Method Translation and Retention Time Locking in Partition GC

Anal. Chem. 1998, 70, 3828-3839

Method Translation and Retention Time Locking in Partition GC Leonid M. Blumberg* and Matthew S. Klee

Hewlett-Packard Company, 2850 Centerville Road, Wilmington, Delaware 19808

Method translation in gas chromatography (GC) is a variation of components (columns, carrier gases, detectors, etc.) and parameters (pressures, temperature programs, etc.) of a method in a way that maintains the peak elution pattern. The concept is based on the fact that the void time can be viewed as a universal time unit in GC; method translation is the scaling of the time axis of the temperature program relative to the void time. Method translation can be used to reduce analysis time, improve resolution, and adopt a method to a different carrier gas or to a different outlet pressure (vacuum for mass spectrometers, ambient pressure for conventional detectors, etc.). It can also be used for retention time locking (RTL). Theoretical and practical aspects of method translation and RTL are analyzed. It is shown that a constantpressure method can be translated between the columns that have the same stationary-phase type and phase ratio. Method translation also implies that optimum temperature ramp rate (expressed in °C per void time) can be the same for a broad class (maybe all) of temperatureprogrammed GC analyses. For partition GC, this rate is ∼10 °C/void time. Suppose that there exists a gas chromatography (GC) method that provides a satisfactory separation of a given mixture, but takes too much time, or uses a column with too low sample capacity, or maybe, there is a need to use a different carrier gas or to change outlet pressure as might be required due to the change from, say, a flame ionization detector (FID) to mass spectrometric detector (MSD). In another scenario known as retention time locking (RTL), it might be necessary to lock retention times in several GC systems, i.e., to make sure that all the systems yield the same retention times for the same solutes. These and similar results can be achieved via method translation1,2 which, in essence, is a rescaling of the pace of the temperature programming in proportion with the change in the void time or, vice versa, rescaling of void time in proportion with the change in the pace of temperature programming. Originally,1 method translation was described as a set of rules to preserve resolution of all peaks in GC while porting a GC method to a smaller column in order to substantially reduce the analysis time. Since then, the technique gained in popularity, (1) Snyder, W. D.; Blumberg, L. M. Proceedings of 14th International Symposium on Capillary Chromatography; 1992; pp 28-38. (2) Snyder, W. D.; Blumberg, L. M. U.S.A. 5,405,432, April 11, 1995.

3828 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

several revisions of method translation software3 have been released, and many applications4-13 have been developed. The scope of method translation has also been expanded. As the following theory shows, the underlying property of the technique is not the preservation of resolution, but the preservation of the peak elution pattern. That means that, while improving or reducing peak resolution and/or analysis time as required, method translation preserves the ratios between retention times of all pairs of peaks corresponding to the same pairs of solutes. This broader view of the technique allowed expansion of its applications to column optimization for the best resolution and resolution per time,10,12,14 adoption of the same method for the detectors requiring different outlet pressure (vacuum for MSD, ambient pressure for FID, etc.),11-13,15 to RTL.7-9,11-13,15-18 (3) Hewlett-Packard Co. GC Method Translation [for Windows]; Internet http:// chem.external.hp.com/cag/servsup/usersoft/main/html. Hewlett-Packard Co.: Wilmington, DE, 1998. (4) Quimby, B. D.; Giarrocco, V.; Klee, M. S. Speed Improvement in Detailed Hydrocarbon Analysis of Gasoline Using 100 µm Capillary Column; Application Note 228-294; Hewlett-Packard Co.: Wilmington, DE, 1995. (5) David, F.; Sandra, P.; Wylie, P. L. Analysis of Essential Oils by Fast Capillary GC Using the HP 6890 Series GC; Application Note 228-335; Hewlett-Packard Co.: Wilmington, DE, 1995. (6) Kolloff, R. H.; Blumberg, L. M.; Snyder, W. D. Pitcon’96. Book of Abstracts; 1996; p 911. (7) Klee, M. S.; Blumberg, L. M.; Giarrocco, V. Eighteen International Symposium on Capillary Chromatography; Hu ¨ ethig: 1996; p 2274. (8) Klee, M. S.; Quimby, B. D.; Blumberg, L. M. Eighteen International Symposium on Capillary Chromatography; Hu ¨ ethig: 1996; p 2273. (9) Blumberg, L. M.; Klee, M. S. Proceedings. Nineteenth International Symposium on Capillary Chromatography and Electrophoresis; 1997; p 354. (10) Klee, M. S.; Giarrocco, V. Predictable Translation of Capillary Gas Chromatography Methods for Fast GC; Application Note 228-373; Hewlett-Packard Co.: Wilmington, DE, 1997. (11) Giarrocco, V.; Quimby, B. D.; Klee, M. S. Retention Time Locking: Concepts and Applications; Application Note 2280392; Hewlett-Packard Co.: Wilmington, DE, 1997. (12) Quimby, B. D.; Blumberg, L. M.; Klee, M. S.; Wylie, P. L. Precise TimeScaling of Gas Chromatographic Methods Using Method Translation and Retention Time Locking; Application Note 228-401; Hewlett-Packard Co.: Wilmington, DE, 1998. (13) Wylie, P. L.; Quimby, B. D. A Method Used to Screen for 567 Pesticides and Suspected Endocrine Disrupters; Application Notes 228-401; Hewlett-Packard Co.: Wilmington, DE, 1998. (14) Blumberg, L. M.; Klee, M. S. Analytica Conference 98, Abstracts; Munich, Germany, 1998; p 164. (15) Quimby, B. D.; Klee, M. S. Proceedings of Nineteenth International Symposium on Capillary Chromatography and Electrophoresis; 1997; pp 348-9. (16) Wylie, P. L.; Quimby, B. D.; Nixon Donald D. Proceedings of Nineteenth International Symposium on Capillary Chromatography and Electrophoresis; 1997; pp 298-9. (17) Giarrocco, V.; Quimby, B. D.; Klee, M. S. Application of Retention Time Locking in Simulated Distillation; Application Note 228-389; Hewlett-Packard Co.: Wilmington, DE, 1997. S0003-2700(97)01141-4 CCC: $15.00

