Article pubs.acs.org/ac
pH-Gradient Reversed-Phase Liquid Chromatography of Ionogenic Analytes Revisited P. Nikitas,* A. Pappa-Louisi, and Ch. Zisi Laboratory of Physical Chemistry, Department of Chemistry, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece S Supporting Information *
ABSTRACT: The work carried out on the linear pH-gradient is critically reviewed in combination with the development of new expressions for the retention time of ionizable analytes. It is shown that under ideal linear pH-gradient conditions the better a model describes the 1/k versus pH plot, where k is the retention factor of the analyte, the better performance it exhibits. However, under real experimental conditions the fitting and prediction performance of a model depend upon its ability to compensate via the regression procedure the various nonidealities, like deviations from the linearity of the pH-gradient profile and the effect of the organic modifier on pK. In general, all models examined exhibit good fitting and prediction performance, and the best model is based on a simple algebraic sigmoid function for 1/k versus pH. However, the main drawback of this model, as well as of all models that are based on the solution of the fundamental equation of gradient elution, is the rather complicated expressions of the retention time. This weakness is overcome by using simple linear models, which give very satisfactory results especially in cases where the retention time varies linearly with the programmed gradient time. The extension of these models so that they are not subject to this restriction is also considered.
T
and his colleagues attempted an approximate solution.6 Note that a very simple approach for retention modeling of solutes under pH-gradient conditions at various organic contents in the mobile phase has been recently proposed.12 The aim of this study is to critically review the work on pHgradients in RP-LC, and to develop better approximate solutions of the fundamental equation of gradient elution. Thus, we examine whether the performance of a model depends upon a realistic description of the elution process or upon its ability to compensate effectively for deviations from idealities like the linearity of the pH-gradient profile. The final target is to indicate the model or models with the best fitting and prediction performance.
he technique of pH-gradients in reversed-phase liquid chromatography (RP-LC) is very useful to separate ionogenic analytes. The presence of these analytes in the mobile phase as a mixture of nondissociated and dissociated forms depending on pH results in the strong dependence of their retention upon the mobile phase pH. Therefore, by manipulating properly the pH, we may achieve an effective separation of a mixture of ionizable compounds, especially if a proper theory that relates the retention time to pH is available. Thus, to benefit from the pH-gradient separation mode, models enabling retention prediction of solutes under pH-gradient conditions are requested, which would allow a computer-aided optimization of the separation. In this area the work carried out by Kaliszan’s group is impressive.1−8 This group has developed a comprehensive theory allowing the prediction of solute retention in the pH-gradient mode as well as the pK determination of monoprotic acids and bases.1−6 Also, a modeling approach allowing the description of retention time and peak width in the combined pH/organic modifier gradient was recently proposed.7 All the above attempts to describe the retention of solutes in the pH-gradient mode theoretically are based on the solution of the fundamental equation of gradient elution.9−11 Although this equation has a solution of the general form f(tR, pH, pK, k0, k1) = 0, where tR is the retention time and k0, k1 are the retention factors of the nonionized and the ionized forms of an ionogenic analyte, this solution, i.e., f(tR, pH, pK, k0, k1) = 0, is not always useful. This is because f(tR, pH, pK, k0, k1) = 0 should be solved numerically with respect to tR, but as shown in the next section, this can be easily done only when k0, k1, and pK are known from isocratic data. We presume that for this reason Kaliszan © 2012 American Chemical Society
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RIGOROUS APPROACH The rigorous approach of pH linear gradients in reversed phase liquid chromatography has been developed by Kaliszan and his colleagues.1,2 A modified approach, which is compatible to the presentation of all models in this paper, is the following. The fundamental equation of gradient elution when changes in the mobile phase pH and/or composition occur may be expressed as9−11,13
∫0
tR − t0
dt =1 t0k
(1)
Received: March 30, 2012 Accepted: June 29, 2012 Published: June 29, 2012 6611
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the last isocratic portion of the pH versus t curve, eq 4 is valid and may be used for the estimation of k0, k1, and pK. However, in the middle increasing/decreasing portion of the pH gradient, for the estimation of k0, k1, and pK we should use eq 5 in a nonlinear least-squares procedure. Therefore, at each step of this procedure eq 5 should be solved numerically with respect to tR, which is at least time-consuming. Note that one may consider the following alternative approach. We may use the experimental tR values in the log function of eq 5, and insert the resulting expression of tR in the sum of squared residuals. However, when k0 → 0, then Br → ∞ and tR calculated from eq 5 becomes equal to the experimental value introduced in the log function of this equation, irrespective of the values of k1 and pK. Thus, this nonlinear least-squares procedure yields the erroneous result that k0 = 0 and arbitrary values for k1 and pK.
