimum heating rates. The range of the useful heating rates, RT, is large due to the dependence of RT on the diversity of method parameters (column dimensions, carrier gas type, stationary phase type, film thickness, etc.) and thermodynamic factors involved in a given separation. It is desirable, therefore, to find a normalization for RT so that the effects of method parameters and thermodynamic factors can be cancelled out, yielding a dimensionless heating rate, r, that is independent of particular method parameters, and hence, has a unique optimum value for a particular optimization criterion. Once an optimum r is found, the optimal RT for a specific set of the method parameters can be found via the transformation known as GC method translation.8-10 As RT is measured in units of temperature/time, one needs to find a properly justified basic time unit and a basic temperature unit for the normalization of RT in order to obtain the dimensionless r. The choice for a basic time unit in chromatography has been discussed in the literature. Giddings1 evaluated the optimum heating rate in terms of the temperature per void time, tM. More recently, in developing GC method translation, we have shown9,10 that tM can be viewed as a basic time unit in chromatography in a very broad sense. Normalization by tM allows one to factor out many method parameters in a scalable (without distortion of a peak elution pattern) way and to reduce the range of uncertainty of the normalized heating rate by several orders of magnitude. To complete the transformation of the heating rate into a dimensionless quantity, r, one needs to find an appropriate basic temperature unit. We refer to this basic temperature unit as the characteristic thermal constant, θchar. Defining the characteristic thermal constant and analyzing the properties of the dimensionless heating rate are the subjects of this study. Our characteristic thermal constant has its roots in a familiar concept1 that can be called a binary thermal constant, θbinary, which (8) Snyder, W. D.; Blumberg, L. M. In Proceedings of the 14th International Symposium on Capillary Chromatography, Baltimore, MD, May 25-29, 1992; ISCC92, Baltimore, 1992; Sandra, P., Ed.; p 28. (9) Blumberg, L. M.; Klee, M. S. Analytica Conference 98, Abstracts, Neue Messe, Munich, Germany, April 21-24, 1998; p 164. (10) Blumberg, L. M.; Klee, M. S. Anal. Chem. 1998, 70, 3828. 10.1021/ac000378f CCC: $19.00
© 2000 American Chemical Society Published on Web 07/28/2000
is a change in the column temperature that causes a change of a factor of 2 in the retention factors of the solutes. Giddings, suggesting1 that the optimum heating rate should directly relate to θbinary, estimated2 that θbinary ranges somewhere between 20 and 40 °C. Similar results follow from the data reported by other workers.4,5 Unfortunately, although the concept of what we call a binary thermal constant, θbinary, is somewhat accepted as a rule of thumb in GC practice, no explicit definition nor particular conditions for its measurement have been proposed to substantiate or test the claim. We found (see below) that, depending on those conditions, the value of θbinary for a particular combination of an analyte and a stationary phase can vary within nearly an order of magnitude. This greatly diminishes the usefulness of θbinary as the basis for constructing a dimensionless heating rate with the unique optimum value. Here, we propose a definition of θchar suitable for any types liquid or solidsof a stationary phase in a linear (nonoverloaded) GC separation. The definition yields a unique value of θchar for each particular combination of analyte and stationary phase. Treating θchar as a fundamental temperature unit in GC together with treating the void time, tM, as a fundamental time unit, we define a characteristic heating rate, RT,char ) θchar/tM, and, using it as a normalization factor, define a dimensionless heating rate as r ) RT/RT,char. Further, we demonstrate that at least one broadly defined class of optimization scenarios leads to a unique optimum r for any specific optimization criterion within that scenario. We also evaluated θchar for about 180 solutes in combination with several silicone stationary phases on the basis of our own experimental data and the data from the literature.11,12 From these evaluations we found that, adjusted for the single reference phase ratio of 250, θchar was somewhere between approximately 23 °C for low-boiling hydrocarbons such as dimethylpropane and about 45 °C for the high molecular weight pesticides such as mirex. We also demonstrated that, for a particular solute and a stationary phase type, an increase of 2 orders of magnitude in the film thickness causes only about a 2-fold increase in the characteristic thermal constant. This preliminary data suggests that, depending on the type of solute and stationary phase (as well as its thickness, in case of a liquid phase), θchar can vary within at least a 4-fold range. This means that (assuming that all factors, other than those affecting the values of θchar, are the same) using θchar for normalization of heating rate reduces the uncertainty of such a normalized optimum heating rate by, at least, a factor of 4. THEORY Thermodynamics of Separation (a Brief Review). This study provides no new model for the thermodynamics of separation, but proposes several new parameters to describe the effect of the column temperature on separation in GC. While the definitions of the new parameters do not rely on a particular thermodynamic model, their relation to known models is used for illustration and interpretation purposes. In view of that, we find it helpful to provide the following brief review. (11) Laub, R. J.; Purnell, J. H. J. High Resolut. Chromatogr. 1988, 11, 649. (12) Snijders, H.; Janssen, H.-G.; Cramers, C. A. J. Chromatogr., A 1995, 718, 339.