© 1998 American Chemical Society Published on Web 09/15/1998

Table 1 symbol

description

units

dc k kˆ K L l N pc pd pi po R Reff Rs r ro S t T TM TM,opt tM Tref tR u u j us x η τ β  γ τR ηr

column internal diameter retention factor apparent retention factor distribution coefficient column length reduced column length column efficiency compressed pressure drop pressure drop inlet pressure outlet pressure propagation factor effective propagation factor peak resolution temperature ramp rate optimum temperature rate translation speed gain time column temperature void temperature, eq 35 optimum void temperature void time reference temperature retention time carrier gas velocity carrier gas average velocity solute velocity distance from column inlet carrier gas viscosity reduced time phase ratio see eqs 13 and 14 permeation reduced retention time reduced viscosity

length 1 1 1 length 1 1 pressure pressure pressure pressure 1 1 1 temperature/time temperature/time 1 time temperature temperature temperature time temperature time length/time length/time length/time length pressure‚time 1 1 1 speed/pressure 1 1

Description of computational mechanics of method translation can be found in ref 1 and in the Help utilities of the publicly available software.3 Similarly, a description of operational procedures for RTL can be found in commercially available software.19 However, the theoretical basis and operational limits of the techniques have not been discussed in the literature and are the main goal for this paper. This paper is written for those who are interested in the theoretical basis for method translation and RTL and for those who are mainly interested in the overview of the techniques and their operational limits. The latter readers might find the most useful information in the Discussion section. Admittedly, the experimental part of this paper is brief. For further details, interested readers are directed to the literature cited in the paper. Nomenclature. In the following analysis, a subscript ref marks the quantities measured at a priori known reference conditions. There is one exception. The symbol tM always indicates a reference void time. However, to simplify this particular notation, the subscript ref is omitted. The list of symbols in this paper is given in Table 1. THEORY Elution Equation and Method Translation. Due to temperature programming, many parameters controlling separation (18) Chang, I.; Treese, C. Enhanced Reliability of Forensic Drug Testing Using Retention Time Locking; Application Note 228-393; Hewlett-Packard Co.: Wilmington, DE, 1998. (19) Hewlett-Packard Co. ChemStation [for Windows]. Hewlett-Packard Co.: Little Falls, DE, 1997.

in GC (retention factors of the solutes, viscosity of carrier gas, etc.) become functions of time, t. Initially, we do not pay attention to the fact that it is the temperature that causes the time dependence. We simply recognize the dependence of these parameters on time. Chromatographic conditions can also change with the location, x, along the column. A chromatographic parameter is uniform if, at any time, its (possibly time-dependent) value is the same at all locations along the column. Otherwise, the parameter is nonuniform. A typical source of the nonuniformity in GC is the nonuniform carrier gas density, velocity, etc. As with the time dependence, we initially ignore the sources of the nonuniformity and only recognize that GC parameters can be functions of distance along the column. In short, we assume that all parameters governing a separation process can be functions of t and x. At any t and x, local migration velocity, us, of a solute can be expressed via local velocity, u, of a carrier gas and the solute’s propagation factor, R, as us ) Ru. (This quantity (R) is also known as retardation factor36-39 or retention ratio.35,40 The first is counterintuitive. Its intuitive meaning is in reverse relation with its defined valuesthe faster the solute migrates the larger the value of its retardation factor. The second term is also not very intuitive. In addition, it can be confused with retention factor, k.39 Let tR be retention time of a solute, η viscosity of a mobile phase, ηref reference viscosity of a mobile phase, and tM reference void time. The term reference in descriptions of the last two quantities means that they are measured at a fixed a priori known reference temperature, Tref. As shown in Appendix 1, under the conditions

(c1) column cross-sectional geometry is independent of time (c2) pressure gradient along the column is independent of time (c3) gas viscosity is uniform (c4) propagation factors of all solutes are uniform

elution of a solute is governed by elution equation

∫

tR

0

ηref η(t)

R(t) dt ) tM

(1a)

Similar equations can be found in the literature.20-22 Equation 1a shows that the time dependence of a gas viscosity acts in a way similar to that of a solute propagation factor. This allows us to view the combined quantity

Reff(t) ) (ηref/η(t))R(t)

(2)

(20) Giddings, J. C. J. Chromatogr. 1960, 4, 11-20. (21) Harris, W. E.; Habgood, H. W. Programmed Temperature Gas Chromatography; John Wiley & Sons: New York, 1966. (22) Zhang, Y. J.; Wang, G. M.; Qian, R. J. Chromatogr. 1990, 521, 71-87.

Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

3829

in eq 1a as a single effective propagation factor of a solute. (Variation of R is the goal of temperature programming. Variation of η is a byproduct of the temperature programming.) Equation 1a becomes

∫

tR

Reff(t) dt ) tM

0

(1b)

The latter can also be written as

∫

tR/S

Reff(St) dt )

0

tM S

(1c)

where S is a positive real number. Equation 1c suggests that increasing the pace of the variation of the solute retention by a factor of S simultaneously with the S-fold reduction in the void time causes S-fold reduction in the peak retention time for all solutes. Since this equally affects all the solutes in the mixture, all the solutes elute S times faster without change in the peak elution pattern. That means that, as a result of the appropriately coordinated scaling of void time and temperature-programming rate, not only does the peak elution order remains unchanged but the ratio between any pair of peaks corresponding to the same pair of solutes remains the same. Equation 1b also suggests that any solute in a mixture elutes from a column when the area under the solute’s effective propagation factor reaches the value of void time. Introducing a reduced time,

τ ) t/tM

(3)

one can also write eq 1b as

∫

τR

0

Reff(tMτ) dτ ) 1

(1d)

where quantity

τR ) tR/tM

(4)

is a reduced retention time. Equation 1d shows that each solute elutes from a column when the area under the function Reff(tMτ) of τ is equal to unity, Figure 1. Those who prefer to view the reduced retention times in the light of a chromatographic concept of retention factors, k, can express τR in eqs 1d and 4 as

τR ) 1 + kˆ

(5)

where kˆ ) (tR - tM)/tM is solute’s apparent retention factor. With these notations, eq 1d becomes

∫

1+kˆ

0

Reff(tMτ) dτ ) 1

(1e)

Method Translation. Suppose that, for a given solute, two chromatographic methods, M1 and M2, result in the peaks that 3830 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

Figure 1. Retention parameters of a solute for temperature programming rates of 10 (solid line), 30 (shorter dashes), and 100 (longer dashes) °C per void time, tM. It is assumed that the solute retention factor, k, depends on the absolute column temperature, T, as k ) ki exp[cTi(Ti/T - 1)] where ki is initial retention factor and Ti is the initial temperature. In these graphs, ki, Ti, and c are, respectively, 100, 300 K, and 0.03. A solute elutes when its elution integral reaches unity. Using as an example the 30 °C/tM curves, it is shown how retention time and other retention parameters of an eluting solute can be found from these graphs.

