Anal. Chem. 2005, 77, 5670-5677
Expressions of the Fundamental Equation of Gradient Elution and a Numerical Solution of These Equations under Any Gradient Profile P. Nikitas* and A. Pappa-Louisi
Laboratory of Physical Chemistry, Department of Chemistry, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
The original work carried out by Freiling and Drake in gradient liquid chromatography is rewritten in the current language of reversed-phase liquid chromatography. This allows for the rigorous derivation of the fundamental equation for gradient elution and the development of two alternative expressions of this equation, one of which is free from the constraint that the holdup time must be constant. In addition, the above derivation results in a very simple numerical solution of the various equations of gradient elution under any gradient profile. The theory was tested using eight catechol-related solutes in mobile phases modified with methanol, acetonitrile, or 2-propanol. It was found to be a satisfactory prediction of solute gradient retention behavior even if we used a simple linear description for the isocratic elution of these solutes. In two recent papers, we have examined certain features of gradient elution in reversed-phase liquid chromatography.1,2 Our occupation with this issue has shown that there are at least two problems related to the fundamental equation for gradient elution. This equation is usually expressed as
∫
tR-t0
0
dt )1 t 0k φ
(1)
where tR is the solute elution time under gradient conditions, t0 is the column holdup time, kφ ) (tφ - t0)/t0 is the solute retention factor, which corresponds to a constant mobile-phase composition equal to φ, and tφ ≡ tR(φ) is the isocratic retention time when the mobile-phase composition is constant and equal to φ. The two problems we are going to discuss in the present paper are related to (a) the column holdup time, t0, and (b) the rigorous derivation of this equation. The presence of t0 in the upper limit of the integral of eq 1 clearly shows that this quantity must be constant. However, there are cases where the column holdup time varies with the variation of the mobile-phase composition.3,4 Therefore, it would be par* To whom correspondence should be addressed. Tel.: +30 2310 997773. Fax: +30 2310 997709. E-mail:
[email protected]. (1) Nikitas, P.; Pappa-Louisi, A.; Papachristos, K. J. Chromatogr., A 2004, 1033, 283. (2) Nikitas, P.; Pappa-Louisi, A. J. Chromatogr., A 2005, 1068, 279. (3) Oumada, F. Z.; Roses, M.; Bosch, E. Talanta 2000, 53, 667. (4) Rimmer, C. A.; Simmons, C. R.; Dorsey, J. G. J. Chromatogr., A 2002, 965, 219.
5670 Analytical Chemistry, Vol. 77, No. 17, September 1, 2005
ticularly useful if we could find an alternative expression of the fundamental equation for gradient elusion, which was not subject to the above constraint. In what concerns the derivation of eq 1, the two most commonly referred derivations of this equation have been presented by Snyder et al.5-9 and Jandera.10 However, a careful examination of these derivations shows that they are at least incomplete. This issue is discussed in detail in the next section. A survey of the available literature shows that the origin of the derivation of eq 1 may be found in the work carried out by Freiling11,12 and Drake.13 Thus, we should address this work, which however should be rewritten in the current language of chromatography. To sum up, in the present paper, we examine the origin of the fundamental equation for gradient elution, eq 1, based on Freiling and Drake’s work. This allows for the rigorous derivation of eq 1 and the development of two alternative expressions of this equation, one of which is free from the constraint that t0 must be constant. Thus, this work has two targets: the first is pedagogical and concerns the rigorous derivation of various expressions of the fundamental equation for gradient elution, and the second is strictly scientific and it is the application of the fundamental equation of gradient elution to systems of variable column holdup time. ON THE FUNDAMENTAL EQUATION OF GRADIENT ELUTION Here, based on the original work carried out by Freiling11,12 and Drake,13 we attempt to derive the fundamental equations describing gradient elution in liquid chromatography. This results in the rigorous derivation of eq 1, the development of two new expressions for the fundamental equation of gradient elution, and finally the clarification of the weaknesses in the derivations of eq 1 proposed by Snyder et al.5-9 and Jandera.10 Freiling’s Approach. For the theoretical study of gradient elution in liquid chromatography, Freiling and Drake have (5) Snyder, L. R. J. Chromatogr. 