where t0 is the column dead volume, and k the retention factor of the analyte. Note that t0 is considered as constant independent of time, t, and as such, it could be put before the integral sign, as a multiplier. Equation 1 exhibits an analytical solution when a monoprotic analyte is eluted under a linear pH-gradient. For monoprotic analytes k is given by14−16
k=
k 0 + k110i(pH − pK ) 1 + 10i(pH − pK )
(2)
where i is an indicator parameter which is equal to 1 for acids and −1 for bases, k0, k1 are the retention factors of the nonionized and the ionized forms of an ionogenic analyte, and pK = −log K, K being the equilibrium constant of the appropriate acid/base equilibrium (see Supporting Information). In addition, the pH-linear gradient profile between an initial value pH0 and a final one pHf may be expressed as
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KALISZAN−WICZLING’S APPROXIMATION To overcome the above problem as well as in order to obtain an analytical expression for tR under pH gradient conditions, Kaliszan and Wiczling approximated eq 2 by the following expression6
⎧ pH 0 t ≤ ts + t D ⎪ ⎨ pH = pH 0 + λ(t − ts − t D) ts + t D < t < ts + tG + t D ⎪ t ≥ ts + tG + tD ⎩ pH f (3)
⎧k0 t ≤ t1K ⎪ k(t ) = ⎨ k 0 + (k1 − k 0)(t − t1k)|λ| /3 t1K < t < t 2K ⎪ ⎪k t ≥ t 2K ⎩ 1
where tG is the gradient duration, λ is the steepness of the gradient, λ = (pHf − pH0)/tG, tD denotes the dwell time, and ts is the pH-gradient starting time, i.e, the time the pH-gradient starts in the pump program. If eqs 2 and 3 are introduced into eq 1, we obtain (see details in the Supporting Information) ⎧ t0 + t0k(pH 0) I1 ≥ 1 ⎪ tR = ⎨t + t + t + t I1 + I2 < 1 s G D ⎪0 (1 ) (pH ) I I t k + − − ⎩ 1 2 0 f
(9)
where
t 2K
(4)
whereas when I1 < 1 and I1 + I2 ≥ 1, tR is calculated from solving the following equation t R = t0 + t D + ts −
(5)
Here, Ar and Br are given by A r = 10i pH0 − λi(ts+ tD) − i pK ,
Br = k1A r /k 0
⎧ t0 + t0k(tA ) I1 ≥ 1 ⎪ ⎪ 3k(tA )e BK − 3k 0 t R = ⎨ t0 + t1K + I1 < 1 & I1 + I2 ≥ 1 |λ|(k1 − k 0) ⎪ ⎪ ⎩ t0 + t 2K + t0k1(1 − I1 − I2) I1 + I2 < 1
(6)
and I1, I2 are calculated from I1 =
∫0
ts + t D
t + tD dt = s ≥1 t0k t0k(pH 0)
(7)
and I2 =
ts + tG + t D
∫t +t s
D
(11)
where tA = t1K when t1K > tD + ts, tA = tD + ts when t1K < tD + ts, and
dt t0k
⎧ ⎡ 1 + B 10 λi(ts+ tG+ tD) ⎤⎫ ⎪ ⎪ 1 A r − Br r ⎢ ⎥⎬ t log =⎨ + G ( ) + i t t λ ⎪ s D ⎢⎣ 1 + Br 10 ⎥⎦⎪ λiBr ⎭ t0k 0 ⎩
(10)
In addition, they introduced eq 9 into eq 1 under the assumptions that both acidic and basic analytes are totally nonionized at the beginning of the gradient, fully ionized at the end of the gradient, and that tD + ts ≤ t1K < t2K ≤ tG + tD + ts. Here, we extended the treatment to include the case t1K ≤ tD + ts, which means that the analytes may be ionized at the beginning of the gradient. The details of the solution of eq 1 when k is given by eq 9 are given in the Supporting Information. We obtain
⎡ 1 + B 10 λi(tR − t0) ⎤ A r − Br r ⎥ log⎢ ⎢⎣ 1 + Br 10 λi(tD+ ts) ⎥⎦ λiBr
+ (1 − I1)t0k 0
pK − i1.5 − pH 0 and λ pK + i1.5 − pH 0 = ts + t D + λ
t1K = ts + t D +
BK = (1 − I1)t0(k1 − k 0)|λ| /3, I2 =
(8)
Equation 4 gives an analytical expression of tR, but eq 5 should be solved numerically with respect to tR. This can be easily done when k0, k1, and pK are known from isocratic measurements. In contrast, the estimation of k0, k1, and pK from pH gradient data needs very careful experimental design for the following reason. If the analyte is eluted in the first or
I1 =
3 ln(k(t 2K )/k(tA )) |λ|t0(k1 − k 0)
tA , t0k(tA ) (12)
Note that if tD + ts ≤ t1K, then k(tA) = k0.