have respective retention times tR1 and tR2 related as

tR1/tR2 ) S

(6)

We will say that these methods are translatable, that M2 is the translation of M1 and vice versa, and that S is the speed gain of translation of M1 into M2, if S is the same for all solutes. We will also say that any two methods are mutually translatable if one is a translation of another. The above definitions suggest that a method is translatable if quantity S in eq 1c can be the same regardless of the particular solute. The latter is true if conditions c1-c4 are satisfied. Hence, conditions c1-c4 are sufficient for a method to be translatable. Furthermore, if conditions c1-c4 are satisfied, then each of the equivalent elution integrals in eqs 1 is valid. Equation 1d shows that two translatable methods, M1 and M2, with the reference void times, tM1 and tM2, are mutually translatable if, for any solute,

their effective propagation factors, Reff1 and Reff2, relate as

Reff2(tM2τ) ≡ Reff1(tM1τ)

(7a)

i.e., are the same for any τ. Equivalently, due to eq 1c, two translatable methods are mutually translatable if

Reff2(t) ≡ Reff1(St)

(8)

(9)

Reff2(t) ≡ Reff1(t)

(10)

tM2 ) tM1

(11)

The remaining material deals with partition GC (gas-liquid chromatography). Effect of Difference in Carrier Gas Viscosity. Can two GC methods utilizing different types of carrier gas be mutually translatable? Earlier we saw, eqs 1b and 2, that elution equation in a temperature-programmed GC is affected by the temperature dependence of a quantity

(12)

(17)

(13)

R ) 1/(1 + k)

0.65 e  e 0.75

(14)

(20)

where K is the distribution coefficient (a solute- and stationaryphase-dependent function of temperature) and β is the phase ratio. Combining eqs 19 and 20, and recognizing that K is a function of temperature, T, that, in turn, is a function of time, t, one has

(21)

Equation 21 implies that two translatable methods M1 and M2 with temperature programs T1(t) and T2(t) are mutually translatable, i.e., eqs 7 are satisfied if

(c5) for any solute, the distribution coefficient is the same function of temperature in both methods

Due to eq 13, reduced viscosity can be expressed as

ηr ) η/ηref ) (T/Tref)

(19)

where k is the solute’s retention factor. The latter can be expressed as

R(t) ) β/(β + K(T(t))) where  is a gas-dependent constant that, for the typical carrier gases (helium, hydrogen, nitrogen) falls within the range of23,24

(18)

This, according to eq 2, indicates that, even after the correction for the difference in the viscosities at the reference temperature, the change from one carrier gas to another can cause up to 0.01% difference in the change of effective propagation factor, Reff, per 1 °C. For retained solutes in partition GC, this impact is several hundred times smaller than the ∼3% per °C25 change in propagation factor due to the change in the temperature alone. From another point of view, the above evaluation means that the impact of the difference in the temperature dependence of viscosities of different carrier gases is roughly equivalent to the impact of a 1 °C difference in the column temperature accumulated over the entire temperature-programming range of several hundreds of degrees centigrade. Stationary Phases and Temperature Programs. The propagation factor, R, of a solute in partition GC can be expressed as

k ) K/β

that can be viewed as reduced viscosity of a carrier gas. Within practical measurement accuracy and within GC temperature range, gas viscosity can be described as

η ) ηref(T/Tref)

∆ ) 2 - 1

ηr2/ηr1 ) (501/500)0.05 ≈ 1.0001

between methods M1 and M2 means that both methods result in the same retention time for any solute. This effect can be referred to as retention time locking between methods M1 and M2. According to eqs 7b, 8, and 9, retention times in translatable methods M1 and M2 are locked when

ηr ) η/ηref

(16)

Assuming that ∆ ) 0.05 (see eq 14), let us evaluate the impact of this difference per 1 °C change at 500 Ksapproximately the middle of typical GC temperature ranges. From eq 16, one has

Retention Time Locking. A unity speed gain,

S)1

ηr2/ηr1 ) (T/Tref)∆ where

(7b)

i.e., if Reff2 evolves S times faster compared to Reff1 where

S ) tM1/tM2

A ratio of the reduced viscosities for two different gases can be expressed as

(15)

(23) Touloukian, Y. S.; Saxena, S. C.; Hestermans, P. Viscosity; IFI/Plenum: New York, 1975. (24) Ettre, L. S. Chromatographia 1984, 18, 243-8.

and

(c6) the phase ratio is the same in both methods Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

3831

Additionally, the temperature programs relate as

T2(tM2τ) ≡ T1(tM1τ)

(22a)

be, respectively, column length, internal diameter, carrier gas average velocity, inlet and outlet pressure, and pressure drop. Void time, tM, can be measured as

j tM ) L/u

or, equivalently,

T2(t) ≡ T1(St)

(22b)

(26)

where average gas velocity, uj , can be found26-28 as

u j ) γapd where S is the same as in eq 8. We will say that two columns that satisfy conditions c5-c6 are mutually translatable. We will also say that the two GC methods with void times tM1 and tM2 have mutually translatable temperature programs, T1(t) and T2(t), respectively, if these programs relate as in eqs 22. It becomes apparent that two methods are mutually translatable if both are translatable (conditions c1-c4), utilize mutually translatable columns (c5-c6), and

(27)

Here

a)

2 3 (pi + po) 4 p2 + pp + p i

i o

2 o

γ ) dc2/(32ηL)

(28) (29)

(c7) have mutually translatable temperature programs. Example: Let T1(t) be a single-ramp temperature program in method M1; i.e., T1(t) ) Tinit + r1t, where Tinit is the initial temperature and r1 is the temperature ramp rate. If M2 is a translation of M1, then, according to eq 22b, the temperature program, T2(t) ) Tinit + r2t, in M2 must be T2(t) ) Tinit + Sr1t. In other words, the ramp rate, r2, in M2 must be r2 ) Sr1. Taking into account eq 8, one has

r1tM1 ) r2tM2

(23)