1964, 13, 415. (6) Snyder, L. R. Chromatogr. Rev. 1965, 7, 1. (7) Snyder, L. R.; Saunders, D. L. J. Chromatogr. Sci. 1969, 7, 145. (8) Snyder, L. R. In High Performance Liquid Chromatography; Horvath, C., Ed.; Academic Press: New York, 1980; Vol. 1, p 207. (9) Quarry, M. A.; Grob, R. L.; Snyder, L. R. Anal. Chem. 1986, 58, 907. (10) Jandera, P.; Churacek, J. Gradient Elution in Liquid Chromatography. Theory and Practice; Elsevier: Amsterdam, 1985. (11) Freiling, E. C. J. Am. Chem. Soc. 1955, 77, 2067. (12) Freiling, E. C. J. Phys. Chem. 1957, 61, 543. (13) Drake, B. Akriv. Kemi 1955, 8, 1. 10.1021/ac0506783 CCC: $30.25
© 2005 American Chemical Society Published on Web 08/04/2005
adopted, in fact, the theory of uniform rectilinear motion. The basic assumption of this approach is that the movement of the eluent chromatographic zone is described by the displacement of the band center and the fronts of the mobile phase are sharp enough. Under this assumption, an arbitrary continuous gradient formed in the mixer of the gradient elution system is approximated by a stepwise gradient composed of a large number n of infinitesimal small steps, δt. This is a reasonable approximation since in the limit n f ∞ the two gradients become identical, irrespective of their shape. All these steps are moved inside the column with a velocity v0 equal to v0 ) L/t0, where L is the total length of the chromatographic column. Therefore, each step occupies a part δLm of the column equal to
δLm ) v0δt ) (L/t0) δt
(2)
The analyte moves inside the column with a velocity va that depends on the mobile-phase composition. If this composition is φ, then va ) L/tφ. This velocity is referred to the column. If we consider the relative movement of the analyte with respect to the mobile phase, then this relative velocity var is given by var ) L/tφ - L/t0. Note that the sign of var is negative, because the relative movement of the analyte with respect to the mobile phase occurs contrary to the movement of the mobile phase. Let us now consider that a step of the mobile phase with composition φ meets the analyte and that the analyte is under the influence of this step for a time period equal to δtc. It is evident that for this time period the analyte is moving from the beginning to the end of this step and therefore we have
δLm ) -var δtc ) L
(
)
1 1 δtc t0 tφ
(3)
From eqs 2 and 3 we readily obtain the following very important equation
δLa δtc δt ) ) L t0kφ t0(1 + kφ)
(4)
where δLa ) va δtc ) (L/tφ) δtc is the distance traveled by the analyte inside the column at a time period equal to δtc. If δt f 0, the differences in the above equation may be replaced by differentials, and by integration of eq 4, we find the fundamental equation of gradient elution, eq 1. A clarification about the integral limits of this equation is given in the following section. Drake’s Graphical Approach. Drake’s approach is, in fact, identical to Freiling’s approach, but the derivation of eq 1 is graphical, as shown in Figure 1. In this figure, the fronts of the mobile phase due to a stepwise variation pattern of φ versus t created in the mixer are indicated by the lines AA′, BB′, CC′, and DD′. Due to this change in φ, the analyte follows inside the column the path Aa′c′E indicated by the corresponding thick line. Therefore, we have the following: OA′ ) eB′ ) t0, OE ) tR, and A′B′ ) B′C′ ) C′D′ ) δt. In addition, from the figure it becomes evident that the following equalities are also valid: bc ) ef ) tφ, d′c′ ) d′′c′′ ) BC ) δt, b′c′ ) b′′c′′ ) δtc, a′b′ ) Bb′′ ) δLa, and B′f ) ef - eB′ ) tφ - t0.
Figure 1. Schematic representation of the movement of the fronts of the mobile phase due to a stepwise variation pattern of φ vs t created in the mixer and the corresponding path of an analyte inside a chromatographic column.