In a linear (nonoverloaded) partition GC, dependence of the retention factor, k, on column temperature, T, can be approximately described as1,3,13-15
k)
∆G 1 exp β RT
∆G ) ∆H - ∆ST
(1)
where R ) 8.3144 J/K/mol is the universal gas constant, ∆G, ∆H, and ∆S are, respectively, the Gibbs free energy, enthalpy, and entropy of a solute evaporation from a stationary phase, and β is the phase ratio. For a capillary column with a circular cross section
β ) dc/(4df)
(2)
where dc and df are, respectively, column diameter and stationaryphase film thickness. To reduce the number of independent parameters in eq 1, one can rewrite it in a well-known compact form, ln k ) a + b/T, or, introducing notations a ) - xchar and b ) Th (see explanation below)
ln k ) - xchar + Th/T
(3)
where
xchar )
∆S ∆H + lnβ, Th ) R R
(4)
In these equations, coefficient Th is measured in units of temperature and can be interpreted as a thermal equivalent of enthalpy (of a solute evaporation from the liquid stationary phase into a carrier gas). A meaning for and the choice of the symbol for the dimensionless coefficient xchar is described later. Elution Temperature and Thermal Constant of a Solute. Conventional thermodynamic parameters, such as xchar and Th in eq 3, allow one to express a solute retention factor, k, via the interaction of the solute with the stationary phase. The question for this study, however, is not how a retention of a solute relates to its interaction with stationary phases, but how to express a retention factor, k, via the chromatographically meaningful parameters, i.e., the ones that directly reflect the performance of a GC separation. So far, parameters xchar and Th are not very helpful in that regard. For example, according to Table 1, xchar for n-alkanes from C10 to C32 in an HP-5 column ranges from 14.7 to 19.3, while Th ranges from 5655 to 11749 K. What of importance do these values tell us about the migration and elution of these solutes in a typical temperature-programmed GC? Notice, first of all, that, being a linear or nearly linear function of 1/T (Figure 1 and Figure 2), lnk is not a linear function of T (Figure 3). In a search for the chromatographically meaningful parameters, we explore a linear approximation of ln k vs T in the vicinity of some a priori known temperature Te. In the case of temperature-programmed GC, we can view Te as a solute elution (13) Martire, D. E.; Boehm, R. E. J. Phys. Chem. 1987, 91, 2433. (14) Martire, D. E. J. Liq. Chromatogr. 1987, 10, 1569. (15) Giddings, J. C. Unified Separation Science; John Wiley & Sons: New York, 1991.
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Table 1. Thermodynamic Parameters of n-Alkanes in Figure 1b parameter
unit
dimensionless characteristic elution temperature, xchar thermal equivalent of enthalpy, Th boiling temperature (from ref 18) characteristic elution temperature, Tchar, eq 22 characteristic thermal constant, θchar, eq 21
K K K °C
C10
C12
C14
C16
C18
C20
C22
C24
C26
C28
C30
C32
14.7
15.0
15.3
15.6
16.0
16.4
16.8
17.2
17.7
18.2
18.8
19.3
5655 447 384 26.1
6209 489 414 27.6
6763 526 442 28.9
7317 560 469 30.0
7871 589 493 30.8
8425 616 515 31.5
8979 642 535 31.9
9533 664 553 32.1
10 087 685 569 32.1
10 641 705 584 32.0
11 195 723 596 31.8
11 749 740 607 31.4
Figure 2. Retention factors, k, of 12 pesticides with HP-5 stationary phase at β ) 250. Shown in order of increase in k are the lines for: unknown, aniline, unknown, dichlorvos, vernolate, lindane, chlorpyriphos-methyl, malathion, dieldrin, mirex, deltamethrin, temephos.
thermal constant of a solute (with respect to a particular stationary phase) as
( | )
θ)Figure 1. Retention factors, k, vs inverse column temperature, 1/T, for even-number n-alkanes from C10 to C32 with HP-5 stationary phase. Shown in (a) are the experimental data points (the symbols are recycled after every three solutes), and the idealized straight lines at the actual phase ratio β ) 320. The idealized straight (b)-lines result from equal upward shift of all (a)-lines by ln(320/250) to adjust for β ) 250 selected in this study as the reference β-value.
temperature. Using Taylor expansion, one can express a linear approximation of lnk vs T in the vicinity of Te as
lnk ) lnke +
|
d(lnk) dT
T)Te
(T - Te)
(5)
where ke is the retention factor at T ) Te, i.e.