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ALTERNATIVE APPROXIMATIONS The main problem with the above approach appears to be the following. The rigorous eq 2 is a sigmoid function, and the 6612
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same is valid for the function 1/k, which is, in fact, used in eq 1. However, approximating k by a linear function, 1/k is not linear any more. This is clearly depicted in Figure 1A, where we observe the poor description by eq 9 of the inflection region of the 1/k versus pH curve. To overcome this problem we may examine the following approximations of 1/k.
If eq 14 is introduced into eq 1 and again under the assumption that t2L ≤ ts+ tD+ tG, we obtain ⎧ t0 + t0k(tA ) ⎪ ⎪ ⎪ − AL + AL2 + 2BL C L tR = ⎨ t0 + ⎪ BL ⎪ ⎪ t + t + t k (1 − I − I ) ⎩0 2L 0 1 1 2
I1 ≥ 1 I1 < 1 & I1 + I2 ≥ 1 I1 + I2 < 1 (17)
where I1 =
tA , t0k(tA )
I2 =
C L = AL tA +
2 − t A2 )/2 AL (t 2L − tA ) + BL (t 2L , t0
BL t A2 + (1 − I1)t0 2
(18)
Sigmoid Approximation of 1/k. Equation 2 is a sigmoid function, and the same is valid for the function 1/k used in eq 1. There are several sigmoid functions, but the simplest algebraic function that leads to an analytical solution of eq 1 may be the following ⎛ 1 = a⎜⎜1 + k ⎝
⎞ ⎟ 2 ⎟ 1 + (pH − c) ⎠ b(pH − c)
(19)
This equation represents quite satisfactorily the 1/k versus pH curve, although some deviations appear at pH < c − 2 and pH > c+2 provided that 0 < pH < 14 (Figure 1B). Note that 1/ k is sigmoidal when pH ∈ (c − 2, c + 2) and becomes practically constant outside this region. If eqs 19 and 3 are introduced into eq 1, we obtain ⎧ t0(1 + k(pH 0)) I1 ≥ 1 ⎪ ⎪ ⎪t I1 < 1 & I1 + I2 > 1 0 ⎪ ⎪ 2 2 2 − + + − − b b A e A eb λ λ 1 ( ) S S tR = ⎨ ⎪+ λ(b2 − 1) ⎪ ⎪ I1 + I2 ≤ 1 ⎪ t0 + ts + tG + t D ⎪ + (1 − I − I )t k(pH ) ⎩ 1 2 0 f
Figure 1. Plots of 1/k vs pH for an analyte with k0 = 30, k1 = 2, pK = 3 calculated from eq 2 (), and (A) eq 9 (−·−), eq 14 (---); (B) eq 19 (---), and eq 24 (+). For eq 19 we used a = 0.2543, b = 1.039, and c = 4.11 obtained from nonlinear fitting of 1/k vs pH data of eq 2 in the range pH = 1−10 with steps of 0.1.