Now the similarity of triangles Bb′′c′′ and abc yields Bb′′/ab ) b′′c′′/bc, which results in
δLa/L ) δtc/tφ
(5)
In addition, from the similarities in triangles Bb′′c′′ and Bef, we obtain d′′c′′/B′f ) Bd′′/BB′ ) Bb′′/Be, which yields
δLa/L ) δt/(tφ - t0)
(6)
These two equations lead directly to eq 4 and therefore to the derivation of eq 1. Note that the integration of eq 6 leads to eq 1 with an upper limit equal to tR - t0, because the sum of all δt is equal to A′E ) OE - OA′ ) tR - t0, as easily arises from Figure 1. Snyder’s Treatment. Snyder’s treatment also approximates a continuous gradient formed in the mixer by a stepwise one with a large number of infinitesimal small steps (dt f 0), but here this stepwise gradient is identical to that experienced by an analyte inside the column. Based on this assumption, Snyder made the additional assumption that when a differential volume element dV of mobile phase passes through the band center of the analyte, it moves this center by
dLa/L ) dV′/V′a
(7)
Here, V′ ) V - V0 and V′a ) Va - V0, where Va is the instantaneous retention volume and V0 the column dead volume. Note that V is interrelated to t through the flow rate F, since V ) tF. Therefore, eq 7 yields
dLa dt dt ) ) L ta - t0 t0kφ
(8)
which results directly in eq 1. Analytical Chemistry, Vol. 77, No. 17, September 1, 2005
5671
The derivation is undoubtedly correct, but its rigorousness is not self-evident. For example, one may wonder why we should use the corrected volumes V′ and V′a in eq 7 and not the uncorrected ones V and Va. That is, this derivation does clarify whether eq 7 or the following eq 9 is correct.
dLa/L ) dV/Va
(9)
Two Alternative Expressions of the Fundamental Equation. Two alternative expressions of the fundamental equation of gradient elusion may be derived as follows. From eq 4 we have
δtc δLa ) L t0(1 + kφ)
(15)
which upon integration yields The validity of eq 7 via eq 8, which is identical to eq 4, arises directly from the treatments of Freiling and Drake. Similarly, these treatments indicate the approximate nature of eq 9. Note that eq 9 yields the following approximate expression for the fundamental gradient elution equation
∫
tR
0
dt )1 t0(1 + kφ)
(10)
Jandera’s Derivation. The rigorousness of Jandera’s derivation of eq 1 could also be questioned. The starting point of Jandera’s derivation of eq 1 is the definition of the retention factor for an infinitesimal small gradient step from
ka )
dt - dt0 dt′ ) dt0 dt0
(11)
Equation 11 may be written as dt0 ) dt′/ka, which upon integration results in
∫
t0
0
dt0 ) t0 )
∫
tR-t0
0
dt ka
(12)
This equation is identical to eq 1 provided that t0 is constant, an assumption that is also adopted in the derivation of eq 1. However, from eq 11 we also obtain dt ) (1 + ka) dt0 that gives dt0 ) dt/ (1 + ka) and therefore
t0 )
∫
tR
0
dt 1 + ka
(13)
which is identical to the approximate eq 10 when t0 is constant. It is seen that the same basic eq 11 yields by simple mathematical transformations two mathematically different equations! Nevertheless, Jandera’s derivation can become rigorous if we take into account the real picture of gradient elution as depicted in Figure 1. From this figure we readily obtain that
ka )
δtc - δt0 b′c′ - b′d′ d′c′ δt ) ) ) δt0 b′d′ b′d′ δt0
(14)
It is seen that the difference δtc - δt0 is, in fact, the time step δt formed in the mixer. For this reason, if we write eq 14 as δt0 ) δt/ka and integrate, we obtain eq 12 and consequently eq 1 with t related to φ through the gradient profile formed in the mixer. Note again that the sum of all δt is equal to tR - t0, and for this reason, the difference tR - t0 is the upper limit of the integral of eq 12. 5672
Analytical Chemistry, Vol. 77, No. 17, September 1, 2005
∫
dtc
tR
0
t0(1 + kφ)
)1
(16)