ke ) k|T)Te
(6)
Since Te is a solute elution temperature in a temperatureprogrammed GC, ke is the elution retention factor of the same solute in a temperature-programmed GC. Back to eq 5, we have a choice of either dealing with the rate, d(lnk)/dT, of the change in lnk or, as it is frequently done in other similar cases, we can choose to deal with a quantity that is the inverse of that rate. We choose the latter approach, and define a 4082
Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
d(lnk) dT
-1
( | )
)-
T)Te
1 dk k dT
-1
T)Te
(7)
The minus sign in this definition is chosen to make θ a positive quantity in view of the decline in lnk with the increase in T. Equation 5 becomes
lnk ) lnke - ∆T/θ, ∆T ) T - Te
(8)
An approximation in eq 8 (as well as the one in eq 5) describes a tangent line at T ) Te, Figure 3, to the function of lnk vs T. In this approximation, θ is a change in the column temperature that causes an e-fold change (e ≈ 2.718 is the base of the natural log) in k. Its inverse, 1/θ, is rate of change in k. Thus, for example, θ ) 30 °C corresponds to 1/θ ≈ 0.033/°C indicating that k declines by 3.3% with each 1 °C increase in the column temperature.16 Sometimes, it is more convenient to assume that ke rather than Te is a known quantity. In that case, Te can be expressed as a function of ke, i.e.
Te ) T|k)ke
(9)
(16) A quantity, known from Giddings,1 that, in the Introduction, we called a binary thermal constant, θbinary, can be found from θ as θbinary ) θ ln 2 ≈ 0.693θ. It is important to point out that 1/θbinary is proportional, but not equal to the rate of change in k vs T. This is one of several reasons for using θ rather than θbinary.
One of the parameters, ke or Te, in this equation (as well as in the approximation, eq 8), is an arbitrary quantity, while the other as well as the quantity θ depends on relations between k and T. In a particular case of a thermodynamic model, described in eq 3, the quantity θ as well as one of the quantities ke or Te, can be deduced from parameters xchar and Th. Thus, when ke is an arbitrary quantity, θ and Te can be found as
θ)
Te )
Th
(12)
(xchar + lnke)2
Th ) (xchar + lnke)θ xchar + lnke
(13)
If, on the other hand, Te is an arbitrary quantity, θ and ke can be found as
( (
θ)-
)| )
Th d -xchar + dT T
T)Te
-1
)
Te2 Th
lnke ) Th/Te - xchar ) Te/θ - xchar
(14) (15)
Sometimes, even when it is desirable to deal with eq 3, it can still be easier to experimentally measure quantities ke, Te, and θ rather than the parameters xchar and Th of eq 3. We shall notice that eqs 12 and 13 allow expression of the relationship between the three temperatures, Th, Te, and θ as
Te2 ) Thθ
Figure 3. Slope variation in lnk vs T curves within the range of the experimental k-values (a), and extrapolated for all k-values within the same temperature range (b) and (c). Conditions are the same as in Figure 1 (b). Dashed are the tangent lines to the lowest and the highest boiling solutes in the mix at lnk ) 0 (a and b), and at the points of the sharpest and the mildest slopes (c). Notice a minor difference between the slopes shown in (a) and in (b), and about 7-fold difference between the slopes shown in (c).
allowing one to express eq 7 as
( | )
θ)-
1 dk k dT
-1
|
dT dk
) -k
k)ke
k)ke
(10)
Because of the one-to-one relation between Te and ke, eqs 7 and 10 are equivalent definitions of θ. Quantities ke, Te, and θ are the chromatographically meaningful parameters that we were looking for to express the relations between k and T. These parameters can be used not only to express a linear approximation, eq 8, for lnk vs T but also to replace parameters xchar and Th in eq 3. The latter can be transformed to become
lnk ) lnke +
1 ∆T , 1 + ∆T/Te θ
(
)
∆T ) T - Te
(11)
(16)
This allows one to find Th from Te and θ. The quantity xchar can be found from eqs 16 and 15. One has
Th ) Te2/θ,
xchar ) Te/θ - lnke
(17)
So far, we have described a thermal constant, θ, corresponding to an arbitrary elution temperature, Te, and an arbitrary elution retention factor, ke. The definitions of these quantities do not depend on a particular thermodynamic model for retention of a solute. For a particular model described in eq 3 via parameters xchar and Th, we derived expressions to find θ, Te, and ke from xchar and Th, and vice versa. Characteristic Elution Temperature and Thermal Constant. A range of the slopes of k vs T functions, and hence, of the values of the thermal constant, θ, can be very broad, Figure 3, if conditions for the measurement of θ are not specified. To identify the appropriate measurement conditions, we notice that, during a temperature ramp in a typical-temperature-programmed GC, all solutes, while eluting at different elution temperatures, Te, elute with nearly the same elution retention factors, ke.1 Evaluating, therefore, θ, for all solutes at the same ke, provides a uniform treatment of all the solutes at the time of their elution. When ke is unity, i.e.
ke ) 1 Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
(18) 4083
many expressions involving θ, including eqs 13 and 12 for calculation of θ for the thermodynamic model in eq 3, become especially simple. We will refer to the thermal constant and elution temperature at ke ) 1, as to the characteristic thermal constant, θchar, and characteristic elution temperature, Tchar, of a solute (in respect to a given stationary phase), i.e.