Similar relations between the ramp rates and the void times can be established for multirate programs. In other words, method translation does not change the product of void time and respective ramp rate. We conclude this analysis with another important observation. We have already established that, in a translated method, each solute elutes sooner compared to the same solute in the original method by the same factor Ssthe speed gain. The same factor S also represents the increase in the pace of the temperature programming in the translated method, eq 1c. As a result, elution temperature of any solute in all mutually translatable methods is the same. Conditions for RTL. As RTL is a method translation with the speed gain, S ) 1, eqs 8 and 22b become, respectively,

T2(t) ≡ T1(t)

pc ) apd

(30)

referred to as compressed pressure drop. (For all combinations of pi and po values, a varies between 3/4, for pi . po, and 1, for |pi - po| , po. Hence, pc always stays within the range 3/4 pd e pc e pd approaching pd when pd is low compared to po and approaching 0.75 pd when pd is large.) (Substitution of eq 30 simplifies eq 27). The latter becomes

u j ) γpc

(31)

expressing uj as a quantity proportional to compressed pressure drop and the column’s pneumatic permeation. Substitution of eqs 29 and 31 in eq 26 yields

tM ) 32l2η/pc

(32)

where l ) L/dc is the reduced length of the column. Equation 32 indicates that the void times in methods M1 and M2 become equal when

pc2/pc1 ) (l2/l1)2η2/η1

(33)

(24)

and tM2 ) tM1. The latter has already been established, eq 11, for a more general case. We conclude that retention times in two mutually translatable methods become locked when

(c7(RTL)) the methods have equal void times and run the same temperature program. How does one achieve equality of void times in two columns? Let L, dc, uj , pi, po and

pd ) pi - po

The latter is a pneumatic permeation of a column. It is convenient to lump quantities a and pd in eq 27 together and view the product apd as a single quantity

(25)

3832 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

The existence of this relation suggests that, if retention times in the two methods can be locked, the adjustment of head pressure on one column is sufficient for locking retention times of all peaks. Method Translation and Peak Resolution. It is shown in Appendix 2 that, for any pair of solutes, the resolutions, Rs1 and Rs2, in mutually translatable methods, respectively, M1 and M2, relate as (25) Littlewood, A. B. Gas Chromatography. Principles, Techniques, and Applications; Academic Press: New York, 1970. (26) Hala´sz, I.; Hartmann, K.; Heine, E. Gas Chromatography 1964; The Institute of Petroleum: London, 1965; pp 38-61. (27) Dose, E. V.; Guiochon, G. Anal. Chem. 1990, 62, 174-81. (28) Blumberg, L. M. Chromatographia 1996, 43, 73-5.

Rs2/Rs1 ) xN2/N1

(34)

where N1 and N2 are column efficiencies (plate numbers) in, respectively, M1 and M2. Equation 34 indicates that resolutions for any solute pair in two mutually translatable methods relate as the square root of the ratio of the column efficiencies. Particularly,1 translation of a method into a new one with the same efficiency preserves resolution of all peak pairs. EXPERIMENTAL SECTION In all experiments, HP 6890 Series GC system was used. Other conditions are described in Tables 2 and 3. RESULTS AND DISCUSSION The underlying idea behind the method translation is very simple. Separation of a mixture by a short, narrow-diameter column runs much faster than the separation of the same mixture in a long and/or wide column. Obviously, faster separation requires faster temperature programming. The idea is to scale the temperature program in the time domainsslower programming for larger columns and faster programming for smaller columns. The idea of the time scaling in GC immediately raises two questions, first, whether the scaling can be accomplished without a change in the peak elution pattern. In other words, can the time scaling be done in a way that preserves not only the peak elution order but also the ratio between any pair of peaks corresponding to the same pair of solutes? If the answer to this question is yes, then the second question is what exactly should govern the scaling. Should the scaling be based on, for example, the changes in column dimension, gas flow rate, or combination of both? The theory developed here shows that, under certain conditions (see below), the scaling of chromatographic time without changing the peak elution pattern is possible. It can be done by compressing or stretching the temperature program along the time axis in proportion with the change in the void time. This is the essence of the method translation. Once this essence is recognized, it becomes clear that the scope of the method translation extends well beyond porting of the methods from one column to another (see below). Given an original method, method translation allows one to find parameters of a new translated method yielding a translated chromatogram where all peaks have the same elution pattern as in the original chromatogram. More specifically, method translation allows one to reduce or increase retention times by the same factor for all solutes. The ratio, S, of analysis time in the original method to that in the translated method is viewed as the translation speed gain. Translation with the speed gain of one is referred to as retention time locking (RTL). As we already mentioned, method translation is possible if certain conditions are satisfied. Analysis of these conditions is the first order of priority in this discussion. Practical Conditions for Method Translation. To be translatable (and, therefore, lockable), a method must satisfy certain conditions. For example, it is probably impossible to

Table 2. Detailed Hydrocarbon Analysis of Gasoline4 a

carrier gas head pressure, psi initial temp, °C initial time, min rate A, °C/min final temp A, °C final time A, min rate B, °C/min final temp B, °C final time B, min rate C, °C/min final temp C, °C final time C, min split ratio

column 1

column 2

helium 39.9 35 13 10 45 15 1 60 15 2 220 5 250:1

hydrogen 77.27 35 3.6 36.12 45 4.15 3.612 60 4.15 7.22 220 1.38 1250:1

a Column 1: 100 m × 250 µm × 0.5 µm HP-1. Column 2: 40 m × 100 µm × 0.2 µm DB-1.

Table 3. Styrene Analysis11 on 60 m × 320 µm × 0.5 µm HP-INNOWax Column: Helium Carrier Gas, 80:1 Split Ratio detector

head pressure, psi initial temp, °C initial time, min rate, °C/min final temp, °C

FID

MSD

MSD

18.2 80 9 5 150

14.7a

8.9b 80 9 5 150

80 7.1 6.4 150

a Causes the same separation efficiency as with FID. b Causes the same void time as with FID.

achieve RTL between two analyses using columns with substantially different types of stationary phase. In the development of a new method, it can be important to make appropriate choices in selection of columns and conditions so that the method can be translated and/or locked if/when it becomes necessary. This boils down to making sure that conditions c1-c4 (see Theory) sufficient for a method to be translatable or conditions c1-c7 for the two methods to be mutually translatable are satisfied. While exact and unambiguous, conditions c1-c7 can be inconvenient in practice. For example, condition c2 (pressure gradient along the column is independent of time) is difficult to verify in practice. However, typically, this condition is satisfied (see below) if column head pressure remains constant during the run. This is a common practice in capillary GC. Below, taking a practical view, we re-interpret conditions c1c7, making them more convenient for everyday use. It should be remembered however, that the true sufficient conditions are c1-c7. If an ambiguity arises, conditions c1-c7 should be used. Translatable Methods. For a method to be translatable, it is sufficient that conditions c1-c4 are satisfied. These conditions came from the derivation of elution equations, eqs 1, on which the method translation is based. Conditions c1-c4 divide chromatographic parameters into two groupssthose that can change only in distance (i.e., can be nonuniform along the column) and those that can change only in time. Conditions c1-c4 dictate that no parameter can change in both distance and time. Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