This equation seems to be similar to the approximate eq 10, but it is totally different due to the different time variables, t in eq 10 and tc in eq 16. The physical meaning of these variables is discussed in the second paragraph. Equation 16 is mathematically equivalent to eq 1, and therefore, it is an alternative expression of the fundamental equation of gradient elusion. This equation derives also from eq 5, and this derivation clarifies that the upper limit of the integral should be tR, because the sum of δtc is equal to OE ) tR. Finally, eq 16 arises from Jandera’s treatment, because from eq 14 we obtain dtc ) (1 + ka) dt0, that is dt0 ) dtc/(1 + ka), which readily yields eq 16. It is seen that eq 16 can be used for prediction of the gradient retention time of the sample solute when there are variations in the holdup time t0, provided that the dependence of kφ upon φ is known. A similar issue has been discussed by Glajch et al.,14 who examined the effect of flow rate variations on the gradient retention time. The whole treatment is based on the linear solvent strength theory8,14 and results in a linear dependence of the difference in tR values due to changes in the flow rate with the logarithm of the flow rate. Note that the time parameter in eq 16 is not the same as that in eq 1. In eq 1, t is related to φ as programmed in the mixer of the gradient elution system. However, this variation profile of φ versus t is no more valid inside the column. Equation 4 shows that if a step lasts in the mixer δt, the same step for a certain analyte lasts longer inside the column, δtc, which means that the analyte feels this step for a time interval δtc, which is always longer than that of δt. This is expected, because the analyte travels inside the column slower than the mobile phase. Thus, the faster the analyte moves, the longer it feels each solvent step. The φ versus tc profile that feels each analyte inside the column can be directly determined from eq 4 if this profile is stepwise. In the case of continuous gradients, the original φ versus t gradient formed in the mixer is transformed to φ versus tc gradient again by means of eq 4, which by integration yields
tc )
∫
t
0
1 + kφ dt kφ
(17)
An alternative expression of the fundamental equation of gradient elution arises from eq 14 if we write it as δt ) kaδt0. However, from eq 14, we also have δt0 ) δtc - δt. Now if we define the new variable (14) Glajch, J. L.; Quarry, M. A.; Vasta, J. F.; Snyder, L. R. Anal. Chem. 1986, 58, 280.
td ) tc - t
(18)
to avoid any confusion with t0, the equality δt ) kaδt0 is written as δt ) kaδtd, which after integration yields
tR ) t0 +
∫
t0
0
ka dtd
(19)
where
td )
∫
t
0
1 + kφ dt - t ) kφ
∫
t
0
dt kφ
(20)
It is seen that there are three mathematically equivalent expressions of the fundamental gradient elution equation, which are eqs 1, 16, and 19. It is interesting to point out that each one uses a different independent variable, t, tc, and td, respectively. In addition, despite their equivalence, each one exhibits its own advantages and drawbacks. Thus, eqs 1 and 19 both presume that t0 is constant. It is also evident that from these two equations eq 1 is simpler because the calculation of tR needs one integration while the calculation of tR through eq 19 requires two integrations. The shortcoming of eqs 1 and 19 to be valid under constant t0 does not hold for eq 16. Therefore, from the three expressions of the fundamental gradient elution equation, eq 16 is the more general, and it is, in fact, equivalent to eqs 1 and 19 only when t0 is constant. Effect of the Dwell Time. In the treatment discussed above the dwell time tD, i.e., the time needed for a certain change in the mixer to reach the beginning of the chromatographic column, has been ignored. If this time is taken into account, Figure 1 is modified to Figure 2. From this figure we readily obtain that eqs 5 and 6 are still valid for all infinitesimal movements of the mobilephase fronts that correspond to the infinitesimal small time steps δt. However, at the very beginning and for a time period equal to tD, the solute covers inside the column a distance L0 ) g (see Figure 2), which is given by