θchar ) θ|ke)1,
Tchar ) Te|ke)1
(19)
Combination of the definitions in eqs 10 and 19 leads to the following more direct definition of θchar that is equivalent to the definition in eq 19
dk (dT | )
-1
θchar ) -
)-
k)1
|
dT dk k)1
(20)
Similarly, a dimensionless tangent line to this dimensionless lnk vs x function can be found from eq 27 as
lnk ) xchar - x
Notice that xchar in the dimensionless expressions, eqs 29 and 30, is the same as that in eq 3. Comparison of eq 24, with the general definition, eq 28, of a dimensionless temperature, indicates that xchar is a dimensionless characteristic elution temperature. For example, according to Table 1, C10 has xchar ) 14.7, meaning that the characteristic elution temperature of C10 is 14.7 θchar units above the absolute zero. Indeed, (due to θchar ) 26.1 °C, Table 1) Tchar ) xcharθchar ) 384 K. Characteristic and Dimensionless Heating Rates. A ramping column temperature, T, in a temperature-programmed GC analysis can be expressed as
T ) Tinit + RT t It should be emphasized that, similarly to the case of eq 10, the definition in eq 20 does not assume any particular thermodynamic model and is valid even if k vs T is other than, say, that shown in eqs 1 and 3. In a particular case of the thermodynamics governed by eq 3, θchar and Tchar can be found from eqs 12 and 13. At ke ) 1, one has
θchar ) Th/x2char
(21)
Tchar ) Th/xchar
(22)
(30)
(31)
where Tinit is the initial temperature preceding the ramp, RT is the heating rate, and t is time. Using the characteristic thermal constant, θchar, as a basic temperature unit and void time, tM, as a basic time unit,10 we introduce a characteristic heating rate
RT,char ) θchar/tM
(32)
Replacing T with dimensionless temperature, x, eq 28, and t with dimensionless time10
τ ) t/tM
(33)
These relations allow the following to be written
Tchar ) xcharθchar,
Th ) xcharTchar ) x2charθchar
(23)
xchar ) Tchar/θchar
(24)
T2char ) Thθchar
(25)
one can write eq 31 in a dimensionless form
1 ∆T , 1 + ∆T/Tchar θ
(
)
∆T ) T - Tchar
lnk ) (Tchar - T)/θchar
(26) (27)
4084
(35)
xinit ) Tinit/θchar
(36)
lnk )
x2char - xchar xinit + rτ
(37)
(28)
and rewrite the temperature-related expressions in a dimensionless form. Thus, for example, taking into account eqs 23, one can rewrite eqs 3 and 26 in a dimensionless form as
lnk )
r ) RT/RT,char
are dimensionless heating rate and dimensionless initial temperature, respectively. Substitution of eq 34 in eq 29 yields, for the retention factor in a single-ramp temperature-programmed GC governed by eq 3
Dimensionless Temperature. Using θchar as a normalization factor, one can introduce a dimensionless temperature
x ) T/θchar
(34)
where
Finally, eq 11 and its tangent, eq 8, at k ) 1 become, respectively
lnk )
x ) xinit + rτ
x2char ∆x - xchar ) , ∆x ) x - xchar (29) x 1 + ∆x/xchar
Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
Similarly, for the first approximation, eq 30, of the above relation, one has
lnk ) ∆X - rτ
(38)
∆X ) xchar - xinit ) (Tchar - Tinit)/θchar
(39)
where
is a dimensionless difference between the solute’s characteristic elution temperature and the initial temperature. Being a function of a solute characteristic elution temperature and of initial temperature in a temperature program, ∆X can be interpreted as the solute’s dimensionless temperature margin in that temperature program. It follows from eq 38 at τ ) 0 that ∆X can also be expressed as
∆X ) lnkinit
(40)
where kinit is the retention factor of a solute at the initial temperature, Tinit.
Figure 4. Characteristic thermal constants of n-alkanes in Figure 1 (b).