Linear Approximation of 1/k. An obvious correction of Kaliszan−Wiczling’s approximation is to linearize 1/k instead of k (Figure 1A). Note that the inflection point of eq 2 is at pHc = pK, but the inflection point of 1/k is at pHc = pK + i log(k 0/k1)
(20)
where
(13)
AS =
Therefore, we may write ⎧1/k 0 t ≤ t1L ⎪ 1 = ⎨ AL + BL t t1L < t < t 2L k(t ) ⎪ t ≥ t 2L ⎩1/k1
(21)
(14)
BL = i(1/k1 − 1/k 0)λ /3
(22)
⎡ b I2 = a⎢tG + ( 1 + (λ(ts + t D + tG) + e)2 ⎣ λ ⎤ − 1 + (λ(ts + t D) + e)2 )⎥ /t0 ⎦
(23)
I1 =
and
(15)
Combined Approximation of 1/k. As pointed out above, eq 19 deviates from the corresponding expression of 1/k of eq 2 at pH < c − 2 and pH > c + 2. Therefore, the performance of eq 19 may be improved if we redefine it as follows
and AL and BL are calculated from the continuity of 1/k at t = t1L and t = t2L. We find AL = 1/k 0 − BL t1L ,
ts + t D t0k(pH 0)
e = pH 0 − λ(ts + t D) − c ,
where pHc − i1.5 − pH 0 t1L = ts + t D + and λ pHc + i1.5 − pH 0 t 2L = ts + t D + λ
t0(1 − I1) b + ts + t D + 1 + (λ(ts + t D) + e)2 λ a
(16) 6613
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⎧ ⎪ a − 2iab/ 5 ⎪ ⎪ ⎪ ⎪ ⎛ b(pH − c) 1 = ⎨ a⎜⎜1 + k ⎪ ⎝ 1 + (pH − c)2 ⎪ ⎪ ⎪ a + 2iab/ 5 ⎪ ⎩
pH ≤ c − 2 acids
A = t0 + ts + t D + γ
pH ≥ c + 2 bases ⎞ ⎟ c − 2 < pH < c + 2 ⎟ ⎠
B=γ
(24)
where ⎛1 1⎞ a = ⎜ + ⎟ /2, k1 ⎠ ⎝ k0
⎞ ⎛ 1 b = i 5⎜ − 1⎟ /2 ⎠ ⎝ ak1
The two branches of eq 24, 1/k0 = a − 2iab/√5 and 1/k1 = a + 2iab/√5, arise directly from eq 19 if we put k = k0 at pH = c − 2i and k = k1 at pH = c + 2i, because the composed function is continuous at the ends of the interval (c − 2i, c + 2i). Figure 1B shows that eq 24 represents the 1/k versus pH curve absolutely satisfactorily. The tR expression arising from eq 24 may be expressed as
I20 = γ
(28)
(32)
ab[pH 0 − c − γ 1 + (pH 0 − c)2 ] t0
EFFECT OF THE MODIFIER CONTENT The analytical expressions for tR presented above hold when the modifier content, φ, is kept constant during the pH-gradient. Therefore, these expressions can be used for prediction and optimization in pH-gradient runs carried out between a given initial and final pH value with different programmed gradient time, tG, and with different organic modifier content, φ, provided that the dependence of k0, k1, and possibly pK upon φ in eqs 11, 17, and 26, and of a, b, c upon φ in eq 20 is known. If the range of φ values is relatively narrow, we may adopt a linear dependence of ln k0, ln k1 upon φ k 0 = exp(k 00 + k 01φ),
and ⎡ b I2 = a⎢t 2C − tA + ( 1 + (λt 2C + e)2 ⎣ λ ⎤ − 1 + (λtA + e)2 )⎥ /t0 ⎦
k1 = exp(k10 + k11φ)
(34)
and keep pK constant, an approximation that has been accepted by Kaliszan’s group.1,6,8 In what concerns parameters a, b, and c of eq 20, we may use for simplicity the expressions a = exp(a0 + a1φ),
(29)
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b = b0 + b1φ
(35)
keeping c constant. At this point it is interesting to note that in a recent paper the effect of the modifier content has been treated in a semiempirical way as follows.12 From the experimental observation that tR depends linearly upon tG and quadratically upon φ
LIMITING CASES A very interesting feature of the pH-gradients at constant modifier content, φ, is that the plots of tR versus tG are linear in the majority of cases.12 This property is predicted from eq 20 provided that |1 − b2| ≪ (ASλ + e)2. Then, after some algebra (see Supporting Information), we find that B λ
(33)
■
t1C = ts + t D +
tR = A +
I21 λ
and γ = j when j(λtG + pH0 − c) > 0. Equation 33, in combination with eq 20, branch I1 + I2 ≤ 1, results in a linear dependence of tR upon 1/λ and thus of tR upon tG. Therefore, the plots tR versus tG may be linear provided that the conditions |1 − b2| ≪ (ASλ + e)2 and/or (λtG + pH0−c)2 ≫ 1 hold. It is also interesting to note that this dependence may consist of two linear branches if the quantity ASλ + e (or λtG + pH0 − c) changes sign within the range of tG values.