L0/L ) g/L )
t/D tφin
)
t0(1 + kφin)
t/D ) tD(1 + kφin)/kφin
(22)
Therefore, in this case, the integral of eq 4 is no longer equal to 1 but equal to 1 - L0/L ) 1 - tD/(t0kφin). Consequently, eq 1 is extended to tR-t0-tD
∫
/ tR-tD
0
dtc t0(1 + kφ)
tD )1 t0kφin
+
(24)
The upper limits arise directly from Figure 2. It is seen that the sum of all δt is equal to B′E ) OE - OA′ - A′B′ ) tR - t0 - tD, and the corresponding sum of all δtc is equal to hE ) OE Oh ) tR - t/D. We should point out here that a simple and elegant derivation of eq 23, which seems to be free from the shortcomings mentioned above for the derivations of the fundamental gradient equation proposed by Snyder and Jandera, has been given by Schoenmakers et al.15 Finally, eq 19 is extended to
∫
t0/
0
kadtd
(25)
and
as it straightforwardly arises from eqs 5 and 6. Here, φin is the initial mobile-phase composition and tD/ ) Oh. It is seen that tD/ is the time interval that is related to tD is the same way that δtc is related to δt. From eq 21 we obtain
0
and eq 16 to
tR ) t 0 + tD +
t/D
tD tD ) (21) L0/L ) tφin - t0 t0kφin
∫
Figure 2. Schematic representation of the movement of an analyte and the fronts of the mobile phase due to a stepwise variation pattern of φ vs t created in the mixer with a dwell time tD.
tD dt + )1 t0kφ t0kφin
(23)
where
t/0 ) t0 - (t/D - tD) ) t0 - tD/kφin
(26)
because here the sum of all δtc plus the initial term t/D - tD is equal to t0. A SIMPLE NUMERICAL SOLUTION OF THE FUNDAMENTAL EQUATIONS OF GRADIENT ELUTION UNDER ANY GRADIENT PROFILE The derivation of the fundamental equations of gradient elution presented above leads almost directly to the following very simple numerical solution of eqs 23 and 24. In practice, all common mixers do not produce an ideally smoothed t versus φ profile. Instead, an arbitrary gradient profile is approximated by small δφ steps at δt time intervals. Under these conditions, an analyte is (15) Schoenmakers, P. J.; Billiet, H. A. H.; Tijssen, R. J. Chromatogr. 1978, 149, 519.
Analytical Chemistry, Vol. 77, No. 17, September 1, 2005
5673
eluted when the sum of δLa, i.e., the sum of the subsequent distances traveled by the analyte inside the column at time intervals equal to δtc, is equal to L. Therefore, from eq 4, we obtain that the analyte is eluted when the following condition is fulfilled
∑
δLa
)
L
tD t0kφin
n
+
δt
∑tk
g1
(27)
i)1 0 φ
where n is the least number of terms of the sum that makes the above inequality valid. Now it is evident that when the condition expressed by eq 27 is fulfilled, then the gradient elution time is given by
tR ) t0 + tD + nδt
(28)
This equation is the numerical solution of eq 23. For eq 24, we may work similarly. But the simpler way is the following. If n is the least integer that validates inequality eq 27, then tR may be calculated from n
tR ) t/D +
∑ i)1
δtc ) t/D + δt
n
1 + kφ
i)1
kφ
∑
(29)
because from Figure 1 we readily obtain that the sum of all δtc plus the time interval t/D is equal to tR. Note that δtc is not a constant, like δt, as readily arises from Figure 1. From eqs 27-29, we readily find that the numerical error in the calculation of tR by means of these equations is less than δt for eq 28 and less than (1 + 1/kφ)δt for eq 29. In our study, we keep this error less than 0.02 min. Finally, eq 25 may be numerically solved using Simpson’s rule for the determination of the integral. In this approach, td is calculated directly from eq 20 using again Simpson’s rule. EXPERIMENTAL SECTION The liquid chromatography system used for the gradient measurements is the same with that described