EXPERIMENTAL SECTION A mixture of 13 even numbered n-alkanes from C10 to C32, and a specially made 12-component mixture of pesticides were analyzed using an HP 6890 GC system equipped with EPC (electronic pneumatic control), an HP 7673 autosampler, 30 m × 320 µm HP-5 columns, and an FID (flame ionization detector). All tests were isothermal at a helium flow rate of 2.6 mL/min. The n-alkanes were tested with a 0.25-µm film column at 50, 100, 150, 200, 250, and 300 °C. The pesticide were tested with a 0.32µm film column at 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, and 325 °C. RESULTS Experimental retention factors, k, vs inverse column temperature, 1/T, are shown in Figure 1 and Figure 2. To construct the idealized-straight-line dependencies, Figure 1a, for lnk vs 1/T for each carbon number, the slope for each n-alkane was calculated from two experimental data points with the lowest positive17 lnk. Next, the dependence of the slope on the carbon number was smoothed by a linear regression. Finally, the intercept for each n-alkane was found from the experimental data and the respective smoothed slopes. This treatment provided an accurate and selfconsistent line fitting in the vicinity of k ) 1sthe region of the greatest interest in this studyswhile causing noticeable departures of the idealized lines from the experimental data for very low k values. However, the region of a very low k is outside of the interest for this study, and the accuracy of the measurements of the very small k was low anyway. All thermodynamic data in this study are compared with each other with respect to the same phase ratio, β ) 250. Because the column used for the n-alkane tests had β ) 320, an adjustment to the straight lines in Figure 1a was made by adding ln(320/250) ≈ 0.247 to each intercept as required by eq 4 for the quantity xchar in eq 3. The adjusted straight lines are shown in Figure 1b. The parameters representing the lines in Figure 1b are shown in Table 1. The same lines as a function of T rather than 1/T are shown in Figure 3. DISCUSSION Evaluation of Characteristic Thermal Constant. Using eq 21, we evaluated characteristic thermal constants, θchar, from xchar and Th parameters in eq 3. To that end, we used a total of about (17) Using only the positive lnk values (i.e. k > 1) allowed us to compute the slopes in the vicinity of k ) 1 while reducing the contribution of the void time measurement errors that are the most pronounced at the very low k.
Figure 5. Characteristic thermal constants at β ) 250 of 147 light (C5 to C10) hydrocarbons (aliphatic, alicyclic, and aromatic) with OV101 stationary phase. Source: ref. [11], Table 2 for xchar and Th in a distribution constant (i.e. at β ) 1).
180 (xchar, Th) data pairs from our experiments and from the literature.11,12 In all cases, before applying eq 21, we adjusted actual values of xchar to their corresponding values at β ) 250 using the formula xchar ) xchar, actual - ln(βactual/250), as required by eq 4. The adjustment provided us a uniform comparison of all θchar values at the same phase ratio, β. A separate discussion of the effect of β on θchar is provided below. For some solutes, we also evaluated characteristic elution temperatures, Tchar, and other related parameters (see below). A complete set of parameters, xchar, Th, θchar, Tchar, and boiling temperatures,18 for our experimental n-alkanes can be found in Table 1. The graphs, Figure 4 and Figure 5, of θchar values for our experimental n-alkanes and for the low-boiling hydrocarbons from the literature11 show a good continuity of θchar values for these solutes over a wide range of the boiling points and the hydrocarbon numbers. More θchar values are shown in Figure 6. One can find from these graphs that the θchar values found in this study fall within a 2-fold range, from about 23 °C for the lowboiling hydrocarbons of ref 11 to about 45 °C for the pesticides in our test mixture. Other statistics are shown in Table 2. Characteristic Thermal Constant and Film Thickness in Partition GC. On the basis of the limited number of about 180 solutes we evaluated so far, we found that characteristic thermal constants, θchar, measured at the same phase ratio, β ) 250, fall (18) Weast, R. C.; Astle, M. J.; Beyer, W. H. CRC Handbook of Chemistry and Physics, 69th ed.; CRC Press: Boca Raton, FL, 1988.
Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
4085
Figure 6. Characteristic thermal constants at β ) 250 of 12 pesticides with HP-5 stationary phase in Figure 2, and of 11component test mix with HP-1 stationary phase. Source for the second group: ref. [12], Table 1 for Exp(-xchar) and Th at β ) 470. Table 2. Average Characteristic Thermal Constants, θChar, and Their Percent Standard Deviations, %STD, for Several Mixtures mix
phase
source
θchar, °C
%STD
C10-C32 hydrocarbons pesticides miscellaneous
HP-5 OV-101 HP-5 HP-1
this study [11] this study [12]
31 25 35 36
8 2.4 13.5 11
within a 2-fold range. (For a capillary column with a circular cross section, β ) 250 corresponds, eq 2, to dc/df ) 1000.) Here we evaluate the effect of β on θchar. As follows from eqs 21 and 4, θchar is a relatively weak function
θchar ) Th/(∆S/R + lnβ)2
(41)
of β, where Th and ∆S are, respectively, thermal equivalent of enthalpy, eqs 4, and entropy of solute evaporation from the stationary phase, and R is the universal gas constant. As shown in Figure 7, an increase of 2 orders of magnitude in the film thickness results in an increase of about a factor of 2 in θchar for some of our experimental n-alkanes. All combinations of the solutes and the stationary phases explored here, Figure 5 and Figure 6, show similar behavior. One can conclude, on the basis of limited data, evaluated so far, that, since typical β values rarely exceed a two-decade range from 10 to 1000
the characteristic thermal constant, θchar, of a solute with a particular stationary phase type (of all practical film thickness values) falls within a 2-fold range, with the highest θchar corresponding to the thickest film Dimensionless Heating Rate. Depending on column dimensions, the type of the carrier gas, the stationary phase, the mixture to be analyzed, etc., a heating rate, RT, in a temperature programmed GC analysis can range19 from fractions of degrees Celsius per minute to many hundreds or even thousands of degrees Celsius per minute. An optimum heating rate, even if based on a very specific criterion or the separation-speed tradeoff, 4086 Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
Figure 7. Characteristic thermal constant vs phase ratio, β, for some n-alkanes in Figure 1(b).