where
t 2C
(31)
atG(b + γ ) , t0
I21 = γ
(26)
pHc − 2i − pH 0 and λ pHc + 2i − pH 0 = ts + t D + λ
b+γ
where
⎧ t0 + t0k(tA) I1 ≥ 1 ⎪ ⎪ ⎪ I1 < 1 & I1 + I2 > 1 ⎪ t0 tR = ⎨ 2 2 2 − + + − − b 1 b ( A e ) A eb λ λ S S ⎪+ ⎪ λ(b2 − 1) ⎪ ⎪ t + t + t k (1 − I − I ) I1 + I2 ≤ 1 ⎩0 2C 0 1 1 2
(27)
b[ 1 + (pH 0 − c)2 − γ(pH 0 − c)]
I2 = I20 +
(25)
t (1 − I1) b + tA + 1 + (λtA + e)2 , AS = 0 λ a tA I1 = t0k(tA )
and
and γ = j when j(ASλ + e) > 0, j being equal to ±1. Equation 30 holds when the analyte is eluted in the middle raising portion between tA and t2K. If the analyte is eluted in the last isocratic portion, tR may be approximated by an expression of the form of eq 30 if (λtG + pH0 − c)2 ≫ 1. In this case I2 from eq 23 may be written as
pH ≥ c + 2 acids pH ≤ c − 2 bases
t0(1 − I1) a(b + γ )
(30)
t R (tG) = c0t + c1ttG
(36)
t R (φ) = c0φ + c1φφ + c 2φφ 2
(37)
the combined effect of tG and φ upon the solute retention, tR(tG, φ), may be expressed through the following equation17
where 6614
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Gradient Data of tR versus tG at Various φ Values Using Known k0, k1 Values. The artificial data examined in the previous section presuppose ideal linear pH versus t dependence, which can be hardly realized in practice. For this reason the next step is the use of experimental gradient data of tR versus tG at various φ values using known k0, k1 values. These data were obtained from a publication of Kaliszan’s group and concern the chromatographic behavior of five bases, aniline, panisidine, 2-amino-5-nitropyridine, 2-methylobenzimidazol, and morphine.1 These analytes have been studied in mobile phases modified by methanol: φ (v/v) = 0.03, 0.07, 0.11, and 0.15. The initial and final pH was 10.5 and 3, respectively, with gradient retention times: tG = 3, 5, 7, 10, 11, 15, 20, 25, and 30 min. For these analytes the dependence of k0, k1 upon φ is known from independent isocratic experiments.1 Therefore, we can fit these data to eqs 34 to obtain analytical expressions for k0(φ) and k1(φ) (Table S-1). When k0(φ), k1(φ) are known, all expressions of tR presented in the Rigorous Approach, Kaliszan−Wiczling’s Approximation, and Alternative Approximation sections depend only upon pK. Therefore, they can be tested either using pK values from the literature or fitting these expressions to gradient tR versus tG data at various φ values to obtain both pK values and the fitting performance of the models. The results are depicted in Table 1 which presents the average and the maximum percentage error in tR when the elution time of the bases under study was estimated by means of eqs 11, 17, 26 and the combination of eqs 1 and 2 which constitute the exact solution for the determination of tR. Equation 20 was not tested because no direct connection between k0, k1, pK and a, b, c is available. It is seen that when pK values are taken from the literature,1 for aniline, p-anisidine, and 2-methylo-benzimidazol the prediction of tR under gradient conditions is satisfactory. In contrast, the error in the predicted tR values for 2-amino-5-nitropyridine and morphine is relatively high. This is to be expected for two reasons: First, the pK values are calculated in pure aqueous solutions, and second deviations from linearity in the experimental pH versus t are possible. Probably for these two reasons all models exhibit almost the same performance in contrast to their performance studied using artificial data above. As expected, the performance of all models is improved when pK is used as an adjustable parameter. In all cases the estimated pK values (Table S-2) are lower than those of the literature. This may again be attributed partly to the fact that the pK values of the literature are calculated in pure aqueous solutions and partly to deviations from linearity in the experimental pH versus t profile. Finally, we should mention that a recent study by Wiczling and Kaliszan showed that when the model parameters are obtained from isocratic data and real measured pH changes are used, then the simulated pH gradient retention times estimated by eq 11 are in excellent agreement with experiment.7 Gradient Data of tR versus tG at Various φ Values with Unknown k0, k1, pK. When k0, k1, as well as pK are unknown, a nonlinear least-squares approach can be applied to fit eqs 11, 17, and 26 to gradient tR versus tG data at various φ values. Similarly, a nonlinear fitting is necessary for eq 20. The algorithms used for this purpose were the RND_LM and R_LM described in ref 18. We used both these algorithms to ensure convergence to the global optimum in all cases. The adjustable parameters are k00, k01, k10, k11, pK for eqs 11, 17, 26 and a0, a1, b0, b1, c for eq 20. In addition, in this case we tested