in refs 1 and 2. It is consisted of a Shimadzu LC-10AD pump, equipped with a lowpressure gradient system (FCV-10AL), a C18 column [250 × 4 mm MZ- Analyzentechnik (5-µm Inertsil ODS-3)] thermostated by a CTO-10AS Shimadzu column oven at 25 °C, and a Gilson electrochemical detector (model 141) equipped with a glassy carbon electrode. The detection of the analytes was performed at 0.8 V versus the Ag/AgCl reference electrode. The flow rate was 1 mL/min and the dwell time tD 4.6 min.1 Eight catechol-related solutes, dopamine (da), serotonin (5ht), 3,4-dihydroxyphenylacetic acid (dopac), 5-hydroxyindole-3-acetic acid (hiaa), vanillylmandelic acid (vma), 5-hydroxytryptophol (htoh), 3,4-dihydroxyphenyl glycol (hpg), and homovanillic acid (hva) at a concentration 5 µg/mL were used to test the gradient equations proposed in this paper. The mobile phases were aqueous phosphate buffers (pH 2.5) modified with methanol, acetonitrile, or 2-propanol. Their total ionic strength was held constant at I ) 0.02 M. All chemicals were used as received from commercial sources. 5674
Analytical Chemistry, Vol. 77, No. 17, September 1, 2005
Table 1. Average Values of Holdup Time (in min) Determined by KBr and NaNO3 water-acetonitrile
water-2-propanol
φ
t0
φ
t0
0 0.02 0.06 0.1 0.14 0.2 0.3
1.82 1.79 1.75 1.7 1.7 1.62 1.55
0 0.02 0.04 0.06 0.1 0.14 0.2 0.3
1.82 1.8 1.78 1.78 1.75 1.73 1.72 1.72
Note that the isocratic behavior of the above analytes as well as the holdup time t0 for every mobile phase has been studied in ref 16. Due to importance of the variation of the holdup time upon φ in this study, we made additional experiments for the determination of t0 as a function of the mobile-phase composition φ. In our previous study in ref 16, we have used water as a marker for the determination of t0. However, it is known that the elution time of neutral species used as markers usually does not depend on pH and ionic strength, whereas for ionic markers, the holdup time may depend on the particular buffer used.3,17 For this reason, here we used the ionic markers KBr and NaNO3. We found again that the holdup time depends on the mobile-phase composition only in the water-acetonitrile and water-2--propanol solutions. The average values of t0 determined by means of the two ionic markers in these mobile phases are given in Table 1. It is seen that they do not exhibit significant deviations from those in ref 16. As a first approximation, the following linear expressions may be used for the dependence of t0 upon φ: t0 ) 1.8 - φ and t0 ) 1.8 - 0.3φ for the water-acetonitrile and water-2-propanol solutions, respectively. Better approximations are the following expressions: t0 ) 1.82 - 0.96φ and t0 ) 1.82 - 0.85φ + 1.76φ2, respectively. DATA ANALYSIS In the present paper, all calculations have been performed at Excel spreadsheets. Figures 3 and 4 show spreadsheets for the calculation of the elution time of a single solute, hva, from eqs 28 and 29, respectively, which solve numerically eqs 23 and 24. All the details concerning the calculations are shown in these figures. In this example application of eqs 23 and 24, the dwell time tD is arbitrarily set equal to 1 min, because this selection gives a great difference between the elution times calculated from the above equations. The gradient is given by the quadratic expression φ ) φin + b1t + c1t2 provided that φ e φmax otherwise φ ) φmax. We have chosen φmax ) 0.3 because this was the maximum φ value used in the isocratic study of hva in water-acetonitrile mobile phases.16 Finally, the isocratic behavior of hva is described by the three-parameter rational function16
ln kφ ) a -
cφ 1 + bφ
(30)
(16) Pappa-Louisi, A.; Nikitas, P.; Balkatzopoulou, P.; Malliakas, C. J. Chromatogr., A 2004, 1033, 29. (17) Roses, M.; Canals, I.; Allemann, H.; Siigur, K.; Bosch, E. Anal. Chem. 1996, 68, 4094.