can fall within a similarly wide range. This complicates the process of determining an optimum heating rate. Fortunately, there are ways to factor out the uncertainties in RT. Previously,9,10 we have shown that RT closely tracks the quantity 1/tM, where tM is void time, and that expressing the time scale in RT via tM allows us to substantially reduce the uncertainty in such a normalized heating rate. Following the same logic and expressing the temperature scale in units of a characteristic thermal constant, θchar, we introduced here a characteristic heating rate, RT,char ) θchar/tM, eq 32, and, using RT,char as a normalization factor, defined dimensionless heating rate, r ) RT/RT,char, eq 35. According to eq 32, characteristic heating rate, RT,char, is the rate of change in the column temperature equal to the change by one characteristic thermal constant per void time. A dimensionless heating rate, r, expresses the heating rate in the units of characteristic thermal constant per void time. Thus, for example, r ) 1 means that the temperature changes by θchar for each tM increment in the analysis time. The range of possible RT,char values is about as wide as the range of appropriate values for the actual heating rate, RT. Example 1. Assume θchar ) 30 °C and the helium carrier is at the SOF (speed-optimized flow rate).19,20 In a 100 m × 530 µm column, tM ≈ 6 min, resulting in RT,char ) 30 °C/(6 min) ) 5 °C/ min. In another extreme of a 1m × 50 µm column, tM ≈ 0.015 min, and RT,char ) 2000 °C/min. The ratio between the two extremes is 400. The range of possible values of θchar, together with the effect of film thickness, the carrier gas type, etc., can boost this factor by another order of magnitude or so. As the characteristic heating rates, RT,char, track the practically useful values of actual heating rates, RT, the range of their ratio, r, is substantially reduced. This happens because the void time component in RT,char cancels out1,9,10 many nonthermodynamic factors (column dimensions, carrier gas type and its flow rate, etc.) in RT, preventing their effect on r. Further, the characteristic thermal constant component in RT,char cancels out thermodynamic properties of the solutes from r. As a result, in a likely optimization scenario, evaluated below, each particular optimization criterion leads to a nearly unique optimum r. (19) Blumberg, L. M.; Klee, M. S. in Proceedings of the 20th International Symposium on Capillary Chromatography (CD ROM); Palazzo dei Congressi, Riva del Garda, Italy, May 26-29, 1998; I. O. P. M. S.; Sandra, P., Rackstraw, A. J., Ed.; Kortrijk, Belgium, 1998. (20) Blumberg, L. M. J. High Resolut. Chromatogr. 1999, 22, 403.