(38)
which yields t R = c0 + c1φ + c 2φ 2 + c3tG + c4φtG
(39)
provided that we keep terms of order up to 2. Equation 39 is subject to the restriction that tR varies linearly with tG. If this requirement is not fulfilled, eq 39 may be extended to t R = c0 + c1φ + c 2tG + c3tG2 + c4φtG
(40)
t R = c0 + c1φ + c 2φ 2 + c3tG + c4tG2 + c5φtG
(41)
or
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TEST OF THE THEORY To evaluate the fitting performance of all models presented above three types of data were adopted: (1) Artificial data of tR versus tG were created by means of eqs 1 and 2 using various k0, k1, and pK values. (2) Gradient data of tR versus tG at various φ values using known values for k0, k1 obtained from isocratic data. (3) Gradient data of tR versus tG at various φ values with unknown values for k0, k1. All experimental data were taken from literature.1,12 Artificial Data. For the creation of artificial tR versus tG, k0 varied from 5 to 50 with steps of 5; at each k0 value k1 took the values 0.5, 1, 2, 3, and at each pair (k0, k1) pK varied by steps of 1 from 3 to 6. In addition, we used ts = 0, t0 = 1 min, tD = 2 min, and four pH-gradient profiles: pH0 − pHf = 1 − 10, 1 − 8, 3 − 10, and 3 − 8. The application of eqs 11, 17, and 26 is straightforward. For the application of eq 20 the correlation between k0, k1, pK and a, b, c should be found. For this purpose at each point (k0, k1, pK) 1/k was calculated from eq 2 for pH values ranging from 1 to 10 by steps of 0.2, and these values of 1/k were fitted to eq 19 to obtain a, b, and c. Figure 2 depicts the absolute percentage average error in tR values estimated from the models under examination. It is seen
Figure 2. Mean absolute percentage error in tR values of the artificial data calculated from eq 11 (■), 17 (□), 20 (▲), and 26 (Δ).
that eq 26 exhibits the best fitting performance. Equation 20 also has a satisfactory performance. Finally, the models of eqs 11 and 17 give overall similar results; their fitting performance may be considered satisfactory, although worse than that of eq 26. 6615
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Table 1. Average and Maximum Absolute Percentage Error in tR Values of Bases aniline model
mean
2-amino-5nitropyridine
p-anisidine max
mean
k0, k1, and pK known from Literature, Reference 1 exact (eqs 1 , 2) 2.65 8.36 3.62 eq 11 2.41 7.76 3.42 eq 17 3.01 9.40 4.19 eq 26 2.71 8.50 3.74 k0, k1 from Literature, Reference 1, pK Adjustable eq 11 2.17 5.88 3.05 eq 17 1.96 5.06 3.60 eq 26 2.04 5.72 3.50 k0, k1, and pK (or a, b, c) Adjustable eq 11 1.83 4.11 2.49 eq 17 1.93 5.32 2.75 eq 20 1.29 3.84 1.20 eq 26 1.57 4.69 2.58 Empirical Models eq 39 (5.22) (15.02) (6.23) eq 40 3.61 8.30 2.33 eq 41 3.28 6.65 2.35
morphine
total error
max
mean
max
mean
max
mean
max
mean
max
12.31 11.24 13.43 12.46
7.68 7.07 9.13 7.91
16.15 16.59 17.52 16.33
4.73 5.30 4.31 4.68
12.45 12.58 12.42 12.42
15.20 16.02 16.67 15.38
24.58 26.86 24.46 24.53
6.78 6.84 7.46 6.88
24.58 26.86 24.46 24.53
10.38 10.79 11.56
5.62 5.77 5.47
14.83 14.22 13.79
5.25 4.34 4.66
14.70 12.69 13.48
6.71 5.09 5.44
16.23 13.13 13.92
4.56 4.15 4.22
16.23 14.22 13.92
7.18 6.86 5.15 6.93
2.08 1.84 1.78 1.91
5.43 5.14 4.82 5.35
4.00 3.47 3.18 3.47
8.42 9.71 6.91 9.37
4.42 3.64 2.46 4.02
15.12 13.66 6.46 14.13
2.96 2.73 1.98 2.71
15.12 13.66 6.91 14.13
(16.26) 6.95 6.66
3.72 2.44 2.36
7.17 5.48 6.34
2.77 4.18 2.63
7.56 9.37 5.09
3.10 4.99 3.10
6.74 9.86 6.87
3.20 3.51 2.74
7.56 9.86 6.87