3833

Typically, it is easier to predict the behavior of all GC parameters if there is no column overloading. In that case, the prime source of temporal changes of the GC parameters is their temperature dependence. On the other hand, the prime source of the nonuniformity of the parameters in distance is pressure drop along the column. Therefore, a practical assumption that

(p0) no column overloading is taking place and the column temperature is uniform means that the pressure gradient along the column is practically the only source of coordinate-dependent changes of GC parameters, and conditions c1-c4 can have the following practically convenient interpretations p1-p4, respectively.

(p1) column cross-sectional geometry is independent of temperature i.e., such effects as the temperature-dependent expansion of the column material must be (and typically are) negligible. Notice that the column cross-sectional geometry does not have to be uniform. For example, a column with a nonuniform diameter, or even a column consisting of several sections having substantially different geometry can be translatable. Condition c2 (the next in collection c1-c4) deals with the need to have a time-independent pressure gradient along the column. The simplest ways to satisfy this condition in practice is to assume that, in addition to p1,

(p2) column’s inlet and outlet pressure do not change during the run (constant-pressure mode) and

(p3) viscosity of carrier gas is pressure-independent Condition p3 suggests that the pressure dependence of the gas viscosity must be (and typically is) negligible. This supports both condition c2 and c3. Finally, condition c4 can be satisfied if

(p4) propagation factors of all solutes are pressure and coordinate independent or, in case of uniform crosssectional dimensions of a column, the stationary phase has uniform and pressure-independent properties.

We can conclude that, in practice, conditions p0-p4 are sufficient for a method to be translatable. The set of condition p0-p4 can be further reduced. Indeed, in a typical temperature-programmed GC, a majority of conditions p0-p4 are always satisfied. Thus, practically speaking, column temperature is always uniform (p0), column dimensions are temperature-independent (p1), gas viscosity is pressure-independent (p3), and the stationary phase has uniform and pressure3834 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

independent properties (p4). The only item not covered by this list is p2, requiring a constant-pressure mode. Therefore, one can conclude that, practically speaking, any constant-pressure method is translatable. Mutually Translatable Methods. In the previous section, we discussed conditions that are sufficient for a GC method to be translatable. A translatable method can be translated into many other methods. It does not mean, however, that any two translatable methods, say, M1 and M2, are mutually translatable; i.e., M2 can be a translation of M1 and vice versa. For example, two translatable methods identical in all aspects (columns, carrier gases, detectors, pressures, etc.) except for their temperature programs can have a different elution order of solutes and, therefore, cannot be translations of each other. Their retention times cannot be locked. As theory shows, two translatable methods are mutually translatable if both utilize mutually translatable columns and have mutually translatable temperature programs. Two columns are mutually translatable if conditions c5-c6 are satisfied. In practice, this takes place when, respectively, the columns have

(p5) chemically identical stationary phases (p6) the same phase ratio Notice that conditions p6 and c6 are identical. The latter has been re-entered here for the completeness of the list of practical conditions. Finally, condition c7 calling for the mutually translatable temperature programs means that the programs should relate as in eqs 22. In practice, for a quantity S in eq 8 to be a translation speed gain,

(p7) in the new method, all temperature set points should be the same as in the original one, but should last for S-fold shorter time, and all temperature ramps should be S-fold faster For RTL, condition c7 takes a more specific form of condition c7(RTL) which has an obvious practical meaning. We restate that retention times in two mutually translatable methods become locked when

(p7(RTL)) the methods have the same void times and the same temperature programs Can the Conditions Be Further Relaxed? We already came to the conclusion that, in practice, a constant-pressure method is translatable and, with the appropriate relation between their temperature programs, two translatable methods utilizing columns with chemically identical stationary phase and the same phase ratio are mutually translatable; their retention times can be locked. Can these conditions be further relaxed? Can, for example, a method utilizing a constant-flow mode or a pressure-programmed method, be translated? Can RTL between several such methods be achieved? To address these questions, we should recall that all the above conditions are sufficient, but not necessary, and, therefore, there can be room for their relaxation.

Figure 2. Detailed hydrocarbon analysis of gasoline: (a) conventional (100 m × 250 µm × 5 µm column, helium carrier gas), (b) translated (40 m × 100 µm × 0.2 µm column, hydrogen carrier gas). Use of the shorter column with smaller diameter, and hydrogen instead of helium as a carrier gas, allowed reduction of analysis time by almost a factor of 4.

The only reason for the selection of the basic set c1-c7 of sufficient conditions in this report was to provide a simple mathematical analysis. More relaxed conditions require more involved study exceeding the scope of this report. However, it is worth mentioning that there are theoretical indications and experimental data confirming that the methods utilizing constantflow mode can be translated. In addition to that, substantial inequality of the phase ratio between the columns can also be overcome in many cases. A way of handling the latter case is implemented in the publicly available method translation software.3 It should also be noticed that our derivation of the elution equations eqs 1 was pointedly made as general as possible. As mentioned earlier, equations similar to eq 1a were known20-22 from the theory of temperature-programmed GC. However, previous theories typically relied on several specific conditions substantially limiting the scope of the equations to the degree where they cannot be used to justify method translation. One of those specifics was an assumption of a specific form of dependence of solute distribution coefficient, K, in eq 20 on temperature, T, typically assuming constant entropy and enthalpy of the distribution. The theory developed in this study does not have this requirement and is valid for any K(T). Another traditionally required condition was that the carrier gas behaved as an ideal gas. In this study, on the other hand, only the pressure independence of the gas viscosity was required. Also it was typically assumed in the previous theories that a column was uniform. Earlier in this discussion, we mentioned that uniformity was not required in this study. As a result, a method utilizing a column with nonuniform cross-sectional dimensions, such as a column consisting of different segments, can be translatable. It appears that, possibly, the only fundamental barrier that prevents the development of two mutually translatable methods or the methods that allow mutual RTL is the case when the methods utilize columns with chemically different stationary phases. Applications of Method Translation. As we mentioned earlier, method translation is based on a simple idea that, in GC under certain conditions (see p1-p7 above), acceleration of temperature programming by the speed gain S combined with the S-fold reduction in the void time can result in the S-fold reduction in the retention times for all peaks. In other words, method translation is equivalent to rescaling the time axes of a chromatogram without changing peak elution pattern.