Figure 3. Spreadsheet for the calculation of the elution time of a single solute, hva, from eqs 27 and 28 which solve numerically eq 23 when t0 is constant (in a water-acetonitrile mobile phase). The details of the calculation are shown on this figure.
The adjustable parameters a, b, and c were determined using the isocratic data16 by means of Solver. The holdup time was kept constant and equal to 1.72 min for the determination of parameters a, b, and c shown in Figure 3 and variable with values given in ref 16 for the corresponding parameters depicted in Figure 4. Note that if we use in Figure 4 t0 ) 1.72 - 0φ ) 1.72 min and replace the a, b, and c values by those of Figure 3, then the two equations, eq 28 and eq 29, give identical results, within the computation error, which can be as small as we want by reducing the time step δt. For the application of eqs 23 and 24 to the calculation of the gradient elution times of the mixture of catecholamines, the spreadsheets of Figures 3 and 4 were properly extended and modified first to include eight solutes and second to use the following gradient
[
φin + b1t + c1t2 when t e ts φ ) a2 + b2t + c2t2 when t > ts φmax when φ > φmax
]
(31)
where from the continuity of φ at t ) ts we obtain that a2 ) φin + (b1 - b2)ts + (c1 - c2)ts2. Note that this gradient is applied after the dwell time. Therefore, the total gradient may consist of four parts: an initial isocratic part due to dwell time, a quadratic part that lasts ts minutes, another quadratic part for t > ts, and finally another isocratic part in the case that φ exceeds a maximum value
φmax. For simplicity, the isocratic behavior of each solute is here described by the linear expression
ln kφ ) a + bφ
(32)
where parameters a and b are determined by a linear fitting using the Linest function of Excel. In this fitting, only two sets of experimental isocratic retention times versus φ were used for each modifier. Particularly, the linear isocratic description of solutes was based on the retention data obtained at φ ) 0.2 and 0.4 for methanol, φ ) 0.1 and 0.2 for acetonitrile, and φ ) 0.04 and 0.14 for 2-propanol. The parameters that should be determined for the optimum separation of a mixture of solutes are the following: φin, ts, b1, c1, b2, and c2. In the present paper, we have not used a certain rigorous optimization technique for the evaluation of these parameters, such as grid search, nonlinear least-squares method, or genetic algorithms. We have estimated them by trial and error. In particular, based on either eq 28 or eq 29, the elution times of all solutes are determined at a spreadsheet. These times are subsequently used for the calculation of the minimum difference δtR,min ) |tR(solute i) - tR(solute j)|, when i * j range all possible values, and the construction of the graph of the calculated chromatogram of the eight catecholamines with vertical lines in place of peaks. The estimation of the best values for parameters φin, ts, b1, c1, b2, and c2 that lead to the optimum separation of the mixture of Analytical Chemistry, Vol. 77, No. 17, September 1, 2005
5675
Figure 4. Spreadsheet for the calculation of the elution time of a single solute, hva, from eqs 27 and 29 which solve numerically eq 24 when t0 varies linearly with φ. In the present example, t0 ) 1.8 - φ in a water-acetonitrile mobile phase. The details of the calculation are shown in the figure.