Table 3. Characteristic Elution Temperatures, Tchar, Eq 22, Thermal Constants, θChar, Eq 21, Dimensionless Temperature Margins, ∆X ) (Tchar - Tinit)/θchar, and Initial Retention Factors, kinit ) e∆x at Initial Temperature, Tinit, in a Single-Ramp Temperature Program, for the Solutes in Figure 8 Tinit 0 °C
-50 °C
-100 °C
solute [source]
Tchar, °C
θchar, °C
∆X
kinit
∆X
kinit
∆X
kinit
dimethylpropane [11] n-C5 [11] n-C10 [this study] n-C12 [this study] mirex [this study] n-C32 [this study]
-22.9 -4.8 109 140 317 336
22.8 23.3 25.8 27.5 45.2 31.6
-1.0 -0.2 4.2 5.1 7.0 10.6
0.37 0.82 67 164 103 4 × 104
1.2 1.9 6.2 6.9 8.1 12.2
3.3 6.7 493 992 3 × 103 2 × 105
3.4 4.1 8.1 8.7 9.2 13.8
30 60 3 × 103 6 × 103 104 106
Recall that, during a typical temperature ramp in a temperatureprogrammed GC analysis, all solutes elute with nearly the same elution retention factors, ke.1 This allows one to formulate the problem of a search for the optimum heating rate in a temperatureprogrammed GC as a problem of finding the optimum ke. This optimization scenario leads to a unique dimensionless heating rate, r, for each particular optimization criterion as long as there is a one-to-one relationship between r and ke. In the following analysis, we evaluated the ke(r) functions in the three single-ramp temperature programs starting at t ) 0 from initial temperatures Tinit ) 0 °C, Tinit ) -50 °C, and Tinit ) -100 °C. Each initial temperature causes its own initial retention factor, kinit. Using Mathematica software,21 dimensionless elution equations (eq 1d from ref 10, see Appendix below for more details) were numerically solved for the most extreme cases of characteristic thermal constant, θchar, and characteristic elution temperatures, Tchar, encountered in this study, Table 3. Although the elution equation allows one to take into account the change in a gas viscosity with the temperature, we ignored that secondary effect in order to focus on the solute retention alone. The results are shown in Figure 8. Figure 8 shows that, at Tinit ) 0 °C, ke(r) functions for all, but dimethylpropane and n-C5 are nearly the same. The departure of ke(r) functions for dimethylpropane and n-C5 from ke(r) functions of other solutes is consistent with the expectations because, with kinit ) 0.37 for dimethylpropane and kinit ) 0.82 for n-C5 at 0 °C (see Table 3), these two solutes are insufficiently retained even at the very beginning of the ramp, providing almost no room for the changes in k. At Tinit ) -50 °C, ke(r) functions for dimethylpropane (kinit ) 3.3) and n-C5 (kinit ) 6.7) are still apart from those of other solutes; however, their departure is much smaller on the relative scale. In fact, depending on the required value of ke, the convergence of n-C5 with the other more retained solutes can be viewed as acceptable. Thus, for example, at ke e 2, for n-C5 to have the same ke(r) as that of the initially more retained solutes, the dimensionless heating rate of n-C5 should be less that 15% lower than the dimensionless heating rates of the other solutes. Finally, at Tinit ) -100 °C, ke(r) for all practically reasonable ke values (ke e 10) converge very well for all solutes including dimethylpropane (kinit ) 30). (21) Wolfram Research Inc. Mathematica 4.0 [for Windows]; Wolfram Research, Inc.: Champaign, IL, 2000.
Figure 8. Elution retention factor, ke, for 6 solutes, Table 3, vs dimensionless heating rate, r, eq 35, in a single ramp temperature program starting from 0 °C, -50 °C and -100 °C. Numerical solution of dimensionless elution equation (eq 1d in ref. [10], see Appendix below for more details) were used to plot these curves. Dependence of the gas viscosity on temperature was ignored.
One can conclude that
in a single-ramp temperature program, the same dimensionless heating rate leads to nearly the same elution retention factor for solutes that are well retained at the beginning of the ramp Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
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The requirement of the notion of being well retained at the beginning of the ramp depends on the requirement for the tightness of the conversion of the ke(r) functions. The tighter the convergence requirement, the larger should be the smallest kinit in the mixture. To make the following discussion more specific, we will assume that all solutes in a mixture are well retained at the beginning of the ramp if kinit g 7. Also, there is an alternative point of view. As show in eq 40, ln kinit ) ∆X where ∆X ) (Tchar Tinit)/θchar is a dimensionless temperature marginsa dimensionless difference between characteristic elution temperature, Tchar, of a solute and the initial temperature, Tinit, in a temperature program. Therefore, the notion of being well retained at the beginning of the ramp can be equivalently expressed via the notion of having a sufficient temperature margin. Because ln 7 ≈ 2, a condition of having a sufficient temperature margin can be expressed as ∆X g 2. In other words, a mixture has a sufficient temperature margin in a single-ramp temperature program, if Tinit of the least retained solute is at least 2θchar lower than Tchar for that solute. For example, to make sure that dimethylpropane is well retained at the beginning of a ramp, Tinit must be at least -68.5 °C or lower, according to Table 3. The following examples illustrate some benefits of using a dimensionless heating rate in the process of temperature optimization. Example 2. Suppose that, on the basis of