Figure 3. Styrene analysis on 60 m × 320 µm × 0.5 µm column: (a) FID chromatogram; (b, c) MSD total ion chromatograms. Method b was a translation of method a with the condition that initial carrier gas mass flow rate be the same in both methods. This translation results in nearly identical resolution of all peak pairs. Method c was a translation of method a with the condition that void times in both methods be the same. This translation results in RTL, i.e., in nearly identical retention times for all peaks (see Table 4).

Originally,1 method translation was described as a tool for porting GC methods without changing resolution between all peak pairs. A majority of published applications4-13 of the technique still fall in this category. However, as we mentioned earlier, the key to method translation is not preservation of resolution, but the preservation of the peak elution pattern. This substantially expands the breadth of applications for the technique. Broad acceptance of method translation has also been helped by the publicly available3 method translation software, briefly described in ref 10. So far, method translation has been found useful in the following groups of application: (a) porting a method from one column to another1,4-6,13 (Figure 2); (b) optimizing a method on the same column (by changing gas flow rate and temperature program to achieve the best separation efficiency,29 the shortest analysis time,30 etc.10,12,14) (c) translation to adapt a method for use with a different carrier gas1,4 (Figure 2); (d) translation to adapt a method to different detectors operating at a different outlet pressure (vacuum for MSD, ambient pressure for FID and other conventional detectors, etc.)11-13,15 (Figure 3); (e) combination of the above, applications (Figure 2); (f) RTL under all of the above conditions7-9,11-13,15,16,18 (Figure 3). Detailed description of applications illustrated in Figures 2 and 3, and of many other applications of method translation, can be found in the aforementioned Application Notes and in ref 1. The (29) Blumberg, L. M. J. High Resolut. Chromatogr. 1997, 20, 597-604. (30) Blumberg, L. M. J. High Resolut. Chromatogr. 1997, 20, 679-87.

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Table 4. Retention Times (min) in Styrene Analysis. RTL of MSD (Total Ion Chromatogram) with FID Figure 4a and c detector

difference

component

FID

MSD

min

%

ethylbenzene p-Xylene m-Xylene isopropylbenzene o-xylene n-propylbenzene R-methylstyrene phenylacetylene β-methylstyrene benzaldehyde

10.315 10.620 10.869 12.038 12.613 13.492 18.276 19.406 21.008 25.475

10.338 10.642 10.890 12.053 12.630 13.508 18.267 19.389 20.987 25.415

0.023 0.022 0.021 0.015 0.017 0.016 -0.009 -0.017 -0.011 -0.060

0.22 0.21 0.19 0.13 0.14 0.12 -0.05 -0.09 -0.05 -0.24

next section is dedicated to RTLsan important case of method translation. Retention Time Locking. RTL is based on a simple idea that, under certain conditions (see p1-p6 and p7(RTL) above), two GC analyses can result in the same retention time for each solute. It is not unusual in an everyday practice that two GC systems running the same method have columns with different length and internal diameter, but are otherwise similar. From the point of view of elution eq 1b, the systems have the same function Reff(t) on the left-hand side but different values of void time, tM, on the right-hand side. As a result, peak retention times, tR, can be different for some or all solutes. A large difference in tM can result in a loss of resolution or even in reversal of elution order for critical pairs. Equation 1b shows that all it takes to make the same retention times for all solutes in two mutually translatable methods, i.e., to lock retention times in these methods, is to make sure that both methods have equal void times measured at the same isothermal conditions. This suggests that the RTL can be viewed as method translation with the speed gain of 1. Analysis of method translation suggests that, practically speaking, RTL is possible between two methods using columns that have the same stationary phase and phase ratio and are running in constant-pressure mode. In this case, adjustment of the void time for the RTL can be achieved by appropriate adjustment of the column head pressure. This analysis suggests that RTL can be used well beyond compensation for the minor difference between the column dimensions in the methods. As we mentioned earlier, RTL can be used, Table 4 and Figure 3a and c, in GC systems using different detectors (requiring different outlet pressure,8,11,12,18 different carrier gas types, columns with substantially different dimensions, etc. In one set of experiments,7 to test the limits of the technique, column length was cut in 10-m steps from 60 to 10 m. A good retention time matching was achieved in all cases. Earlier we found that, for any two mutually translatable methods, adjustment of the column pressure in one system is sufficient for locking all retention times between the systems. This leads to a simple conclusion that, if retention times in two chromatograms can be locked, then locking one peak, locks all peaks. This opens the door for many implementations of RTL. Thus, although the key to the RTL is equality of void times in the two 3836 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

methods, there is no need to measure the void time. The fact that locking one peak locks all peaks suggests that it does not matter how one finds the right pressure for the locking of one peak. The RTL pressure can be empirically found using only one peak.15 And, once that peak is locked, the entire chromatogram is locked. The RTL in currently available commercial GC system19 is based on this approach. Void Time Is a Universal Time Unit in GC. Method translation points to several interesting concepts. The key to method translation is realization of the fact that void time can be viewed as a universal time unit of a separation process in chromatography in general and in GC in particular. From that point of view, translation of a GC method is nothing but the scaling up or down of the void time accompanied by the proportional stretching or compression of the temperature program in time. This observation follows from elution equation eq 1b, which shows that, in temperature-programmed GC, the effect of a process leading to elution of a solute at retention time, tR, can be reduced to one combined parametersthe solute’s time-dependent effective propagation factor, Reff(t). A solute elutes from the column when, Figure 1, the area under the function Reff(t) becomes tMsthe void time measured at some reference temperature, Tref. (Tref can be any temperature such as 0 °C, room temperature, or a medium temperature in a given chromatographic run. However, it must be the same for measurement of both tM and ηref in eq 2.) It can be shown (see Theory) that, in a more generic, reduced form, Reff(t) can be expressed as Reff(tMτ), where τ is reduced time defined, eq 3, as τ ) t/tM. This indicates that the reduced time, τ, can be viewed as absolute time, t, expressed in units of tM. The fact that, due to eq 1d, tM can be the only time-scaling factor affecting reduced retention time, τR, of every solute indicates that void time can be viewed as universal time unit in GC. We also established, eq 23, that method translation does not change the value of the product rtM where r is the temperature ramp rate. In other words, quantity