catecholamines is facilitated by the fact that the first four catecholamines, da, hpg, 5ht, and vma, are always eluted under isocratic conditions and only the last four, dopac, htoh, hiaa, and hva, under gradient conditions. Thus. the gradient profile should, in fact, shift the elution times of dopac, htoh, hiaa, and hva toward lower values. We found out that after some trials we could easily determine a good gradient characterized by a reasonably high δtR,min value, usually δtR,min > 0.4 min, when the maximum elution time was ∼10 min. RESULTS AND DISCUSSION We first examine whether the assumption of a constant t0 value in a system where t0 varies with the composition of the mobile phase affects significantly the value of tR calculated from eq 23. For this reason, we used a variety of tD values and gradient profiles for the water-acetonitrile system. We found that for the particular systems studied the effect of this assumption is in general weak but there are cases where the error of this assumption may become significant. Such a case is depicted in Figures 3 and 4. It is seen that in a system with tD ) 1 min, if we take into account the variation of t0, hva is eluted at tR ) 4.99 min, whereas the assumption that t0 is constant and equal to the average value of all t0 values yields the imprecise result tR ) 5.51 min, i.e., an error of 10%. Due to the above observation, the calculation of the gradient elution times of the mixture of catecholimines was based on both 5676 Analytical Chemistry, Vol. 77, No. 17, September 1, 2005
Figure 5. ED chromatograms of an eight-component mixture composed of (1) da, (2) hpg, (3) 5ht, (4) vma, (5) dopac, (6) htoh, (7) hiaa, and (8) hva. They are recorded under gradient conditions using methanol as modifier and the following parameter: φin ) 0.2, ts ) 2 min, b1 ) 0.25, c1 ) -0.1, b2 ) 0.2, and c2 ) 0. The dotted vertical lines indicate the predicted retention times by means of eqs 27 and 28 using t0 ) 1.84 min, whereas the dash-dotted line shows the variation pattern of φ when it reaches the electrochemical detector.
eqs 28 and 29, which solve numerically eqs 23 and 24, respectively. Figures 5-7 show chromatograms recorded under optimum separation conditions. The dotted vertical lines indicate the predicted retention times calculated from eqs 27 and 29 using variable t0, except for the case of methanol-water solution, where
Figure 6. As in Figure 5, but for acetonitrile instead of methanol using φin ) 0.1, ts ) 2 min, b1 ) 0.05, c1 ) 0.01, b2 ) 0, and c2 ) 0. The dotted vertical lines indicate the predicted retention times by means of eqs 27 and 29 using t0 ) (1.82 - 0.96φ) min.
t0 is constant, whereas the dash-dotted line shows the variation pattern of φ when it reaches the electrochemical detector. It is seen that the separation is quite effective and that there is a reasonably good prediction of the retention times from eq 29. In fact, the average error between experimental and calculated tR values is lower than 2.5% as shown in Table 2. In what concerns eq 28, if we assume as a constant t0 value the average value of t0 in the φ region used for the linear description of the isocratic elution of solutes, then this equation can be used effectively in prediction of tR values. Thus, Table 2 depicts that the average deviations between experimental and predicted under gradient conditions by eq 28 with a value of t0 equal to 1.84 for methanol, 1.69 for acetonitrile, and 1.75 min for 2-propanol, respectively, are slightly higher than those obtained by eq 29 and a variable t0 value. Finally, the choice of the variation model for t0 has a rather small effect on the predicted retention times. We found that the average error between the predicted values of tR when we use (a) t0 ) 1.8 - φ and t0 ) 1.82-0.96φ for the water-acetonitrile solutions and (b) t0 ) 1.8-0.3φ and t0 ) 1.82-0.85φ + 1.76φ2 for the water2-propanol solutions is less than 1 and 0.2%, respectively, whereas the corresponding maximum error is 1.4 and 0.4%.
Figure 7. As in Figure 5, but for 2-propanol instead of methanol using φin ) 0.04, ts ) 2 min, b1 ) 0.05, c1 ) 0, b2 ) 0.02, and c2 ) 0. The dotted vertical lines indicate the predicted retention times by means of eqs 27 and 29 using t0 ) (1.82 - 0.85φ + 1.76φ2) min. Table 2. Percentage Average Deviation between Experimental and Predicted Retention Times of Solutes Eluted under Gradient Conditions Describing in Figures 5-7 modifier
eq 28
eq 29
methanol acetonitrile 2-propanol
2.1 2.2 2.5
1.4 1.6
Therefore, with the proper choice of t0, the fundamental eq 23 can be used in gradient optimization techniques, although the most secure procedure is based on eq 24. Take into account that this satisfactory prediction of solute gradient retention behavior was achieved using a linear description for the isocratic elution of the solutes, which requires only two sets of experimental ln t versus φ data for each modifier. Received for review April 20, 2005. Accepted July 7, 2005. AC0506783
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