fittings are given in Tables S-3−8 in the Supporting Information. It is seen that all models give good fitting results. However, the best performance is now presented by eq 20. It is also interesting to note that the simple model of eq 39 gives very satisfactory results provided that the basic requirement for its application, which is the linearity of the tR versus tG plots at constant φ, is fulfilled. Note that this requirement holds for the 17 OPA/2-mercaptoethanol derivatives as well as for 2-amino5-nitropyridine, 2-methylo-benzimidazol, and morphine. In cases where this restriction is not fulfilled, eq 40 or 41 may be tested. We observe that eq 41 gives very satisfactory results under all circumstances. A drawback of the simple linear models of eqs 39−41 seems to be the fact that their adjustable parameters, c0−c4 (c5), are strictly valid for a certain pH-gradient profile. However, exactly the same problem is met in practice also in the case of the models based on the solution of the fundamental equation for gradient elution, eq 1. This becomes clear if we use the adjustable parameters which were determined in the pHgradient 3.2−9 to predict the retention times of the analytes under the pH-gradient 2.8−10.7 (Table S-9). We observe that the error in the predicted tR values is increased considerably. Similarly, the performance of the models deteriorates when we use all data from both 3.2−9 and 2.8−10.7 pH-gradients to fit the models (Table S-9). These results clearly suggest that the good performance of the various models does not entail that they describe equally well the elution process but it may be due to their ability to cope effectively with the various nonidealities via the regression procedure. The basic nonidealities in a pHgradient process are deviations of the pH-gradient from linearity and the effect of the organic modifier content in the mobile phase on the pK of the ionizable analytes. Linearity of tR versus tG Plots. As already mentioned, many experimental systems exhibit linear tR versus tG plots when φ is kept constant,1,12 and this property is predicted theoretically under certain conditions examined in the Limiting Cases section. Analyzing the artificial data we found that indeed in the majority of cases tR varies linearly with tG. In particular,
the simple eqs 39, 40, and 41 using multilinear regression to obtain the adjustable parameters c0−c5. The experimental data adopted were the five bases already used above as well as the following 17 OPA/2-mercaptoethanol derivatives of amino acids taken from ref 12: (1) L-arginine (Arg), (2) L-asparagine (Asn), (3) L-glutamine (Gln), (4) Lserine (Ser), (5) L-aspartic acid (Asp), (6) L-glutamic acid (Glu), (7) L-threonine (Thr), (8) β-(3,4-dihydroxyphenyl)-Lalanine (Dopa), (9) L-alanine (Ala), (10) L-tyrosine (Tyr), (11) 4-aminobutyric acid (GABA), (12) lmethionine (Met), (13) Lvaline (Val), (14) L-tryptophan (Trp), (15) L-isoleucine (Ile), (16) L-phenylanine (Phe), and (17) L-leucine (Leu). These analytes have been studied in two series of pH-gradients, one from pH = 2.8 to 10.7 and another from pH = 3.2 to 9.0. At each gradient the concentration of acetonitrile in the mobile phase was kept constant at volume fractions φ = 0.25, 0.27, 0.3, or 0.35. The fitting results concerning the performance of the models are given in Tables 1 and 2. Differences between experimental Table 2. Mean and Maximum Absolute Percentage Error in tR Values of the 17 OPA/2-Mercaptoethanol Derivatives of Amino Acids pH-gradient: 3.2−9.0
pH-gradient: 2.8−10.7
model
mean
max
mean
max
eq eq eq eq eq
1.57 1.77 1.36 1.67 1.30
8.91 8.91 5.66 8.93 4.03
1.24 1.25 0.93 1.23 1.34
5.76 6.97 5.08 6.50 7.20
11 17 20 26 39
2-methylobenzimidazol
and predicted retention times calculated from eq 20 of the 17 OPA/2-mercaptoethanol derivatives of amino acids studied under pH-gradients from pH = 2.8 to 10.7 as well as the absolute percentage error between experimental and predicted retention times calculated from eqs 11, 17, 20, and 26 for Dopa are indicatively shown in Figures S-1 and S-2 in the Supporting Information. The adjustable parameters obtained from the 6616
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Figure 3. Plots of tR vs tG for (left) artificial data with pH0 = 3, pHf = 10, ts = 0, t0 = 1 min, tD = 2 min, and (o) k0 = 5, k1 = 0.5, pK = 3, (●) k0 = 5, k1 = 1, pK = 6, and (Δ) k0 = 10, k1 = 2, pK = 3, and (right) (■) aniline, (▲) p-anisidine, (o) 2-amino-5-nitropyridine, (Δ) 2-methylo-benzimidazol, (●) morphine at φ = 0.11. Curves from p-anisidine to morphine have been shifted by 1, 3, 4, 4 along the y-axis to make the picture clearer.