TM ) rtM

(35)

is the same for all mutually translatable methods. That means that, within a class of mutually translatable methods represented by a given value of TM, the actual value of the ramp rate, r, for a particular method with a particular value of tM can be found as

r ) TM/tM

(36)

Giddings31 referred to the product rtM as the void temperature. (Actually, Giddings used the term “dead time” for tM and “dead temperature” for the product rtM.) Equation 36 suggests that the void temperature, TM, can be viewed as a normalized temperature ramp rate. Measured in units of temperature (e.g., °C) per void time, it provides a generic single-value representation for the ramp rates for the entire class of mutually translatable methods. We will use the terms void temperature and normalized temperature ramp rate as synonyms. (31) Giddings, J. C. Gas Chromatography; Academic Press: New York, 1962; pp 57-77.

Suppose now that, based on some criteria, ro is the optimum ramp rate for a given tM. Hence, the product TM,opt ) rotM is the optimum void temperature (i.e., optimum normalized ramp rate) for that tM. In other words, TM,opt is the optimum void temperature for a particular complex mixture analyzed on a particular column with a particular carrier gas, its pressure, etc. For example, ro can be the ramp rate in a single-ramp temperature program corresponding to the largest number of resolved peaks in the shortest time. Once such optimum ro has been found, further optimization of the method can be possible via its translation to accommodate the most suitable carrier gas, its pressure, etc. As method translation does not disturb the peak elution pattern and affects the peak resolution in a predictable, uniformly scalable manner, eq 34, it can be used to find a better optimum ramp rate for the new pneumatic conditions of the column. It is remarkable, however, that this additional optimization, while possibly substantially changing the value of ro and substantially improving peak resolution and/or analysis time, does not affect the value of the product rotM, and, therefore, the value of the optimum void temperature, TM,opt. This leads to the conclusion that a normalized temperature ramp rate, TM,opt, that is optimum for one method is optimum for all its translations. This fact further suggests that, based on some criteria, for each class of mutually translatable methods, there can exist a single value of optimum normalized temperature ramp rate. This observation can be taken even further. Suppose that for some class of mutually translatable GC methods an optimum void temperaturesan optimum normalized ramp ratesis known. Can this temperature be substantially different for other complex mixtures? Can it be substantially different for other liquid stationary phases or for other phase ratio values? Searching for the answer to these questions, we are not interested in a temperature ramp rate or rates that are the most effective in resolving particular critical peak pairs. Rather, we are interested in the possibility of a single void temperature that can be used for the calculation of something like a suggested (default) void temperature for a broad class of methods (possibly, all methods) before particular critical pairs in a particular mixture to be analyzed by the method are identified. We already established the existence of a single optimum void temperature for any class of mutually translatable methods. The question now is, how different can these optimums be for different combinations of solutes and stationary phases? Apparently, not very much different. It is widely known that, for most of the solutes in combination with the most of the liquid stationary phases of any thickness, the same change in the column temperature causes roughly the same relative rate of change in the distribution coefficients, K. According to Littlewood,25 this rate is ∼3%/°C; a similar rate was experimentally confirmed by Goedert and Guiochon.32 Giddings33 assumed that K doubles per every 20 °C to 40 °C reduction in column temperature (i.e., 1.8 to 3.5%/ °C). So far, we established two facts: (a) A void temperature, TM,opt, that is optimum for one method is optimum for the entire class of mutually translatable methods. (b) There is a narrow range of values of temperature-related rates of change of distribution (32) Goedert, M.; Guiochon, G. Anal. Chem. 1970, 42, 962-8. (33) Giddings, J. C. J. Chem. Educ. 1962, 39, 569-73.

coefficient for the wide range of solutes and stationary phases. These two facts suggest that if some value of the void temperature leads to the best compromise between resolution and time in one GC analysis of a complex mixture, then it is very likely that roughly the same void temperature leads to the same optimum in a very broad class (maybe all) of analyses of a broad class (maybe all) of complex mixtures. In other words, there must exist a single value of void temperature that is roughly optimum for a broad class (maybe all) of temperature-programmed GC analyses. This optimum void temperature can be recommended as a default void temperature for all temperature-programmed GC analyses. Example. Suppose that the optimum void temperature, TM,opt, for the temperature-programmed GC analyses is 10 °C; i.e., the optimum temperature program rate for any GC analysis is 10 °C per void time. Based on eq 36, the following two ramp rates, ro, are optimum for their respective void times, tM: (a) For tM ) 2.5 min (e.g., 0.6 mL/min at 25 °C helium in 25 m × 250 µm column), ro ) 4 °C/min. (b) For tM ) 0.2 min. (e.g., 1 mL/min at 25 °C hydrogen in 10 m × 100 µm column), ro ) 50 °C/min. Once the existence of a single optimum void temperature for a broad range of GC analyses has been established, it becomes important to find the value of that optimum. At this point, there are no sufficient data to support a precise value of the optimum void temperature for any class of solutes and liquid phases. However, some estimate is possible. In our opinion, based on the assumption25 of 3% change in distribution coefficient per 1 °C change in column temperature, and on the analysis of many chromatograms, in partition GC, ramp rates of ∼3-10 °C/void time provide the best compromise between the number of resolved peaks in a complex mixture and the analysis time. Based on this observation, the value

TM,opt ) 10 °C

(37)

can be recommended as a default void temperature for partition GC. Guidelines similar to these were known before. Giddings suggested that (see closing remarks before Conclusion in ref 31), “[void] temperature should probably not exceed 20-30 °C, and might preferably be lower”. Giddings also recommended following the guideline of rtM ) RR‚50 °C,31 or rtM ) RR‚45 °C,33 where RR is the propagation factor (see Theory) of an eluting solute. On the basis of the latter recommendation and on the assumption that RR ≈ 0.3, Pauschmann recommended34 use of rtM ) 13 °C. A broad range of the recommended void temperature values can be explained by the lack of a coherent theory for its calculation and by the insufficiency of experimental data. However, there is a theoretical justification to expect that a substantial change in the carrier gas density along the column causes reduction in the optimum void temperature by a factor of ∼2 compared to the case when the compressibility of the carrier gas can be ignored. As in the above-mentioned studies,31,33 the compressibility of the carrier gas was ignored; it can be one reason the Giddings and (34) Pauschmann, H. H., letter to L. M. Blumberg, April 1998.