some optimization criterion of choice for the analysis of a given mixture, one determines that ke,opt ) 1 is the optimal value of ke. Figure 8 suggests that ropt ) 0.6 is the optimal dimensionless heating rate for that analysis. This means that the optimum heating rate, RT,opt ) roptRT,char, for a 100 m × 530 µm column as in Example 1 is 0.6 × 5 °C/min ) 3 °C/min. For a 1 m × 50 µm column as in Example 1, RT,opt ) 0.6 × 2000 °C/min ) 1200 °C/min. Example 3. An optimization problem can be formulated in the inverse form, yielding ropt for a given optimization criterion. Suppose ropt ) 1; Figure 8 suggests that, under the optimum conditions, all solutes should elute with ke ) 0.5. CONCLUSION We introduced a characteristic thermal constant, θchar ) -dT/dk at k ) 1, a change in a column temperature, T, that causes a e-times change (e ) 2.718 is the base of the natural log) in retention factor, k, in the vicinity of k ) 1. Analyzing our experimental data and the data from the literature (total of about 180 solutes in respect to several types of stationary phase) for partition GC, we found that, at a phase ratio β ) 250, θchar ranged from about 23 °C for low molecular weight hydrocarbons such as dimethylpropane to about 45 °C for high molecular weight pesticides such as mirex. We also found that a 2 orders of magnitude increase in β causes about a factor of 2 increase in θchar. Depending on the method parameters (column dimensions, carrier gas type and flow rate, etc.) and thermodynamic properties of the analytes, a column heating rate, RT, in a temperatureprogrammed analysis can range from fractions of degrees Celsius per minute to thousands of degrees Celsius per minute. Using θchar as a normalization factor for temperature along with the void time, tM, as a normalization factor for time, we introduced the characteristic heating rate, RT,char ) θchar/tM, a normalization factor for RT, and the dimensionless heating rate, r ) RT/RT,char. Being 4088
Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
independent of the above-mentioned method parameters and thermodynamic factors, r is confined within the range that is substantially narrower than that of RT. We also introduced a characteristic elution temperature, Tchar, of a solutesa column temperature that causes the solute to migrate at k ) 1 in an isothermal GC or the elution temperature of a solute eluting with k ) 1 in a temperature-programmed GC. To demonstrate the utility of r, we have shown that, in a singleramp temperature program starting at an initial temperature Tinit, there is nearly a one-to-one relation between r and elution retention factor, ke, for all solutes, as long as Tinit is about 2θchar (about 50 °C) below Tchar of the least retained solute. Convergence of ke vs r relations for all solutes suggests that, in any optimization scenario that can be formulated as a search for the optimum ke in a temperature programmed GC, a particular optimization criterion leads to a unique optimum dimensionless heating rate, ropt. Once ropt is known for a certain optimization scenario, the optimum value, RT,opt, of the actual heating rate, RT, for a particular set of method parameters can be found as
RT,opt ) roptRT,char ) roptθchar/tM To transform an optimum temperature program developed for one particular column to the optimum temperature program for another column, a transformation know as a GC method translation can be used. APPENDIX. DIMENSIONLESS ELUTION EQUATIONS FOR A SINGLE-RAMP PARTITION GC With the temperature effect on the gas viscosity ignored (η ) ηref in eq 2 of ref 10), the dimensionless elution equation ( eq 1d of ref 10) becomes
∫
dτ )1 1+k
τR
0
(42)
where τR is the dimensionless retention timesa retention time expressed in the units of void time, tM, as in eq 33. For the partition GC with a single-ramp temperature program, eq 42 yields, due to eq 37
∫
dτ
τR
0
1+
Exp(x2char/(xinit
+ rτ) - xchar)
)1
(43)
Solving eq 43 for τR allows τR to be found as a function of the reduced heating rate, r; dimensionless elution temperature, xchar; and dimensionless initial temperature, xinit. Once τR is found, ke can be calculated from eq 37 as
lnke )
x2char - xchar xinit + rτR
(44)
NOMENCLATURE Symbol
Description (Units)
dc
column diameter (length)
df
film thickness (length)
∆G
Gibbs free energy of solute evaporation (energy/mole)
∆H
enthalpy of solute evaporation (energy/mole)
tR
retention time (time)
k
retention factor of a solute (1)
x
dimensionless temperature eq 28 (1)
ke
elution retention factor of a solute (1)
∆x
dimensionless temperature margin, eq 39 (1)
kinit
initial retention factor of a solute in a temperature programmed GC (1)
xchar
dimensionless characteristic elution temperature, eqs 4 and 24 (1)
R
universal gas constant, 8.3144 J/K/mol
xinit
r
dimensionless heating rate, eq 35 (1)
dimensionless initial temperature in a temperature program, eq 36 (1)
RT
heating rate (temp/time)
β
phase ratio (1)
RT,char
characteristic heating rate, eq 32 (temp/time)
θ
thermal constant, eq 10 (temp)
∆S
entropy of solute evaporation (energy/temp/mole)
θchar
characteristic thermal constant, eq 20 (temp)
T
temperature (temp)
τ
dimensionless time eq 33 (1)
Te
elution temperature (temp)
Tchar
characteristic elution temperature, eq 19 (temp)
Th
thermal equivalent of enthalpy, eq 4 (temp)
Tinit
initial temperature in a temperature program (temp)
Received for review March 30, 2000. Accepted June 20, 2000.
tM
void time (time)
AC000378F
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