for the pH-gradient profiles pH0−pHf = 1−10 and 3−10, 75% of the tR versus tG plots exhibit linear plots with R2 > 0.99 and this number increases to 90% for plots of R2 > 0.95. In addition, there are about 9% of the plots that exhibit two linear branches due to change in the sign of ASλ + e within the range of tG values. Finally, there are only two cases (1%) with R2 < 0.95 (R2 = 0.94 and 0.90) in which the criteria |1 − b2| ≪ (ASλ + e)2, (λtG + pH0−c)2≫1 do not hold. An interesting point is that deviations from the linearity, which appear either with two linear branches or with a small curvature, are always favored by a relatively small value in k0. High values in k0 always yield linear plots of tR versus tG with R2 > 0.99. Examples of tR versus tG plots are given in Figure 3. The plot of tR versus tG on the left of this figure concerns artificial data, whereas the plots on the right are experimental data of the bases aniline, p-anisidine, 2-amino-5-nitropyridine, 2-methylobenzimidazol, and morphine at φ = 0.11. Note that the k0 values of the analytes under these conditions are 5.17, 6.15, 9.39, 13.68, 19.74 from aniline to morphine.1
least-squares fitting is necessary to achieve convergence to the global optimum. From this point of view the simple model of eq 39 is quite interesting. It gives very satisfactory results provided that the plot of tR versus tG at constant φ is linear and its application is easy because its adjustable parameters can be determined by linear regression. As already mentioned, the linear dependence of tR upon tG when φ is kept constant is predicted theoretically from the model of eq 20 and it appears in the majority of the experimental cases. Deviations from this property may appear if k0, the retention factor of the nonionized form of the analyte, is relatively small. In any case the check of this requirement is straightforward, and if it is fulfilled, the application of eq 39 offers the best solution. Otherwise the modifications of this equation, eq 40 or 41, may be adopted.
DISCUSSION AND CONCLUSIONS The work carried out mainly by Kaliszan’s group up to now has revealed that the pH-gradient technique offers special advantages for chromatographic analysis of ionizable analytes.1−7 It provides faster and better separations of most analytes of interest in medicinal and environmental chemistry. In addition, it may offer new means of determining the physicochemical properties of analytes. In this paper we have critically reviewed this work. In addition, we have provided new expressions for the retention time of ionizable analytes. It is shown that under ideal linear pH-gradient conditions the better a model describes the 1/k versus pH plot, the better performance it exhibits. However, under real experimental conditions the performance of a model depends upon its ability to compensate effectively for deviations from linearity of the pH-gradient profile as well as the effect of φ on pK via the regression procedure. In general all models based on the fundamental eq 1 exhibit good fitting performance, but the best of them is the one described by eq 20. For this model both the average and the maximum error in tR values is very low. However, the rather complicated nature of the expressions of tR as a function of φ and pH, eqs 11, 17, 20, and 26, makes their application rather difficult, so proper software for nonlinear
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ASSOCIATED CONTENT
S Supporting Information *
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Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.
AUTHOR INFORMATION
Corresponding Author
*Phone: +30 2310 997773. Fax: +30 2310 997709. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research has been cofinanced by the European Union (European Social FundESF) and Greek National Funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)Research Funding Program: Heracleitus II, Investing in Knowledge Society through the European Social Fund.
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REFERENCES
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