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Pauschmann recommendations are higher than our estimate of 3-10 °C. It is important, however, that, while the accuracy of the known values of the optimum void temperature is highly uncertain, the existence of that optimum is certain. We believe that the research to refine the value of the optimum void temperature should continue. Some of the unanswered questions are, What are the optimum void temperatures for the most widely used combinations of sample mixtures and liquid phases? How large is the difference between these optimums for the different mixture-liquid-phase combinations? How practically sufficient is the difference? CONCLUSION Method translation allows one to generate several GC methods resulting in the same peak elution pattern while utilizing columns with different dimensions, different types of carrier gas, and different outlet pressures (as required by different detectors). The technique can be used to port a method from one set of conditions to another allowing for substantial improvement in the speed of analysis, peak resolution, and sample capacity. Retention time locking allows one to maintain equal retention times among several mutually translatable methods. A theory of method translation and RTL in partition GC (gasliquid chromatography) has been developed. A substantial part of the theory is dedicated to the analysis of conditions sufficient for the translatability of a chromatographic method and for two methods to be translations of each other. It has been shown that any constant-pressure method is translatable. It can be translated to the same or a different column as long as both columns have the same type of stationary phase and equal phase ratio. RTL between any two mutually translatable methods can be achieved by adjusting column head pressure. In addition to its direct utility, method translation has many useful implications. It implies, for example, that void time can be viewed as a universal time unit in chromatography. It also provides a strong indication that the optimum temperature ramp rate expressed in units of temperature per void time can be the same for a broad class (maybe all) of temperature-programmed GC analyses. This rate can be used as a default ramp rate if no information is available about specific critical pairs in a sample. As a default rate for partition GC, we recommend 10 °C/void time. APPENDIX 1. Derivation of Elution Equation. Local velocity, us ) dx/ dt, of a solute at a distance x from the column inlet can be expressed as

dx/dt ) Ru

(38)

where u and R are, respectively, the local velocity of a mobile phase and propagation factor of the solute. The former, according (35) Giddings, J. C. Unified Separation Science; John Wiley & Sons: New York, 1991. (36) Consden, R.; Gordon, A. H.; Martin, A. J. P. Biochem. J. 1944, 38, 224-32. (37) Keulemans, A. I. M. Gas Chromatography, 2nd ed.; Reinhold Publishing Corp.: New York, 1959. (38) Dal Nogare, S.; Juvet, R. S. Gas-Liquid Chromatography. Theory and Practice; John Wiley & Sons: New York, 1962. (39) Ettre, L. S. Pure Appl. Chem. 1993, 65, 819-72. (40) Fuller, E. N.; Giddings, J. C. J. Gas Chromatogr. 1965, 3, 222-7.

3838 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998

to Darcy’s law can be expressed as

u ) -(a/η)gp

(39)

where η is the local viscosity of a mobile phase, gp ) ∂p/∂x is the local pressure gradient along the column, and a is a quantity that depends on the column type and cross-sectional geometry. (For a capillary column with circular cross section, e.g., a ) dc2/32, where dc is the column internal diameter.) Combining eqs 38 and 39, one can have

dx/dt ) - (Ra/η)gp

(40)

This equation can also be written as

R dt/η ) - dx/(agp)

(41)

If conditions c1-c4 (main text) are met, then all the variables on the lhs of eq 41 are independent of x while all the ones on the rhs are independent of t. Separation of the variables in eq 41 allows for the independent integration of its lhs and rhs. It is convenient, however, to introduce a reference viscosity, ηref, prior to the integration of eq 41. Let Tref be some a priori known fixed reference temperature for some class of chromatographic analyses. Viscosity, ηref, measured at Tref can be viewed as a reference viscosity of the mobile phase. Equation 41 can be written as

Rηref dt ηref dx )η agp

(42)

Comparing the rhs of this expression with eq 39, one can note that eq 42 can be written as

Rηref dt dx ) η uref

(43)

uref ) - agp/ηref

(44)

where

Notice that, due to conditions c1 and c2, neither quantity in eq 44 can be a function of t. Hence, the rhs of eq 43 is still independent of t while its lhs remains independent of x. Hence, eq 43 can be integrated as

∫

tR

0

Rηref dt ) η

∫

L

0

dx uref

(45)

One can recognize that the rhs in this expression is void time, tM, in the isothermal run under the reference column temperature, Tref. That void time calculated as

tM )

∫

L

0

dx uref

(46)

can be viewed as a reference void time. Taking in account this relation and recognizing that R and η in eq 45 are function of time allows us to rewrite eq 45 as eq 1a in the main text. 2. Method Translation and Peak Resolution. Resolution, Rs, of two closely eluting peaks a and b with peak a preceding peak b can be found as35

Rs ) (tb - ta)/(4σ)

(48)

where tM is the void time, N is the column efficiency, and R is the propagation factor of the peak at the time of its elution. Consider a ratio Rs2/Rs1 of resolutions of the pair of peaks corresponding to the same pair of solutes in mutually translatable methods M1 and M2, respectively. In view of eq 47, one can write

(49)

which after the substitution of eq 48 for σ1 and σ2 becomes

Rs2 tb2 - ta2 R2(ta2) tM1 ) N /N Rs1 tb1 - ta1 R1(ta1) tM2 x 2 1

(47)

where ta and tb are retention times of the peaks and σ is their standard deviation. (In the following analysis we use standard deviation, σa, of the first elutant as the measure of the standard deviation, σ, for both peaks. The result would be the same if the standard deviation, σb, of the second elutant or, what is more appropriate, the average (σa + σb)/2 was used instead.) The latter can be found as

σ ) tM/(RxN)

Rs2 tb2 - ta2 σ1 ) Rs1 tb1 - ta1 σ2

(50)

As M1 and M2 are mutually translatable, eq 6 can be used to express retention times in M2 via retention times in M1. One can write

tM2 ) tM1/S, ta2 ) ta1/S, tb2 ) tb1/S

(51)

where speed gain S is the same for all solutes (including the nonretained ones). It has also been established earlier that, in mutually translatable methods, elution temperature of any solute is the same, indicating that, in eq 50, R2(ta2) ) R1(ta1). Substitution of this relation together with eqs 51 in eq 50 yields eq 34 in the main text. Received for review October 15, 1997. Accepted June 30, 1998. AC971141V

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